Paul's pages

2011-11-16T09:00:00.000-08:00

Slightly revised Nov. 16 2011

Explosive traces in 9/11 dust?
Media ducked claim by experts
July 14, 2011
The "criminal-media nexus" under fire in London draws attention to major news events that have gone unreported by the mainstream press on both sides of the Atlantic.

In 2009, for example, a team of scientists reported that evidence of an advanced explosive, favored by the Pentagon, had been found in the dust of the collapsed World Trade Center towers. This paper came after an admission by the National Institute for Standards and Technology that it had done no forensic tests in its investigation of the collapses.

Yet a search today of Google News for the name of one of the investigators, Niels Harrit, a chemistry professor at the University of Copenhagen, turned up no mainstream news organizations, other than Berliner Umschau, which cited Harrit in an opinion piece.

In the paper, published by the Open Chemical Physics Journal, the scientists told of finding red-gray flakes in various samples of dust and determining that the flakes were from a material similar to that in advanced TBX weaponry.

The team had sought submissions of samples of dust from the public and received containers submitted by people who had decided to save such samples. The only samples used in the study came from the five persons who agreed to let themselves be identified publicly.

There has been no public statement from the FBI on the work of the scientists. However, it may be assumed that the evidence can be ignored based on the fact that the chain of custody is broken. There is no way to be sure that the samples weren't doctored.

And yet, such tampering would seem to have required a technically advanced conspiracy, wherein volunteers working for conspirators either submit doctored material or are able to intercept and switch samples. In other words, a tampering conspiracy would require a sophisticated intelligence operation.

But, the question then arises: if intelligence units were behind the 9/11 attacks, why didn't they intercept the samples and switch them for non-incriminating dust. It seems plausible that honest agents had made such a switch too risky, and that conspirators counted on what Britain's former prime minister, Gordon Brown, denounced as a "criminal-media nexus" that includes not only the Murdoch press, but other news organizations as well.

The report, Active Thermitic Material Discovered in Dust from the 9/11 World Trade Center Catastrophe, told of igniting the chips and watching them flame. "The evidence for active, highly energetic thermitic material in the WTC dust is compelling," the scientists wrote.
http://www.diexx88blog.com/wordpress/wp-content/uploads/2011/05/activethermitic_911.pdf

[Also see 9/11 probers skipped key forensic tests
http://www.angelfire.com/ult/znewz1/trade7.html]

They said the residues from ignited chips were "strikingly similar" to the chemical signature of commercial thermite. The scientists believed that the thermite residues were consistent with "super-thermite," also known as "nano-thermite," They cited a 2001 report on Defense Department research into "nano-energetics" and thermobaric (TBX) weapons.
Here is such a report:
http://www.p2pays.org/ref/34/33115.pdf

"Super-thermite electric matches" had been devloped by Los Alamos National Laboratory for such applications as triggering explosives for demolitions, the experts noted.

The authors said their tests ruled out the possibility that the red chips were flakes of ordinary paint.

However, one of the authors, physicist Steven E. Jones, had already been given the Murdoch treatment for having raised questions over the reliability of official accounts, and it is quite possible that journalists and politicians alike shrank from covering the report out of fear of the "criminal-media nexus" blackball.

Harrit works alongside Thomas Bjorn Holm, head of the Nano-Science Center at the Department of Chemistry at the University of Copenhagen and may have been able to consult with Bjorn Holm, whose name does not appear on the report.

Harrit's Facebook page:
http://www.facebook.com/pages/Niels-Harrit/32509864153

Jones was a professor at Brigham Young University before being pressured to retire as a result of his 9/11 criticism,
http://www.physics.byu.edu/research/energy/

The scientists used advanced technology, including scanning electron microscopy, X-ray energy dispersive spectroscopy and differential scanning calorimetry.

One of the authors, Jeffrey Farrer, manages the electron microscopy facility for Brigham Young University's Department of Physics and Astronomy. His research includes nano-particle and thermitic reactions.
http://www.physics.byu.edu/directory.aspx?personID=23

Another author is Kevin R. Ryan, terminated by Underwriters Laboratory after raising technical issues concerning the official 9/11 narrative.
http://www.mindfully.org/Reform/2004/Kevin-R-Ryan22nov04.htm

Physicist Daniel E. Farnsworth, as with a number of experts critical of the government claims about 9/11, is retired and presumably beyond the reach of career retribution incited by the mendacious, criminalized press. Like Jones, Farnsworth taught at Brigham Young.
http://www.physics.byu.edu/research/energy/currvitaApril09.htm

Another author, Frank Legge, is an Australian chemist who serves as a co-editor at the Journal of 9-11 Studies.
http://www.scientistsfor911truth.org/mempages/Legge.html

Gregg Roberts, a 9/11 activist, is also listed, but a Google search gives no inkling of his scientific background.
http://world911truth.org/tag/gregg-roberts/

James R. Gourley identifies himself as a chemical engineer in an extensive criticism he submitted to the National Institute of Standards and Technology 9/11 investigation.
http://911research.wtc7.net/letters/nist/WTC7Comments.html

Bradley R. Larsen is another author. His firm, S&J Scientific, does not appear to have a web page trackable by Google and his scientific background did not show up in a Google search.*

The authors acknowledge conversations with a number of 9/11 critics, including retired naval physicist David L. Griscom
http://www.impactglassresearchinternational.com/
and former University of Iowa physicist Crockett Grabbe.
http://www.sealane.org/speak/index1.html
-----------------------------------------------------
*CORRECTION: A previous version of this page linked to an incorrect web site for Larsen.

First published Thursday, July 12, 2007

The prosecutor's fallacy

There are various forms of the "prosecutor's paradox" or "the prosecutor's fallacy," in which probabilities are used to assign guilt to a defendant. But probability is a slippery subject.

For example, a set of circumstances may seem or be highly improbable. But defense attorneys might wish to avail themselves of something else: the more key facts there are in a string of facts, the higher the probability that at least one fact is false. Of course, that probability is difficult to establish unless one knows either the witnesses' rates of observational error or some standard rates of observational error, such as the rate typical of an untrained observer versus an error rate typical of a police officer.

(For a non-rigorous but useful example of likelihood of critical misstatement, please see the post Enrico Fermi and a 9/11 plausibility test. In that post we are testing plausibility which is far different from ironclad guilt or innocence. Also, for a discussion of probabilities of wrongful execution, please search Fatal flaws at Znewz1.blogspot.com.)

Suppose an eyewitness is tested for quick recall and shows a success rate of 96 percent and a 4 percent error rate. If the witness is testifying to 7 things he saw or heard with no prior knowledge concerning these things, the likelihood that the testimony is completely accurate is about 75 percent. So does the 25 percent probability of error constitute reasonable doubt -- especially if no fact can be expunged without forcing a verdict of not guilty? (Of course, this is why the common thread by several witnesses tends to have more accuracy; errors tend to cancel out.)

The prosecutor's paradox is well illustrated by The people v.
Collins, a case from 1964 in which independent probabilities were incorrectly used, the consequence being dismissal of the conviction on appeal.

To summarize, a woman was shoved to the ground and her purse snatched. She and a nearby witness gave a description to Los Angeles police which resulted in the arrest of a white woman and a black man. I do not intend to treat the specifics of this case, but rather just to look at the probability argument.

The prosecutor told the jury that the arrested persons matched the description given to police so closely that the probability of their innocence was about 1 in 12 million.

The prosecutor gave these probabilities:

Yellow auto, 1/10; mustached man, 1/4; woman with ponytail, 1/10; woman with blonde hair, 1/3; black man with beard, 1/10; interracial couple in car, 1/1000. With a math professor serving as an expert witness, these probabilities were multiplied together and the result was the astoundingly high probability of "guilt."

However, the prosecutor did not conduct a comparable test of witness error rate. Suppose the witnesses had an average observational error rate of 5 percent. The probability that at least one fact is wrong is about 26 percent. Even so, if one fact is wrong, the computed probability of a correct match remains very high. Yet, if that fact was essential to the case, then a not guilty verdict is still forced, probability or no.

But this is not the only problem with the prosecutor's argument. As the appellate court wrote, there seems to be little or no justification for the cited statistics, several of which appear imprecise. On the other hand, the notion that the reasoning is never useful in a legal matter doesn't tell the whole story.

Among criticisms leveled at the Los Angeles prosecutor's reasoning was that conditional probabilities weren't taken into account. However, I would say that conditional probabilities need not be taken into account if a method is found to randomize the collection of traits or facts and to limit the intrusion of confounding bias.

But also the circumstances of arrest are critical in such a probability assessment. If the couple was stopped in a yellow car within minutes and blocks of the robbery, a probability assessment might make sense (though of course jurors would then use their internalized probability calculators, or "common sense"). However, if the couple is picked up on suspicion miles away and hours later, the probability of a match may still be high. But the probability of error increases with time and distance.

Here we run into the issue of false positives. A test can have a probability of accuracy of 99 percent, and yet the probability that that particular event is a match can have a very low probability. Take an example given by mathematician John Allen Paulos. Suppose a terrorist profile program is 99 percent accurate and let's say that 1 in a million Americans is a terrorist. That makes 300 terrorists. The program would be expected to catch 297 of those terrorists. However, the program has an error rate of 1 percent. One percent of 300 million Americans is 3 million people. So a data-mining operation would turn up some 3 million "suspects" who fit the terrorist profile but are innocent nonetheless. So the probability that a positive result identifies a real terrorist is 297 divided by 3 million, or about one in 30,000 -- a very low likelihood.

But data mining isn't the only issue. Consider biometric markers, such as a set of facial features, fingerprints or DNA patterns. The same rule applies. It may be that if a person was involved in a specific crime or other event, the biometric "print" will finger him or her with 99 percent accuracy. Yet context is all important. If that's all the cops have got, it isn't much. Without other information, the odds are still tens of thousands to one that the cops or Border Patrol have the wrong person.

The probabilities change drastically however if the suspect is connected to the crime scene by other evidence. But weighing those probabilities, if they can be weighed, requires a case-by-case approach. Best to beware of some general method.

Turning back to People v. Collins: if the police stopped an interracial couple in a yellow car near the crime scene within a few minutes of the crime, we might be able to come up with a fair probability assessment. It seems certain that statistics were available, or could have been gathered, about hair color, facial hair, car color, hair style, and race. (Presumably the bandits would have had the presence of mind to toss the rifled purse immediately after the robbery.)

So let us grant the probabilities for yellow car at 0.1; woman with ponytail, 0.1; and woman with blonde hair, 0.333. Further, let us replace the "interracial couple in car" event with an event that might be easier to quantify. Instead we estimate the probability of two people of different races being paired. We'd need to know the racial composition of the neighborhood in which they were arrested. Let's suppose it's 60 percent white, 30 percent black, 10 percent other. If we were to check pairs of people in such a neighborhood randomly, the probability of such a pair is 0.6 x 0.3 = 0.18 or 18 percent. Not a big chance, but certainly not negligible either.

Also, we'll replace the two facial hair events with a single event: Man with facial hair, using a 20 percent estimate (obviously, the actual statistic should be easy to obtain from published data or experimentally).

So, the probability that the police stopped the wrong couple near the crime scene shortly after the crime would be 0.1 x 0.1 x 0.333 x 0.18 x 0.2 = about 1.2^-4, or about 1 chance in 8300 of a misidentification. Again, this probability requires that all the facts given to police were correct.

But even here, we must beware the possibility of a fluke. Suppose one of the arrestees had an enemy who used lookalikes to carry out the crime near a point where he knew his adversary would be. Things like that happen. So even in a strong case, the use of probabilities is a dicey proposition.

However, suppose the police picked up the pair an hour later. In that situation, probability of guilt may still be high -- but perhaps that probability is based in part on inadmissible evidence. Possibly the cops know the suspects' modus operandi and other traits and so their profiling made sense to them. But if for some reason the suspects' past behavior is inadmissible, then the profile is open to a strong challenge.

Suppose that a test is done of the witnesses and their averaged error rate is used. Suppose they are remarkably keen observers and their rate of observational error is an amazingly low 1 percent. Let us, for the sake of argument, say that 2 million people live within an hour's drive of the crime scene. How many people are there who could be mistakenly identified as fitting the profile of one of the assailants? One percent of 2 million is 20,000. So, absent other evidence, the probability of wrongful prosecution is in the ballpark of 20,000 to 1.

It's possible that the male or female associate of the innocent suspect's partner is guilty, of course. So one could be an innocent member of a pair while the other member is guilty.

It's possible the crime was by two people who did not normally associate, which again throws off probability analysis. But, let's assume that for some reason the witnesses had reason to believe that the two assailants were well known to each other. We would then look at the number of heterosexual couples among the 2 million. Let's put it at 500,000. Probability is in the vicinity of 5000 to 1 in favor of of wrong identification of the pair. Even supposing 1 in 1000 interracial couples among the 2 million, that's 2000 interracial couples. A one percent error rate turns up roughly 20 couples wrongly identified as suspects.

Things can get complicated here. What about "fluke" couples passing through the area? Any statistics about them would be shaky indeed, tossing all probabilities out the window, even if we were to find two people among the 20 who fit the profile perfectly and went on to multiply the individual probabilities. The astoundingly low probability number may be highly misleading -- because there is no way to know whether the real culprits escaped to San Diego.

If you think that sounds unreasonable, you may be figuring in the notion that police don't arrest suspects at random. But we are only using what is admissible here.

On the other hand, if the profile is exacting enough -- police have enough specific details of which they are confident -- then a probability assessment might work. However, these specific details have to be somehow related to random sampling.
After all, fluke events really happen and are the bane of statistical experiments everywhere. Not all probability distributions conform to the normal curve (bell curve) approximation. Some data sets contain extraordinarily improbable "ouliers." These flukes may be improbable, but they are known to occur for this specified form of information.

Also, not all events belong to a set containing ample statistical information. In such cases, an event may intuitively seem wonderfully unlikely, but the data are insufficient to do a statistical analysis. For example, the probability that three World Trade Center buildings -- designed to withstand very high stresses -- would collapse on the same day intuitively seems unlikely. In fact, if we only consider fire as the cause of collapse, we can gather all recorded cases of U.S. skyscraper collapses and all recorded cases of U.S. skyscraper fires. Suppose that in the 20th Century, there were 2,500 skyscraper fires in the United States. Prior to 9/11 essentially none collapsed from top to bottom as a result of fire. So the probability that three trade center buildings would collapse as a result of fire is 2,500^-3
or one chance in 156 billion.

Government scientists escape this harsh number by saying that the buildings collapsed as a result of a combination of structural damage and fire. Since few steel frame buildings have caught fire after being struck by aircraft, the collapses can be considered as flukes and proposed probabilities discounted.

Nevertheless, the NIST found specifically that fire caused the principle structural damage, and not the jet impacts. The buildings were well designed to absorb jet impact stresses, and did so, the NIST found. That leaves fire as the principle cause. So if we ignore the cause of the fires and only look at data concerning fires, regardless of cause, we are back to odds of billions to one in favor of demolition by explosives.

Is this fair? Well, we must separate the proposed causes. If the impacts did not directly contribute significantly to the collapses, as the federal studies indicate (at least for the twin towers), then jet impact is immaterial as a cause and the issue is fire as a cause of collapse. Causes of the fires are ignored. Still, one might claim that fire cause could be a confounding factor, introducing bias into the result. Yet, I suspect such a reservation is without merit.

Another point, however, is that the design of the twin towers was novel, meaning that they might justly be excluded from a set of data about skyscrapers. However, the NIST found that the towers handled the jet impacts well; still, there is a possibility the buildings were well-designed in one respect but poorly designed to withstand fire. Again, the NIST can use the disclaimer of fluke events by saying that there was no experience with fireproofing (reputedly) blown off steel supports prior to 9/11.

A general continuing fraction recursion algorithm for square roots

2011-11-10T15:58:00.001-08:00

A very minor result that happens to please me:

The continuing real fraction

J + 1/(J + 1/(J + 1/(J + 1/ ...

= J + [(J^2 + 4)^.5]/2

is a special case of a recursion function yielding that limit. That general function is

Xsub(n+1) = (Xsub(n) + C)^(-1) + C

Setting Xo = 0 and C = J, we see (where sub(-1) is not an initial value but a designation for the constant prior to application of the function):

Convergent ; Our function; Continued fraction

0 ; Xsub(-1) = J; J

1 ; Xsub(1) = (J^-1) + J; J + J^-1

2 ; Xsub(2) = (J + J^-1)^-1 + J; as above

Of course, we needn't set Xo = 0. In fact, the curious thing is that this recursion function arrives at the same limit no matter what real initial value is chosen (other than Xo = -C, which must be excluded).

That is, (lim n-->inf)Xsub(n+1) = (lim n-->inf)Ysub(n+1)

when Xsub(1) = (Xo + C)^-1 + C and Ysub(1) = (Yo + C)^-1 + C. It is the constant C that determines the limit, which is the limit of the continuing fraction

1 + 1/C...

That is, beginning with any real but -C for Xo and any real but -C for Yo, we obtain the limit above because we find that

(lim n-->inf)(Xsub(n) - Ysub(n)) = 0,

where (Xsub(n) - Ysub(n)) alternates sign by n.

A bit of perfunctory algebra, which I omit, establishes these facts.

So, this algorithm yields an infinity of approaches to any square root. That is, Xsub(n) =/= Ysub(n) for finite n.

An example: (lim n-->inf)X(sub n) = (2 + 8^.5)/2 = 1 + 2^.5

For Xo = 1 and C = 2, some recursive (calculator) values are:

2.333...

2.428571429

2.411764706

2.414634146

For Xo = 1/2 and C = 2

2.5 2.4 2.416...6...

2.413793103

2.414285714

For Xo = -31 and C = 2

-29.0

1.965517241

2.50877193

2.398601399

2.416909621

For Xo = 31 and C = 2

33.0

2.03...03...

2.492537313

2.401197605

2.416458853

For Xo = 1/31 and C = 2 2.032258065

2.492063492

2.401273885

2.416445623

2.413830955

For Xo = -1/31 and C = 2

1.967741935

2.508196721

2.39869281

2.416893733

Note the pattern of alternately too high--too low.

2011-11-10T15:55:00.001-08:00

The Monty Hall problem -- over easy

The 9/11 collapses -- an unlikely sequence
Tests for divisibility by 9 and 11

The Monty Hall problem

A player is shown three closed doors. One hides a nice prize and the other two hide booby prizes. The player picks a door. Monty then opens another door, to reveal a booby prize. The player is asked whether he or she would like to stand pat or switch his choice to the remaining closed door. Should he or she switch?

The reason this problem is so popular is that, for most of us, the answer is counterintuitive. Of course, in the original TV game show, Let's Make a Deal, the player was not given the option to switch.

My mathematician son, Jim Conant, almost instinctively knew the right answer: switch! He saw immediately that in 1/3 of cases standing pat fails so that in 2/3 of cases, switching must succeed.

But to the duller-witted of us, that reasoning doesn't seem to account for the apparent point that once the set is whittled down to two choices, switching gives a 50-50 chance of success.

However, probabilities are about information, and the information the player obtains by Monty's opening of the door affects the probabilities.

I confess I found this very difficult to grasp (and retain!) until I was able to come up with the neat proof below.

I arrived at the proof by thinking: Suppose we divide the doors into subsets {A} and {B,C} and Monty doesn't open a door once a player selects A. There is a 1/3 probability of success if he chooses set A and 2/3 for {B,C}. Now if someone tells him he can't choose, say, element B, that doesn't affect the 2/3 probability for {B,C}. So if he switches to {B,C} he must take C, which then has a 2/3 probability of success.

However, here's the proof over easy:

           A     B    C   case 1.    0    (0    1) case 2.    0    (1    0) case 3.    1    (0    0)   case 1: Monty opens B, switch succeeds case 2: Monty opens C, switch succeeds case 3: Monty opens B or C, switch fails

In 2/3 of cases, switch succeeds.

I suggest that a source of confusion is the "or" in case 3. Despite there being two options, Monty opens only one door, as in cases 1 and 2.

100-door Monty

If you're still unhappy, Jim Conant suggests a light bulb might go on if you consider this scenario:

We have 100 doors, with a prize behind only one. You choose a door. Monty now opens 98 doors, none of which hides a prize. Should you switch? Of course, since the probability that the other door hides the prize is influenced by the knowledge that 98 other doors hid nothing. What is your chance of having chosen the winner? Still, 1 percent. What is the chance that the prize is behind one of the other 99 doors? Still 99 percent. So the knowledge you are given squeezes that probability onto the other closed door.

Enter information theory

Let's have a bit of fun with information theory.

The base-2 information content, or value, of the three closed doors is simply log₂3 = 1.58 bit. Each door's information value is of course 1/3 log 3 = 0.528 bit. When one of the doors is opened, the remaining two doors still have information values of 0.528 bit each.

On the other hand, in the case of two closed doors, the information content of each door is 1/2 log 2 = 0.5, which corresponds to maximum uncertainty. Hence the 0.028 difference in information corresponds with your awareness of an asymmetric change in probabilities that occurs once a door is opened. Because one door is opened, the information content of that door, once you get past your surprise, becomes 0. In this case, the information content of the unopened door then becomes 1.06 bit.

Now if we have 100 doors, the scenario opens with a total information content of log 100 = 2 log 10 = 6.64 bit. A single door has an information content of 0.07 bit.

The information content of the 98 doors Monty opens is 6.51, which is far greater than 0.07. Indeed, I suspect you feel much more confident that the prize is behind the other remaining closed door in the 100-door scenario.

We also have that the information content of the set of 99 doors is 6.57. Once 98 of those doors are opened, the information content of those doors becomes 0 and hence the information content of the remaining door in the set is 6.57 versus the measly 0.07 content of the door you chose.

The Monty Hall problem is easily solved via the rules of conditional probability. I recommend Afra Zomorodian's proof, which is best accessed by Googling.
This page has been deleted from Wikipedia and blacklisted from the Wiki system. Not sure why.

2011-11-10T15:53:00.001-08:00

Null set uniqueness theorem

John Allen Paulos. Probability and politics
D.J. Velleman. Set theory and logic
Herbert B. Enderton. Set theory and logic
Axioms of Zermelo-Fraenkel set theory

Note: This page was corrected in August 2006 to include a missing not-symbol and a missing parenthesis. Also, a redundant statement was deleted and some other minor changes were made.

Comment: Some are puzzled by the standard set theoretic fact that even when set A does not equal set B, A - A equals B - B (also written A\A and B\B). To beginners, it is sometimes counterintuitive that the empty subset of A is indistinct from the empty subset of B.

In the Zermelo-Fraenkel version of set theory, the unique empty set is given as an axiom. However, using the rules of logic, it is possible to derive the empty set from other axioms of ZF. See the link above for all the ZF axioms.

The theorem below is not strictly necessary, but hopefully may still prove of use.

We accept these rules of inference, letting "~" stand for "not":

   The expression i)    x ® y       means "x implies y", which is equivalent to  ii)   if x is true, then y is true        which is equivalent to  iii)  ~x or y       which "means either x is false, or y is true".  Example:  i)    A person's being alive implies that he or she has       a beating heart  ii)   If a person is alive, then he or she has a       beating heart  iii)  Either a person has a beating heart, or       he or she is dead

This permits us to write "x ® y" as "~x or y" (although English idiom often has "y or ~x" but the statements are equivalent as a truth table check shows).

We take "x « y" to mean "x is true only if y is true and y is true only if x is true" which can be written (x ® y) and (y ® x)

We accept these definitions and conditions a priori:

A set is defined by its elements, not by its descriptions. For example, let A = {r|r is a real root of x² -5x + 6 = 0} and B = {2,3}. In that case, A = B.

Every element is unique. That is, if A = {2} and B = {2,2}, then A = B.

A set is permitted to be an element of another set, (but in standard theory it can't be a member of itself). By this we see that every set is unique.

A subset of a set is itself a set.

B subset of A means (using "e" for "element of"):

x e B ® x e A

Equality means

x e B « x e A, meaning B is a subset of A and A is a subset of B.

First, let us see that every set has a an empty subset, meaning you cannot remove the null subset from a set.

Let us provisionally call B a subset of A and then require that B have no elements.

By definition, x e B ® x e A, which is equivalent to

x ~e B or x e A

which is true, since x is not a member of B. Even if x is not a member of A (perhaps A is empty), the statement remains true.

That is, a subset with no elements satisfies the definition of subset.

Now suppose our presumption that any set A has a null subset is false. Using { }_A to mean null subset of A, we have:

i. ~(x e { }_A ® x e A), or,

ii. ~(x ~e { }_A or x e A), or,

iii. x e { }_A and x ~e A,

This last is false simply because x ~e { }_A. That is, "x e { }_A or x ~e A" is true, but the "and" makes the statement wrong.

Hence our suggestion is correct.

Now to prove uniqueness we simply need show that { }_A = { }_B for arbitrary sets A and B. We have

x e { }_A « x e { }_B, or,

(x ~e { }_A or x e { }_B) and (x ~e { }_B or x e { }_A).

To the left of the "and" we have the truthfulness of x not being an element of { }_Aand similarly to the right.

Hence { }_A = { }_B

Remembering that any subset is a set, our claim that only one null set exists is verified.

It is also instructive to verify the uniqueness of the null set by considering complement sets.

Adopting the notation for the complement set, we have

x e A\B ® x ~e B

We know that A = A and so A is a subset of A. This fact allows us to write:

x e A\A ® x ~e A

which is true. Look at the rewritten statement:

~(x ~e A) ® ~(x e A\A), or

x e A ® x ~e A\A.

Now this holds for any variable. Hence A\A is a set with no elements.

Now suppose B ~= A. Nevertheless,

x e B\B « x e A\A

as we see from

(x ~e B\B or x e A) and (x e B\B or x ~e A),

which is true.

So let us denote A\A with an empty set symbol { } and consider the cases { }\{ }, ~{ }\{ }, and { }\~{ }.

i.    x e { }\{ } --> x e { } and x e { }           which holds vacuously [note transformations above]  ii.   x e ~{ }\{ } --> x e ~{ } and x ~{ }           which is true.  iii.  x e { }\~{ } --> x e { } and x ~e ~{ }           which means       x e {}\~{} --> x e {} and x e {}           which is false, meaning that the expression           {}\~{} is not defined in standard set

theory.

Now we have that A\A = B\B = { }\{ } = { }, establishing that the null subset of any set is indistinct from the null subset of any other set, meaning that there is exactly one null set.

Note on 'exclusive or' symbolism

Today's preferred symbol for the exclusive or operation is "XOR". Yet even today the plus sign "+" is used to denote exclusive or. This symbol derives from "<--/-->", meaning "does not strictly imply." The truth table for nonexclusive or is

 i)     A  B     ----     T  F   T     F  T   T     T  T   T     F  F   F

whereas the truth table for exclusive or is

ii)     A  B     ----     T  F   T     F  T   T     T  T   F     F  F   F

By transforming the statement A <--/--> B, we will arrive at table ii. To wit: A <--/--> B

= not-((A --> B) & (B --> A))

= not-((not-A v B) & (not-B v A))

applying de Morgan's negation law, we get

not-(not-A v B) v not-(not-B v A)

A second application of that law yields

(A v not-B) v (not-A v B)

This formalism includes the possible expression:

(A & not-B) and (B and not-A), which is equavalent to

(A & not-A) & (B & not-B)

which, as a contradiction, must be disregarded, leaving the two remaining possibilities:

A & not-B

B & not-A

, conforming to table ii.

Thanks to John Peterson, an alert reader who caught a point of confusion that was due to two typographical errors, since corrected.

2011-11-10T15:50:00.000-08:00

Tests for divisibility by 9 and 11

If one adds the digits of a number and the sum is divisible by 9, then the number is divisible by 9. Similarly, if one alternates the signs of a multi-digit number's digits and the sum is divisible by 11, then the number is divisible by 11.

Example

The sum of the digits of 99 is 18, which is divisible by 9. Likewise, the sum 1+8 is also divisible by 9.

The sum of the alternately signed digits of 99 is 9 + (-9) = 0, and 0 is divisible by 9.

And, of course, 99 = 9(11).

How do these tests of divisibility work?

This description is for people with no background in number theory.

Proposition I

A number is divisible by 9 (with a remainder of 0) if and only if the sum of its digits, in base-10 notation, is divisible by 9.

That is, if n is divisible by 9, the sum of its digits is divisible by 9 and if the sum of its digits is divisible by 9, n is divisible by 9.

So our method of proof can either begin with the assumption that n is divisible by 9 or with the assumption that the sum of n's digits is divisble by 9. Below we have chosen to assume n is divisible by 9. But, first some background.

What is base 10?

It is customary to use a place system for numbers of any base. The base-10 system, with its 10 digits, uses digit position to tell us what multiple of 10 we have.

When we write, say 231, this tells us that we are to add 200 + 30 + 1. Each place represents a power of 10. That is, 200 + 30 + 1 = 2 · 10² + 3 · 10¹ + 1 · 10⁰ (where any number with exponent 0 is defined as equal to 1).

If we wish to write 23 in binary, or base-2, notation, we first write:

1 · 2⁴ + 0 · 2³ + 1 · 2² + 1 · 2¹ + 1 · 2⁰.

Then, limiting ourselves to the digit set {0,1}, we write 10111, knowing that the place signifies a power of 2.

Sets of numbers differing by multiples of q

Now we want to think about some sets of numbers, each of which is divisible by some number n.

For example, let's consider the series

{...-17, -10, -3, 4, 11...} where any two members differ by some multiple of 7.

As we see,

(-17) - (-10) = -7

(-17) - 11 = -28

Another such series is

{...-15, -8, -1, 6, 13, 20...}

where subtraction of any two numbers in the series also yields a number divisible by 7.

Once we know one member of such a set, we know them all. That is, the set is writable:

{-3 + 7k|k e K}, with K the set of integers.

It is customary to express such a set thus:

[-3]₇. We can also express this set as [-10]₇. In fact, in this notation,

[-3]₇ = [-10]₇

In general, a series denoted [n]_q is identical to the series denoted [n + qk]_q, where k is any integer.

The set [n]_q is known as congruence class n modulo q.

Note that

[0]_q = [0 + qk]. Every member of this series is divisible by q for all k. That is, every element of this series, when divided by q, equals an integer k.

[q]_q = [q + qk]. Every member of this series is divisible by q. That is, every member, when divided by q, equals k+1.

What happens if we add elements of [m]_q and [n]_q?

We have m + qk + n + qj = m+n + q(k+j), letting j be an integer.

If we set k+j to 0, this gives m+n, which we use to establish the series [m+n]_q = [m+n + qk]_q.

For example, we have [-3]₇ + [-1]₇ = [-4]₇,

which means,

{...-17,-10,-3,4,11...} + {...-15,-8,-1,6,13,20...} = {...-18,-11,-4,3,10,17...}

We see that -18 - 3 = -21, which is indeed divisible by 7.

By kindred reasoning we can show that it is possible to multiply an element from each of two similar series to obtain a third series similar to the other two. By 'similar' is meant a series in which elements differ by an integer multiple of q.

That is, [p]_q · [r]_q = [pr]_q.

Proof of Proposition I

Let d_n, d_n-1 ... d₁, d₀ represent the decimal expansion of some number N.

Then, by definition, we have N = d_n · 10ⁿ + d_n-1 · 10^n-1+...+d₀ · 10⁰

[See explanation above.]

Since both sides of the expression are equal, each must belong to the same congruence class.

That is, putting N as a member of the series [N]_q means that the right side of the equation is also a member of [N]_q.

That is, [N]_q = [d_n · 10ⁿ + d_n-1 · 10^n-1 +...+ d₀ · 10⁰]_q.

Now our condition is that N be divisible by 9. In that case N is a member of the congruence class [0]₉. That is, [N]₉ = [0]₉. Likewise, the decimal expansion is also a member of [0]₉.

That is, [N]₉ = [0]₉ = [N's decimal expansion]₉.

Because [p+r]_q = [p]_q+[r]_q and [pr]_q = [p]_q · [r]_q, we may write:

[N]₉ = [0]₉ = [d_n]₉ · [10]₉ⁿ + [d_n-1] · [10]₉^n-1 + ... + [d₁]₉ · [10]₉⁰

Now we know that [10]₉ = [1]₉, since 10-9 = 1.

Hence we can substitute the number 1 for the number 10, obtaining

[0]₉ = [d_n]₉ · [1]₉ⁿ + [d_n-1]₉ · [1]₉^n-1+...+ [d₀]₉ · [1]₉⁰ ...

Obviously 1^m = 1. So we may write

[N]₉ = [d_n]₉ + [d_n-1]₉+...+[d₀]₉

which equals

[d_n+d_n-1+...+d₀]₉, which equals [N]₉, which equals [0]₉.

Since the sum of the digits is equal to congruence class [0]₉, it must be divisible by 9.

QED

Proposition II

A number is divisible by 11 only if the alternating sum of its digits is divisible by 0.

That is, d_n - d_n-1 + d_n-2 etc. must equal a sum divisible by 11.

The proof for 9 can be used for 11, noting that [10]₁₁ = [-1]₁₁, since 10 - (-1) = 11. And it should be remembered that (-1)²ⁿ is positive, while (-1)^{2n ± 1}

is negative.

First published ca. 2002

2011-11-10T15:47:00.000-08:00

First published Monday, June 14, 2010

Freaky facts about 9 and 11

It's easy to come up with strange coincidences regarding the numbers 9 and 11. See, for example,

http://www.unexplained-mysteries.com/forum/index.php?showtopic=56447

How seriously you take such pecularities depends on your philosophical point of view. A typical scientist would respond that such coincidences are fairly likely by the fact that one can, with p/q the probability of an event, write (1-p/q)ⁿ, meaning that if n is large enough the probability is fairly high of "bizarre" classically independent coincidences.

But you might also think about Schroedinger's notorious cat, whose live-dead iffy state has yet to be accounted for by Einsteinian classical thinking, as I argue in this longish article:

http://www.angelfire.com/ult/znewz1/qball.html

Elsewhere I give a mathematical explanation of why any integer can be quickly tested to determine whether 9 or 11 is an aliquot divisor.

http://www.angelfire.com/az3/nfold/iJk.html

Here are some fun facts about divisibility by 9 or 11.

# If integers k and j both divide by 9, then the integer formed by stringing k and j together also divides by 9. One can string together as many integers divisible by 9 as one wishes to obtain that result.

Example:

27, 36, 45, 81 all divide by 9

In that case, 27364581 divides by 9 (and equals 3040509)

# If k divides by 9, then all the permutations of k's digit string form integers that divide by 9.

Example:

819/9 = 91

891/9 = 99

198/9 = 22

189/9 =21

918/9 = 102

981/9 = 109

# If an integer does not divide by 9, it is easy to form a new integer that does so by a simple addition of a digit.

This follows from the method of checking for factorability by 9. To wit, we add all the numerals, to see if they add to 9. If the sum exceeds 9, then those numerals are again added and this process is repeated as many times as necessary to obtain a single digit.

Example a.:

72936. 7 + 2 + 9 + 3 + 6 = 27. 2 + 7 = 9

Example b.:

Number chosen by random number generator:

37969. 3 + 7 + 9 + 6 + 9 = 34. 3 + 4 = 7

Hence, all we need do is include a 2 somewhere in the digit string.

372969/9 = 4144

Mystify your friends. Have them pick any string of digits (say 4) and then you silently calculate (it looks better if you don't use a calculator) to see whether the number divides by 9. If so, announce, "This number divides by 9." If not, announce the digit needed to make an integer divisible by 9 (2 in the case above) and then have your friend place that digit anywhere in the integer. Then announce, "This number divides by 9."

In the case of 11, doing tricks isn't quite so easy, but possible.

We check if a number divides by 11 by adding alternate digits as positive and negative. If the sum is zero, the number divides by 11. If the sum exceeds 9, we add the numerals with alternating signs, so that a sum 11 or 77 or the like, will zero out.

Let's check 5863.

We sum 5 - 8 + 6 - 3 = 0

So we can't scramble 5863 any way and have it divide by 11.

However, we can scramble the positively signed numbers or the negatively signed numbers how we please and find that the number divides by 11.

6358 = 11*578

We can also string numbers divisible by 11 together and the resulting integer is also divisible by 11.

253 = 11*23, 143 = 11*13

143253 = 11*13023

Now let's test this pseudorandom number:

70517. The sum of digits is 18 (making it divisible by 9).

We need to get a -18. So any digit string that sums to -18 will do. The easiest way to do that in this case is to replicate the integer and append it since each positive numeral is paired to its negative.

7051770517/11 = 641070047

Now let's do a pseudorandom 4-digit number:

4556. 4 - 5 + 5 - 6 = - 2. Hence 45562 must divide by 11 (obtaining 4142).

Sometimes another trick works.

5894. 5 - 8 + 9 - 4 = 2. So we need a -2, which, in this case can be had by appending 02, ensuring that 2 is found in the negative sum.

Check: 589402/11 = 53582

Let's play with 157311.

Positive digits are 1,7,1
Negative digits are 5, 3, 1

Positive permutations are

117, 711, 171

Negative permutations are

531, 513, 315, 351, 153, 135

So integers divisible by 11 are, for example:

137115 = 11*12465

711315 = 11*64665

Sizzlin' symmetry
There's just something about symmetry...

To form a number divisible by both 9 and 11, we play around thus:

Take a number, say 18279, divisible by 9. Note that it has an odd number of digits, meaning that its copy can be appended such that the resulting number 1827918279 yields a pattern pairing each positive digit with its negative, meaning we'll obtain a 0. Hence 1827918279/11 = 166174389 and that integer divided by 9 equals 20312031. Note that 18279/9 = 2031,

We can also write 1827997281/11 = 166181571 and that number divided by 9 equals 203110809.

Suppose the string contains an even number of digits. In that case, we can write say 18271827 and find it divisible by 9 (equaling 2030203). But it won't divide by 11 in that the positives pair with positive clones and so for negatives. This is resolved by using a 0 for the midpoint.

Thence 182701827/11 = 16609257. And, by the rules given above, 182701827 is divisible by 9, that number being 20300203.

Ah, wonderful symmetry.

2011-11-10T15:45:00.001-08:00

A set of 'gamma' constants

The Euler constant designated g is defined:

lim _b->inf (å_(1,b) 1/i - S_(1,b) 1/x dx) =

lim _b->inf (å_(1,b) 1/i - In b)

We say that a constant is a member of the "gamma" set if there is a formula

lim _b->inf (å_(a,b) f(i) - S_(a,b) f(x) dx) = c, with c non-zero.

A curious thing about this limit is that it differs from lim_(b->inf)å_(a,b)f(i) - lim_(b->inf)S_(a,b)f(x) dx.

That last formula always goes to zero in the limit. Visualize a series of bar graphs with h = 1. So, for i^-1 we put each value side by side. The continuous curve x^-1 intersects the bar graph at integer values. So what we have is the area under the bar graph curve minus the area under the smooth curve. The difference is a sequence of ever-smaller, by percentage, pieces of the bars. Their total area equals gamma.

There is an infinity of formulas whereby the difference between a pair of converging curves yields a constant.

For example, lim _(b->inf) (å_(0,b) 1/i² - S_(0,b) 1/x² dx) = constant less than or equal to p²/6. And in general, for r > 1, lim _(b->inf) (å_(0,b) 1/i^r - S_(0,b) 1/x^r dx = constant. That is, there is a set of left-over pieces of the bars. This area is less than or equal to z(r). We do not know whether any particular r, including r = 2s, corresponds to a rational number.

However, for divergent curves other than the euler constant curves, I have not found a formula which produces a nonzero constant.

The family of Euler constants -- also known as Stieltjes constants -- follows this pattern:

c = lim _(n->inf) (Sum _(1,n) (In m)^r/r - Integral_(1,n) (In x)^m /x dx)

Other cases:

lim _(b->inf) (å_(0,b) i - S_(0,b) x dx). Now at every value of b, we have the b(b-1)/2 - b²/2 = -b/2 -- and lim_(0,b)-b/2 diverges.

Also:

lim_(b->inf) (å_(0,b) i² - S_(0,b) x² dx) = -b²/4, which diverges. Negative divergence also holds for f(i) = i³ and f(x) = x³ and also for exponent 4.

We also have

lim_(b->inf)(å_(0,b) eⁱ - S_(0,b) e^x)dx) =

lim_(b->inf)(å eⁱ - (e^x - 1)) =

lim_(b->inf)(å_(0,b) e^(i-1)) - 1 -- which diverges.

2011-11-10T15:40:00.001-08:00

An interesting zero?

Posted Oct. 16 2009 by Paul Conant     We choose arbitrarily A and B as positive reals, neither of which is proportional to the number e.  We have     A^x + B^x - C^x = 0  We rewrite this as     e^xlnA + e^xlnB - e^xlnC  This gives     1 + xlnA + (xlnA)²/2! + (xlnA)³/3! + ...     1 + xlnB + (xlnB)²/2! + (xlnB)³/3! + ...   - 1 - xlnC - (xlnC)²/2! - (xlnC)³/3! - ...  which equals     1 + x(lnAB/C) + x²[(lnA)² + (lnB)² - (lnC)²]/2! + ...  That is,

0 = 1 + å^¥_j=0 x^j/j![(lnA)^j + (lnB)^j - (lnC)^j]  So we see the sigma sum equals -1 = e^ip  So we see that -1 can be expressed by an infinite family of infinite series.  Further, we may write

x = -åx^j+1/j![(lnA)^j + (lnB)^j - (lnC)^j]

and so any x may be so expressed.

This also holds for z = x + iy = re^iu  And of course, we also have     c = 2/x + [lnAB] + x/2![(lnA)² + (lnB)²] + x²/3![(lnA)³ + (lnB)³] ...       = 2/x + å_j=1^¥x^j-1/j![(lnA)^j + (lnB)^j]

2011-11-10T15:31:00.001-08:00

An objection to proposition 1 of Wittgenstein's 'Tractatus'

From Wittgenstein's 'Tractatus Logico-Philosophicus,' proposition 1:

1. The world is all that is the case.

1.1 The world is the totality of facts, not of things.

1.11 The world is determined by the facts, and by their being ALL the facts.

1.12 For the totality of facts determines what is the case, and also whatever is not the case.

1.13. The facts in logical space are the world.

1.2 The world divides into facts.

1.21 Each item can be the case or not the case while everything else remains the same.

We include proposition 2.0, which includes a key concept:

2.0 What is the case -- a fact -- is the existence of states of affairs [or, atomic propositions].

According to Ray Monk's astute biography, 'Ludwig Wittgenstein, the Duty of Genius' (Free Press division of Macmillan, 1990), Gottlob Frege aggravated Wittgenstein by apparently never getting beyond the first page of 'Tractatus' and quibbling over definitions.

However, it seems to me there is merit in taking exception to the initial assumption, even if perhaps definitions can be clarified (as we know, Wittgenstein later repudiated the theory of pictures that underlay the 'Tractatus'; nevertheless, a great value of 'Tractatus' is the compression of concepts that makes the book a goldmine of topics for discussion).

Before doing that, however, I recast proposition 1 as follows:

1. The world is a theorem.

1.1 The world is the set of all theorems, not of things [a thing requires definition and this definition is either a 'higher' theorem or an axiom]

1.12 The set of all theorems determines what is accepted as true and what is not.

1.13 The set of theorems is the world [redundancy acknowledged]

2. It is a theorem -- a true proposition -- that axioms exist.

This world view, founded in Wittgenstein's extensive mining of Russell's 'Principia' and fascination with Russell's paradox is reflected in the following:

Suppose we have a set of axioms (two will do here). We can build all theorems and anti-theorems from the axioms (though not necessarily solve basic philosophical issues).

With p and q as axioms (atomic propositions that can't be durther divided by connectives and other symbols except for vacuous tautologies and contradictions), we can begin:

1. p, 2. ~p

3. q, 4. ~q

and call these 4 statements Level 0 set of theorems and anti-theorems. If we say 'it is true that p is a theorem' or 'it is true that ~p is an anti-theorem' then we must use a higher order system of numbering. That is, such a statement must be numbered in such a way as to indicate that it is a statement about a statement.

We now can form set Level 1:

5. p & q [theorem]

6. ~p & ~q [anti-theorem]

7. p v q

8. ~p & ~q

9. p v ~q

10. ~p & q

11. ~p v q

12. p & ~q

Level 2 is composed of all possible combinations of p's, q's and connectives, with Level 1 statements combined with Level 2 statements, being a subset of Level 2.

By wise choice of numbering algorithms, we can associate any positive integer with a statement. Also, the truth value of any statement can be ascertained by the truth table method of analyzing such statements. And, it may be possible to find the truth value of statement n by knowing the truth value of sub-statement m, so that reduction to axioms can be avoided in the interest of efficiency.

I have no objection to trying to establish an abstract system using axioms. But the concept of a single system as having a priori existence gives pause.

If I am to agree with Prop 1.0, I must qualify it by insisting on the presence of a human mind, so that 1.0 then means that there is for each mind a corresponding arena of facts. A 'fact' here is a proposition that is assumed true until the mind decides it is false.

I also don't see how we can bypass the notion of 'culture,' which implies a collective set of beliefs and behaviors which acts as an auxiliary memory for each mind that grows within that culture. The interaction of the minds of course yields the evolution of the culture and its collective memory.

Words and word groups are a means of prompting responses from minds (including one's own mind). It seems that most cultures divide words into noun types and verb types. Verbs that cover common occurrences can be noun-ized as in gerunds.

A word may be seen as an auditory association with a specific set of stimuli. When an early man shouted to alert his group to imminent danger, he was at the doorstep of abstraction. When he discovered that use of specific sounds to denote specific threats permitted better responses by the group, he passed through the door of abstraction.

Still, we are assuming that such men had a sense of time and motion about like our own. Beings that perceive without resort to time would not develop language akin to modern speech forms.

In other words, their world would not be our world.

Even beings with a sense of time might differ in their perception of reality. The concept of 'now' is quite difficult to define. However, 'now' does appear to have different meaning in accord with metabolic rate. The smallest meaningful moment of a fly is possibly below the threshold of meaningful human perception. A fly might respond to a motion that is too short for a human to cognize as a motion.

Similarly, another lifeform might have a 'now' considerably longer than ours, with the ultimate 'now' being, theoretically, eternity. Some mystics claim such a time sense.

The word 'deer' (perhaps it is an atomic proposition) does not prove anything about the phenomenon with which it is associated. Deer exist even if a word for a deer doesn't.

Or does it? They exist for us 'because' they have importance for us. That's why we give it a name.

Consider the eskimo who has numerous words for phenomena all of which we English-speakers name 'snow.' We assume that each of these phenomena is an element of a class named 'snow.' But it cannot be assumed that the eskimo perceives these phenomena as types of a single phenomenon. They might be as different as sails and nails as far as he is concerned.

These phenomena are individually named because they are important to him in the sense that his responses to the sets of stimuli that 'signal' a particular phenomenon potentially affect his survival. (We use 'signal' reservedly because the mind knows of the phenomenon only through the sensors [which MIGHT include unconventional sensors, such as spirit detectors].

Suppose a space alien arrived on earth and was able to locomote through trees as if they were gaseous. That being might have very little idea of the concept of tree. Perhaps if it were some sort of scientist, using special detection methods, it might categorize trees by type. Otherwise, a tree would not be part of its world, a self-sevident fact.

What a human is forced to concede is important, at root, is the recurrence of a stimuli set that the memory associates with a pleasure-pain ratio. The brain can add various pleasure-pain ratios as a means of forecasting a probable result.

A stimuli set is normally, but not always, composed of elements closely associated in time. It is when these elements are themselves sets of elements that abstraction occurs.

Much more can be said on the issue of learning. perception and mind but the point I wish to make is that when we come upon logical scenarios, such as syllogisms, we are using a human abstraction or association system that reflects our way of learning and coping with pleasure and pain. The fact that, for example, some pain is not directly physical but is 'worry' does not materially affect my point.

That is, 'reality' is quite subjective, though I have not tried to utterly justify the solipsist point of view. And, if reality is deeply subjective, then the laws of form which seem to describe said reality may well be incomplete.

I suggest this issue is behind the rigid determinism of Einstein, Bohm and Deutsch (though Bohm's 'implicate order' is a subtle and useful concept).

Deutsch, for example, is correct to endorse the idea that reality might be far bigger than ordinarily presumed. Yet, it is his faith that reality must be fully deterministic that indicates that he thinks that 'objective reality' (the source of inputs into his mind) can be matched point for point with the perception system that is the reality he apprehends (subjective reality).

For example, his reality requires that if a photon can go to point A or point B, there must be a reason in some larger scheme whereby the photon MUST go to either A or B, even if we are utterly unable to predict the correct point. But this 'scientific' assumption stems from the pleasure-pain ratio for stimuli sets in furtherance of the organism's probability of survival. That is, determinism is rooted in our perceptual apparatus. Even 'unscientific' thinking is determinist. 'Causes' however are perhaps identified as gods, demons, spells and counter-spells.

Determinism rests in our sense of 'passage of time.'

In the quantum area, we can use a 'Russell's paradox' approach to perhaps justify the Copenhagen interpretation.

Let's use a symmetrical photon interferometer. If a single photon passes through and is left undetected in transit, it reliably exits only in one direction. If, detected in transit, detection results in a change in exit direction in 50 percent of trials. That is, the photon as a wave interferes with itself, exiting in a single direction. But once the wave 'collapses' because of detection, its position is irrevocably fixed and so exits in the direction established at detection point A or detection point B.

Deutsch, a disciple of Hugh Everett who proposed the 'many worlds' theory, argues that the universe splits into two nearly-identical universes when the photon seems to arbitrarily choose A or B, and in fact follows path A in Universe A and path B in Universe B.

Yet, we might use the determinism of conservation to argue for the Copenhagen interpretation. That is, we may consider a light wave to have a minimum quantum of energy, which we call a quantum amount. If two detectors intercept this wave, only one detector can respond because a detector can't be activated by half a quantum unit. Half a quantum unit is effectively nothing. Well, why are the detectors activated probablistically, you say? Shouldn't some force determine the choice?

Here is where the issue of reality enters.

From a classical standpoint, determinism requires ENERGY. Event A at time(0) is linked to event B at time(1) by an expenditure of energy. But the energy needed for 'throwing the switch on the logic gate' is not present.

We might argue that a necessary feature of a logically consistent deterministic world view founded on discrete calculations requires that determinism is also discrete (not continuous) and hence limited and hence non-deterministic at the quantum level.

[The hit counters on all of Paul Conant's pages have been behaving erratically, going up and in numbers with no apparent rhyme or reason.]

[This page first posted January 2002]

2011-11-10T15:29:00.001-08:00

Most Hamiltonian circuit families grow polynomially

Draft

Statement (i) A method exists for obtaining a set of optimal 'traveling salesman' routes that is a member of a family that grows no faster than 0(In2)2ⁿ. (ii) However, even for large n, the method yields a polynomial rate for most sets of fields of nodes.

Remark 1

We operate on a euclidean plane using nodes (cities) that can be specified with cartesian coordinates. We ignore graphs where all points lie on a line.

We adopt the convention that a route begins and ends at an origin city.

Remark 2

We assume that every road linking two cities is straight. This assumption is justified by the fact that a graph composed of curvilinear roads can be deformed into another one that preserves distances -- provided the roads don't cross outside the cities. This proviso is one of the criteria adopted below.

Proof

(i) Our aim is to show that a pretzel circuit can always be replaced by a shorter simple loop (Hamiltonian circuit).

We begin by showing that route backtrackings and crossings are unnecessary.

It is straightforward that if a route covers the same link twice, a shorter route can be drawn.

Suppose a node p is inside a field of nodes and the route enters p from the left, proceeds to x and then backtracks to p exiting toward the right.

                     x                                         n                  p                  m

For non-trivial cases, there must exist some node m to the immediate right of p and some node n to the immediate left. Since mpx is not straight, it is longer than mx. Hence mxpn is shorter than mpxpn.

Similar reasoning holds if p is outside a field.

We show that a route that intersects at a city can be replaced by a shorter route.

Consider any two two-dimensional fields of n nodes and have them intersect at point p.

       [F  I  E  L  D  1]          m          n                p         q         r        [F  I  E  L  D  2]

The optimal route for the new field F₁ U F₂ will not cross at p because of the option mpq linking F₁ and F₂ and nr linking the fields, with nr < npr; or because of the option npr and mq.

So then no route drawn on a graph need cross at a city.

We now show that a route that intersects outside a city can be replaced by a shorter route.

We have some field of four nodes.

        a                        d    b      x                  c

The route acdba crosses at a non-nodal point we call x. Because bxc is not straight, bxc is longer than bc and hence abxcdxa is longer than abcda.

Now we take another four-node field F₂ and attach it to the graph above. But we consider only two-point attachments, as we already know that one-point attachments are not optimal. So it is evident that if F₂'s route does not intersect x, the composite graph's route is shorter than if the alternative holds. An induction proof can be used for any number of fields, of course.

Now suppose we have a ray segment that holds more than two nodes (ignoring origin) adjacent to another ray with at least two.

                     a                       b                                           c                                               O       f       d

How might we link these rays with an un-loop? Suppose cfdba. We will find that more than one link between two adjacent rays is wasteful. The proof is left for the reader.

Also, we know that backtracking is wasteful and so we would not enter the vertical ray at b. In fact our route enters a ray at top or bottom node and leaves at bottom or top. Intermediate nodes can be ignored.

We will not always require that our rays have more than one non-origin node.

(ii)

Our aim is to show a means of obtaining the set of simple loops on any (non-trivial) field of nodes.

We take a field of n nodes and select any node as origin of a set of rays, with each ray drawn through one or more nodes in the field such that all nodes are intersected by rays.

Now our route must begin at origin and link every ray before returning to origin. At origin, we pick any ray to begin and (b = bottom node, t = top node) our path on that ray is determined: 0bt. However, after that there exists a choice. For ray 2 our path may be bt or tb and likewise for subsequent rays. Now if there are 2 nodes per ray, there are n/2 rays and there are about 2^n/2 acceptable paths.

Repeating this procedure for every node in the field, it will be seen that we have established the set of all n-gons, regular or irregular, that can be drawn on the field. Since any route that is not an n-gon is not a simple loop, such a route intersects itself at least once, which we know can be replaced by a shorter circuit.

The set of simple loops would then have an upper limit of n(2^n/2), though it is evident that n is much too high since changes of reference frame will slide nodes out of ray alignment and there is no reason to expect that an equal number of new alignments will occur.

Now suppose we have a field with all rays intersecting two nodes. If we add a point to that field but place it on a ray, the ray's interior point drops out and there is no significant change.

This tends to show that though a set of 0 2ⁿ may occur, it is very unlikely to occur on the next step. But, at this point, we do not rule out the possibility of a set of fields corresponding to an exponential growth rate 0(In2)2^m for m < n.

However, in most cases, a field's upper limit of acceptable routes is given by the number of rays per origin times the number of nodes in the field, or n²(n-1), which gives 3n² - 2n, or 0n², as the rate of change.

Remark 3

It seems apparent that a computer program needs to identify the nodes that form edges of the regular or irregular polygons associated with each node as origin and then to measure only edge distances. Hence we can expect that, for a random field of n nodes, the program will very likely, but perhaps not certainly, run efficiently.

Remark 4

It may be that, for some families of loops, a hard rate continues from n through infinity; or that a hard rate alternates with an efficient rate over infinity; or that the hard rate always zeroes out. The third statement would prove that NP=P.

Assuming that for most sets of fields P-time computer programs can be derived from our method, it may be that encypherment experts may wish to consider their options.

Primes up to 50,000

2011-11-10T15:27:00.001-08:00

When algorithms collide: An infinite sum that isn't (or is it?)

This page went online Dec. 26, 2001

Peano, Hilbert and others devised space-filling curves about a century ago, helping to initiate the field of topology. So, from a topological standpoint, the result below is unremarkable.

Yet, philosophically it seems curious that the same structure can yield two different values.

It seems quite reasonable that an area under a curve is ever better approximated by summing areas defined by rectangular strips that fit the area ever more closely. At the limit of width w = 0, the area is defined as the infinite sum of vertical heights f(x) between x=a and x=b. This calculus approximation algorithm is standard.

Yet it is possible to devise an algorithm which seems to yield a sum of y heights that clashes with the area algorithm.

We create a sawtooth path between some curve f(x) and a finite x axis interval by this means: Divide the interval by h, keeping h a positive odd integer. Then trace the path a to f(a) to f(a+1/h) to f(a+2/h) to a+2/h to a+3/h to f(a+3/h) ... to f(a+n/h) to a+n/h = b. The length of this path is always longer than the path connecting the discrete points of f(x). [There are more than two points between f(a) and f(a+1/h) for the sawtooth path but only two points for the other path.]

As h approaches infinity, the number of sawteeth approaches infinity. At the limit of h=infinity, the sawteeth have narrowed to the infinitesimal area f(x), where the route is presumably twice f(x). But the infinite sum of f(x) cannot be infinite if the area is finite (fits inside a finite square). Quite often, the infinite sum (the integral) is less than the continuous arc length for f(x).

So this would seem to indicate that the sawtooth function must be considered undefined at h = infinity.

The special case of the unit square makes this plain:

Use the unit square lying on the positive x axis between origin and x=1. Divide 1 by the odd (for convenience) positive integer h. Trace a sawtooth path from 0,0 to 0,1 to 1/h,1 to 1/h,0 to 2/h,0 to 2/h,1 and so on to 1,0. The sawtooth path length is h+2. As h approaches infinity, the sawtooth path length also nears infinity.

It would seem that at the limit, the 'infinitesimal entity' is twice the height f(x). The sum is the integral of f(x), which, by the fundamental theorem of calculus, is the area quantified as 1.

So that at infinity, if the sawtooth function exists, the sawtooth route length collapses to distance 1.

But would a long-lived or very speedy salesman traveling through all the points of a sawtooth curve find that the sawtooth route is optimal at infinity?

The area is rarely directly proportional to some x interval or radius. For example, take any finite radius circle and define its radius as 1. Then the area is less than the circumference. But take exactly the same circle and define its radius as 3, and now the area is greater than the circumference. Now make a sawtooth path through half of the same circle.

The salesman will find, as h increases, that his path length exceeds the area quantity for every h after some h=k. If the sawtooth algorithm is completable (an 'actual infinity' is agreed), then he would appear to find that the optimal route is the sawtooth route.

But, there is an infinity of areas that can be assigned to the semicircle. For every area quantity there exists a k after which h means that the sawtooth path length is longer.

This observation can be generalized.

So we conclude that the sawtooth function is either unbounded or undefined at h's infinite limit.

If we say it is unbounded, however, we would need to explain why we are forbidden to sum the f(x) heights.

As my son Jim, a Cornell topologist, points out, sawtooth functions are well-known for producing curves with infinite perimeters that contain a finite area, the Koch snowflake being a famous example.

Of course the issue here is not that such functions occur but that the logical conclusion of this particular sawtooth algorithm yields an anomolous result.

2011-11-10T15:24:00.000-08:00

Disjoint nondenumerable sets of irrationals

[This page went online Feb. 28, 2002.]

I wish to thank Amherst logician Dan Velleman, author of 'How to Prove It,' for not permitting any slack or insecure arguments. He has not vouched for the final version of this theorem and proof. The theorem itself is nothing special. But the proof is amusing in that it gives an inkling of how vast is the sea of irrationals with respect to the rationals.

Theorem: There exists a denumerable infinite family of sets such that each family member is a nondenumerable set of irrationals that is disjoint from every other member.

Proof:

We establish a monotonically climbing differentiable real function h(x), where x is continuous, and where the first derivative exists and is neither a constant nor a periodic (as in cos(ax)). We use only the integer values h(n) in our proof.

We denote a rational q's infinite decimal digit string by S_0 and its recurring finite substring by s. We see that the first digit of s, which has k digit places, recurs every (nk)th digit place. We denote the initial digit place of s by p. We pair p with d_2, which symbolizes a specific numeral digit. Initially, we are assuming a base 10 system.

We then establish the erase-and-replace function f by locating the h(n)th (p, d_2) and replacing d_2 with d_3.

The altered string, S_x, is forever aperiodic.

Example:

s = (01), k = 2, h = n^2

S_0 = 0.010101010101010101...

S_x = 0.210101210101010121...

We know the set of h functions is infinite and we presume that that set, and hence the set of f functions, is denumerable. So we form the indexed set U f_i.

However, this permits us to form a 'diagonal' function g, such that g is formed by the sequence f_1(1), f_2(2), f_3(3) ... ad infinitum. We then alter the g(n)th (p,d_1) by use of a third digit symbol, writing (p,d_3).

Since g is aperiodic and cannot be indexed, we have established a nondenumerable set of irrationals, which we denote R.

In general, we will write R_a to mean the set R derived from the rational q_a.

Indexing the denumerable set of rationals by j, we question whether R_1 intersection R_2 is nonempty. If empty, we proceed consecutively until we find -- if ever -- R_1 intersection R_j is nonempty.

In that event, we form a new function t_i and require use of (p, d_4) on the subset of R_j strings, denoted R'_j, that are identical to R_1 strings. R_j U R'_j we call T_j (which still holds if R_j U R'_j = R_j). We then proceed to the next instance where R_j intersection R_(j+n) is nonempty. Again, we insert digit d_5, obtaining T_(j+n).

Obviously, with a base 10 system, this procedure can be repeated 9 times.

As the number of digit places in s is irrelevant, we can increase our set of available digits with a higher base number system. In that case, we apply induction: If a base m system yields m-1 replacement digits, then a base m+1 system, yields m replacement digits.

With m e M = N, and N denumerable, we find that the family U( T_j) exists, where J = N, requiring that U( T) is denumerable and that for all j e J, n e N(T_j is disjoint from T_(j+n).

Proof concluded.

Remark: It is curious that the set of writable functions is considered denumerable because each expression is composed of a finite string of symbols. For example, 2 + log x, contains 9 symbols, including spaces. Hence, each such expression can be counted, and the entire set of such expressions must also be countable. Since our diagonal function is writable as g(n) = f_n(n), we have shown a nondenumerable subset of writable functions.

2011-11-10T15:20:00.001-08:00

A geometric note on Russell's paradox

A lengthier discussion of Russell's paradox was found on my web page on vacuous truth before it vanished into the vacuum left by the demise of Geocities. It is there that I discuss banishing 'actual infinity' and replacing it with construction algorithms. However, such an approach, though countering other paradoxes, does not rid us of Russell's, which is summed up with the question: If sets are sorted into two types: those that are elements of themselves ('S is the name of a set that contains sets named for letters of the alphabet' is an example) and those that are not, then what type of set is the set that contains sets that are elements of themselves?

Here we regard the null set as the initial set and build sets from there, as in:

Step 0: { }

Step 1: { { { } },{ } }

Using an axiom of infinity, we can continue this process indefinitely, leading directly to a procedure for building an abstraction of all countable sets; indirectly, noncountable sets can also be justified.

In Russell's conundrum, some sets are elements of themselves.

So let us regard a plane as pre-existent (null space or some such) and regard Set sub 0, the empty set, as a circle surrounding nothing. Set sub 1 we picture as a concentric circle around set 0. Now those two circles express the element represented as {{ }}.

We might also say the two circles express the set that contains the element { }.

In that case, we may wish to say that the set {{ }} = the element {{ }}.

Suppose we establish two construction algorithms for sets of concentric circles. In algorithm A, the outer circle is a solid line, so that the null space container is not construed to be part of the set. In algorithm B, the outer circle is a dashed line, so that the null space container is construed to be part of the set.

At each step we add a new enclosing circle and require that it be either solid or dashed.

So as step n is taken to infinity, we find that the null space container vanishes. That is, the circles fill the plane.

In that case, the container is non-existent or undefined, so that the rule that requires that it be either part of the set or not part of the set is not applicable.

We have assumed that the initial circle surrounds something that is not an element -- as with { }. But we really must give a rule that says the inmost area is not an element. We could have a rule that permits not only the outmost but the inmost circle to be dashed. In that case the inner void would be considered a part (element) of the set. Though the void in a null set is, for sound reasons, not permitted to be construed as an element, it is still useful to see the relationship of admitting the inner void to Russell's paradox.

Russell's paradox is usually disposed of by resort to an axiom prohibiting a set from being an element of itself. But we can see that sets and elements can be identical except for the rule prohibiting a set belonging to itself.

For example, element {{ }} in a set-building algorithm X is identical to the set containing the element { } in set-building algorithm Y.

It seems that the axiom while necessary remains questionable from the standpoint of logical consistency.

First published ca. 2000

An algorithm to imply the set of reals

2011-11-10T15:14:00.000-08:00

I have reconsidered my initial idea that the algorithm I give proves non-denumerability of the reals, an idea which I based on the fact that there is no n for which a cellular diagonal exists that intercepts every row of cells. However, despite intuition, it does not immediately follow that no such diagonal exists for the infinite set.

An algorithm to imply the set of reals

We present an algorithm for "constructing" the entire set of reals and find that our intuition suggests that no diagonal can exist, which accords with Cantor's proof.

Cantor's diagonalization proof says that if it were possible that all reals were listed, then one could create an anti-diagonal number from the diagonal that intersects all the listed numbers. Hence, no such diagonal exists and the set of reals is not 1-to-1 with the set of naturals.

However, we are assuming that there is a set N containing all naturals. We are also assuming that because a finite n x n square always has a (non-Pythagorean diagonal of n cells) that intersects every row, then the same holds for the tiled quarter-plane.

However, the infinity axiom says that N exists, in which case we may say that the set of X axis integers is 1-to-1 with the set of Y axis integers, and further that the cellular diagonal with a cell vertex at (0,0) has cardN and intersects every horizontal string of cells bounded by y and y+1. In that case, we may use Cantor's argument to say that the reals can't be mapped 1-to-1 onto this tiling.

Interestingly, our algorithm for constructing all the reals intuitively says that such a set has no diagonal. But, that intuition is simply an intuition until we apply Cantor's proof and the infinity axiom.

We write an algorithm for determining the set of reals, thus:

We have the set of 1x1 square cells in a quadrant of the Cartesian grid. Each square is eligible to contain a digit. We consider only the horizontal strings of cells.

Suppose we use a base 2 system. We begin at step 0 using 2 consecutive spaces and obtaining 4 possible combinations, to wit: 00, 11, 10, 01. For step 1 (3 spaces), the number of combinations is 2*4 and for step n the number of combinations is 2ⁿ⁺².

At lim_n->inf. 2ⁿ⁺², we have established all base 2 infinite digit strings, expressing the entire set of reals greater than an arbitrary j e Z and less than or equal to j+1.

Remark: Our algorithm does not compute reals sequentially. It only computes 'pre-reals,' since a real is defined here as an infinite digit string. No element "materializes" prior to the entire set's establishment at (denumerable) infinity.

The algorithm above requires that for any step n, the set of digit strings is a rectangle n2ⁿ⁺².

But lim_n->inf. n/2ⁿ = 0, meaning the rectangle elongates and narrows toward a limiting form of a half-line.

For an integer diagonal to include an element of all horizontal digit strings, we must have a square of n columns and n rows. But such a square of the reals is never attainable. It would then seem safe to say that the set of reals, expressed as infinite digit strings, has no diagonal, which is equivalent to saying the set of reals is non-denumerable.

However, our intuition that the set of reals so constructed should have no diagonal is provable by agreement to the infinity axiom, which permits the cardN diagonal, and by Cantor's anti-diagonal result.

It also follows that no exponentially defined set of tiles n x kⁿ has a cellular diagonal at the tiling algorithm's infinite limit.

On the other hand, a tiling defined by n^k can be said to have k cellular diagonals, such that collectively the k diagonals intersect every horizontal cellular string. It then can be shown that such a tiling is one-to-one with N.

Interestingly, the power set of any n e N has card2ⁿ, which corresponds to step n of our algorithm, in which we have a set of 2ⁿ⁺² pre-reals.

Additional remark:

Lemma: Any set with lim_n->inf kⁿ elements has the cardinality of the reals, with k =/= 0, -1, 1 and k e Z.

Proof:

The set of reals is 1-to-1 with a set that has lim_n->inf 2ⁿ elements. Hence the cardinality of each set is identical. Similarly, the algorithm above can be rewritten as ckⁿ, with c a nonzero integer constant, meaning that all real digit strings are established at lim_n->inf ckⁿ.

Theorem: Some non-enumerable reals can be approximated with explicit rationals to any degree of accuracy in a finite number of steps.

Proof:

Construct n Turing machines consecutively, truncating the initial integer, and compute each one's output as a digit string n digits long. Use some formula to change each diagonal digit.

The infinite diagonal cannot be encoded as a Turing machine number, so it is not enumerable. Yet a computer can compute its approximation as a rational up to n. (The accuracy of this approximation is the same as the accuracy obtainable, in principle, for an enumerable irrational.)

Comment: The denumerable set of computables implies an extension of the concept of denumerability.

Justification:

We give these instructions for diagonalizing Turing computables:

Up to and including the nth diagonal space, follow this rule: if a digit is not 0, replace it with 0; if 0, replace it with 1. After the nth diagonal space, follow this rule: if a digit is not 2, replace it with 2; if it is 2, replace it with 3.

None of these diagonals is enumerable with respect to the Turing numbers. Yet we have a countably infinite set of diagonals. Hence, non-denumerability implies the existence of two denumerable sets of reals which are not denumerable with respect to each other.

If we diagonalize the diagonals, it is not apparent to me that this real is not a member of the computables.

Definition of choice sequence:

If f(n) gives a_n of cauchy sequence {a_n}, then {a_n} is a choice sequence if f(n) --> a_{o_n+1} or a_{1_n+1} or . . . or a_{m_n+1}.

Note i.Since a choice sequence is cauchy |a_m - a_n| <= 1/k for all m and n after some n_o. However, the rule for determining step n+1 means that more than one choice sequence is possible for every n after some n_o. That is, a choice sequence's limiting value must fall within an upper and lower bound.

Note ii: It may be that a_{x_n+1} is non-randomly determined. Yet, there exists an infinity of choice sequences such that the limiting value of {a_n} is an effectively random element of some infinite subset of reals (known as a 'spread') bounded by a least upper bound and a greatest lower bound.

Remark: Though choice sequences are primarily of interest to intuitionists, here we require that they be governed by ZFC.

Theorem: The question of whether the set of choice sequences contains an element with a non-enumerable limiting value is not decidable.

Proof:

We first prove (Lemma i) that within a spread (x,y), with x the GLB and y the LUB, a non-enumerable exists.

Use a diagonalization formula on the digit string outputs from the set of Turing machines, obtaining one non-enumerable real. Prefix, in turn, every rational digit string to this real and then move the decimal point to the front of each new string. Lemma i is proved.

So then suppose x and y are irrational enumerables. Rationals arbitrarily close to x from above and to y from below can be found.

Let x < p and q < y. So calling the choice sequence limit L, we have

Case i: (x < p < L < q < y).

Case ii: The possibility x < L < p exists, but then a smaller rational can be found between x and L.

Case iii: Likewise for the possibility q < L < y.

It is now straightforward that if L is a choice function limit, there is an effectively random possibility that L is enumerable or non-enumerable. This possibility is in principle undecidable.

Though probability laws suggest that the set of choice sequences includes a sequence with a non-enumerable limit, this suggestion is undecidable.

The cosmos cannot be fully modeled as a Turing machine or Turing computation

2011-11-10T14:55:00.001-08:00

Dave Selke, an electrical engineer with a computer background, has made a number of interesting comments concerning this page, spurring me to redo the argument in another form. The new essay is entitled "On Hilbert's sixth problem" and may be found at

http://paulpages.blogspot.com/2011/11/first-published-tuesday-june-26-2007-on.html

Draft 3 (Includes discussion of Wolfram cellular automata)
Comments and suggestions welcome

Note: The word "or" is usually used in the following discussion in the inclusive sense.

Many subscribe to the view that the cosmos is essentially a big machine which can be analyzed and understood in terms of other machines.

A well-known machine is the general Turing machine, which is a logic system that can be modified to obtain any discrete-input computation. Richard Feynman, the brilliant physicist, is said to have been fascinated by the question of whether the cosmos is a computer -- originally saying no but later insisting the opposite. As a quantum physicist, Feynmen would have realized that the question was difficult. If the cosmos is a computer, it certainly must be a quantum computer. But what does that certainty mean? Feynmen, one assumes, would also have concluded that the cosmos cannot be modeled as a classical computer, or Turing machine [see footnote below].

Let's entertain the idea that the cosmos can be represented as a Turing machine or Turing computation. This notion is equivalent to the idea that neo-classical science (including relativity theory) can explain the cosmos. That is, we could conceive of every "neo-classical action" in the cosmos to date -- using absolute cosmic time, if such exists -- as being represented by a huge logic circuit, which in turn can be reduced to some instance (computation) of a Turing algorithm. God wouldn't be playing dice.

A logic circuit always follows if-then rules, which we interpret as causation. But, as we know, at the quantum level, if-then rules only work (with respect to the observer) within constraints, so we might very well argue that QM rules out the cosmos being a "classical" computer.

On the other hand, some would respond by arguing that quantum fuzziness is so miniscule on a macroscopic (human) scale, that the cosmos can be quite well represented as a classical machine. That is, the fuzziness cancels out on average. They might also note that quantum fluctuations in electrons do not have any significant effect on the accuracy of computers -- though this may not be true as computer parts head toward the nanometer scale. (My personal position is that there are numerous examples of the scaling up or amplification of quantum effects. "Schrodinger's cat" is the archetypal example.)

Of course, another issue is that the cosmos should itself have a wave function that is a superposition of all possible states -- until observed by someone (who?). (I will not proceed any further on the measurement problem of quantum physics, despite its many fascinating aspects.)

Before going any further on the subject at hand, we note that a Turing machine is finite (although the set of such machines is denumerably infinite). So if one takes the position that the cosmos -- or specifically, the cosmic initial conditions (or "singularity") -- are effectively infinite, then no Turing algorithm can model the cosmos.

So let us consider a mechanical computer-robot, A, whose program is a general Turing machine. A is given a program that instructs the robotic part of A to select a specific Turing machine, and to select the finite set of initial values (perhaps the "constants of nature"), that models the cosmos.

What algorithm is used to instruct A to choose a specific cosmos-outcome algorithm and computation? This is a typical chicken-or-the-egg self-referencing question and as such is related to Turing's halting problem, Godel's incompleteness theorem and Russell's paradox.

If there is an algorithm B to select an algorithm A, what algorithm selected B? -- leading us to an infinite regression.

Well, suppose that A has the specific cosmic algorithm, with a set of discrete initial input numbers, a priori? That algorithm, call it T_c, and its instance (the finite set of initial input numbers and the computation, which we regard as still running), imply the general Turing algorithm T_g. We know this from the fact that, by assumption, a formalistic description of Alan Turing and his mathematical logic result were implied by T_c. On the other hand, we know that every computable result is programable by modifying T_g. All computable results can be cast in the form of "if-then" logic circuits, as is evident from Turing's result.

So we have

T_c <--> T_g

Though this result isn't clearly paradoxical, it is a bit disquieting in that we have no way of explaining why Turing's result didn't "cause" the universe. That is, why didn't it happen that T_g implied Turing who (which) in turn implied the Big Bang? That is, wouldn't it be just as probable that the universe kicked off as Alan Turing's result, with the Big Bang to follow? (This is not a philisophical question so much as a question of logic.)

Be that as it may, the point is that we have not succeeded in fully modeling the universe as a Turing machine.

The issue in a nutshell: how did the cosmos instruct itself to unfold? Since the universe contains everything, it must contain the instructions for its unfoldment. Hence, we have the T_c instructing its program to be formed.

Another way to say this: If the universe can be modeled as a Turing computation, can it also be modeled as a program? If it can be modeled as a program, can it then be modeled as a robot forming a program and then carrying it out?

In fact, by Godel's incompleteness theorem, we know that the issue of T_c "choosing" itself to run implies that the T_c is a model (mathematically formal theory) that is inconsistent or incomplete. This assertion follows from the fact that the T_c requires a set of axioms in order to exist (and hence "run"). That is, there must be a set of instructions that orders the layout of the logic circuit. However, by Godel's result, the Turing machine is unable to determine a truth value for some statements relating to the axioms without extending the theory ("rewiring the logic circuit") to include a new axiom.

This holds even if T_c = T_g (though such an equality implies a continuity between the program and the computation which perforce bars an accurate model using any Turing machines).

So then, any model of the cosmos as a Boolean logic circuit is inconsistent or incomplete. In other words, a Turing machine cannot fully describe the cosmos.

If by "Theory of Everything" is meant a formal logico-mathematical system built from a finite set of axioms [though, in fact, Zermelo-Frankel set theory includes an infinite subset of axioms], then that TOE is either incomplete or inconsistent. Previously, one might have argued that no one has formally established that a TOE is necessarily rich enough for Godel's incompleteness theorem to be known to apply. Or, as is common, the self-referencing issue is brushed aside as a minor technicality.

Of course, the Church thesis essentially tells us that any logico-mathematical system can be represented as a Turing machine or set of machines and that any logico-mathematical value that can be expressed from such a system can be expressed as a Turing machine output. (Again, Godel puts limits on what a Turing machine can do.)

So, if we accept the Church thesis -- as most logicians do -- then our result says that there is always "something" about the cosmos that Boolean logic -- and hence the standard "scientific method" -- cannot explain.

Even if we try representing "parallel" universes as a denumerable family of computations of one or more Turing algorithms, with the computational instance varying by input values, we face the issue of what would be used to model the master programer.

Similarly, one might imagine a larger "container" universe in which a full model of "our" universe is embedded. Then it might seem that "our" universe could be modeled in principle, even if not modeled by a machine or computation modeled in "our" universe. Of course, then we apply our argument to the container universe, reminding us of the necessity of an infinity of extensions of every sufficiently rich theory in order to incorporate the next stage of axioms and also reminding us that in order to avoid the paradox inherent in the set of all universes, we would have to resort to a Zermelo-Frankel-type axiomatic ban on such a set.

Now we arrive at another point: If the universe is modeled as a quantum computation, would not such a framework possibly resolve our difficulty?

If we use a quantum computer and computation to model the universe, we will not be able to use a formal logical system to answer all questions about it, including what we loosely call the "frame" question -- unless we come up with new methods and standards of mathematical proof that go beyond traditional Boolean analysis.

Let us examine the hope expressed in Stephen Wolfram's New Kind of Science that the cosmos can be summarized in some basic rule of the type found in his cellular automata graphs.

We have no reason to dispute Wolfram's claim that his cellular automata rules can be tweaked to mimic any Turing machine. (And it is of considerable interest that he finds specific CA/TM that can be used for a universal machine.)

So if the cosmos can be modeled as a Turing machine then it can be modeled as a cellular automaton. However, a CA always has a first row, where the algorithm starts. So the algorithm's design -- the Turing machine -- must be axiomatic. In that case, the TM has not modeled the design of the TM nor the specific initial conditions, which are both parts of a universe (with that word used in the sense of totality of material existence).

We could of course think of a CA in which the first row is attached to the last row and a cylinder formed. There would be no specific start row. Still, we would need a CA whereby the rule applied with aribitrary row n as a start yields the same total output as the rule applied at arbitrary row m. This might resolve the time problem, but it is yet to be demonstrated that such a CA -- with an extraordinarily complex output -- exists. (Forgive the qualitative term extraordinarily complex. I hope to address this matter elsewhere soon.)

However, even with time out of the way, we still have the problem of the specific rule to be used. What mechanism selects that? Obviously it cannot be something from within the universe. (Shades of Russell's paradox.)

Footnote

Informally, one can think of a general Turing machine as a set of logic gates that can compose any Boolean network. That is, we have a set of gates such as "not", "and," "or," "exclusive or," "copy," and so forth. If-then is set up as "not-P or Q," where P and Q themselves are networks constructed from such gates. A specific Turing machine will then yield the same computation as a specific logic circuit composed of the sequence of gates.

By this, we can number any computable output by its gates. Assuming we have less than 10 gates (which is more than necessary), we can assign a base-10 digit to each gate. In that case, the code number of the circuit is simply the digit string representing the sequence of gates.

Note that circuit A and circuit B may yield the same computation. Still, there is a countable infinity of such programs, though, if we use any real for an input value, we would have an uncountable infinity of outputs. But this cannot be, because an algorithm for producing a real number in a finite number of steps can only produce a rational approximation of an irrational. Hence, there is only a countable number of outputs.

*************************

Thanks to Josh Mitteldorf, a mathematician and physicist, for his incisive and helpful comments. Based upon a previous draft, Dr. Mitteldorf said he believes I have shown that, if the universe is finite, it cannot be modeled by a subset of itself but he expressed wariness over the merit of this point.

2011-11-10T14:50:00.001-08:00

Alternate proof

of the Schroeder-Bernstein theorem

PlanetMath's standard proof

Wikipedia's standard proof

Draft 2   The following proof of the Schroeder-Bernstein theorem is, as far as I know, new as of April 2006. The standard proof can be found at the links above.   Notation:  "cardA" denotes the cardinal number of a set A. "<=" means "less than or equal to." "~" means "equinumerous to."

Schroeder-Bernstein theorem: If, and only if, A and B are sets such that card A <= cardB and card B <= card A, then cardA = cardB.

Proof:

Remark: We grant that the transitivity of subsets is established from axioms. That is, A Í B Í C implies A Í C. Further, for nonempty sets, we accept that if A Ì B, then B Ì A is false. This follows from the definition of subset: for all x, x element of X implies x element of Y. But, if A is a proper subset of A, then, for some b, b element of B implies b is not an element of A. Hence B does not fit the definition of subset of A.

We rewrite cardA <= cardB and card B <= card A, thus:

A ~ C Í B and B ~ D Í A

Suppose A ~ C = B. Then we are done. Likewise for B ~ D = A. So we test the situation for nonempty sets with: A ~ C Ì B and B ~ D Ì A. (We note that if C ~ D, then A ~ B, spoiling our interim hypothesis. So we assume C is not equinumerous to D.)

We now form the sets A U C and B U D, permitting us to write (A U C) Ì (A U B) and (B U D) Ì (A U B).

We now have two options:

i) (A U C) Ì (A U B) Ì (B U D) Ì (A U B)

This is a contradiction since A U B cannot be a proper subset of itself.

ii) (A U C) Ì (A U B) É (B U D) Ì (A U B)

Also a contradiction, since B U D cannot be a proper subset of A U B and also properly contain A U B.

Only if

We let B = C and D = A.

This proof, like the standard modern proof, does not rely on the Axiom of Choice.

2011-11-10T14:47:00.001-08:00

In search of a blind watchmaker

Richard Dawkins' web site
Wikipedia article on Dawkins
Wikipedia article on Francis Crick
Abstract of David Layzer's two-tiered adaptation
Joshua Mitteldorf's home page
Do dice play God? A book review

A discussion of The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design by Richard Dawkins  First posted Oct. 5, 2010 and revised as of Oct. 8, 2010  Please notify me of errors or other matters at "krypto78...at...gmail...dot...com"

By PAUL CONANT

Surely it is quite unfair to review a popular science book published years ago. Writers are wont to have their views evolve over time.¹ Yet in the case of Richard Dawkins' The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design (W.W. Norton 1986), a discussion of the mathematical concepts seems warranted, because books by this eminent biologist have been so influential and the "blind watchmaker" paradigm is accepted by a great many people, including a number of scientists.

Dawkins' continuing importance can be gauged by the fact that his most recent book, The God Delusion (Houghton Mifflin 2006), was a best seller, and by the links above. In fact, Watchmaker, also a best seller, was re-issued in 2006.

I do not wish to disparage anyone's religious or irreligious beliefs, but I do think it important to point out that non-mathematical readers should beware the idea that Dawkins has made a strong case that the "evidence of evolution reveals a universe without design."

There is little doubt that some of Dawkins' conjectures and ideas in Watchmaker are quite reasonable. However, many readers are likely to think that he has made a mathematical case that justifies the theory(ies) of evolution, in particular the "modern synthesis" that combines the concepts of passive natural selection and genetic mutation.

Dawkins wrote his apologia back in the eighties when computers were becoming more powerful and accessible, and when PCs were beginning to capture the public fancy. So it is understandable that, in this period of burgeoning interest in computer-driven chaos, fractals and cellular automata, he might have been quite enthusiastic about his algorithmic discoveries.

However, interesting computer programs may not be quite as enlightening as at first they seem.

Cumulative selection

Let us take Dawkins' argument about "cumulative selection," in which he uses computer programs as analogs of evolution. In the case of the phrase, "METHINKS IT IS LIKE A WEASEL," the probability -- using 26 capital letters and a space -- of coming up with such a sequence randomly is 27^-28 (the astonishingly remote 8.3 x 10^-41). However, that is also the probability for any random string of that length, he notes, and we might add that for most probability distributions. when n is large, any distinct probability approaches 0.

Such a string would be fantastically unlikely to occur in "single step evolution," he writes. Instead, Dawkins employs cumulative selection, which begins with a random 28-character string and then "breeds from" this phrase. "It duplicates it repeatedly, but with a certain chance of random error -- 'mutation' -- in the copying. The computer examines the mutant nonsense phrases, the 'progeny' of the original phrase, and chooses the one which, however slightly most resembles the target phrase, METHINKS IT IS LIKE A WEASEL."

Three experiments evolved the precise sentence in 43, 64 and 41 steps, he wrote.

Dawkins' basic point is that an extraordinarily unlikely string is not so unlikely via "cumulative selection."

Once he has the readers' attention, he concedes that his notion of natural selection precludes a long-range target and then goes on to talk about "biomorph" computer visualizations (to be discussed below).

Yet it should be obvious that Dawkins' "methinks" argument applies specifically to evolution once the mechanisms of evolution are at hand. So the fact that he has been able to design a program which behaves like a neural network, really doesn't say much about anything. He has achieved a proof of principle that was not all that interesting, although I suppose it would answer a strict creationist, which was perhaps his basic aim.

But which types of string are closer to the mean? Which ones occur most often? If we were to subdivide chemical constructs into various sets, the most complex ones -- which as far as we know are lifeforms -- would be farthest from the mean.(Dawkins, in his desire to appeal to the lay reader, avoids statistics theory other than by supplying an occasional quote from R.A. Fisher.)

Let us, like Dawkins, use a heuristic analog². Suppose we take the set of all grammatical English sentences of 28 characters. The variable is an English word rather than a Latin letter or space. What would be the probability of any 28-character English sentence appearing randomly?

My own sampling of a dictionary found that words with eight letters appear with the highest probability of 21%. So assuming the English lexicon to contain 500,000 words, we obtain about 105,000 words of length 8.

Now let us do a Fermi-style rough estimate. For the moment ignoring spaces, we'll posit average word length of 2 to 9 as covering virtually all combinations. That is, we'll pretend there are sentences composed of only two-letter words, only three-letter and so on up to nine letters. Further, we shall put an upper bound of 10⁵ on the set of words of any relevant length (dropping the extra 5,000 for eight-letter words as negligible for our purposes).

This leads to a total number of combinations of (10⁵)² + 10⁸ + ... + 10¹⁴, which approximates 10¹⁴.

We have not considered spaces nor (directly) combinations of words of various lengths. It seems overwhelmingly likely that any increases would be canceled by the stricture that sentences be grammatical, something we haven't modeled. But, even if the number of combinations were an absurd 10 orders of magnitude higher, the area under the part of some typical probability curve that covers all grammatical English sentences of length 28 would take up a miniscule percentage of a tail.

Analogously, to follow Dawkins, we would suspect that the probability is likewise remote for random occurrence of any information structure as complex as a lifeform.

To reiterate, the entire set of English sentences of 28 characters is to be found far out in the tail of some probability distribution. Of course, we haven't specified which distribution because we have not precisely defined what is meant by "level of complexity." This is also an important omission by Dawkins.

We haven't really done much other than to underscore the lack of precision of Dawkins' analogy.

Dawkins then goes on to talk about his "biomorph" program, in which his algorithm recursively alters the pixel set, aided by his occasional selecting out of unwanted forms. He found that some algorithms eventually evolved insect-like forms, and thought this a better analogy to evolution, there having been no long-term goal. However, the fact that "visually interesting" forms show up with certain algorithms again says little. In fact, the remoteness of the probability of insect-like forms evolving was disclosed when he spent much labor trying to repeat the experiment because he had lost the exact initial conditions and parameters for his algorithm. (And, as a matter of fact, he had become an intelligent designer with a goal of finding a particular set of results.)

Again, what Dawkins has really done is use a computer to give his claims some razzle dazzle. But on inspection, the math is not terribly significant.

It is evident, however, that he hoped to counter Fred Hoyle's point that the probability of life organizing itself was equivalent to a tornado blowing through a junkyard and assembling from the scraps a fully functioning 747 jetliner, Hoyle having made this point not only with respect to the origin of life, but also with respect to evolution by natural selection.

So before discussing the origin issue, let us turn to the modern synthesis.

The modern synthesis

I have not read the work of R.A. Fisher and others who established the modern synthesis merging natural selection with genetic mutation, and so my comments should be read in this light.

Dawkins argues that, although most mutations are either neutral or harmful, there are enough progeny per generation to ensure that an adaptive mutation proliferates. And it is certainly true that, if we look at artificial selection -- as with dog breeding -- a desirable trait can proliferate in very short time periods, and there is no particular reason to doubt that if a population of dogs remained isolated on some island for tens of thousands of years that it would diverge into a new species, distinct from the many wolf sub-species.

But Dawkins is of the opinion that neutral mutations that persist because they do no harm are likely to be responsible for increased complexity. After all, relatively simple life forms are enormously successful at persisting.

And, as Stephen Wolfram points out (A New Kind of Science, Wolfram Media 2006), any realistic population size at a particular generation is extremely unlikely to produce a useful mutation because the ratio of possible mutations to the number of useful ones is some very low number. So Wolfram also believes neutral mutations drive complexity.

We have here two issues:

1. If complexity is indeed a result of neutral mutations alone, increases in complexity aren't driven by selection and don't tend to proliferate.

2. Why is any species at all extant? It is generally assumed that natural selection winnows out the lucky few, but does this idea suffice for passive filtering?

Though Dawkins is correct when he says that a particular mutation may be rather probable by being conditioned by the state of the organism (previous mutation), we must consider the entire chain of mutations represented by a species.

If we consider each species as representing a chain of mutations from the primeval organism, then we have a chain of conditional probability. A few probabilities may be high, but most are extremely low. Conditional probabilities can be graphed as trees of branching probabilities, so that a chain of mutation would be represented by one of these paths. We simply multiply each branch probability to get the total probability per path.

As a simple example, a 100-step conditional probability path with 10 probabilities of 0.9 and 60 with 0.7 and 30 with 0.5 yields a cumulative probability of 1.65 x 10^-19.

In other words, the more mutations and ancestral species attributed to an extanct species, the less likely it is to exist via passive natural selection. The actual numbers are so remote as to make natural selection by passive filtering virtually impossible, though perhaps we might conjecture some nonlinear effect going on among species that tends to overcome this problem.

Dawkins' algorithm demonstrating cumulative evolution fails to account for this difficulty. Though he realizes a better computer program would have modeled lifeform competition and adaptation to environmental factors, Dawkins says such a feat was beyond his capacities. However, had he programed in low probabilities for "positive mutations," cumulative evolution would have been very hard to demonstrate.

Our second problem is what led Hoyle to revive the panspermia conjecture, in which life and proto-life forms are thought to travel through space and spark earth's biosphere. His thinking was that spaceborne lifeforms rain down through the atmosphere and give new jolts to the degrading information structures of earth life. (The panspermia notion has received much serious attention in recent years, though Hoyle's conjectures remain outside the mainstream.)

From what I can gather, one of Dawkins' aims was to counter Hoyle's sharp criticisms. But Dawkins' vigorous defense of passive natural selection does not seem to square with the probabilities, a point made decades previously by J.B.S. Haldane.

Without entering into the intelligent design argument, we can suggest that the implausible probabilities might be addressed by a neo-Lamarkian mechanism of negative feedback adaptations. Perhaps a stress signal on a particular organ is received by a parent and the signal transmitted to the next generation. But the offspring's genes are only acted upon if the other parent transmits the signal. In other words, the offspring embryo would not strengthen an organ unless a particular stress signal reached a threshold.

If that be so, passive natural selection would still play a role, particularly with respect to body parts that lose their role as essential for survival.

Dawkins said Lamarkianism had been roundly disproved, but since the time he wrote the book molecular biology has shown the possibility of reversal of genetic information (retroviruses and reverse transcription). However, my real point here is not about Lamarkianism but about Dawkins' misleading mathematics and reasoning.

Joshua Mitteldorf, an evolutionary biologist with a physics background and a Dawkins critic, points out that an idea proposed more than 30 years ago by David Layzer is just recently beginning to gain ground as a response to the cumulative probabilities issue. Roughly I would style Layzer's proposal a form of neo-Lamarckianism. The citation³ is found at the bottom of this essay and the link is posted above.

On origins

Dawkins concedes that the primeval cell presents a difficult problem, the problem of the arch. If one is building an arch, one cannot build it incrementally stone by stone because at some point, a keystone must be inserted and this requires that the proto-arch be supported until the keystone is inserted. The complete arch cannot evolve incrementally. This of course is the essential point made by the few scientists who support intelligent design.

Dawkins essentially has no answer. He says that a previous lifeform, possibly silicon-based, could have acted as "scaffolding" for current lifeforms, the scaffolding having since vanished. Clearly, this simply pushes the problem back. Is he saying that the problem of the arch wouldn't apply to the previous incarnation of "life" (or something lifelike)?

Some might argue that there is a possible answer in the concept of phase shift, in which, at a threshold energy, a disorderly system suddenly becomes more orderly. However, this idea is left unaddressed in Watchmaker. I would suggest that we would need a sequence of phase shifts that would have a very low cumulative probability, though I hasten to add that I have insufficient data for a well-informed assessment.

Cosmic probabilities

Is the probability of life in the cosmos very high, as some think? Dawkins argues that it can't be all that high, at least for intelligent life, otherwise we would have picked up signals. I'm not sure this is valid reasoning, but I do accept his notion that if there are a billion life-prone planets in the cosmos and the probability of life emerging is a billion to one, then it is virtually certain to have originated somewhere in the cosmos.

Though Dawkins seems to have not accounted for the fact that much of the cosmos is forever beyond the range of any possible detection as well as the fact that time gets to be a tricky issue on cosmic scales, let us, for the sake of argument, grant that the population of planets extends to any time and anywhere, meaning it is possible life came and went elsewhere or hasn't arisen yet, but will, elsewhere.

Such a situation might answer the point made by Peter Ward and Donald Brownlee in Rare Earth: Why Complex Life Is Uncommon in the Universe (Springer 2000) that the geophysics undergirding the biosphere represents a highly complex system (and the authors make efforts to quantify the level of complexity), meaning that the probability of another such system is extremely remote. (Though the book was written before numerous discoveries concerning extrasolar planets, thus far their essential point has not been disproved. And the possibility of non-carbon-based life is not terribly likely because carbon valences permit high levels of complexity in their compounds.)

Now some may respond that it seems terrifically implausible that our planet just happens to be the one where the, say, one-in-a-billion event occurred. However, the fact that we are here to ask the question is perhaps sufficient answer to that worry. If it had to happen somewhere, here is as good a place as any. A more serious concern is the probability that intelligent life arises in the cosmos.

The formation of multicellular organisms is perhaps the essential "phase shift" required, in that central processors are needed to organize their activities. But what is the probability of this level of complexity? Obviously, in our case, the probability is one, but, otherwise, the numbers are unavailable, mostly because of the lack of a mathematically precise definition of "level of complexity" as applied to lifeforms.

Nevertheless, probabilities tend to point in the direction of cosmically absurd: there aren't anywhere near enough atoms -- let alone planets -- to make such probabilities workable. Supposing complexity to result from neutral mutations, probability of multicellular life would be far, far lower than for unicellular forms whose speciation is driven by natural selection. Also, what is the survival advantage of self-awareness, which most would consider an essential component of human-like intelligence?

Hoyle's most recent idea was that probabilities were increased by proto-life in comets that eventually reached earth. But, despite enormous efforts to resolve the arch problem (or the "jumbo jet problem"), in my estimate he did not do so.

(Interestingly, Dawkins argues that people are attracted to the idea of intelligent design because modern engineers continually improve machinery designs, giving a seemingly striking analogy to evolution. Something that he doesn't seem to really appreciate is that every lifeform may be characterized as a negative-feedback controlled machine, which converts energy into work and obeys the second law of thermodynamics. That's quite an "arch.")

The intelligent design proponents, however, face a difficulty when relying on the arch analogy: the possibility of undecidability. As the work of Godel, Church, Turing and Post shows, some theorems cannot be proved by tracking back to axioms. They are undecidable. If we had a complete physical description of the primeval cell, we could encode that description as a "theorem." But, that doesn't mean we could track back to the axioms to determine how it emerged. If the "theorem" were undecidable, we would know it to be "true" (having the cell description in all detail), but we might be forever frustrated in trying to determine how it came to exist.

In other words, a probabilistic argument is not necessarily applicable.

The problem of sentience

Watchmaker does not examine the issue of emergence of human intelligence, other than as a matter of level of complexity.

Hoyle noted in The Intelligent Universe (Holt, Rhinehart and Winston 1984) that over a century ago, Alfred Russel Wallace was perplexed by the observation that "the outstanding talents of man... simply cannot be explained in terms of natural selection."

Hoyle quotes the Japanese biologist S. Ohno:

"Did the genome (genetic material) of our cave-dwelling predecessors contain a set or sets of genes which enable modern man to compose music of infinite complexity and write novels with profound meaning? One is compelled to give an affirmative answer...It looks as though the early Homo was already provided with the intellectual potential which was in great excess of what was needed to cope with the environment of his time."

Hoyle proposes in Intelligent that viruses are responsible for evolution, accounting for mounting complexity over time. However, this seems hard to square with the point just made that such complexity doesn't seem to occur as a result of passive natural winnowing and so there would be no selective "force" favoring its proliferation.

At any rate, I suppose that we may assume that Dawkins in Watchmaker saw the complexity inherent in human intelligence as most likely to be a consequence of neutral mutations.

Another issue not addressed by Dawkins (or Hoyle for that matter) is the question of self-awareness. Usually the mechanists see self-awareness as an epiphenomenon of a highly complex program (a notion Roger Penrose struggled to come to terms with in The Emperor's New Mind (Oxford 1986) and Shadows of the Mind (Oxford 1994).)

But let us think of robots. Isn't it possible in principle to design robots that multiply replications and maintain homeostasis until they replicate? Isn't it possible in principle to build in programs meant to increase probability of successful replication as environmental factors shift?

In fact, isn't it possible in principle to design a robot that emulates human behaviors quite well? (Certain babysitter robots are even now posing ethics concerns as to an infant's bonding with them.)

And yet there seems to be no necessity for self-awareness in such designs. Similarly, what would be the survival advantage of self-awareness for a species?

I don't suggest that some biologists haven't proposed interesting ideas for answering such questions. My point is that Watchmaker omits much, making the computer razzle dazzle that much more irrelevant.

Conclusion

In his autobiographical What Mad Pursuit (Basic Books 1988) written when he was about 70, Nobelist Francis Crick expresses enthusiasm for Dawkins' argument against intelligent design, citing with admiration the "methinks" program.

Crick, who trained as a physicist and was also a panspermia advocate (see link above), doesn't seem to have noticed the difference in issues here. If we are talking about an analog of the origin of life (one-step arrival at the "methinks" sentence), then we must go with a distinct probability of 8.3 x 10^-41. If we are talking about an analog of some evolutionary algorithm, then we can be convinced that complex results can occur with application of simple iterative rules (though, again, the probabilities don't favor passive natural selection).

One can only suppose that Crick, so anxious to uphold his lifelong vision of atheism, leaped on Dawkins' argument without sufficient criticality. On the other hand, one must accept that there is a possibility his analytic powers had waned.

At any rate, it seems fair to say that the theory of evolution is far from being a clear-cut theory, in the manner of Einstein's theory of relativity. There are a number of difficulties and a great deal of disagreement as to how the evolutionary process works. This doesn't mean there is no such process, but it does mean one should listen to mechanists like Dawkins with care.

******************

1. In a 1996 introduction to Watchmaker, Dawkins wrote that "I can find no major thesis in these chapters that I would withdraw, nothing to justify the catharsis of a good recant."  2. My analogy was inadequately posed in previous drafts. Hopefully, it makes more sense now.  3. Genetic Variation and Progressive Evolution David Layzer The American Naturalist Vol. 115, No. 6 (Jun., 1980), pp. 809-826 (article consists of 18 pages) Published by: The University of Chicago Press for The American Society of Naturalists Stable URL: http://www.jstor.org/stable/2460802

Note: An early draft contained a ridiculous mathematical error that does not affect the argument but was very embarrassing. Naturally, I didn't think of it until after I was walking outdoors miles from an internet terminal. It has now been put right.

2011-11-10T14:44:00.001-08:00

Do dice play God? A review of Irreligion

Wikipedia article on Chaitin-Kolmogorov complexity
In search of a blind watchmaker (by Paul Conant)
Wikipedia article on runs test
Eric Schechter on Wolfram vs intelligent design
On Hilbert's sixth problem (by Paul Conant)
The scientific embrace of atheism (by David Berlinski)
John Allen Paulos' home page

A review of Irreligion: a mathematician explains why the arguments for God just don't add up (Hill and Wang division of Farrar, Straus and Giroux 2008) posted Nov 9, 2010.  Please contact Conant at krypto...at...gmail...dot....com to report errors or make comments. Thank you.

By PAUL CONANT

John Allen Paulos has done a service by compiling the various purported proofs of the existence of a (monotheistic) God and then shooting them down in his book Irreligion: a mathematician explains why the arguments for God just don't add up.

Paulos, a Temple University mathematician who writes a column for ABC News, would be the first to admit that he has not disproved the existence of God. But, he is quite skeptical of such existence, and I suppose much of the impetus for his book comes from the intelligent design versus accidental evolution controversy.¹

Really, this review isn't exactly kosher, because I am going to cede most of the ground. My thinking is that if one could use logico-mathematical methods to prove God's existence, this would be tantamount to being able to see God, or to plumb the depths of God. Supposing there is such a God, is he likely to permit his creatures, without special permission, to go so deep?

This review might also be thought rather unfair because Paulos is writing for the general reader and thus walks a fine line on how much mathematics to use. Still, he is expert at describing the general import of certain mathematical ideas, such as Gregory Chaitin's retooling of Kurt Goedel's undecidability theorem and its application to arguments about what a human can grasp about a "higher power."

Many of Paulos' counterarguments essentially arise from a Laplacian philosophy wherein Newtonian mechanics and statistical randomness rule all and are all. The world of phenomena, of appearances, is everything. There is nothing beyond. As long as we agree with those assumptions, we're liable to agree with Paulos.

Just because...

Yet a caveat: though mathematics is remarkably effective at describing physical relations, mathematical abstractions are not themselves the essence of being (though even on this point there is a Platonic dispute), but are typically devices used for prediction. The deepest essence of being may well be beyond mathematical or scientific description -- perhaps, in fact, beyond human ken (as Paulos implies, albeit mechanistically, when discussing Chaitin and Goedel).²

Paulos' response to the First Cause problem is to question whether postulating a highly complex Creator provides a real solution. All we have done is push back the problem, he is saying. But here we must wonder whether phenomenal, Laplacian reality is all there is. Why shouldn't there be something deeper that doesn't conform to the notion of God as gigantic robot?

But of course it is the concept of randomness that is the nub of Paulos' book, and this concept is at root philosophical, and a rather thorny bit of philosophy it is at that. The topic of randomness certainly has some wrinkles that are worth examining with respect to the intelligent design controversy.

One of Paulos' main points is that merely because some postulated event has a terribly small probability doesn't mean that event hasn't or can't happen. There is a terribly small probability that you will be struck by lightning this year. But every year, someone is nevertheless stricken. Why not you?

In fact, zero probability doesn't mean impossible. Many probability distributions closely follow the normal curve, where each distinct probability is exactly zero.

Paulos applies this line of reasoning to the probabilities for the origin of life, which the astrophysicist Fred Hoyle once likened to the chance of a tornado whipping through a junkyard and leaving a fully assembled jumbo jet in its wake. (Nick Lane in Life Ascending: The Ten Great Inventions of Evolution (W.W. Norton 2009) relates some interesting speculations about life self-organizing around undersea hydrothermal vents. So perhaps the probabilities aren't so remote after all, but, really, we don't know.)

Shake it up, baby

What is the probability of a specific permutation of heads and tails in say 20 fair coin tosses? This is usually given as 0.5²⁰, or about one chance in a million. What is the probability of 18 heads followed by 2 tails? The same, according to one outlook.

Now that probability holds if we take all permutations, shake them up in a hat and then draw one. All permutations in that case are equiprobable.⁴

However, intuitively it is hard to accept that 18 heads followed by 2 tails is just as probable as any other ordering. In fact, there are various statistical methods for challenging that idea.⁵

One, which is quite useful, is the runs test, which determines the probability that a particular sequence falls within the random area of the related normal curve. A runs test of 18H followed by 2T gives a z score of 3.71, which isn't ridiculously high, but implies that the ordering did not occur randomly with a confidence of 0.999.

Now compare that score with this permutation: HH TTT H TT H TT HH T HH TTT H. A runs test z score gives 0.046, which is very near the normal mean.

To recap: the probability of drawing a number with 18 ones (or heads) followed by 2 zeros (or tails) from a hat full of all 20-digit strings is on the order of 10^-6. The probability that that sequence is random is on the order of 10^-4. For comparison, we can be highly confident the second sequence is, absent further information, random. (I actually took it from irrational root digit strings.)

Again, those permutations with high runs test z scores are considered to be almost certainly non-random.³

At the risk of flogging a dead horse, let us review Paulos' example of a very well-shuffled deck of ordinary playing cards. The probability of any particular permutation is about one in 10⁶⁸, as he rightly notes. But suppose we mark each card's face with a number, ordering the deck from 1 to 52. When the well-shuffled deck is turned over one card at a time, we find that the cards come out in exact sequential order. Yes, that might be random luck. Yet the runs test z score is a very large 7.563, which implies effectively 0 probability of randomness as compared to a typical sequence. (We would feel certain that the deck had been ordered by intelligent design.)

Does not compute

The intelligent design proponents, in my view, are trying to get at this particular point. That is, some probabilities fall, even with a lot of time, into the nonrandom area. I can't say whether they are correct about that view when it comes to the origin of life. But I would comment that when probabilities fall far out in a tail, statisticians will say that the probability of non-random influence is significantly high. They will say this if they are seeking either mechanical bias or human influence. But if human influence is out of the question, and we are not talking about mechanical bias, then some scientists dismiss the non-randomness argument simply because they don't like it.

Another issue raised by Paulos is the fact that some of Stephen Wolfram's cellular automata yield "complex" outputs. (I am currently going through Wolfram's A New Kind of Science (Wolfram Media 2002) carefully, and there are many issues worth discussing, which I'll do, hopefully, at a later date.)

Like mathematician Eric Schechter (see link above), Paulos sees cellular automaton complexity as giving plausibility to the notion that life could have resulted when some molecules knocked together in a certain way. Wolfram's Rule 110 is equivalent to a Universal Turing Machine and this shows that a simple algorithm could yield any computer program, Paulos points out.

Paulos might have added that there is a countable infinity of computer programs. Each such program is computed according to the initial conditions of the Rule 110 automaton. Those conditions are the length of the starter cell block and the colors (black or white) of each cell.

So, a relevant issue is, if one feeds a randomly selected initial state into a UTM, what is the probability it will spit out a highly ordered (or complex or non-random) string versus a random string. Runs test scores would show the obvious: so-called complex strings will fall way out under a normal curve tail.

Grammar tool

I have run across quite a few ways of gauging complexity, but, barring an exact molecular approach, it seems to me the concept of a grammatical string is relevant.

Any cell, including the first, may be described as a machine. It transforms energy and does work (as in W = 1/2mv²). Hence it may be described with a series of logic gates. These logic gates can be combined in many ways, but most permutations won't work (the jumbo jet effect).

For example, if we have 8 symbols and a string of length 20, we have 125,970 different arrangements. But how likely is it that a random arrangement will be grammatical?

Let's consider a toy grammar with the symbols a,b,c. Our only grammatical rule is that b may not immediately follow a.

So for the first three steps, abc and cba are illegal and the other four possibilities are legal. This gives a (1/3) probability of error on the first step.

In this case, the probability of error at every third step is not independent of the previous probability as can be seen by the permutations:

 abc  bca  acb  bac  cba  cab

That is, for example, bca followed by bac gives an illegal ordering. So the probability of error increases with n.

However, suppose we hold the probability of error at (1/3). In that case the probability of a legal string where n = 30 is less than (2/3)¹⁰ = 1.73%. Even if the string can tolerate noise, the error probabilities rise rapidly. Suppose a string of 80 can tolerate 20 percent of its digits wrong. In that case we make our n = 21.333. That is the probability of success is (2/3)^21.333 = 0.000175.

And this is a toy model. The actual probabilities for long grammatical strings are found far out under a normal curve tail.

This is to inform you

A point that arises in such discussions concerns entropy (the tendency toward decrease of order) and the related idea of information, which is sometimes thought of as the surprisal value of a digit string. Sometimes a pattern such as HHHH... is considered to have low information because we can easily calculate the nth value (assuming we are using some algorithm to obtain the string). So the Chaitin-Kolmogorov complexity is low, or that is, the information is low. On the other hand a string that by some measure is effectively random is considered here to be highly informative because the observer has almost no chance of knowing the string in detail in advance.

However, we can also take the opposite tack. Using runs testing, most digit strings (multi-value strings can often be transformed, for test purposes, to bi-value strings) are found under the bulge in the runs test bell curve and represent probable randomness. So it is unsurprising to encounter such a string. It is far more surprising to come across a string with far "too few" or far "too many" runs. These highly ordered strings would then be considered to have high information value.

This distinction may help address Wolfram's attempt to cope with "highly complex" automata. By these, he means those with irregular, randomlike stuctures running through periodic "backgrounds." If a sufficiently long runs test were done on such automata, we would obtain, I suggest, z scores in the high but not outlandish range. The z score would give a gauge of complexity.

We might distinguish complicatedness from complexity by saying that a random-like permutation of our grammatical symbols is merely complicated, but a grammatical permutation, possibly adjusted for noise, is complex. (We see, by the way, that grammatical strings require conditional probabilities.)

A jungle out there

Paulos' defense of the theory of evolution is precise as far as it goes but does not acknowledge the various controversies on speciation among biologists, paleontologists and others.

Let us look at one of his counterarguments:

The creationist argument "goes roughly as follows: A very long sequence of individually improbable mutations must occur in order for a species or a biological process to evolve. If we assume these are independent events, then the probability that all of them will occur in the right order is the product of their respective probabilities" and hence a speciation probability is miniscule. "This line of argument," says Paulos, "is deeply flawed."

"Note that there are always a fantastically huge number of evolutionary paths that might be taken by an organism (or a process), but there is only one that actually will be taken. So, if, after the fact, we observe the particular evolutionary path actually taken and then calculate the a priori probability of its having been taken, we will get the miniscule probability that creationists mistakenly attach to the process as a whole."

Though we have dealt with this argument in terms of probability of the original biological cell, we must also consider its application to evolution via mutation. We can consider mutations to follow conditional probabilities. And though a particular mutation may be rather probable by being conditioned by the state of the organism (previous mutation and current environment), we must consider the entire chain of mutations represented by an extant species.

If we consider each species as representing a chain of mutations from the primeval organism, then we have for each a chain of conditional probability. A few probabilities may be high, but most are extremely low. Conditional probabilities can be graphed as trees of branching probabilities, so that a chain of mutation would be represented by one of these paths. We simply multiply each branch probability to get the total probability per path.

As a simple example, a 100-step conditional probability path with 10 probabilities of 0.9 and 60 with 0.7 and 30 with 0.5 yields a cumulative probability of 1.65 x 10^-19. In other words, the more mutations and ancestral species attributed to an extanct species, the less likely that species is to exist via passive natural selection. The actual numbers are so remote as to make natural selection by passive filtering virtually impossible, though perhaps we might conjecture some nonlinear effect going on among species that tends to overcome this problem.

Think of it this way: During an organism's lifetime, there is a fantastically large number of possible mutations. What is the probability that the organism will happen upon one that is beneficial? That event would, if we are talking only about passive natural selection, be found under a probability distribution tail (whether normal, Poisson or other). The probability of even a few useful mutations occurring over 3.5 billion years isn't all that great (though I don't know a good estimate).

A 'botific vision

Let us, for example, consider Wolfram's cellular automata, which he puts into four qualitative classes of complexity. One of Wolfram's findings is that adding complexity to an already complex system does little or nothing to increase the complexity, though randomized initial conditions might speed the trend toward a random-like output (a fact which, we acknowledge, could be relevant to evolution theory).

Now suppose we take some cellular automata and, at every nth or so step, halt the program and revise the initial conditions slightly or greatly, based on a cell block between cell n and cell n+m. What is the likelihood of increasing complexity to the extent that a Turing machine is devised? Or suppose an automaton is already a Turing machine. What is the probability that it remains one or that a more complex-output Turing machine results from the mutation?

I haven't calculated the probabilities, but I would suppose they are all out under a tail.

Paulos has elsewhere underscored the importance of Ramsey theory, which has an important role in network theory, in countering the idea that "self-organization" is unlikely. Actually, with sufficient n, "highly organized" networks are very likely.⁶ Whether this implies sufficient resources for the self-organization of a machine is another matter. True, high n seem to guarantee such a possibility. But, the n may be too high to be reasonable.

Darwin on the Lam?

However, it seems passive natural selection has an active accomplice in the extraordinarily subtle genetic machinery. It seems that some form of neo-Lamarckianism is necessary, or at any rate a negative feedback system which tends to damp out minor harmful mutations without ending the lineage altogether (catastrophic mutations usually go nowhere, the offspring most often not getting a chance to mate).

It may be that the in's and out's of evolution arguments were beyond the scope of Irreligion, but I don't think Paulos has entirely refuted the skeptics in this matter.

Nevertheless, the book is a succinct reference work and deserves a place on one's bookshelf.

1. Paulos finds himself disconcerted by the "overbearing religiosity of so many humorless people." Whenever one upholds an unpopular idea, one can expect all sorts of objections from all sorts of people, not all of them well mannered or well informed. Comes with the territory. Unfortunately, I think this backlash may have blinded him to the many kind, cheerful and non-judgmental Christians and other religious types in his vicinity. Some people, unable to persuade Paulos of God's existence, end the conversation with "I'll pray for you..." I can well imagine that he senses that the pride of the other person is motivating a put-down. Some of these souls might try not letting the left hand know what the right hand is doing. 2. Paulos recounts this amusing fable: The great mathematician Euler was called to court to debate the necessity of God's existence with a well-known atheist. Euler opens with: "Sir, (a + bⁿ)/n = x. Hence, God exists. Reply." Flabbergasted, his mathematically illiterate opponent walked away, speechless. Yet, is this joke as silly as it at first seems? After all, one might say that the mental activity of mathematics is so profound (even if the specific equation is trivial) that the existence of a Great Mind is implied. 3. We should caution that the runs test, which works for n₁ > 7 and n₂ > 7, fails for the pattern HH TT HH TT... This failure seems to be an artifact of the runs test assumption that a usual number of runs is about n/2. I suggest that we simply say that the probability of that pattern is less than or equal to H T H T H T..., a pattern whose z score rises rapidly with n. Other patterns such as HHH TTT HHH... also climb away from the randomness area slowly with n. With these cautions, however, the runs test gives striking results.

4. Thanks to John Paulos for pointing out an embarrassing misstatement in a previous draft. I somehow mangled the probabilities during the editing. By the way, my tendency to write flubs when I actually know better is a real problem for me and a reason I need attentive readers to help me out. 5. I also muddled this section. Josh Mitteldorf's sharp eyes forced a rewrite. 6. Paulos in a column writes: 'A more profound version of this line of thought can be traced back to British mathematician Frank Ramsey, who proved a strange theorem. It stated that if you have a sufficiently large set of geometric points and every pair of them is connected by either a red line or a green line (but not by both), then no matter how you color the lines, there will always be a large subset of the original set with a special property. Either every pair of the subset's members will be connected by a red line or every pair of the subset's members will be connected by a green line. If, for example, you want to be certain of having at least three points all connected by red lines or at least three points all connected by green lines, you will need at least six points. (The answer is not as obvious as it may seem, but the proof isn't difficult.) For you to be certain that you will have four points, every pair of which is connected by a red line, or four points, every pair of which is connected by a green line, you will need 18 points, and for you to be certain that there will be five points with this property, you will need -- it's not known exactly - between 43 and 55. With enough points, you will inevitably find unicolored islands of order as big as you want, no matter how you color the lines.

2011-11-10T14:42:00.001-08:00

Drunk and disorderly:

the inexorable rise of entropy

Some musings about entropy posted Nov. 20, 2010

By PAUL CONANT

One might describe the increase of the entropy⁰ of a gas to mean that the net vector -- sum of vectors of all particles -- at between time t₀ and t_n tends toward 0 and that once this equilibrium is reached at t_n, the net vector stays near 0 at any subsequent time.

One would expect a nearly 0 net vector if the individual particle vectors are random. This randomness is exactly what one would find in an asymmetrical n-body scenario, where the bodies are close together and about the same size. The difference is that gravity isn't the determinant, but rather collisional kinetic energy. It has been demonstrated that n-body problems can yield orbits that become extraordinarily tangled. The randomness is then of the Chaitin-Kolmogorov variety: determining future position of a particular particle becomes computationally very difficult. And usually, over some time interval, the calculation errors increase to the point that all predictability for a specific particle is lost.

But there is also quantum randomness at work. The direction that an excited photon exits an atom is probabilistic only, meaning that the recoil is random. This recoil vector must be added to the other electric charge recoil vector associated with particle collision -- though its effect is very slight and usually ignored.

Further, if one were to observe one or more of the particles, the observation would affect the knowledge of the momentum or position of the observed particles.

Now supposing we keep the gas at a single temperature in a closed container attached via a closed valve to another evacuated container, when we open the valve, the gas expands to fill both containers. This expansion is a consequence of the effectively random behavior of the particles, which on average "find less resistance" in the direction of the vacuum.

In general, gases tend to expand by inverse square, or that is spherically (or really, as a ball), which implies randomization of the molecules.

The drunkard's walk

Consider a computerized random walk (aka "drunkard's walk") in a plane. As n increases, the area covered by the walk tends toward that of a circle. In the infinite limit, there is probability 1 that a perfect circle has been covered (though probability 1 in such cases does not exclude exceptions).

So the real question is: what about the n-body problem yields pi-randomness? It is really a statistical question. When enough collisions occur in a sufficiently small volume (or area), the particle vectors tend to cancel each other out.

Let's go down to the pool hall and break a few racks of balls. It is possible to shoot the cue ball in such a way that the rack of balls scatters symmetrically. But in most situations, the cue ball strikes the triangular array at a point that yields an asymmetrical scattering. This is the sensitive dependence on initial conditions associated with mathematical chaos. We also see Chaitin-Kolmogorov complexity enter the picture, because the asymmetry means that for most balls predicting where one will be after a few ricochets is computationally very difficult.

Now suppose we have perfectly inelastic, perfectly spherical pool balls that encounter idealized banks. We also neglect friction. After a few minutes, the asymmetrically scattered balls are "all over the place" in effectively random motion. Now such discrete systems eventually return to their original state: the balls coalesce back into a triangle and then repeat the whole cycle over again, which implies that in fact such a closed system, left to its own devices, requires entropy to decrease, a seeming contradiction of the second law of thermodynamics. But the time scales required mean we needn't hold our breaths waiting. Also, in nature, there are darned few closed systems (and as soon as we see one, it's no longer closed at the quantum level), allowing us to conclude that in the ideal of 0 friction, the pool ball system may become aperiodic, implying the second law in this case holds.

(According to Stephen Wolfram in A New Kind of Science a billiard ball launched at any irrational angle to the banks of an idealized, frictionless square pool table must visit every bank point [I suppose he excludes the corners]. Since each point must be visited after a discrete time interval, it would take eternity to reach the point of reversibility.)

And now, let us exorcize Maxwell's demon, which, though meant to elucidate, to this day bedevils discussions of entropy with outlandish "solutions" to the alleged "problem." Maxwell gave us a thought experiment whereby he posited a little being controlling the valve between canisters. If (in this version of his thought experiment) the gremlin opened the valve to let speedy particles past in one direction only, the little imp could divide the gas into a hot cloud in one canister and a cold cloud in the other. Obviously the energy the gremlin adds is equivalent to adding energy via a heating/cooling system, but Maxwell's point was about the very, very minute possibility that such a bizarre division could occur randomly (or, some would say, pseudo-randomly).

This possibility exists. In fact, as said, in certain idealized closed systems, entropy decrease MUST happen. Such a spontaneous division into hot and cold clouds would also probably happen quite often at the nano-nano-second level. That is, when time intervals are short enough, quantum physics tells us the usual rules go out the window. However, observation of such actions won't occur for such quantum intervals (so there is no change in information or entropy), and as for the "random" chance of observing an extremely high-ordering of gas molecules, even if someone witnessed such an occurrence, not only does the event not conform to a repeatable experiment, no one is likely to believe the report, even if true.

Truly universal?

Can we apply the principle of entropy to the closed system of the universe? A couple of points: We're not absolutely sure the cosmos is a closed system (perhaps, for example, "steady state" creation supplements "big bang" creation). If there is a "big crunch," then, some have speculated, we might expect complete devolution to original states (people would reverse grow from death to birth, for example). If space curvature implies otherwise, the system remains forever open or asymptotically forever open.

However, quantum fuzziness probably rules out such an idealization. Are quantum systems precisely reversible? Yes and no. When one observes a particle collision in an accelerator, one can calculate the reverse paths. However, in line with the Heisenberg uncertainty principle one can never be sure of observing a collision with precisely identical initial conditions. And if we can only rarely, very rarely, replicate the exact initial conditions of the collision, then the same holds for its inverse.

Then there is the question of whether perhaps a many worlds (aka parallel universes) or many histories interpretation of quantum weirdness holds. In the event of a collapse back toward a big crunch, would the cosmos tend toward the exact quantum fluctuations that are thought to have introduced irregularities in the early universe that grew into star and galactic clustering?Or would a different set of fluctuations serve as the attractor, on grounds both sets were and are superposed and one fluctuation is as probable as the other? And, do these fluctuations require a conscious observer, as in John von Neumann's interpretation?

In A New Kind of Science, thinking in terms of computer-like algorithms, writes that it is unclear whether the "basic rules of the universe are really reversible," arguing that it could be that apparent reversibility arises due to effects of an attractor (he does not specify gravitational). He writes that "if pieces of the universe can break off but not reconnect, then there will be inevitably loss of information," thus increasing entropy.

Of course, we face such difficulties when trying to apply physical or mathematical concepts to the entire cosmos. It seems plausible that any system of relations we devise to examine properties of space and time may act like a lens that increases focus in one area while losing precision in another. I.e., a cosmic uncertainty principle.

Conservation of information?

A cosmic uncertainty principle would make information fuzzy. As the Heisenberg uncertainty principle shows, information about a particle's momentum is gained at the expense of information about its position. But, you may respond, the total information is conserved.

But wait! Is there a law about the conservation of information? In fact, information cannot be conserved -- in fact can't exist -- without memory, which in the end requires the mind of an observer. In fact, the "law" of increase of entropy says that memories fade and available information decreases. In terms of pure Shannon information, entropy expresses the probability of what we know or don't know.² Thus entropy is introduced by noise entering the signal. In realistic systems, supposing enough time elapses, noise eventually overwhelms the intended signal. For example, what would you say is the likelihood that this essay will be accessible two centuries from now? (I've already lost a group of articles I had posted on the now defunct Yahoo Geocities site.) Or consider Shakespeare's plays. We cannot say with certainty exactly how the original scripts read.

In fact, can we agree with some physicists that a specified volume of space contains a specific quantity of information? I wonder. A Shannon transducer is said to contain a specific quantity of information, but no one can be sure of that prior to someone reading the message and measuring the signal-to-noise ratio.

And quantum uncertainty qualifies as a form of noise, not only insofar as random jiggles in the signal, but also insofar as what signal was sent. If two signals are "transmitted" in quantum superposition, observation randomly determines which signal is read.

So one may set up a quantum measurement experiment and say that for a specific volume, the prior information describes the experiment. But quantum uncertainty still says that the experiment cannot be exactly described in a scientifically sensible way. So if we try to extrapolate information about a greater volume from the experiment volume, we begin to lose accuracy until the uncertainty reaches maximum. We see that quantum uncertainty can progressively change the signal-to-noise ratio, meaning entropy increases until the equilibrium level of no knowledge.

This of course would suggest that, from a human vantage point, there can be no exact information quantity for the cosmos.

So this brings us to the argument about whether black holes decrease the entropy of the universe by making it more orderly (i.e., simpler). My take is that a human observer in principle can never see anything enter a black hole. If one were to detect, at a safe distance, an object approaching a black hole, one would observe that its time pulses (its Doppler shift) would get slower and slower. In fact, the time pulses slow down asymptotic to eternity.

So the information represented by the in-falling object is, from this perspective, never lost.

But suppose we agree to an abstraction that eliminates the human observer -- as opposed to a vastly more gifted intelligence. In that case, perhaps the cosmos has an exact quantity of information at t_a. It then makes sense to talk about whether a black hole affects that quantity.

Consider a particle that falls into a black hole. It is said that all the information available about a black hole is comprised of the quantities for its mass and its surface area. Everything this super-intelligence knew about the particle, or ever could know, seemingly, is gone. Information is lost and the cosmos is a simpler, more orderly place, higher in information and in violation of the second law... maybe.

But suppose the particle is a twin of an entangled pair. One particle stays loose while the other is swallowed by the black hole. If we measure, say, the spin of one such particle we would ordinarily automatically know the spin of the other. But who's to tell what the spin is of a particle headed for the gravitational singularity at the black hole's core? So the information about the particle vanishes and entropy increases. This same event however means the orderliness of the universe increases and the entropy decreases. So, which is it? Or is it both. Have no fear, this issue is addressed in the next section.

Oh, and of course we mustn't forget Hawking radiation, whereby a rotating black hole slowly leaks radiation as particles every now and then "tunnel" through the gravitational energy barrier and escape into the remainder cosmos. The mass decreases over eons and eons until -- having previously swallowed everything available -- it eventually evaporates, Hawking conjectures.

A question: suppose an entangled particle escapes the black hole? Is the cosmic information balance sheet rectified? Perhaps, supposing it never reached the singularity. But, what of particles down near the singularity? They perhaps morph as the fields transform into something that existed close to the cosmic big bang. So it seems implausible that the spin information is retained. But, who knows?

Where's that ace?

There is a strong connection between thermodynamic entropy and Shannon information entropy. Consider the randomization of the pool break on the frictionless table after a few minutes. This is the equivalent of shuffling a deck of cards.

Suppose we have an especially sharp-eyed observer who watches where the ace of spades is placed in the deck as shuffling starts. We then have a few relatively simple shuffles. After the first shuffle, he knows to within three cards how far down in the deck the ace is. On the next shuffle he knows where it is with less accuracy. Let's say to a precision of (1/3)(1/3) = 1/9. After some more shuffles his potential error has reachs 1/52, meaning he has no knowledge of the ace's whereabouts.

The increase in entropy occurs from one shuffle to the next. But at the last shuffle, equilibrium has been reached. Further shuffling can never increase his knowledge of where the ace is, meaning the entropy won't decrease.

The runs test gives a measure of randomness¹ based on the normal distribution of numbers of runs, with the mean at n/2, "Too many" runs are found in one tail and "too few" in another. That is, a high z score implies that the sequence is non-random or "highly ordered."

What however is meant by order? (This is where we tackle the conundrum of a decrease in one sort of cosmic information versus an increase in another sort.)

Entropy is often defined as the tendency toward decrease of order, and the related idea of information is sometimes thought of as the surprisal value of a digit string. Sometimes a pattern such as HHHH... is considered to have low information because we can easily calculate the nth value (assuming we are using some algorithm to obtain the string). So the Chaitin-Kolmogorov complexity is low, or that is, the information is low. On the other hand a string that by some measure is effectively random is considered here to be highly informative because the observer has almost no chance of knowing the string in detail in advance.

So, once the deck has been sufficiently shuffled the entropy has reached its maximum (equilibrium). What is the probability of drawing four royal flushes? If we aren't considering entropy, we might say it is the same as that for any other 20-card deal. But, a runs test would give a z score of infinity (probability 1 that the deal is non-random) because drawing all high cards is equivalent to tossing a fair coin and getting 20 heads and no tails. If we don't like the infinitude we can posit 21 cards containing 20 high cards and 1 low card. The z score still implies non-randomness with a high degree of confidence.

Negative entropy?

Our discussion should not ignore the impact of Ramsey theory, an important subdiscipline of network theory. "Self-organizing" possibilities are inevitable with sufficient number of nodes in a network. In fact, one might argue that Ramsey theory implies negative entropy. Suppose we had n poker players. The probability that among them there is a royal flush skyrockets quite rapidly. So as n increases, the probability of a specific set of cards increases and the information surprisal value decreases.³

0.Taken from a Wikipedia article: The dimension of thermodynamic entropy is energy divided by temperature, and its SI unit is joules per kelvin. In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication."
1. We should caution that the runs test, which works for n1 > 7 and n2 > 7, fails for the pattern HH TT HH TT... This failure seems to be an artifact of the runs test assumption that a usual number of runs is about n/2. I suggest that we simply say that the probability of that pattern is less than or equal to H T H T H T..., a pattern whose z score rises rapidly with n. Other patterns such as HHH TTT HHH... also climb away from the randomness area slowly with n. With these cautions, however, the runs test gives striking results.
2.Taken from Wikipedia: In information theory, entropy is a measure of the uncertainty associated with a random variable. In this context, the term usually refers to the Shannon entropy, which quantifies the expected value of the information contained in a message, usually in units such as bits. Equivalently, the Shannon entropy is a measure of the average information content one is missing when one does not know the value of the random variable. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication." Shannon's entropy represents an absolute limit on the best possible lossless compression of any communication, under certain constraints: treating messages to be encoded as a sequence of independent and identically-distributed random variables, Shannon's source coding theorem shows that, in the limit, the average length of the shortest possible representation to encode the messages in a given alphabet is their entropy divided by the logarithm of the number of symbols in the target alphabet. A fair coin has an entropy of one bit. However, if the coin is not fair, then the uncertainty is lower (if asked to bet on the next outcome, we would bet preferentially on the most frequent result), and thus the Shannon entropy is lower. Mathematically, a coin flip is an example of a Bernoulli trial, and its entropy is given by the binary entropy function. A long string of repeating characters has an entropy rate of 0, since every character is predictable. The entropy rate of English text is between 1.0 and 1.5 bits per letter,[1] or as low as 0.6 to 1.3 bits per letter, according to estimates by Shannon based on human experiments.
3. John Allen Paulos on Ramsey theory: 'A more profound version of this line of thought can be traced back to British mathematician Frank Ramsey, who proved a strange theorem. It stated that if you have a sufficiently large set of geometric points and every pair of them is connected by either a red line or a green line (but not by both), then no matter how you color the lines, there will always be a large subset of the original set with a special property. Either every pair of the subset's members will be connected by a red line or every pair of the subset's members will be connected by a green line. If, for example, you want to be certain of having at least three points all connected by red lines or at least three points all connected by green lines, you will need at least six points. (The answer is not as obvious as it may seem, but the proof isn't difficult.) For you to be certain that you will have four points, every pair of which is connected by a red line, or four points, every pair of which is connected by a green line, you will need 18 points, and for you to be certain that there will be five points with this property, you will need -- it's not known exactly - between 43 and 55. With enough points, you will inevitably find unicolored islands of order as big as you want, no matter how you color the lines.'

2011-11-10T14:39:00.001-08:00

Einstein, Sommerfeld and the twin paradox

Topologist Jeff Weeks on the twin paradox
Reach Michel Janssen's paper
Toward a signal model of perception by Paul Conant

This paper has been updated as of Dec. 10, 2009

The commentator, Paul Conant, is a science-minded journalist with no science degrees. Though he is able to follow the technicalities of special relativity, he is not conversant with differential geometry, and hence untutored in the field equations of general relativity.

This essay is in no way intended to impugn the important contributions of Einstein or other physicists. Everyone makes errors.

Comments and suggestions welcome. Please write to krypto78@gmail.com.

The paradox

Einstein's groundbreaking 1905 relativity paper, "On the electrodynamics of moving bodies," contained a fundamental inconsistency which was not addressed until 10 years later, with the publication of his paper on gravitation.

Many have written on this inconsistency, known as the "twin paradox" or the "clock paradox" and more than a few have not understood that the "paradox" does not refer to the strangeness of time dilation but to a logical inconsistency in what is now known as the special (for "special case") theory of relativity.

Among those missing the point: Max Born in his book on special relativity¹, George Gamow in an essay and Roger Penrose in Road to Reality², and, most recently, Leonard Susskind in The Black Hole War.³

Among those who have correctly understood the paradox are topologist Jeff Weeks (see link above) and science writer Stan Gibilisco⁴, who noted that the general theory of relativity resolves the problem.

As far back as the 1960s, the British physicist Herbert Dingle⁵ called the inconsistency a "regrettable error" and was deluged with "disproofs" of his assertion from the physics community. (It should be noted that Dingle's 1949 attempt at relativistic physics left Einstein bemused.⁶) Yet every "disproof" of the paradox that I have seen uses acceleration, an issue not addressed by Einstein until the general theory of relativity. It was Einstein who set himself up for the paradox by favoring the idea that only purely relative motions are meaningful, writing that various examples "suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest." [Electrodynamics translated by Perett and Jeffery and appearing in a Dover (1952) reprint]. In that paper, he also takes pains to note that the term "stationary system" is a verbal convenience only.⁷

But later in Elect., Einstein offered the scenario of two initially synchronized clocks at rest with respect to each other. One clock then travels around a closed loop, and its time is dilated with respect to the at-rest clock when they meet again. In Einstein's words: "If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result: If one of two synchronous clocks at A is moved in a journey lasting t seconds, then by the clock which has remained at rest the traveled clock on its arrival at A will be 1/2tv²/c² slow."

Clearly, if there is no preferred frame of reference, a contradiction arises: when the clocks meet again, which clock has recorded fewer ticks?

Both in the closed loop scenario and in the polygon-path scenario, Einstein avoids the issue of acceleration. Hence, he does not explain that there is a property of "real" acceleration that is not symmetrical or purely relative and that that consequently a preferred frame of reference is implied, at least locally.

The paradox stems from the fact that one cannot say which velocity is higher without a "background" reference frame. In Newtonian terms, the same issue arises: if one body is accelerating away from the other, how do we know which body experiences the "real" force? No answer is possible without more information, implying a background frame.

In comments published in 1910, the physicist Arnold Sommerfeld, a proponent of relativity theory, "covers" for the new paradigm by noting that Einstein didn't really mean that time dilation was associated with purely relative motion, but rather with accelerated motion; and that hence relativity was in that case not contradictory.

Sommerfeld wrote: "On this [a time integral and inequality] depends the retardation of the moving clock compared with the clock at rest. The assertion is based, as Einstein has pointed out, on the unprovable assumption that the clock in motion actually indicates its own proper time; i.e. that it always gives the time corresponding to the state of velocity, regarded as constant, at any instant. The moving clock must naturally have been moved with acceleration (with changes of speed or direction) in order to be compared with the stationary clock at world-point P. The retardation of the moving clock does not therefore actually indicate 'motion,' but 'accelerated motion.' Hence this does not contradict the principle of relativity." [Notes appended to Space and Time, a 1908 address by Herman Minkowski, Dover 1952, Note 4.]

However, Einstein's 1905 paper does not tackle the issue of acceleration and more to the point, does not explain why purely relative acceleration would be insufficient to meet the facts. The principle of relativity applies only to "uniform translatory motion" (Elect. 1905).

Neither does Sommerfeld's note address the issue of purely relative acceleration versus "true" acceleration, perhaps implicitly accepting Newton's view (below).

And, a review of various papers by Einstein seems to indicate that he did not deal with this inconsistency head-on, though in a lecture-hall discussion ca. 1912, Einstein said that the [special] theory of relativity is silent on how a clock behaves if forced to change direction but argues that if a polygonal path is large enough, accelerative effects diminish and (linear) time dilation still holds.

On the other hand, of course, he was not oblivious to the issue of acceleration. In 1910, he wrote that the principle of relativity meant that the laws of physics are independent of the state of motion, but that the motion is non-accelerated. "We assume that the motion of acceleration has an objective meaning," he said. [The Principle of Relativity and its Consequences in Modern Physics, a 1910 paper reproduced in Collected Papers of Albert Einstein, Hebrew University, Princeton University Press].

In that same paper Einstein emphasizes that the principle of relativity does not cover acceleration. "The laws governing natural phenomena are independent of the state of motion of the coordinate system to which the phenomena are observed, provided this system is not in accelerated motion."

Clearly, however, he is somewhat ambiguous about small accelerations and radial acceleration, as we see from the lecture-hall remark and from a remark in Foundation of the General Theory of Relativity (1915) about a "familiar result" of special relativity whereby a clock on a rotating disk's rim ticks slower than a clock at the origin.

General relativity's partial solution

Finally, in his 1915 paper on general relativity, Einstein addressed the issue of acceleration, citing what he called "the principle of equivalence." That principle (actually, introduced prior to 1915) said that there was no real difference between kinematic acceleration and gravitational acceleration. Scientifically, they should be treated as if they are the same.

So then, Einstein notes in Foundation, if we have system K and another system K' accelerating with respect to K, clearly, from a "Galilean" perspective, we could say that K was accelerating with respect to K'. But, is this really so?

Einstein argues that if K is at rest relative to K', which is accelerated, the oberserver on K cannot claim that he is being accelerated -- even though, in purely relative terms, such a claim is valid. The reason for this rejection of Galilean relativity: We may equally well interpret K' to be kinematically unaccelerated though the "space-time territory in question is under the sway of a gravitational field, which generates the accelerated motion of the bodies" in the K' system.

This claim is based on the principle of equivalence which might be considered a modification of his previously posited principle of relativity. By the relativity principle, Einstein meant that the laws of physics can be cast in invariant form so that they apply equivalently in any unformly moving frame of reference. (For example, |v_b - v_a| is the invariant quantity that describes an equivalence class of linear velocities.)

By the phrase "equivalence," Einstein is relating impulsive acceleration (for example, a projectile's x vector) to its gravitational acceleration (its y vector). Of course, Newton's mechanics already said that the equation F = mg is a special case of F = ma but Einstein meant something more: that local spacetime curvature is specific for "real" accelerations -- whether impulsive or gravitational.

Einstein's "equivalence" insight was his recognition that one could express acceleration, whether gravitational or impulsive, as a curvature in the spacetime continuum (a concept introduced by Minkowski). This means, he said, that the Newtonian superposition of separate vectors was not valid and was to be replaced by a unitary curvature. (Though the calculus of spacetime requires specific tools, the concept isn't so hard to grasp. Think of a Mercator map: the projection of a sphere onto a plane. Analogously, general relativity projects a 4-dimensional spacetime onto a Euclidean three-dimensional space.)

However, is this "world-line" answer the end of the problem of the asymmetry of accelerated motion?

The Einstein of 1915 implies that if two objects have two different velocities, we must regard one as having an absolutely higher velocity than the other because one object has been "really" accelerated.

Yet one might conjecture that if two objects move with different velocities wherein neither has a prior acceleration, then the spacetime curvature would be identical for each object and the objects' clocks would not get out of step. But such a conjecture would violate the limiting case of special relativity (and hence general relativity); specifically, such a conjecture would be inconsistent with the constancy of the vacuum velocity of light in any reference frame.

So then, general relativity requires that velocity differences are, in a sense, absolute. Yet in his original static and eternal cosmic model of 1917, there was no reason to assume that two velocities of two objects necessarily implied the acceleration of one object. Einstein introduced the model, with the cosmological constant appended in order to contend with the fact that his 1915 formulation of GR apparently failed to account for the observed mass distribution of the cosmos.

Despite the popularity of the Big Bang model, a number of cosmic models hold the option that some velocity differences needn't imply an acceleration, strictly relative or "real."

Einstein's appeal to spacetime curvature to address the frame of reference issue is similar to Newton's assertion that an accelerated body requires either an impulse imputed to it or the gravitational force. There is an inherent local physical asymmetry. Purely relative motion will not do.

Einstein also brings up the problem of absolute relative motion in the sense of Newton's bucket. Einstein uses two fluid bodies in space, one spherical, S₁ and another an ellipsoid of revolution, S₂. From the perspective of "Galilean relativity," one can as easily say that either body is at rest with respect to the other.

But, the radial acceleration of S₂ results in a noticeable difference: an equatorial bulge. Hence, says Einstein, it follows that the difference in motion must have a cause outside the system of the two bodies.

Of course Newton in Principia Mathematica first raised this point, noting that the surface of water in a rapidly spinning bucket becomes concave. This, he said, demonstrated that force must be impressed on a body in order for there to be a change in acceleration. Newton also mentioned the issue of the fixed stars as possibly of use for a background reference frame, though he does not seem to have insisted on that point. He did however find that absolute space would serve as a background reference frame.

It is interesting to note here that Einstein's limit c can be used as an alternative to the equatorial bulge argument. If we suppose that a particular star is sufficiently distant, then the x component of its radial velocity (which is uniform and linear) will exceed the velocity of light. Such a circumstance being forbidden, we are forced to conclude that the earth is spinning, rather than the star revolving around the earth. We see that, in this sense, the limit c can be used to imply a specific frame of reference. At this point, however, I cannot say that such a circumstance suffices to resolve the clock paradox of special relativity.

Interestingly, the problem of Newton's bucket is quite similar to the clock paradox of special relativity. In both scenarios, we note that if two motions are strictly relative, what accounts for a property associated with one motion and not the other? In both cases, we are urged to focus on the "real" acceleration.

Newton's need for a background frame to cope with "real" acceleration predates the 19th century refinement of the concept of energy as an ineffable, essentially abstract "substance" which passes from one event to the next. That concept was implicit in Newton's Principia but not explicit and hence Newton did not appeal to the "energy" of the object in motion to deal with the problem. That is, we can say that we can distinguish between two systems by examining their parts. A system accelerated to a non-relativistic speed nevertheless discloses its motion by the fact that the parts change speed at different times as a set of "energy transactions" occur. For example, when you step on the accelerator, the car seat moves forward before you do; you catch up to the car "because" the car set imparts "kinetic energy" to you.

But if you are too far away to distinguish individual parts or a change in the object's shape, such as from equatorial bulge, your only hope for determining "true" acceleration is by knowing which object received energy prior to the two showing a relative change in velocity.

Has the clock paradox gone away?

Now does GR resolve the clock paradox?

GR resolves the paradox non-globally, in that Einstein now holds that some accelerations are not strictly relative, but functions of a set of curvatures. Hence one can posit the loop scenario given in Electrodynamics and say that only one body can have a higher absolute angular velocity with respect to the other because only one must have experienced an acceleration that distorts spacetime differently from the other.

To be consistent, GR must reflect this asymmetry. That is, suppose we have two spaceships separating along a straight line whereby the distance between them increases as a constant velocity. If ship A's TV monitor says B's clock is ticking slower than A's and ship B's TV monitor says A's clock is ticking slower than B's, there must be an objective difference, nevertheless.

The above scenario is incomplete because the "real" acceleration prior to the opening of the scene is not given. Yet, GR does not tell us why a "real" acceleration must have occurred if two bodies are moving at different velocities.

So yes, GR partly resolves the clock paradox and, by viewing the 1905 equations for uniform motion as a special case of the 1915 equations, retroactively removes the paradox from SR, although it appears that Einstein avoided pointing this out in 1915 or thereafter.

However, GR does not specify a global topology (cosmic model) of spacetime, though Einstein struggled with this issue. The various solutions to GR's field equations showed that no specific cosmic model followed from GR. The clock paradox shows up in the Weeks model of the cosmos, with local space being euclidean (or rather Minkowskian). As far as this writer knows, such closed space geodesics cannot be ruled out on GR grounds alone.

Jeff Weeks, in his book The Shape of Space, points out that though physicists commonly think of three cosmic models as suitable for GR, in fact there are three classes of 3-manifolds that are both homogenous and isotropic (cosmic information is evenly mixed and looks about the same in any direction). Whether spacetime is mathematically elliptic, hyperbolic or euclidean, there are many possible global topologies for the cosmos, Weeks says.

One model, described by Weeks in the article linked above, permits a traveler to continue straight in a closed universe until she arrives at the point of origin. Again, to avoid contradiction, we are required to accept a priori that an acceleration that alters a world line has occurred.

Other models have the cosmic time axis following hyperbolic or elliptical geometry. Originally, one suspects, Einstein may have been skeptical of such an axis, in that Einstein's abolishment of simultaneity effectively abolished the Newtonian fiction of absolute time. But physicist Paul Davies, in his book About Time, argued that there is a Big Bang oriented cosmic time that can be approximated quite closely.

Kurt Goedel's rotating universe model left room for closed time loops, such that an astronaut who continued on a protracted space flight could fly into his past. This result prompted Godel to question the reality of time in general relativity. Having investigated various solutions of GR equations, Goedel argued that a median of proper times of moving objects, which James Jeans had thought to serve as a cosmic absolute time, was not guaranteed in all models and hence should be questioned in general.

Certainly we can agree that Goedel's result shows that relativity is incomplete in its analysis of time.

Mach's principles

Einstein was influenced by the philosophical writings of the German physicist Ernst Mach, whom he cites in Foundations.

According to Einstein (1915) Mach's "epistomological principle" says that observable facts must ultimately appear as causes and effects. Mach believed that the brain organizes sensory data into knowledge and that hence data of scientific value should stem from observable, measurable phenomena. This philosophical viewpoint was evident in 1905 when Einstein ruthlessly ejected the Maxwell-Lorentzian ether from physics.

Mach's "epistomological principle" led Mach to reject Newtonian absolute time and absolute space as unverifiable and made Einstein realize that the Newtonian edifice wasn't sacrosanct. However, in 1905 Einstein hadn't replaced the edifice with something called a "spacetime continuum." Curiously, later in his career Einstein impishly but honestly identified this entity as "the ether."

By rejecting absolute space and time, Mach also rejected the usual way of identifying acceleration in what is known as Mach's principle:

Version A. Inertia of a ponderable object results from a relationship of that object with all other objects in the universe.

Version B. The earth's equatorial bulge is not a result of absolute rotation (radial acceleration) but is relative to the distant giant mass of the universe.

For a few years after publication of Foundations, Einstein favored Mach's principle, even using it as a basis of his "cosmological constant" paper, which was his first attempt to fit GR to a cosmic model, but was eventually convinced by the astronomer Wilem de Sitter (see Janssen above) to abandon the principle. In 1932 Einstein adopted the Einstein-de Sitter model that posits a cosmos with a global curvature that asymptotically zeroes out over eternity. The model also can be construed to imply a Big Bang, with its ordered set of accelerations.

A bit of fine-tuning

We can fine-tune the paradox by considering the velocity of the center of mass of the twin system. That velocity is m₁v/m₁ + m₂.

So the CM velocity is larger when the moving mass is the lesser and the converse. Letting x be a real greater than 1 we have two masses xm and m. The algebra reveals there is a factor (x/x+1) > 1/(x+1). The CM velocity for earth moving at 0.6c with respect to a 77kg astronaut is very close to 0.6c. For the converse, however, that velocity is about 2.3 meters per femto-second.

If we like, we can use the equation

E = mc²(1-v²/c²)^1/2

to obtain the energies of each twin system.

If the earth is in motion and the astronaut at rest, my calculator won't handle the quantity for the energy. If the astronaut is in motion with the earth at rest, then E = 5.38*10⁴¹J.

But the paradox is restored as soon as we set m₁ equal to m₂.

Notes on the principle of equivalence

Now an aside on the principle of equivalence. Can it be said that gravitational acceleration is equivalent to kinematic acceleration? Gravitational accelerations are all associated with the gravitational constant G and of the form g = Gm/r². Yet it is easy to write expressions for accelerations that cannot be members of the gravitational set. If a is not constant, we fulfill the criterion. If in r^x, x =/= 2, there will be an infinity of accelerations that aren't members of the gravitational set.

At any rate, Einstein's principle of equivalence made a logical connection between a ponderable object's inertial mass and its gravitational mass. Newton had not shown a reason that they should be exactly equal, an assumption validated by acute experiments. (A minor technicality: Einstein and others have wondered why these masses should be exactly equal, but, properly they meant why should they be exactly proportional? Equality is guaranteed by Newton's choice of a gravitational constant. But certainly, m_in = km_gr, with k equaling one because of Newton's choice.)

Also, GR's field equations rest on the premise (Foundation) that for an infinitesimal region of spacetime, the Minkowskian coordinates of special relativity hold. However, this 1915 assumption is open to challenge on the basis of the Heisenberg uncertainty principle (ca. 1925), which sets a finite limit on the precision of a measurement of a particle's space coordinate given its time coordinate.

Einstein's Kaluza-Klein excursion

In Subtle is the Lord Pais tells of a period in which Einstein took Klein's idea for a five-dimensional spacetime and reworked it. After a great deal of effort, Einstein offered a paper which took Klein's ideas presented as his own, on the basis that he had found a way to rationalize obtaining the five-dimensional effect while sticking to the conventional perceptual view of space and time denoted 3D+T (making one wonder what he thought of his own four-dimensional spacetime scheme).

A perplexed Abraham Pais notes that a colleague dismissed Einstein's work as unoriginal, and Einstein then quickly dropped it⁷. But reformulation of the ideas of others is exactly what Einstein did in 1905 with the special theory. He presented the mathematical and physical ideas of Lorenz, Fitzgerald and Poincare, whom he very likely read, and refashioned them in a manner that he thought coherent, most famously by rejecting the notion of ether as unnecessary.

Yet it took decades for Einstein to publicly acknowledge the contribution of Poincare, and even then, he let the priority matter remain fuzzy. Poincare's work was published in French in 1904, but went unnoticed by the powerful German-speaking scientific community. As a French-speaking resident of Switzerland, it seems rather plausible that the young patent attorney read Poincare's paper.

But, as Pais pointed out, it was Einstein's interpretation that made him the genius of relativity. And yet, that interpretation was either flawed, or incomplete, as we know from the twin paradox.

Footnotes

Apologies for footnotes being out of order. Haven't time to fix.

1. Einstein's Theory of Relativity by Max Born (Dover 1962).

2. Road to Reality by Roger Penrose (Random House 2006).

3. The Black Hole War by Leonard Susskind (Little Brown 2009).

4. Understanding Einstein's Theories of Relativity by Stan Gibilisco (Dover reprint of the 1983 edition).

7. In his biography of Einstein, Subtle is the Lord (Oxford 1983), physicist Abraham Pais mentions the "clock paradox" in the 1905 Electrodynamics paper but then summarily has Einstein resolve the contradiction in a paper presented to the Prussian Academy of Physics after the correct GR paper of 1915, with Einstein arguing that acceleration ends the paradox, which Pais calls a "misnomer." I don't recall the Prussian Academy paper, but it should be said that Einstein strongly implied the solution to the contradiction in his 1915 GR paper. Later in his book, Pais asserts that sometime after the GR paper, Einstein dispatched a paper on what Pais now calls the "twins paradox" but Pais uncharacteristically gives no citation.

5. Though Dingle seems to have done some astronomical work, he was not -- as a previous draft of this page said -- an astronomer, according to Harry H. Ricker III. Dingle was a professor of physics and natural philosophy at Imperial College before becoming a professor of history and the philosophy of science at City College, London, Ricker said. "Most properly he should be called a physicist and natural philosopher since his objections to relativity arose from his views and interpretations regarding the philosophy of science."

6. Dingle's paper Scientific and Philosophical Implications of the Special Theory of Relativity appeared in 1949 in Albert Einstein: Philosopher-Scientist, edited by Paul Arthur Schilpp. Dingle used this forum to propound a novel extension of special relativity which contained serious logical flaws. Einstein, in a note of response, said Dingle's paper made no sense to him.

8. See for example Max Von Laue's paper in Albert Einstein: Philosopher-Scientist edited by Paul Arthur Schilpp (1949).

2011-11-10T14:36:00.001-08:00

First published Thursday, November 3, 2011

Draft 04

The knowledge delusion
Essay by Paul Conant

Reflections on The God Delusion (Houghton Mifflin 2006) by the evolutionary biologist Richard Dawkins.

Preliminary remarks:
Our discussion focuses on the first four chapters of Dawkins' book, wherein he makes his case for the remoteness of the probability that a monolithic creator and controller god exists.

Alas, it is already November 2011, some five years after publication of Delusion. Such a lag is typical of me, as I prefer to discuss ideas at my leisure. This lag isn't quite as outrageous as the timing of my paper on Dawkins' The Blind Watchmaker, which I posted about a quarter century after the book first appeared.

I find that I have been quite hard on Dawkins, or, actually, on his reasoning. Even so, I have nothing but high regard for him as a fellow sojourner on spaceship Earth. Doubtless I have been unfair in not highlighting positive passages in Delusion, of which there are some (1). Despite my desire for objectivity, it is clear that much of the disagreement is rooted in my personal beliefs (see the link Zion below).

Summary:
Dawkins applies probabilistic reasoning to etiological foundations, without defining probability or randomness. He disdains Bayesian subjectivism without realizing that that must be the ground on which he is standing. In fact, nearly everything he writes on probability indicates a severe lack of rigor. This lack of rigor compromises his other points.

Relevant links:

In search of a blind watchmaker
http://www.angelfire.com/az3/nfold/watch.html

Do dice play God?
http://www.angelfire.com/az3/nfold/dice.html

Toward a signal model of perception
http://www.angelfire.com/ult/znewz1/qball.html

On Hilbert's sixth problem
http://kryptograff.blogspot.com/2007/06/on-hilberts-sixth-problem.html

The world of null-H
http://kryptograff.blogspot.com/2007/06/world-of-null-h.html

The universe cannot be modeled as a Turing machine
http://www.angelfire.com/az3/nfold/turing.html

Drunk and disorderly: the inexorable rise of entropy
http://www.angelfire.com/az3/nfold/entropy.html

Biological observer-participation and Wheeler's 'law without law'
by Brian D. Josephson
http://arxiv.org/abs/1108.4860

Where is Zion?
http://www.angelfire.com/az3/newzone/zion1.html

Other Conant pages
http://conantcensorshipissue.blogspot.com/2011/11/who-is-paul-conant-paul-conants-erdos.html

Essay

Richard Dawkins argues that he is no proponent of simplistic "scientism" and yet there is no sign in Delusion's first four chapters that in fact he isn't a victim of what might be termed the "scientism delusion." But, as Dawkins does not define scientism, he has plenty of wiggle room.

From what I can gather, those under the spell of "scientism" hold the, often unstated, assumption that the universe and its components can be understood as an engineering problem, or set of engineering problems. Perhaps there is much left to learn, goes the thinking, but it's all a matter of filling in the engineering details. (http://en.wikipedia.org/wiki/Scientism).

Though the notion of a Laplacian cosmos that requires no god to, every now and then, act to keep things stable is officially passe, nevertheless many scientists seem to be under the impression that the model basically holds, though needing a bit of tweaking to account for the effects of relativity and of quantum fluctuations.

Doubtless Dawkins is correct in his assertion that many American scientists and professionals are closet atheists, with quite a few espousing the "religion" of Einstein, who appreciated the elegance of the phenomenal universe but had no belief in a personal god (2).

Interestingly, Einstein had a severe difficulty with physical, phenomenal reality, objecting strenuously to the "probabilistic" requirement of quantum physics, famously asserting that "god" (i.e., the cosmos) "does not play dice." He agreed with Erwin Schroedinger that Schroedinger's imagined cat strongly implies the absurdity of "acausal" quantum behavior (3). It turns out that Einstein was wrong, with statistical experiments in the 1980s demonstrating that "acausality" -- within constraints -- is fundamental to quantum actions.

Many physicists have decided to avoid the quantum interpretation minefield, discretion being the better part of valor. Even so, Einstein was correct in his refusal to play down this problem, recognizing that modern science can't easily dispense with classical causality. We speak of energy in terms of vector sums of energy transfers (notice the circularity) but no one has a good handle on what the it is behind that abstraction.

A partly subjective reality at a fundamental level is anethema to someone like Einstein -- so disagreeable, in fact, that one can ponder whether the great scientist deep down suspected that such a possibility threatened his reasoning in denying a need for a personal god. Be that as it may, one can understand that a biologist might not be familiar with how nettlesome the quantum interpretation problem really is, but Dawkins has gone beyond his professional remit and taken on the roles of philosopher and etiologist. True, he rejects the label of philosopher, but his basic argument has been borrowed from the atheist philosopher Bertrand Russell.

Dawkins recapitulates Russell thus: "The designer hypothesis immediately raises the question of who designed the designer."

Further: "A designer God cannot be used to explain organized complexity because a God capable of designing anything would have to be complex enough to demand the same kind of explanation... God presents an infinite regress from which we cannot escape."

Dawkins' a priori assumption is that "anything of sufficient complexity to design anything, comes into existence only as the end product of an extended process of gradual evolution."

If there is a great designer, "the designer himself must be the end product of some kind of cumulative escalator or crane, perhaps a version of Darwinism in its own universe."

Dawkins has no truck with the idea that an omnipotent, omniscient (and seemingly paradoxical) god might not be explicable in engineering terms. Even if such a being can't be so described, why is he/she needed? Occam's razor and all that.

Dawkins does not bother with the results of Kurt Goedel and its implications for Hilbert's sixth problem: whether the laws of physics can ever be -- from a human standpoint -- both complete and consistent. Dawkins of course is rather typical of those scientists who pay little heed to that result or who have tried to minimize its importance in physics. A striking exception is the mathematical physicist Roger Penrose who saw that Goedel's result was profoundly important (though mathematicians have questioned Penrose's interpretation).

A way to intuitively think of Goedel's conundrum is via the Gestalt effect: the whole is greater than the sum of its parts. But few of the profound issues of phenomenology make their way into Dawkins' thesis. Had the biologist reflected more on Penrose's The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics (Oxford 1989), perhaps he would not have plunged in where Penrose so carefully trod.

Penrose has referred to himself, according to a Wikipedia article, as an atheist. In the film A Brief History of Time, the physicist said, "I think I would say that the universe has a purpose, it's not somehow just there by chance ... some people, I think, take the view that the universe is just there and it runs along -- it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."

By contrast, we get no such ambiguity or subtlety from Dawkins. Yet, if one deploys one's prestige as a scientist to discuss the underpinnings of reality, more than superficialities are required. The unstated, a priori assumption is, essentially, a Laplacian billiard ball universe and that's it, Jack.

Dawkins embellishes the Russellian rejoinder with the language of probability: What is the probability of a superbeing, capable of listening to millions of prayers simultaneously, existing? This follows his scorning of Stephen D. Unwin's The Probability of God (Crown Forum 2003), which cites Bayesian methods to obtain a high probability of god's existence.
http://www.stephenunwin.com/

Dawkins is uninterested in Unwin's subjective prior probabilities, all the while being utterly unaware that his own probability assessment is altogether subjective. Heedless of the philosophical underpinnings of probability theory, he doesn't realize that by assigning a probability of "remote" at the extremes of etiology, he is engaging in a subtle form of circular reasoning.

The reader deserves more than an easy putdown of Unwin in any discussion of probabilities. Dawkins doesn't acknowledge that Bayesian statistics is a thriving school of research that seeks to find ways to as much as possible "objectify" the subjective assessments of knowledgeable persons. There has been strong controversy concerning Bayesian versus classical statistics, and there is a reason for that controversy: it gets at foundational matters of etiology. Nothing on this from Dawkins.

Without a Bayesian approach, Dawkins is left with a frequency interpretation of probability (law of large numbers and so forth). But we have very little -- in fact Dawkins would say zero -- information about the existence or non-existence of a sequence of all powerful gods or pre-cosmoses. Hence, there are no frequencies to analyze. Hence, use of a probability argument is in vain.

Dawkins elsewhere says (4) that he has read the great statistician Ronald Fisher, but one wonders whether he appreciates the meaning of statistical analysis. Fisher, who also opposed the use of Bayesian premises, is no solace when it comes to frequency-based probabilities. Take Fisher's combined probability test, a technique for data fusion or "meta-analysis" (analysis of analyses): What are the several different tests of probability that might be combined to assess the probability of god?

Dawkins is quick to brush off William A. Dembski, the intelligent design advocate who uses statistical methods to argue that the probability is cosmically remote that life originated in a random manner. And yet Dawkins himself seems to have little or no grasp of the basis of probabilities.

In fact, Dawkins makes no attempt to define randomness, a definition routinely brushed off in elementary statistics texts but which represents quite a lapse when getting at etiological foundations (5) and using probability as a conceptual, if not mathematical, tool.

But, to reiterate, the issue goes yet deeper. If, at the extremes, causation is not nearly so clear-cut as one might naively imagine, then at those extremes probabilistic estimates may well be inappropriate.

Curiously, Russell discovered Russell's paradox, which was ousted from set theory by fiat (axiom). Then along came Goedel who proved that axiomatic set theory (a successor to the theory of types propounded by Russell and Alfred North Whitehead in their Principia Mathematica) could not be both complete and consistent. That is, Goedel jammed Russell's paradox right down the old master's throat, and it hurt. It hurt because Goedel's result makes a mockery of the fond Russellian illusion of the universe as giant computerized robot. How does a robot plan for and build itself? Algorithmically, it is impossible. Dawkins handles this conundrum, it seems, by confounding the "great explanatory power" of natural selection -- wherein lifeform robots are controlled by robotic DNA (selfish genes) -- with the origin of the cosmos.

But the biologist, so focused on this foundational issue of etiology, manages to avert his eyes from the Goedelian "frame problem." And yet even atheistic physicists sense that the cosmos isn't simplistically causal when they describe the overarching reality as a "spacetime block." In other words, we humans are faced with some higher or other reality -- a transcendent "force" -- in which we operate and which, using standard mathematical logic, is not fully describable. This point is important. Technically, perhaps, we might add an axiom so that we can "describe" this transcendent (topological?) entity, but that just pushes the problem back and we would then need another axiom to get at the next higher entity.

Otherwise, Dawkins' idea that this higher dimensional "force" or entity should be constructed faces the Goedelian problem that such construction would evidently imply a Turing algorithm, which, if we want completeness and consistency, requires an infinite regress of axioms. That is, Dawkins' argument doesn't work because of the limits on knowledge discovered by Goedel and Alan Turing. This entity is perforce beyond human ken.

One may say that it can hardly be expected that a biologist would be familiar with such arcana of logic and philosophy. But then said biologist should beware superficial approaches to foundational matters (6).

At this juncture, you may be thinking: "Well, that's all very well, but that doesn't prove the existence of god." But here is the issue: One may say that this higher reality or "power" or entity is dead something (if it's energy, it's some kind of unknown ultra-energy) or is a superbeing, a god of some sort. Because this transcendent entity is inherently unknowable in rationalistic terms, the best someone in Dawkins' shoes might say is that there is a 50/50 chance that the entity is intelligent. I hasten to add that probabilistic arguments as to the existence of god are not very convincing (7).

A probability estimate's job is to mask out variables on the assumption that with enough trials these unknowns tend to cancel out. Implicitly, then, one is assuming that a god has decided not to influence the outcome (8). At one time, in fact, men drew lots in order to let god decide an outcome. (One of the reasons that some see gambling as sinful is because it dishonors god and enthrones Lady Randomness.)

Curiously, Dawkins pans the "argument from incredulity" proffered by some anti-Darwinians but his clearly-its-absurdly-improbable case against a higher intelligence is in fact an argument from incredulity, being based on his subjective expert estimate.

Dawkins' underlying assumption is that mechanistic hypotheses of causality are valid at the extremes, an assumption common to modern naive rationalism.

Another important oversight concerns the biologist's Dawkins-centrism. "Your reality, if too different from mine, is quite likely to be delusional. My reality is obviously logically correct, as anyone can plainly see." This attitude is quite interesting in that he very effectively gives some important information about how the brain constructs reality and how easily people might suffer from delusions, such as being convinced that they are in regular communication with god.

True, Dawkins jokingly mentions one thinker who posits a Matrix-style virtual reality for humanity and notes that he can see no way to disprove such a scenario. But plainly Dawkins rejects the possibility that his perception and belief system, with its particular limits, might be delusional.

In Dawkins' defense, we must concede that the full ramifications of quantum puzzlements have yet to sink into the scientific establishment, which -- aside from a distaste for learning that, like Wile E. Coyote, they are standing on thin air -- has a legitimate fear of being overrun by New Agers, occultists and flying saucer buffs. Yet, by skirting this matter, Dawkins does not address the greatest etiological conundrum of the 20th century which, one would think, might well have major implications in the existence-of-god controversy.

Dawkins is also rather cavalier about probabilities concerning the origin of life, attacking the late Fred Hoyle's "jumbo jet" analogy without coming to grips with what was bothering Hoyle and without even mentioning that scientists of the caliber of Francis Crick and Joshua Lederberg were troubled by origin-of-life probabilities long before Michael J. Behe and Dembski touted the intelligent design hypothesis.

Astrophysicist Hoyle, whose steady state theory of the universe was eventually trumped by George Gamow's big bang theory, said on several occasions that the probability of life assembling itself from some primordial ooze was equivalent to the probability that a tornado churning through a junkyard would leave a fully functioning Boeing 747 in its wake. Hoyle's atheism was shaken by this and other improbabilities, spurring him toward various panspermia (terrestrial life began elsewhere) conjectures. In the scenarios outlined by Hoyle and Chandra Wickramasinghe, microbial life or proto-life wafted down through the atmosphere from outer space, perhaps coming from "organic" interstellar dust or from comets.

One scenario had viruses every now and again floating down from space and, besides setting off the occasional pandemic, enriching the genetic structure of life on earth in such a way as to account for increasing complexity. Hoyle was not specifically arguing against natural selection, but was concerned about what he saw as statistical troubles with the process. (He wasn't the only one worried about that; there is a long tradition of scientists trying to come up with ways to make mutation theory properly synthesize with Darwinism.)

Dawkins laughs off Hoyle's puzzlement about mutational probabilities without any discussion of the reasons for Hoyle's skepticism or the proposed solutions.

There are various ideas about why natural selection is robust enough to, thus far, prevent life from petering out (9). In my essay Do dice play God? (link above), I touch on some of the difficulties and propose a neo-Lamarckian mechanism as part of a possible solution, and at some point I hope to write more about the principles that drive natural selection. At any rate, I realize that Dawkins may have felt that he had dealt with this subject elsewhere, but his four-chapter thesis omits too much. A longer, more thoughtful book -- after the fashion of Penrose's The Emperor's New Mind -- is, I would say, called for when heading into such deep waters.

Hoyle's qualms, of course, were quite unwelcome in some quarters and may have resulted in the Nobel prize committee bypassing him. And yet, though the space virus idea isn't held in much esteem, panspermia is no longer considered a disrespectable notion, especially as more and more extrasolar planets are identified. Hoyle's use of panspermia conjectures was meant to account for the probability issues he saw associated with the origin and continuation of life. (Just because life originates does not imply that it is resilient enough not to peter out after X generations.)

Hoyle, in his own way, was deploying panspermia hypotheses in order to deal with a form of the anthropic principle. If life originated as a prebiotic substance found across wide swaths of space, probabilities might become reasonable. It was the Nobelist Joshua Lederberg who made the acute observation that interstellar dust particles were about the size of organic molecules. Though this correlation has not panned out, that doesn't make Hoyle a nitwit for following up.

In fact, Lederberg was converted to the panspermia hypothesis by yet another atheist (and Marxist), J.B.S. Haldane, a statistician who was one of the chief architects of the "modern synthesis" merging Mendelism with Darwinism.

No word on any of this from Dawkins, who dispatches Hoyle with a parting shot that Hoyle (one can hear the implied chortle) believed that archaeopteryx was a forgery, after the manner of Piltdown man. The biologist declines to tell his readers about the background of that controversy and the fact that Hoyle and a group of noted scientists reached this conclusion after careful examination of the fossil evidence. Whether or not Hoyle and his colleagues were correct, the fact remains that he undertook a serious scientific investigation of the matter.

http://www.chebucto.ns.ca/Environment/NHR/archaeopteryx.html

Another committed atheist, Francis Crick, co-discoverer of the doubly helical structure of DNA, was even wilder than Hoyle in proposing a panspermia idea in order to account for probability issues. He suggested in a 1970s paper and in his book Life Itself: Its Origin and Nature (Simon & Schuster 1981) that an alien civilization had sent microbial life via rocketship to Earth in its long-ago past, perhaps as part of a program of seeding the galaxy. Why did the physicist-turned-biologist propose such a scenario? Because the DNA helixes of all earthly life twist in the same direction. That seemed staggeringly unlikely to Crick, who thought we should find some DNA screws turning left and some right.

I don't bring this up to argue with Crick, but to underscore that Dawkins plays Quick-Draw McGraw with serious people without discussing the context. I.e., his book comes across as propagandistic, rather than fair-minded. It might be contrasted with John Allen Paulos' book Irreligion (see Do dice play god? above), which tries to play fair and which doesn't make duffer logico-mathematical blunders (10).

Though Crick and Hoyle were outliers in modern panspermia conjecturing, the concept is respectable enough for NASA to take seriously.

The cheap shot method can be seen in how Dawkins deals with Carl Jung's claim of an inner knowledge of god's existence. Jung's assertion is derided with a snappy one-liner that Jung also believed that objects on his bookshelf could explode spontaneously. That takes care of Jung! -- irrespective of the many brilliant insights contained in his writings, however controversial. (Disclaimer: I am neither a Jungian nor a New Ager).

Granted that Jung was talking about what he took to be a paranormal event and granted that Jung is an easy target for statistically minded mechanists and granted that Jung seems to have made his share of missteps, we make three points:

1. There was always the possibility that the exploding object occurred as a result of some anomalous, but natural event.

2. A parade of distinguished British scientists have expressed strong interest in paranormal matters, among them officers of paranormal study societies. The American Brian Josephson, who received a Nobel prize for the quantum physics behind the Josephson junction, speaks up for the reality of mental telepathy (for which he has been ostracized by the "billiard ball" school of scientists).

3. If Dawkins is trying to debunk the supernatural using logical analysis, then it is not legitimate to use belief in the supernatural to discredit a claim favoring the supernatural.

Getting back to Dawkins' use of probabilities, the biologist contends with the origin-of-life issue by invoking the anthropic principle and the principle of mediocrity, along with a verbal variant of Drake's equation http://en.wikipedia.org/wiki/Drake_equation

The mediocrity principle says that astronomical evidence shows that we live on a random speck of dust on a random dustball blowing around in a (random?) mega dust storm.

The anthropic principle says that, if there is nothing special about Earth, isn't it interesting how Earth travels about the sun in a "Goldilocks zone" ideally suited for carbon based life and how the planetary dynamics, such as tectonic shift, seem to be just what is needed for life to thrive (as discussed in the book Rare Earth: Why Complex Life is Uncommon in the Universe by Peter D. Ward and Donald Brownlee (Springer Verlag 2000))? Even further, isn't it amazing that the seemingly arbitrary constants of nature are so exactly calibrated as to permit life to exist, as a slight difference in the index of those constants known as the fine structure constant would forbid galaxies from ever forming? This all seems outrageously fortuitous.

Let us examine each of Dawkins' arguments.

Suppose, he says, that the probability of life originating on Earth is a billion to one or even a billion billion to one (10^-9 and 10^-18). If there are that many Earth-like planets in the cosmos, the probability is virtually one that life will arise spontaneously. We just happen to be the lucky winner of the cosmic lottery, which is perfectly logical thus far.

Crick, as far as I know, is the only scientist to point out that we can only include the older sectors of the cosmos, in which heavy metals have had time to coalesce from the gases left over from supernovae -- i.e., second generation stars and planets (by the way, Hoyle was the originator of this solution to the heavy metals problem). Yet still, we may concede that there may be enough para-Earths to answer the probabilities posed by Dawkins.

Though careful to say that he is no expert on the origin of life, Dawkins' probabilities, even if given for the sake of argument, are simply Bayesian "expert estimates." But, it is quite conceivable that those probabilities are far too high (though I candidly concede it is very difficult to assign any probability or probability distribution to this matter).

Consider that unicellular life, with the genes on the DNA (or RNA) acting as the "brain," exploits proteins as the cellular workhorses in a great many ways. We know that sometimes several different proteins can fill the same job, but that caveat doesn't much help what could be a mind-boggling probability issue.

Suppose that, in some primordial ooze or on some undersea volcanic slope, a prebiotic form has fallen together chemically and, in order to cross the threshold to lifeform, requires one more protein to activate. A protein is the molecule that takes on a specific shape, carrying specific electrochemical properties, after amino acids fold up. Protein molecules fit into each other and other constituents of life like lock and key (though on occasion more than one key fits the same lock).

The amino acids used by terrestrial life can, it turns out, be shuffled in many different ways to yield many different proteins. How many ways? About 10^60, which exceeds the number of stars in the observable universe by 24 orders of magnitude! And the probability of such a spark-of-life event might be in that ball park. If one considers the predecessor protein link-ups as independent events and multiplies those probabilities, we would come up with numbers even more absurd.

But, Dawkins has a way out, though he loses the thread here. His way out is that a number of physicists have posited, for various reasons, some immense -- even infinite -- number of "parallel" universes, which have no or very weak contact with this one and are hence undetectable. This could handily account for our universe having the Goldilocks fine structure constant and, though he doesn't specify this, might well provide enough suns in those universes that have galaxies to account for even immensely improbable events.

I say Dawkins loses the thread because he scoffs at religious people who see the anthropic probabilities as favoring their position concerning god's existence without, he says, realizing that the anthropic principle is meant to remove god from the picture. What Dawkins himself doesn't realize is that he mixes apples and oranges here. The anthropic issue raises a disturbing question, which some religious people see as in their favor. Some scientists then seize on the possibility of a "multiverse" to cope with that issue.

But now what about Occam's razor? Well, says Dawkins, that principle doesn't quite work here. To paraphrase Einstein, once one removes all reasonable explanations the remaining explanation, no matter how absurd it sounds, must be correct.

And yet what is Dawkins' basis for the proposition that a host of undetectable universes is more probable than some intelligent higher power? There's the rub. He is, no doubt unwittingly, making an a priori assumption that any "natural" explanation is more reasonable than a supernatural "explanation." Probabilities really have nothing to do with his assumption.

But perhaps we have labored in vain over the "multiverse" argument, for at one point we are told that a "God capable of calculating the Goldilocks values" of nature's constants would have to be "at least as improbable" as the finely tuned constants of nature, "and that's very improbable indeed." So at bottom, all we have is a Bayesian expert prior estimate.

Well, say you, perhaps a Wolfram-style algorithmic complexity argument can save the day. Such an argument might be applicable to biological natural selection, granted. But what selected natural selection? A general Turing machine can compute anything computable, including numerous "highly complex" outputs programed by easy-to-write inputs. But what probability does one assign to a general Turing machine spontaneously arising, say, in some electronic computer network? Wolfram found that "interesting" celullar automata were rare. Even rarer would be a complex cellular automaton that accidentally emerged from random inputs.

I don't say that such a scenario is impossible, but rather to assume that it just must be so is little more than hand-waving.

Dawkins tackles the problem of the outrageously high information values associated with complex life forms by conceding that a species, disconnected from information about causality, has only a remote probability of occurrence by random chance. But, he counters, there is in fact a non-random process at work: natural selection.

I suppose he would regard it a quibble if one were to mention that mutations occur randomly, and perhaps so it is. However, it is not quibbling to question how the powerful process of natural selection first appeared on the scene. In other words, the information values associated with the simplest known form (least number of genes) of microbial life is many orders of magnitude greater than the information values associated with background chemicals -- which was Hoyle's point in making the jumbo jet analogy.

And then there is the probability of life thriving. Just because it emerges, there is no guarantee that it would be robust enough not to peter out in a few generations (9).

Dawkins dispenses with proponents of intelligent design, such as biologist Michael J. Behe, author of Darwin’s Black Box: The Biochemical Challenge to Evolution (The Free Press 1996), by resort to the conjecture that a system may exist after its "scaffolding" has vanished. This conjecture is fair, but, at this point, the nature of the scaffolding, if any, is unknown. Dawkins can't give a hint of the scaffolding's constituents because, thus far, no widely accepted hypothesis has emerged. Natural selection is a consequence of an acutely complex mechanism. The "scaffolding" is indeed a "black box" (it's there, we are told, but no one can say what's inside).

Though it cannot be said that intelligent design advocate Behe has proved "irreducible complexity," the fact is that the magnitude of organic complexity has even prompted atheist scientists to look far afield for plausible explanations.

Biologists, Dawkins writes, have had their consciousnesses raised by natural selection's "power to tame improbability" and yet that power has very little to do with the issues of the origins of life or of the universe and hence does not bolster his case against god. I suppose that if one waxes mystical about natural selection -- making it a mysterious, ultra-abstract principle, then perhaps Dawkins makes sense. Otherwise, he's amazingly naive.

1. We don't claim that none of his criticisms are worth anything. Plenty of religious people, Martin Luther included, would heartily agree with some of his complaints, which, however, are only tangentially relevant to his main argument.

Anyone can agree that vast amounts of cruelty have occurred in the name of god. Yet, it doesn't appear that Dawkins has squarely faced the fact of the genocidal rampages committed under the banner of godlessness (Mao, Pol Pot, Stalin).

What drives mass violence is of course an important question. As an evolutionary biologist, Dawkins would say that such behavior is a consequence of natural selection, a point underscored by the ingrained propensity of certain simian troops to war on members of the same species. No doubt Dawkins would concede that the bellicosity of those primates had nothing to do with beliefs in some god.

So it seems that Dawkins may be placing too much emphasis on beliefs in god as a source of violent strife, though we should grant that it seems perplexing as to why a god would permit such strife.

Still, it appears that the author of Climbing Mount Improbable (W.W. Norton 1996) has confounded correlation with causation.

2. Properly this footnote, like the previous one, does not affect Dawkins' case against god's existence, which is the reason for the placement of these remarks.

In a serious lapse, Dawkins has that "there is something to be said" for treating Buddhism and Confucianism not as religions but as ethical systems. In the case of Buddhism, it may be granted that Buddhism is atheistic in the sense of denying a personal, monolithic god. But, from the perspective of a materialist like Dawkins, Buddhism certainly purveys numerous supernaturalistic ideas, with followers espousing ethical beliefs rooted in a supernatural cosmic order -- which one would think qualifies Buddhism as a religion.

True, Dawkins' chief target is the all-powerful god of Judaism, Christianity and Islam (Zoroastrianism too), with little focus on pantheism, hentheism or supernatural atheism. Yet a scientist of his standing ought be held to an exacting standard.

3. As well as conclusively proving that quantum effects can be scaled up to the "macro world."

4. The Blind Watchmaker: Why the Evidence of Evolution Reveals a Universe without Design (W.W. Norton 1986).

5. The same might be said of Dembski.

6. A fine, but significant, point: Dawkins, along with many others, believes that Zeno's chief paradox has been resolved by the mathematics of bounded infinite series. However, quantum physics requires that potential energy be quantized. So height H above ground is measurable discontinuously in a finite number of lower heights. So a rock dropped from H to ground must first reach H', the next discrete height down. How does the rock in static state A at H reach static state B at H'? That question has no answer, other than to say something like "a quantum jump occurs." So Zeno makes a sly comeback.

This little point is significant because it gets down to the fundamentals of causality, something that Dawkins leaves unexamined.

7. After the triumphs of his famous theorems, Goedel stirred up more trouble by a finding a solution to Eistein's general relativity field equations which, in Goedel's estimation, demonstrated that time (and hence naive causality) is an illusion. A rotating universe, he found, could contain closed time loops such that if a rocket traveled far enough into space it would eventually reach its own past, apparently looping through spacetime forever. Einstein dismissed his friend's solution as inconsistent with physical reality.

Before agreeing with Einstein that the solution is preposterous, consider the fact that many physicists believe that there is a huge number of "parallel," though undetectable, universes.

And we can leave the door ajar, ever so slightly, to Dawkins' thought of a higher power fashioning the universe being a result of an evolutionary process. Suppose that far in our future an advanced race builds a spaceship bearing a machine that resets the constants of nature as it travels, thus establishing the conditions for the upcoming big bang in our past such that galaxies, and we, are formed. Of course, we then are faced with the question: where did the information come from?

8. Unless one assumes another god who is exactly contrary to the first, or perhaps a group of gods whose influences tend to cancel.

9. Consider a child born with super-potent intelligence and strength. What are the probabilities that the traits continue?

A. If the child matures and mates successfully, the positive selection pressure from one generation to the next is faced with a countervailing tendency toward dilution. It could take many, many generations before that trait (gene set) becomes dominant, and in the meantime, especially in the earlier generations, extinction of the trait is a distinct possibility.

B. In social animals, very powerful individual advantages come linked to a very powerful disadvantage: the tendency of the group to reject as alien anything too different. Think of the recent tendency of white mobs to lynch physically superior black males. Or of the early 19th century practice of Australian tribesmen to kill mixed race offspring born to their women.

10. I have also made more than my share of those.

Plato and Cantor v. Wittgenstein and Brouwer

2011-11-10T14:30:00.001-08:00

Axiomatic thought realms and the foundations of mathematics

Pertinent N-fold pages
A geometric note on Russell's paradox
When algorithms collide
Disjoint nondenumerable sets of irrationals
Thoughts on the axiom of choice
Math resources on the web
An algorithm for implying the reals

Prove all things. Hold fast to that which is good.

--I Thes 5:21

[This page was begun in January 2002; as of Aug. 1, 2002, it remains a work in progress. Correction added Aug. 13, 2004, to include an inadvertently omitted "undecidable of the third kind."]

Integers and intuition

Without going into an extensive examination of phenomenology and the psychology of learning, perception and cognition, let us consider the mind of a child.

Think of Mommy controlling a pile of lollipops and crayons, some of which are red. In this game, the child is encouraged to pick out the red objects and transfer them to 'his' pile.

The child employs a mental act of separation (some might call this 'intuition') to select out an item, in this case by direct awareness of the properties of redness and of ease of holding with his hands. This primal separation ability is necessary for the intuition of replication. Crayon and lollipop are 'the same' by virtue of redness. In turn, this intuition of replication, or iteration, requires a time sense, whereby if the child hears 'more' he associates the word with an expectation of a craving being satisfied ('more milk').

The child becomes able to associate name-numbers with iteration, such that 'one thing more for me' becomes 'one thing,' which in time is abstracted to 'one.' A sequence of pulses is not truly iterative, because there is no procedure for enumeration. The enumeration procedure is essentially a successor function, with names (integers) associated with each act of selection by replication intuition. Likewise, we must have amorphous 'piles' before we can have sets. As adults we know that the 'mine' pile and the 'Mommy' pile have specific, finite numbers of elements. But we cannot discern the logico-mathematical objects of set and element without first having a concept of counting.

That is, in the minds of small children and of adults of primitive cultures, integers are associated with intuitively replicable material objects, such as apples and oranges. But the names are so useful that it is possible to mentally drop the associated objects in a process of abstraction. That is, we might consider an integer to be quite similar in spirit to an 'operator.' Whatever objects are associated with operators, a common set of rules of manipulation applies to the operators alone.

Platonism vs. intuitionism

Cantor's acceptance of 'actual infinities' seems to me to require a platonic concept of ideals: forms or formalisms that count as existing a priori.

The intuitionists, led by Brouwer and partially supported by Wittgenstein (and Kronecker before them), would object to a set or, possibly, a number that 'cannot be constructed. Related to this division is the dispute as to whether a theorem or mathematical form is discovered, as the platonists, see it, or invented, as the intuitionists see it.

I don't intend to inspect every wrinkle of these controversies but rather to focus on the concept of existence of mathematical statements and forms.

Forthwith, let's dismiss the concept of 'potential infinity' that was in vogue in the 19th century as a means of describing a successor operation. 'Potential' invokes the thought of 'empowered to achieve an end.' To say that 'potential infinity' is conceptually acceptable is to say that 'actual infinity' is also permissible.

Let us accept the Zermelo-Fraenkel successor set axiom as underpinning proof by induction and as underpinning open-ended successor functions, such as the function f(n) = n+1, which describes the natural numbers. This axiom is often known as the infinity axiom, but we are not, without further thought, entitled to take that leap. The successor axiom says that a recursion function needn't have a specific stop order. We may visualize a computer that spits out a stream of discrete outputs nonstop. (An issue here is that in thinking of a nonstop successor algorithm, we presuppose units of time being 1:1 with N, the set of natural numbers. Alas, our mental picture is inadequate to overcome the interdependence of primitive concepts.)

So at this point the successor axiom permits us to build ever-larger finite entities but does not permit us to assume some 'limiting value' associated with a particular successor function. Yet such limits are highly desirable.

Let us consider irrational reals. In the case of square roots, a geometric form -- the hypotenuse of a right triangle -- can be measured by a ruler (we neglect the issue of accuracy of the ruler, an issue that also applies to rationals) in less than a minute. Most would agree that the distance expressed by a square root exists and can be plotted on a number line, justifying the naming of that distance by a symbol such as x^0.5.

However, other irrationals, such as 2^0.2 can only ever-better 'approximated' as a rational by some successor function, such as Newton's method or an 'infinite' series. Because the nested interval in which such an irrational is found grows smaller and smaller, we might through careless thinking suppose that we can justify some limiting distance from origin because we believe we are getting closer and closer to an interval of zero length. But our successor function requires eternity to exactly locate that point. So in human terms, such a distance is unmeasurable and might be said by some to be nonexistent. Still, the difference between two nonstop successor functions, unequal in every finite output, may still be held to grow ever smaller, helping to justify existence of such a point.

Now, should we regard this distance/number a fait accompli or should we regard it as impossible to achieve?

Consider the circle. Is the circle an ideal thought form that axiomatically pre-exists geometry or is it an artifact of human ingenuity which in fact doesn't exist because a 'true circle' requires a nonstop algorithm -- perhaps the positing of a a set of n-gons of evermore facets? (Then of course the straight line, the point and the plane must be accepted a priori.)

Daniel J. Velleman (Philosophical Review,1993) proposed that constructivism be 'liberalized' to accept countable infinities (but not uncountable ones) on the grounds that 'performing infinitely many computations is not logically impossible, but only "medically impossible".'

Yet, an intuitionist or constructionist might disagree that such a performance is logically possible, but rather argue that the term 'countable infinity' is simply a phrase for describing extension by induction. That is,

If X is infinite and countable, then x e X <--> (x+n_o e X.

Still, we are implicitly presupposing that time is already divided into a countably infinite number of unit 1 intervals. That is, we face a circularity problem.

Nevertheless, the inductive model of X does not require that X ever be complete. That is, we can write [we use 'All' for the universal quantifier and '$' for the existential quantifier]

$n e N All m e N All t e T ( f(t_m)--> (x e X <--> x+n_o e X))

We would say that T is an ideal in P-space and not a result of a performable algorithm.

It seems quite evident that the pure constructivist program falters on the issue of time.

The issue is interesting because the successor axiom brings us to the issue of paradoxes (or antimonies), in particular those of Russell and Cantor. Though such paradoxes may be ruled axiomatically out of order, such an approach leaves a mild sense of disquiet, though I fear we must, if pressed, always resort to axioms.

At any rate, the value of a fundamental contradiction is that it demonstrates that a system of rules of thought based on form alone is insufficient to express the 'stuff' of being. And, of course, such a contradiction may pose serious questions as to the usefulness of a theory; I have in mind Cantor's paradox to the effect that the cardinality of U, the set of all sets, is unstable.

Following the Brouwerian path, Wittgenstein, who disliked such self-referencing anomalies, tried to dispose of them through a philosphical appeal to constructionist ideas. Cantor's champion, Hilbert, tried to limit the use of 'ideals,' but was nevertheless pushed to defend the notion of infinite totalities, at least implicitly. Without infinite totalities, or actual infinities, Cantor's paradise would fall.

The dispute between, essentially, platonists and constructionists is not resolvable without further elucidation, and is unlikely ever to be fully resolvable.

I suggest introduction of two axiomatic, or, primitive, concepts: a realm of thought assigned a timelike property, which, for short, we might dub T-space, for time-controlled, or Turing, space; a realm of thought with the property of timelessness, which we might dub P-space, for Platonic space. These spaces, or realms, are not topologically definable.

Now we are in a position to say that rules of mathematical thought that exist in T-space do not exist in P-space. There are no set-theoretic rules or relations in P-space, because no timelike operations.

We are permitted to collect all P-space objects into a set, but P-space itself is axiomatically not a mathematical set. The set of P-space objects is however an ideal and a resident of P-space.

Now, a P-space ideal may be exported to T-space and used in operations. Even so, if a P-space ideal is related to a T-space recursion function, the recursive's successor rule may not be applied to the ideal (no self-referencing permitted).

For example, a limiting value -- no finite nth step of an algorithm can go above it, or below it -- is is considered to exist in P-space. Likewise, the 'construction' of a circle occurs in T-space. The limiting form, a pure circle, is assigned to P-space.

At this juncture, it is necessary to point out that some constructions are purely logical, while others require repetitive computation. A recursion function, such as an algorithm to obtain pi, cannot yield an output value without the previous output value as an input value. That is, computation follows F_n o F_(n-1) o F_(n-2) ... F_0, where F_0 is the initial step of a composite function. Here construction occurs by 'building' one brick at a time. In the case of, for example, [lim n--> inf.] n/(n+1), a recursive computation does not occur. However, an inductive logical operation does occur. That is, we mean that n/(n+1) < (n+1)/(n+2) < 1 for any finite n. Does such a logical relation imply 'construction'? We may say that it can be thought of as a secondary form of construction, since values of n are constructed by

f(n+1) = n + 1.

Still, whether we have direct recursive construction or only indirect recursion, we place such mathematical operations in T-space. Because T-space is considered to be timelike, we avoid the issue of which comes first, the algorithm or the ideal.

Relations between T-space and P-space

An actual infinity can be defined as nondenumerable if it cannot be produced by a nonstop n-step algorithm. The algorithm's logical form is inductive: If property p holds for step n, then property p holds for step n+1.

In the case of a denumerable infinity, it is always possible to relate this platonic-space ideal to a Turing-space induction condition. However, simple induction of course is insufficient to justify a nondenumerable infinity. Cantor's diagonal proof of the nondenumerability of the reals uses the contradiction of the possibility of simple induction. Here we have a situation where the set of irrationals exists in P-space but the rule of inference in T-space is not induction alone.

So at this point we assert that if a relation between a P-space ideal and a T-space procedure cannot be justified to the satisfaction of the mathematical community, then we would say that the ideal is not recognized as a mathematical object, even if it be in some way numerical. For example, an infinite digit string which is a priori random would seem to have no direct relation to a T-space procedure, though perhaps an indirect relation might be found. However, if we set up a no-halt order procedure for pseudorandomly assigning at step n a digit to the nth digit space, the P-space ideal of an infinite pseudorandomly digit string would be held to exist as a mathematical object in P-space, being justified by an inductive claim: no matter how great n, there is no step at which the entire digit string inclusive of the nth digit can be known in advance.

In the case of a strict induction model for a geometric ideal, such as a curve, we can partly justify analytic methods here but the issue of the real continuum must also be addressed (see below). That is, we can say that if an n-gon with all facet endpoints equidistant from a centerpoint can be drawn, then a like (n+1)-gon can also be drawn. We relate this induction model to the ideal of a true circle by saying that 0 is the 'limiting value' of n-gon facet length.

Likewise, we can authorize the 'area under a curve' using a numerical 'approximation' induction model. [However, see the page above, 'When algorithms collide.']

A significant analytic issue here is raised by the induction model of obtaining arc length as a sum of approximated line segments. As n is increased, the difference in facet length decreases, so that at the limit of 0 length, all points are of equal length. Yet, each point on the arc is 1:1 with a point on the axis, which are also of 0 length. Yet the infinite sum of the points of the arc may be unequal to the infinite sum of the points on the axis interval. Does this mean arc zeroes are unequal to interval zeroes? Anyway, isn't 0 x infinity equal to 0?

We can always write off such a puzzlement as 'counterintuitive' and leave it at that. But I think it might help to say that the ideal associated with the T-space arc formula A is not identical with the ideal of the T-space arc formula B. We cannot in this case 'compare zeroes.' But we can say that ideal A is a quantum-like ideal where the zero is related to a sub-ideal which we call an infinitesimal quantity. And infinitesimal quantities may be unequal.

Perhaps you accept that the epsilon-delta proofs of analysis have killed off the dread infinitesimal. By that you mean that the induction method obtains a numerical limit but that pure geometric forms are not in fact mathematical objects. The number exists but the curve is not 'constructible.' The 'actual infinity' that makes 'all' points of a curve 1:1 with 'all' points of a line does not exist in this scenario.

Yet if ideals are sometimes necessary in mathematics, why arbitrarily rule out a particular ideal, such as an infinitesimal?

I do not wish to assert that fundamental issues are now, voila!, all resolved. I simply say that the concepts of T-space and P-space may make us more comfortable from the standpoint of consistency. Yet, more reflection is needed as to what these concepts mean with respect to Godel's incompleteness theorems.

Godel's incompleteness theorems say that for a consistent formal system F based, for example, on Peano arithmetic, there is always a true statement P that cannot be proved in F.

If we extend F (call it F₁) by adding P as a nonlogical axiom of F, then there is a statement P_F that is true but not provable in F₁.

We can define a set of systems such that F_n+1 is the formal system obtained by adding P_{F_n} as a nonlogical axiom to F_n.

So we have a T-space construction routine, or recursion algorithm, for compiling formal systems such that F_n --> F_n+1 --> P_{F_n+1 is true but not provable in F_n+1.}

If we define a P-space ideal lim_n->inf. F_{n, we see that Godel's result does not apply, since constructive activity is not permitted in P space, in which case the Godel sentence P_{F_n} is not defined.}

On a more fundamental level, we may wish to address the issue of belief that a theorem is true, based on our particular algebra, as against the theorem being a priori true, regardless of what one believes.

Consider what Wittgenstein saw as 'Moore's paradox,' which he obtained by coupling the statements 'There is a fire in the room' and 'I believe there is no fire in the room.' If statement A is a priori true, then, according to some, we would face a fundamental paradox.

You respond perhaps that one does not say 'There is a fire in the room' without either believing or not believing the assertion, whether or not there is an a priori truth to support it. That is, the truth value of a 'fact' is meaningless without a mind to review it. A cognitive act is not precluded by the notion that 'experience tells one' that previous sensory impressions (beliefs) about fire leads one to anticipate (believe) that one's current sensory impression about fire is valid.

So then, does a mathematical ideal require belief (perhaps justified by T-space inference) in order to exist? We come down to the definition of 'exist.' Certainly such an ideal cannot be apprehended without cognition. If, by cognition, we require a sense of time, then we would say that ideals are 'pointed to' from T-space thought patterns but also might exist independently of human minds in P-space, though of course the realms of mentation designated platonic space and turing space presuppose existence of some mind.

Of course, we must beware considering T-space to be a domain and P-space a range. These spaces are a priori mental conditions that cannot be strictly defined as sets or as topological objects.

Coping with paradoxes

Consider Cantor's paradox. The definition of power set permits us to compute the quantity of all elements of a finite power set. In every case, there area 2^n elements. But does an actual infinity, U, the set of all sets, exist? Since U is a set, shouldn't it have a corresponding actually infinite power set? What of the contradiction (with C the subset symbol and U' meaning power set) expressed:

U C U' C U'' C U''' ...?

Our response is that U, as a P-space ideal, may not have its 'shadow' generation rule applied to it. We can also accept U', as a P-space ideal, which also cannot have its shadow generation rule applied to it. Though we might export U or U' to T-space for some logico-mathematical operation, we cannot do the operation U C U', which requires application of set-building rules on U, a banned form of self-referencing.

In general, we prohibit a successor rule from being applied non-vacuously to an actually infinite ideal. For example, [lim n->inf.](n) + 1 = n is simply the vacuous application of a successor rule.

Similarly for Russell's paradox: R, the set of 'all' (here assuming 'all' signifies an infinitude) sets that contain themselves as members, and S, which is R's complement, exist as P-space ideals. If exported to T-space a successor rule cannot be applied. So the question of whether R e R is prohibited. However the T-space operation R u S = U is permitted.

An infinite (or open-ended) set 'generated' by a successor rule requires a concept of time, which includes the concept of 'rate of change' (even if the rate is an unobtrusive 1). If we talk about a completed denumerably infinite set, we are saying that 0 is the limiting value of the generation algorithm's rate of change.

Let A be a finite set and P(A) be the power set of A. We now specify

P(A)->P(P(A)), which we may express P^[0](A)-> P^[1](A).

So, in general, we have P^[n](A).

Now to indicate the power set of the set of all power sets, we write

lim_{n-> ¥} P^[n](A) The usual way to dispose of this paradox is to say that though a collection is an extension of the set concept, a collection is not necessarily a set. Hence the collection of all sets would not itself be a set -- a theorem stemming from the ZF axioms.

However, here we address the paradox by saying that, if the denumerably infinite set is construed as completed, then the generation algorithm's rate of change is 0, as in

lim_{n-> ¥}P^[n+1](A) = lim_{n-> ¥}P^[n](A).

In other words, lim_{n-> ¥}P^[n](A) exists in P-space and a 'self-referencing' T-space algorithm is prohibited.

The principle of the excluded middle

The principle of the excluded middle -- which is often read to mean a logico-mathematical statement is either true or false, with no third possibility -- was strongly challenged by Brouwer, who argued that the principle is unreliable for infinities. Our rule of prohibiting 'self-referencing' operations on ideals helps address that concern.

The reliability of the principle of the excluded middle is a concern in, for example, the Goldbach conjecture.

Let us define the Goldbach conjecture inductively as

i) Q = ((P(2x) --> P(2(x+1)))

A disproof requires

ii) ~Q = ~((P(2x) --> P(2(x+1)))

There is also the possibility that neither i) nor ii) is decidable [using '+' for the exclusive 'or']:

iii) ~(Q + ~Q)

Here we see a point where platonists and intuitionists clash. The platonists, rejecting iii) as a way of writing 'Q is undecidable' would claim that merely because we cannot know whether Q or ~Q is true does not mean that it is false that either has a truth value. The intuitionists would argue that Q's alleged truth value is of no mathematical interest.

If we accept iii), we must require that De Morgan's law not apply to the exclusive 'or,' even though truth tables for ~P v Q and ~P + Q are identical.

De Morgan's law transforms iii) into ~Q & Q, which is false by contradiction.

However,

P v Q = ((P & Q) + (P + Q))

But if P = ~Q, we would have

~Q v Q = ((~Q & Q) + (~Q + Q))

Yet the contradiction ~Q & Q is disallowed.

A similar philosophical perplexity arises from the question of whether Euler's constant is rational or irrational. The constant g is considered to be a number on the basis of at least one induction model. To wit:

lim _{-> ¥} å1/n - In1/n = g

where g is an ideal constant.

It is quite plausible that gamma's rationality is undecidable, that there is insufficient data to determine rationality. So the statement 'gamma is rational' may not have a knowable truth value. Does it have an a priori truth value? Many mathematicians would assert that if a truth value is unknowable, the issue of a priori truth value is irrelevant.

Still, undecidability is most satisfactory if proved. Our position would be that, in the case of gamma, the irrationality conjecture would be proved undecidable if rationality could never be decided without application of a successor rule on gamma.

In the case of the continuum hypothesis, we see a case where a logico-mathematical statement has a 0 truth value, validating the warning against unrestricted use of the principle of the excluded middle. Godel and Cantor have collectively shown that Cantorian and ZF set theory contain insufficient information for a yes or no answer to the conjecture, which says that there is no Cantorian cardinal number between cardN (or Aleph_null, if you like) and cardP(N) (another Aleph).

The implicit flaw in the continuum conjecture is the expectation that the conjecture is either true or false. If you draw a playing card face down from a well-shuffled deck an do not turn it over, the proposition 'the card is a face card' is either true or false -- even if you do not examine the obverse before shuffling the card back into the deck. Though the truth value remains forever undecidable, it is presumed to have an a priori truth value. I call such a proposition an undecidable statement of the first kind.

The continuum conjecture is then an undecidable statement of the second kind -- undecidable because the statement has no truth value in some logic system, whether that system be sharply or fuzzily defined.

[Thanks to Paul Kassebaum for drawing my attention to a difficulty with my categorization of undecidables. It seems that I inadvertently omitted the category of undecidable statements of the third kind, which would cover questions that are notionally answerable but which are computationally too difficult. For example, it is computationally imposssible to even name most numbers, let alone compute with them. However, Paul had a good point in noting that computational difficulty seems to fit my "first kind" category, in that, from a platonist perspective, both categories have a priori answers that are inaccessible.

In addition, a "fourth kind" seems in order: the obvious one stemming from Godel's incompleteness theorems: a sufficiently rich complete system contains at least one undecidable statement.]

There is of course the issue of the provability of the assertion that the playing card is either a face card or not; taking a cue from the Copenhagen interpretation of quantum mechanics, we cannot be sure that the two realities are not combined into a superposed state, with neither reality in existence until an observation is made. Though such an interpretation is normally applied to the nanometer world, the thought experiment about Schrodinger's cat shows that quantum weirdness can be scaled up to the macro-world. We cannot be sure that 'reality' does not work that way. (See 'The resurrection of Schrodinger's cat' at the link above.)

I have been unable to think of a logico-mathematical statement that is an undecidable of the first kind and I conjecture that such a statement cannot be proposed.

Of course, propositions of the second kind are common in mathematics, as in: 'The nth integer is prime.'

Godel and Cohen have proved the continuum hypothesis to be such a 'meaningless' statement; similarly our scheme makes the paradoxes of Russell and Cantor equivalent to an undecidable of the second kind.

In our model, we would say that cardN

and cardP(N) are P-space ideals but that a cardX such that cardN less than cardX less than cardP(N) is not a mathematical object in P-space because no inference rule exists relating a T-space procedure to a P-space ideal.

In a 1930 paper, Heyting

(appearing in 'From Brouwer to Hilbert,' compiled by Paolo Mancosu, Oxford, 1998), says the intuitionists replace the concept 'p is false' with 'p implies a contradiction.' So then, ~p is a 'new proposition expressing the expectation of being able to reduce p to a contradiction' and '|- ~p will mean "it is known how to reduce ~p to a contradiction".' Hence comes the possibility that neither |- p nor |- ~p is decidable.

Heyting notes that |- ~~p means 'one knows how to reduce to a contradiction the supposition that p implies a contradiction,' and that |- ~~p can occur without |- ~p 'being fulfilled,' thus voiding double-negation and the principle of the excluded middle.

Through tables of such inferences, Heyting derives the logico-mathematical inference states of proved, contradictory, unsolvable, unprovable, not unprovable, not contradictory and not decided.

On the continuum

The obvious way to define the reals is to posit 'any' infinite digit string and couple it to every integer. Of course, Cantor's diagonal argument proves by contradiction that the reals cannot be enumerated. Since the rationals can be counted, it is the irrationals that cannot be.

It is curious that the if f is a function that yields a unique real, the family of such functions, Uf, is considered denumerable. That is, we might try to list such writable functions by i e I. We could then write an antidiagonal function g = f_i(i) + 1. But logicians disdain this type of paradox by requiring that f be written in a language L that imposes a finite set of operations on a finite set of symbols. It is found that the function g cannot be written without resort to an extended language L'.

(It is however possible to establish a nondenumerable subset of irrationals that is 1:1 with a subset of writable functions f if we permit the extended language L'. See 'Disjoint nondenumerable sets of irrationals' above.) So then we find that Cantor's diagonal proof reduces to an existence theorem for a subset of reals undefinable in some language L. To put it another way, such a real is 'unreachable.' We cannot order such an r by the inequality p/q < r < s/t (with p,q,s,t e Z) because we cannot ascertain p/q or s/t.

In my view, such unreachable reals are ideals. They exist in relation to the ideal of the totality of reals. But their shadows do not exist in T-space. So they have low relevance to mathematics. In fact, we cannot even say that the subset of such reals contains individual elements, raising a question as to whether such a subset exists. Again, the subset is a P-space ideal, and the T-space method of defining this subset by individual elements does not apply.

So then we have that cardR (whatever aleph that is after aleph-null) becomes a shorthand way of saying that there exists a set of irrationals whose elements are undefinable in L.

The concept of nondenumerability is useful in that it conveys that the set of irrationals is much 'denser' than the set of rationals. This gives rise to the thought of ordering of infinities with transfinite numbers. But 'X is nondenumerable' means X is ~cardN. The continuum conjecture could be expressed cardN < cardM < cardR. But it can be better expressed cardN < cardM < ~cardN. This last expression succinctly illustrates the result of the labors of Godel and Cohen: that the conjecture is 'independent' of set theoretic logic.

Arithmetic recursives

Consider

å_{i_o} i

We might call this a paradigm iterative function, where the domain and range intersect except at possibly initial and final (or limit) values. Such a function is considered 'non-chaotic' because the sequence is considered informative. That is the naming routine of å i is the same as the naming routine for i. The digit-place system is considered informative because we can order a number y less than x less than z in accord with this naming routine. That is, we know that 52 > 5 because of the rules of the digit-place system.

As shown below, a recursive function may be construed as non-chaotic if the recursive f is writable as some simple well-known function, such as n or n!

An arithmetic recursive can be written:

f(n) = h(n)f(n-1) + g(n)

We note that h(n) and-or g(n) can also be recursive, for a composite of the type: h(n) =k(n)h(n-1) + l(n)

and that it is also possible to have such expressions as

f(n) = h(n)f(n-1) + f(n-2)

where f(n-2) kicks in after a kth step of f(n).

A few recursive sequences:

h(n) = n, g(n) = 0

f(0) = 1, f(n) = n!

In general, if g(n) = 0, then f(n) = Õh(n).

h(n) = (n+1)/n, g(n)=n

f(0) =1, f(n) = 4, 9, 16 = n²; f(0) = 0, f(n) = 3/2, 3, 19/4...

h(n) = 1/n, g(n) = 1

f(0) = 1, f(n) = 1, 3/2, 11/8...

h(n) = 1/n, g(n) = n

f(0) = 1, f(n) = 2, 3, 4... = n

h(n) = 1/n, g(n) =1/ n

f(0) = 1, f(n) = 2, 5/2, 7/6, 13/24...

h(n) = 1, g(n) = 1/n

f(0) = 0, f(n) = 1, 3/2, 5/6 ...

f(0) = 1, f(n) = 1, 2, 5/2...

h(n) = n, g(n) = n

f(0) = 1, f(n) = 2, 6, 21...

h(n) = -n, g(n) = -n

f(0) = 0, f(n) = 0, -3, 8, -45 ...

h(n) = -n, g(n) = n

f(0) = 0, f(n) = 1, 0, 3, -8, 45...[recheck]

h(n) = -n, g(n) = -1

f(0) = 0, f(n) = -1, 1, -4, 15, -66...

if g(n) = 1, we get f(n) = 1, -1, 4, -15, 66...

Basic manipulations of such recursives:

Let f_k express h(n)f(n-1) + g(n) where f(0) = k and k is any real initial value.

If g(n) = 0, then f_k/f_j = (Õ h(n)k)/(Õ h(n)j) = k/j.

f_k - f_j = Õh(n)k - Õ h(n)j = Õ h(n)(k-j)

This last expression yields a function

m_(k-j) = h(n)f(n-1) + 0g(n)

Denoting the discrete first derivative of f as f', we have, if h = 1:

f'(n) = f(n) - f(n-1) = g(n).

If h does not equal 1 at all values, we can write f' as an inequality, as in

h'(n) £ f'(n) £ g'(n), or g'(n) £ f'(n) £ h'(n)

Example:

f(n) = (n² + (1)f(n-1) + n³

g'(n) is polynomial but Õ k(n^2 + 1) changes exponentially. Hence after some finite value of n, we have, assuming k is a positive integer:

3n² < f'(n) < [Õ k(n² + 1) - Õ k((n-1)² + 1)]

Example:

f(n) = (n² + 1)f(n-1) + Õ (n³ + 1)

Because g(n)'s exponential rate of change is higher than f(n)'s, we have, after some finite n:

h'(n) < f'(n) < g'(n)

Such inequalities are found in recursive ratios f_a/f_b, where f_a =/= f_b

For example, the irrational z(-2) can be written:

f_a(n)/f_b(n) = [n²f_a(n-1) - (n-1)²!]/n²!

At step n+1, the ratio has a numerator that contains an integer greater than the numerator integer for step n; likewise for the denominator.

We see that ever-greater integers are required. Assuming the limit of f_a/f_b is constant, when this ratio at step n is converted into a decimal string, the string lengthens either periodically or aperiodically.

So we might say that [lim n-> inf]f_a is an 'infinite integer' -- or a pseudo-integer. An irrational can be then described as a ratio whereby two pseudo-integers are relatively prime. Or, we might say that there exists a proof that if the ratio is relatively prime at step n, it must be relatively prime at step n+1.

Obviously, the set of pseudo-integers is 1:1 with the reals, a pseudo-integer being an infinite digit string sans decimal point.

If we require that a real be defined by a writable function based on the induction requirement, then the set of reals is countable. But if we divorce the function from the general induction requirement, then the set of reals is nondenumerable, as discussed here.

Do your best to present yourself to God as one approved, a workman who has no need to be ashamed, rightly handling the word of truth.

--2 Tim 2:15

2011-11-10T14:20:00.000-08:00

The axiom of choice

and non-enumerable reals

Plato and Cantor vs. Wittgenstein and Brouwer
Thoughts on diagonal reals
Eric Schechter's page on the Axiom of Choice (plenty of links)

[Posted online March 13, 2002; revised July 30, 2002, Aug. 28, 2002, Oct. 12, 2002, Oct. 24, 2002; June 2003]

The following proposition is presented for purposes of discussion. I agree that, according to standard Zermelo-Fraenkel set theory, the proposition is false.

Proposition: The Zermelo-Fraenkel power set axiom and the axiom of choice are inconsistent if an extended language is not used to express all the reals.

Discussion:

The power set axiom reads: 'Given any set X there is a set Y which has as its members all the subsets of X.' The axiom of choice reads: 'For any nonempty set X there is a set Y which has precisely one element in common with set X.'*

*Definitions taken from 'Logic for Mathematicians,' A.G. Hamilton, Cambridge, revised 1988.

It is known that there is a denumerable set X of writable functions f such that f defines r e R, where R is the nondenumerable set of reals. By writable, we mean f can be written in a language L that has a finite set of operations on a finite set of symbols. In other words, X contains all computable reals.

[This theorem stems from the thought that any algorithm for computing a number can be encoded as a single unique number. So, it is argued, since the set of algorithms is denumerable, so is the set of computable reals. However, we must be cautious here. It is possible for a brouwerian choice rule (perhaps using a random number generator) to compute more than one real.]

X is disjoint from a nondenumerable set Y, subset of R, that contains all noncomputable and hence non-enumerable reals.

P(Y) contains all the subsets of Y, and, like Y, is a nondenumerable infinity. Yet y e Y is not further definable. We cannot distinguish between elements of Y since they cannot be ordered, or even written. Hence, we cannot identify a 'choice' set Z that contains one element from every set in P(Y).

[However, it is important to note that some non-enumerables can be approximated as explicit rationals to any degree of accuracy in a finite number of steps, though such numbers are not Turing computable. See 'Thoughts on diagonal reals' above.]

Remark: It may be that an extended language L' could resolve this apparent inconsistency.

The basic criticism from two mathematicians is that merely because a choice set Z cannot be explicitly identified by individual elements does not prevent it from existing axiomatically.

Dan Velleman, an Amherst logician, remarked: 'But the axiom of choice does not say that 'we can form' [I later replaced 'form' with 'identify'] a choice set. It simply says that the choice set exists. Most people interpret AC as asserting the existence of certain sets that we cannot explicitly define.'

My response is that we are then faced with the meaning of 'one' in the phrase 'one element in common.' The word 'one' doesn't appear to have a graspable meaning for the set Z. Clearly, the routine meaning of 'choice' is inapplicable, there being nothing that can be selected. The set Z must be construed as an abstract ideal that is analogous to the concept of infinitesimal quantity, which is curious since set theory arose as an answer to the philosophical objection to such entities.

It is amusing to consider two types of vacuous truth (using '$' for the universal quantifier and '#' for the existential quantifier):

I. $w e W Fw & W = { }.

II. Consider the countable set X containing all computable reals and the noncountable set Y containing all noncomputable reals.

The statement

$r e R Fr --> $y e Y Fy

even though no y e Y can be specified from information given in Y's definition.

That is, I. is vacuously true because W contains no elements, whereas II. is vacuously true because Y contains no elements specified by Y's definition.

It is just the set Y that the intuitionist opposes, of course. Rather than become overly troubled by the philosophy of

existence

, it may be useful to limit ourselves to specifiability, which essentially means the ability to pair an element with a natural number.

Those numbers in turn can be paired with a successor function, such as S...S(O).

We should here consider the issue of transitivity, whereby the intuitionist admits to

A --> {A}

but does not accept A --> {{A}}

without first specifying, defining or expressing {A}.

That is, the intuitionist says

A --> {{A}} only if {{A}} --> {A}, which is only true if {A} has been specified, which essentially means paired with n e N.

In their book, Philosophies of Mathematics (Blackwell, 2002), Alexander George and Velleman offer a deliberately weak proof of the theorem that says that every infinite set has a denumerable subset:

'Proof: Suppose A is an infinite set. Then A is certainly not the empty set, so we can choose an element a₀ e A. Since A is infinite, A =/= {a₀}, so we can choose some a₁ e A such that a₁ =/= a₀. Similarly, A =/= {a₀,a₁}, so we can choose a₂ e A such that a₂ =/= a₀ and a₂ =/= a₁. Continuing in this way, we can recursively choose a_n e A such that a_n ~e {a₀,a₁,...,a_n-1}. Now let R = {<0,a₀>, <1,a₁>, <2,a₂>...} . Then R is a one-to-one correspondence between N and the set {a₀,a₁,a₂...}, which is a subset of A. Therefore, A has a denumerable subset.'

The writers add, 'Although this proof seems convincing, it cannot be formalized using the set theory axioms that we have listed so far. The axioms we have discussed guarantee the existence of sets that are explicitly specified in various ways -- for example, as the set of all subsets of some set (Axiom of Power Sets), or as the set of all elements of some set that have a particular property (Axiom of Comprehension). But the proof of [the theorem above] does not specify the one-to-one correspondence R completely, because it does not specify how the choices of the elements a₀,a₁,a₂... are to be made. To justify the steps in the proof, we need an axiom guaranteeing the existence of sets that result from such arbitrary choices:'

The writers give their version of the axiom of choice:

'Axiom of choice. Suppose F is a set of sets such that Æ =/= F. Then there is a function C with domain F such that, for every X e F, C(X) e X.'

They add, 'The function C is called a choice function because it can be thought of as choosing one element C(X) from each X e F.'

The theorem cited seems to require an abstracted construction algorithm. However, how does one select a₀,a₁,a₂... if the elements of Y are individually nondefinable? AC now must be used to justify counting elements that can't be identified. So now AC is used to assert a denumerable subset by justifying a construction algorithm that can, in principle, never be performed.

Suppose we define a real as an equivalence class of cauchy sequences. If [{a_n}] e X, then [{a_n}] is computable and orderable. By computable, we mean that there is some rule for determining, in a finite number of steps, the exact rational value of any term a_n and that this rule must always yield the same value for a_n.

A brouwerian choice sequence fails to assure that a_n has the same value on every computation, even though {a_n} is cauchy. Such numbers are defined here as 'non-computable,' though perhaps 'non-replicable' is a better characterization. A brouwerian cauchy sequence {a_n}, though defined, is not orderable since, in effect, only a probability can be assigned to its ordering between 1/p and 1/q.

Now we are required by AC to say that either {a_n} is equivalent to {b_n} and hence that [{a_n}] = [{b_n}] or that the two sequences are not equivalent and that the two numbers are not equal.

Yet, a brouwerian choice sequence defines a subset W of Y, whereby the elements of W cannot be distinguished in a finite number of steps. Yet AC says that the trichotomy law applies to w₁ and w₂.

We should note that W may contain members that coincide with some x e X. For example, we cannot rule out that a random-number generator might produce all the digits in pi.

In a 1993 Philosophical Review article, Constructivism liberalized Velleman defends the notion that only denumerable sets qualify as actual infinities. In that case, AC would, I suppose, not apply to a nondenumerable set X since the choice function could only apply to a denumerable subset of X. One can't apply the choice function to something that doesn't exist.

Essentially, Velleman 1993 is convinced that Cantor's reducto ad absurdum proof of nondenumerability of the reals should be interpreted: 'If a set of all reals exists, that set cannot be countable.' By this, we avoid the trap of assuming, without definition, that a set of all reals exists.

He writes that 'to admit the existence of completely unspecifiable reals would violate our principle that if we want to treat real numbers as individuals, it is up to us to individuate them.'

'As long as we maintain this principle, we cannot accept the classical mathematician's claim that there are uncountably many completely unspecifiable real numbers. Rather, the natural conclusion to draw from Cantor's proof seems to be that any scheme for specifying reals can be extended to a more inclusive one, and therefore the reals form an indefinitely extensible totality.'

He favors use of intuitionist logic for nondenumerable entities while retaining classical logic for denumerable sets.

'The arguments of the constructionists have shaken my faith in the classical treatment of real numbers, but not natural numbers,' Velleman 1993 writes in his sketching of a philisophical program he calls 'liberal constructivism.' Unlike strict constructivists, he accepts 'actual infinities,' but unlike classical mathematicians, he eschews uncountable totalities.

For example, he doubts that 'the power set operation, when applied to an infinite set, results in a well-defined totality.'

His point can be seen by considering the Cantorian set of reals. We again form the denumerable set X of all computable, and enumerable, reals and then write the complement set R-X. Now if we apply AC to R-X in order to form a subset Y, does it not seem that Y ought to be perforce denumerable, especially if we are assuming that Y may be constructed? That is, the function C(R-X) seems to require some type of instruction to obtain a relation uRv. If an instruction is required in order to pair u and v, then Y would be denumerable, the set of instructions being denumerable. But does not AC imply that no instruction is required?

Of course, we can then write (R-X)-Y to obtain a nondenumerable subset of R.

We can think of two versions of AC¹: the countable version and the noncountable. In the countable version, AC says that it is possible to select one element from every set in a countable collection of sets. In the noncountable version, AC says that the choice function may be applied to a nondenumerable collection of sets.

In strong AC, we must think of the elements being chosen en masse, rather than in a step-by-step process.

The wildness implicit in AC is further shown by the use of a non-formulaic function to pair noncomputables in Y with noncomputables in a subset of Y, as in f:Y->Y. That is, suppose we take all the noncomputables in the interval (0,1/2) and pair each with one noncomputable in (1/2,1), without specifying a means of pairing, via formula or algorithm. Since we can do the same for every other noncomputable in (1/2,1), we know there exists a nondenumerable set of functions pairing noncomputables.

This is strange. We have shown that there is a nondenumerable set of nonformulaic functions to pair non-individuated members of domY with a non-individuated member of ranY. If x e domY, we say that x varies, even though it is impossible to say how it varies. If y_o e ranY, we can't do more than approximate it on the real line. We manipulate quantities that we can't grasp. They exist courtesy of AC alone.

In an August 2002 email, Velleman said that though still attracted to this modified intuitionism, he is not committed to a particular philosophy.

Jim Conant, a Cornell topologist, commented that the reason my exposition is not considered to imply a paradox is that 'the axioms of set theory merely assert existence of sets and never assert that sets can be constructed explicitly.' The choice axiom 'in particular is notorious for producing wild sets that can never be explicitly nailed down.'

He adds, 'A platonist would say that it is a problem of perception: these wild sets are out there but we can never perceive them fully since we are hampered by a denumerable language. Others would question the meaning of such a statement.'

Also, the ZF axioms are consistent only if the ZF axioms + AC are consistent, he notes, adding that 'nobody knows whether the ZF axioms are consistent.' (In fact, his former adviser, topologist Mike Freedman, believes there are ZF inconsistencies 'so complicated' that they have yet to be found.)

'Therefore I take the point of view that the axiom of choice is simply a useful tool for proving down to earth things.'

Yet it is hard to conceive of Y or P(Y)\Y as 'down to earth.' For example, because Y contains real numbers, they are axiomatically 'on' the real number line. Yet no element of Y can be located on that line. y is a number without a home.

[See the 'Plato' link above.]

Note added in April 2006: Since arithmetic can be encoded in ZFC, we know from Kurt Godel that ZFC is either inconsistent or incomplete. That is, there is at least one true statement in ZFC that cannot be proved from axioms or ZFC contains a contradiction.

We also know that ZFC is incomplete in the sense that the continuum hypothesis can be expressed in ZFC, its truth status is independent of ZFC axioms, as Godel and Paul Cohen have shown.

1. Jim Conant brought this possibility to my attention.

2011-11-10T14:17:00.001-08:00

An algorithm for implying all reals

Thoughts on diagonalization

An algorithm for constructing the reals
Approximating some non-enumerable reals with rationals
Extending the concept of non-denumerability
Does the set of choice sequences contain a non-enumerable limit?
Have algorithm, will travel (sets of optimal graphs)

Thoughts on the axiom of choice
Constructing non-denumerable sets of reals

An algorithm to construct the set of reals

We present an algorithm for constructing the entire set of reals and find that our intuition suggests that no diagonal can exist, which accords with Cantor's proof.

We write an algorithm for determining the set of reals, thus:

We have the set of 1x1 square cells in a quadrant of the Cartesian grid. Each square is eligible to contain a digit. We consider only the horizontal strings of cells.

At lim_n->inf. 2ⁿ⁺², we have established all base 2 infinite digit strings, expressing the entire set of reals greater than an arbitrary j e Z and less than or equal to j+1.

The algorithm above requires that for any step n, the set of digit strings is a rectangle n2ⁿ⁺².

But lim_n->inf. n/2ⁿ = 0, meaning the rectangle elongates and narrows toward a limiting form of a half-line.

It also follows that no exponentially defined set of tiles n x kⁿ has a cellular diagonal at the tiling algorithm's infinite limit.

Interestingly, the power set of any n e N has card2ⁿ, which corresponds to step n of our algorithm, in which we have a set of 2ⁿ⁺² pre-reals.

Additional remark:

Lemma: Any set with lim_n->inf kⁿ elements has the cardinality of the reals, with k =/= 0, -1, 1 and k e Z.

Proof:

Theorem: Some non-enumerable reals can be approximated with explicit rationals to any degree of accuracy in a finite number of steps.

Proof:

Construct n Turing machines consecutively, truncating the initial integer, and compute each one's output as a digit string n digits long. Use some formula to change each diagonal digit.

Comment: The denumerable set of computables implies an extension of the concept of denumerability.

Justification:

We give these instructions for diagonalizing Turing computables:

If we diagonalize the diagonals, it is not apparent to me that this real is not a member of the computables.

Definition of choice sequence:

If f(n) gives a_n of cauchy sequence {a_n}, then {a_n} is a choice sequence if f(n) --> a_{o_n+1} or a_{1_n+1} or . . . or a_{m_n+1}.

Remark: Though choice sequences are primarily of interest to intuitionists, here we require that they be governed by ZFC.

Theorem: The question of whether the set of choice sequences contains an element with a non-enumerable limiting value is not decidable.

Proof:

We first prove (Lemma i) that within a spread (x,y), with x the GLB and y the LUB, a non-enumerable exists.

So then suppose x and y are irrational enumerables. Rationals arbitrarily close to x from above and to y from below can be found.

Let x < p and q < y. So calling the choice sequence limit L, we have

Case i: (x < p < L < q < y).

Case ii: The possibility x < L < p exists, but then a smaller rational can be found between x and L.

Case iii: Likewise for the possibility q < L < y.

It is now straightforward that if L is a choice function limit, there is an effectively random possibility that L is enumerable or non-enumerable. This possibility is in principle undecidable.

Though probability laws suggest that the set of choice sequences includes a sequence with a non-enumerable limit, this suggestion is undecidable.

2011-11-10T14:15:00.001-08:00

Time thought experiments

Godel's theorem and a time travel paradox

In How to Build a Time Machine (Viking 2001), the physicist Paul Davies gives the 'most baffling of all time paradoxes.' Writes Davies:

'A professor builds a time machine in 2005 and decides to go forward ... to 2010. When he arrives, he seeks out the university library and browses through the current journals. In the mathematics section he notices a splendid new theorem and jots down the details. Then he returns to 2005, summons a clever student, and outlines the theorem. The student goes away, tidies up the argument, writes a paper, and publishes it in a mathematics journal. It was, of course, in this very journal that the professor read the paper in 2010.'

Davies finds that, from a physics standpoint, such a 'self-consistent causal loop' is possible, but, 'where exactly did the theorem come from?... it's as if the information about the theorem just came out of thin air.'

Davies says many worlds proponent David Deutsch, author of The Fabric of Reality and a time travel 'expert,' finds this paradox exceptionally disturbing, since information appears from nowhere, in apparent violation of the principle of entropy.

This paradox seems well suited to Godel's main incompleteness theorem, which says that a sufficiently rich formal system if consistent, must be incomplete.

Suppose we assume that there is a formal system T -- a theory of physics -- in which a sentence S can be constructed describing the mentioned time travel paradox.

If S strikes us as paradoxical, then we may regard S as the Godel sentence of T. Assuming that T is a consistent theory, we would then require that some extension of T be constructed. An extension might, for example, say that the theorem's origin is relative to the observer and include a censorship, as occurs in other light-related phenomena. That is, the professor might be required to forget where he got the ideas to feed his student.

But, even if S is made consistent, there must then be some other sentence S', which is not derivable from T'.

Of course, if T incorporates the many worlds view, S would likely be consistent and derivable from T. However, assuming T is a sufficiently vigorous mathematical formalism, there must still be some other sentence V that may be viewed as paradoxical (inconsistent) if T is viewed as airtight.

How old is a black hole?

Certainly less than the age of the cosmos, you say.

The black hole relativistic time problem illustrates that the age of the cosmos is determined by the yardstick used.

Suppose we posit a pulsar pulsing at the rate T, and distance D from the event horizon of a black hole. Our clock is timed to strike at T/2, so that pulse A has occurred at T=0. We now move the pulsar closer to the event horizon, again with our clock striking at what we'll call T'/2. Now because of the gravitational effect on observed time, the time between pulses is longer. That is T' > T, and hence T'=0 is farther in the past than T=0.

Of course, as we push the pulsar closer to the event horizon, the relative time T^N becomes asymptotic to infinity (eternity). So, supposing the universe was born 15 billion years ago in the big bang, we can push our pulsar's pulse A back in time beyond 15 billion years ago by pushing the pulsar closer to the event horizon.

No matter how old we make the universe, we may always obtain a pulse A that is older than the cosmos.

Yes, you say, but a real pulsar would be ripped to shreds and such a happening is not observable. Nevetherless, the general theory of relativity requires that we grant that time calculations can yield such contradictions.

Anthropic issues

A sense of awe often accompanies the observation: 'The conditions for human (or any) life are vastly improbable in the cosmic scheme of things.'

This leads some to assert that the many worlds scenario answers that striking improbability, since in most other universes, life never arose and never will.

I point out that the capacity for the human mind to examine the cosmos is perhaps 2.5 x 10⁴ years old, against a cosmic time scale of 1.5 x 10⁹. In other words, we have a ratio of 2.5(10⁴)/1.5(10⁹) = 1.6/10⁵.

In other words, humanity is an almost invisible drop in the vast sea of cosmic events.

Yet here we are! Isn't that amazing?! It seems as though the cosmos conspired to make our little culture just for us, so we could contemplate its vast mysteries.

However, there is the problem of the constants of nature. Even slight differences in these constants would, it seems, lead to universes where complexity just doesn't happen. Suppose that these constants depend on initial cosmic conditions which have a built-in random variability. In that case, the existence of a universe with just the right constants for life (in particular, humanity) to evolve is nothing short of miraculously improbable. Some hope a grand unified theory will resolve the issue. Others suggest that there is a host of bubble universes, most of which are not conducive to complexity, and hence the issue of improbability is removed (after all, we wouldn't be in one of the barren cosmoses). For more on this issue, see the physicist-writers John Barrow, Frank Tipler and Paul Davies.

At any rate, it doesn't seem likely that this drop will last long, in terms of cosmic scales, and the same holds for other such tiny drops elsewhere in the cosmos.

Even granting faster-than-light 'tachyon radio,' the probability is very low that an alien civilization exists within communications range of our ephemeral race. That is, the chance of two such drops existing 'simultaneously' is rather low, despite the fond hopes of the SETI crowd.

On the other hand, Tipler favors the idea that once intelligent life has evolved, it will find the means to continue on forever.

Anyway, anthropomorphism does seem to enter into the picture when we consider quantum phenomena: a person's physical reality is influenced by his or her choices.

2011-11-10T14:13:00.001-08:00

Einstein, Sommerfeld and the twin paradox

Other Conant pages
Topologist Jeff Weeks on the twin paradox
Reach Michel Janssen's paper
Toward a signal model of perception by Paul Conant

This paper has been updated as of Dec. 10, 2009

This essay is in no way intended to impugn the important contributions of Einstein or other physicists. Everyone makes errors.

Comments and suggestions welcome. Please write to krypto78@gmail.com.

The paradox

Clearly, if there is no preferred frame of reference, a contradiction arises: when the clocks meet again, which clock has recorded fewer ticks?

Neither does Sommerfeld's note address the issue of purely relative acceleration versus "true" acceleration, perhaps implicitly accepting Newton's view (below).

General relativity's partial solution

However, is this "world-line" answer the end of the problem of the asymmetry of accelerated motion?

Despite the popularity of the Big Bang model, a number of cosmic models hold the option that some velocity differences needn't imply an acceleration, strictly relative or "real."

Has the clock paradox gone away?

Now does GR resolve the clock paradox?

Certainly we can agree that Goedel's result shows that relativity is incomplete in its analysis of time.

Mach's principles

Einstein was influenced by the philosophical writings of the German physicist Ernst Mach, whom he cites in Foundations.

By rejecting absolute space and time, Mach also rejected the usual way of identifying acceleration in what is known as Mach's principle:

Version A. Inertia of a ponderable object results from a relationship of that object with all other objects in the universe.

Version B. The earth's equatorial bulge is not a result of absolute rotation (radial acceleration) but is relative to the distant giant mass of the universe.

A bit of fine-tuning

We can fine-tune the paradox by considering the velocity of the center of mass of the twin system. That velocity is m₁v/m₁ + m₂.

If we like, we can use the equation

E = mc²(1-v²/c²)^1/2

to obtain the energies of each twin system.

If the earth is in motion and the astronaut at rest, my calculator won't handle the quantity for the energy. If the astronaut is in motion with the earth at rest, then E = 5.38*10⁴¹J.

But the paradox is restored as soon as we set m₁ equal to m₂.

Notes on the principle of equivalence

Einstein's Kaluza-Klein excursion

But, as Pais pointed out, it was Einstein's interpretation that made him the genius of relativity. And yet, that interpretation was either flawed, or incomplete, as we know from the twin paradox.

Footnotes

Apologies for footnotes being out of order. Haven't time to fix.

1. Einstein's Theory of Relativity by Max Born (Dover 1962).

2. Road to Reality by Roger Penrose (Random House 2006).

3. The Black Hole War by Leonard Susskind (Little Brown 2009).

4. Understanding Einstein's Theories of Relativity by Stan Gibilisco (Dover reprint of the 1983 edition).

8. See for example Max Von Laue's paper in Albert Einstein: Philosopher-Scientist edited by Paul Arthur Schilpp (1949).

2011-11-10T14:09:00.001-08:00

First published Friday, October 20, 2006

On infinitely long statements

This note addresses a point I raised elsewhere: Is there a set of noncomputable but grammatical strings that are inherently impossible to cryptanalyze?

Again, we are assigning a digit to each symbol in some logic language (agreeing to first make sure we start out with a sufficiently high base number system). A string of digits then represents a string of symbols.

A grammatical string of symbols is one whereby certain substrings are barred as ungrammatical. But this does not mean we rule out logical contradictions or "false" statements. For example the string (A and not-A) is permitted. However, we see that the set of all proofs (defining proof as a statement verifying another statement) is a subset of our set of grammatical strings.

Whether an infinitely long grammatical string represents a proof, or a true or false, or undecidable, statement is a matter of philosophical preference.

But, to the matter at hand:

Can an infinitely long string be noncomputable but grammatical? The answer depends on the "reasonableness" of the grammatical rules. Note that in routine first-order logic notation our biggest concerns as to grammar are the right and left parentheses. If we had a set of 30 symbols, we would still have nearly 28 random choices for step n+1. So let's be generous and suggest that for language L, half the symbol set is disallowed.

[Note: I have been told that there is a proof that some such strings are satisfiable (have a truth value) and that others are undecidable.]

Now, using Zermelo-Frankel set theory's infinity axiom to permit use of induction, we consider the set of all n-length strings of base K digits (there are K^n strings).
By induction we see that we obtain the set of all possible strings, and this must be bijective with the set of reals.

Now suppose we add the proviso that at any n, we permit only (K^n)/2 strings. Yet by the ZF infinity axiom and induction we obtain half the reals, which is still a nondenumerable infinity. Since the computables have a denumerable cardinality, there must be a nondenumerable set of noncomputable but grammatical strings.

However, for grammatical rules that increasingly limit the number of choices for n, this theorem is not valid.

Related pages by Conant:
http://www.angelfire.com/az3/nfold/diag.html
http://www.angelfire.com/az3/nfold/choice.html
http://www.angelfire.com/az3/nfold/qcomp.html

2011-11-10T14:04:00.000-08:00

First published Thursday, October 12, 2006

Information theory and intelligent design

Draft 3

Before his trail-blazing paper on information theory (or "communication theory"), Claude Shannon wrote a confidential precursor paper during World War II on the informational and transmission issues inherent in cryptography, an indication of how closely intertwined are information theory and cryptology.

In this post, we digress from cryptology a bit to approach the issue of "meaning" in information theory, an issue Shannon quite properly avoided by ignoring it. We are going to avoid the philosophical depths of "meaning" also while addressing a continuing concern, the fact that some information is more useful or compelling or relevant than other information. We might think of Shannon's work as a complete generalization of communicative information, whereas our idea is to draw some distinctions. (I have only a modest familiarity with information theory and so I have no idea of whether any of what follows is original.)

For convenience we limit ourselves to the lower case alphabet and assign equal probability to the occurrence of letters in a letter string. We also use the artificially short string n=4. In that case, the Shannon information content of the gibberish string abbx equals 18.8 bit. The Shannon information value of the word goal is likewise 18.8 bit.

Now we ask the probability that a four-letter string is an English word. Let us suppose there are 3,000 four-letter English words (I haven't checked). In that case, the probability that a string belongs to the set of English words would be 3000/26^4, or 0.0065, which we now characterize as equivalent to a structured information content of 0.0095 bit. Of course, the alphabet provides the primary (axiomatic?) structure. In this case, an English dictionary provides the secondary structure.

The number of gibberish strings is then 1-0.00656, or 0.9934, which we say is equivalent to a structured information content of 0.0095 bit. We see that these values are closer to our intuitive notion of information and also fits well with the Shannonist notion that a piece of information carries a surprisal value.

Here we say that we are not particularly surprised at the string abbx because it is a member of a lawless set and because background noise is, in many circumstances, ubiquitous. We say that for our purposes the information value of any member of the lawless set is identical with the information value of the set, as is the case for any member of the structured set and the structured set. On the other hand, the surprisal value of the string goal is fairly high because of the likelihood that it was not generated randomly and hence stems from a structured or designed set. That is, the chances are fairly good that a mind originated the string goal but the chances that a mind originated a string such as abbx are harder to determine. Clearly, our confidence tends to increase with length of string and with the number of set rules.

We see how our concept of structured information fits well with cryptography, though we will not dwell on that here.

Another way to deal with the structure issue here is to ignore the gibberish strings and simply say that goal has a probability of (say) 1/3000, with an equivalent information content of 11.55 bit.

What we are doing here is getting at a principle. We are not bothering to assign exact probabilities to individual letters, letter pairs, letter triplets or letter quadruplets. We are not assigning an empirical frequency to the word goal.
Rather, what we are doing, is closing on the problem of assigning an alternative information value to patterns that show a specified structure.

Above, we have used a streamlined alphabet. But a set of some logic language's symbols can be treated like an alphabet. Importantly, we assign only grammatical symbol strings to the structured set, using rules such as ")" cannot be used to begin a sentence. We can then use the process sketched above to assign a structured information value to any string.

Clearly this method can be used for all sorts of sets divided into lawless and lawful subsets, where "law" is a pairing rule or relation. (For example, by this, we could arrange that a non-computable irrational number have a much lower information value than a computable irrational.)

We see that the average information of a gibberish string (as defined via the structured set) is far less than that of the string matching elements of the structured set. For example, the string abbx rsr is a member of the gibberish set and gibberish, in this case (assuming as a wild guess 5,000 three-letter English words) has a probability of 1-(3,000/26^4) + 1-(5,000/26^3), for an information average of 0.4925 bit. Compare the string goal new (disregarding word order), which has the complementary probability, with an information average of 50 bit.

Hence, if one saw the message goal new one could have a strong degree of confidence that the string was not random but stemmed from a designed set.

A design inference?
William Dembski, the scholar who advocates a rational basis for inferring design by intelligence, uses the SETI example to buttress his cause. The hunters of extraterrestrial intelligence, in a fictional account, are astounded by a sequence of radioed 'zeroes and ones' that matches the prime number sequence for the first 100 primes. One must assume that such a low entropy (and high average information) content must be by design, he says, and uses that as a basis for justifying the inference of an intelligent designer behind the creation of life.

However, it should be noted that it seems imperative that in order to have a set of low entropy elements, there must be a human mind to organize that set (not the physical aspects, but the cognitively appreciated set). So, such a bizarre signal from the stars would be recognized as other than background noise because of a centuries-long human effort to distill certain mathematical relations into concise form. Hence, human receivers would recognize a similar intelligence behind the message.

But does that mean one can detect a signal from amid noise without falling into the problem whereby one sees all sorts of "things" in an atmospheric cloud?
That is, when one says that the formation of the first life forms is highly improbable, what does one mean? Can we be sure that the designer set (using human assumptions) has been sufficiently defined? (I am not taking sides here, by the way.)

However, as noted above, the question of computability enters the picture here. Following prevailing scientific opinion (Penrose being an exception), every organism
can be viewed as a machine and every machine responds to a set of algorithms. Hence every machine can be assigned a unique and computable number. One simply assigns numbers to each element of the logic language in use and puts together an algorithm for the machine. The machine's integer number corresponds to some right-left or left-right sequence of symbols (to avoid confusion, the computation may require a high-base number system).

So then, the first organic machines -- proto-cell organisms perhaps -- must be regarded as part of a larger machine, the largest machine of all being the cosmos. But, the cosmos cannot be modeled as a classical machine or computer. See link in sidebar.

A sea of unknowable 'designs'
Also, the set of algorithmic machines (the set of algorithms) is bijective with a subset of the computable reals (some algorithmic substrings are disallowed on grammatical grounds).

Now a way to possibly obtain a noncomputable real is to use a random lottery for choice of the nth digit in the string and, notionally, to continue this process over denumerable infinity. Because the string is completely random we do not know whether it is a member of the computable or noncomputable reals (which set, following Cantor's diagonal proof and other proofs, has a higher infinite cardinality than the set of computable reals).

So there is no reason to conclude that a grammatical string might not be a member of the noncomputables. In fact, there must be a nondenumerable infinity of such strings.
Nevertheless, a machine algorithm is normally defined as always finite. On the other hand, one could imagine that a machine with an eternal time frame might have an infinite-step algorithm.

That is, what we have arrived at is the potential for machine algorithms that cannot possibly have been arrived at by human ken. Specifically, we have shown that there exists a nondenumerable infinity of grammatical statements of infinite length. One might then argue that there is a vast sea of infinite designs that the human mind cannot apprehend.

2011-11-10T14:03:00.001-08:00

First published Wednesday, November 01, 2006

Does math back 'intelligent design'?

Two of the main arguments favoring "intelligent design" of basic biotic machines:

. Mathematician William A. Dembski (Science and Evidence for Design in the Universe) says that if a pattern is found to have an extraordinarily low probability of random occurrence -- variously 10^(-40) to 10^(-150) -- then it is reasonable to infer design by a conscious mind. He points out that forensics investigators typically employ such a standard, though heuristically.

. Biochemist Stephen C. Meyer (Darwin's Black Box) says that a machine is irreducibly complex if some parts are interdependent. Before discussing intricate biological mechanisms, he cites a mousetrap as a machine composed of interdependent parts that could not reasonably be supposed to fall together randomly.

Meyer is aware of the work of Stuart Kauffman, but dismisses it because Kauffman does not deal with biological specifics. Kauffman's concept of self-organization via autocatalysis however lays the beginnings of a mathematical model demonstrating how systems can evolve toward complexity, including sudden phase transitions from one state -- which we might perceive as "primitive" -- to another state -- which we might perceive as "higher." (Like the word "complexity," the term "self-organization" is sometimes used rather loosely; I hope to write something on this soon.)

Kauffman's thinking reflects the work of Ilya Prigogine who made the reasonable point that systems far from equilibrium might sometimes become more sophisticated before degenerating in accordance with the "law of entropy."

This is not to say that Meyer's examples of "irreducible complexity" -- including cells propelled by the cilium "oar" and the extraordinarily complex basis of blood-clotting -- have been adequately dealt with by the strict materialists who sincerely believe that the human mind is within reach of grasping the essence of how the universe works via the elucidation of some basic rules.

One such scientist is Stephen Wolfram whose New Kind of Science examines "complexity" via iterative cellular automaton graphs. He dreams that the CA concept could lead to such a breakthrough. (But I argue that his hope, unless modified, is vain; see sidebar link on Turing machines.)

Like Kauffman, Wolfram is a renegade on evolution theory and argues that his studies of cellular atomata indicate that constraints -- and specifically the principle of natural selection -- have little impact on development of order or complexity. Complexity, he finds, is a normal outcome of even "simple" sets of instructions, especially when initial conditions are selected at random.

Thus, he is not surprised that complex biological organisms might be a consequence of some simple program. And he makes a convincing case that some forms found in nature, such as fauna pigmentation patterns, are very close to patterns found according to one or another of his cellular automatons.

However, though he discusses n-dimensional automata, the findings are sketchy (the combinatorial complexity is far out of computer range) and so cannot give a three-dimensional example of a complex dynamical system emerging gestalt-like from some simple algorithm.

Nevertheless, Wolfram's basic point is strong: complexity (highly ordered patterns) can emerge from simple rules recursively applied.

Another of his claims, which I have not examined in detail, is that at least one of his CA experiments produced a graph, which, after sufficient iterations, statistically replicated a random graph. That is, when parts of the graph were sampled, the outcome was statistically indistinguishable from a graph generated by computerized randomization. This claim isn't airtight, and analysis of specific cases needs to be done, but it indicates the possibility that some structures are somewhat more probable than a statistical sampling would indicate. However, this possibility is no disproof of Dembski's approach. (By the way, Wolfram implicitly argues that "pseudorandom" functions refer to a specific class of generators that his software Mathematica avoids when generating "random" numbers. Presumably, he thinks his particular CA does not fall into such a "pseudorandom" set, despite its being fully deterministic.)

However, Wolfram also makes a very plausible case (I don't say proof because I have not examined the claim at that level of detail) that his cellular automata can be converted into logic languages, including ones that are sufficiently rich for Godel's incompleteness theorem to apply.

As I understand Godel's proof, he has demonstrated that, if a system is logically consistent, then there is a class of statements that cannot be derived from axioms. He did this through an encipherment system that permits self-referencing and so some have taken his proof to refer only to an irrelevant semantical issue of self-referencing (akin to Russell's paradox). But my take is that the proof says that statements exist that cannot be proved or derived.

So, in that case, if we model a microbiotic machine as a statement in some logic system, we see immediately that it could be a statement of the Godel type, meaning that the statement holds but cannot be derived from any rules specifying the evolution of biological systems. If such a statement indeed were found to be unprovable, then many would be inclined to infer that the machine specified by this unprovable statement must have been designed by a conscious mind. However, such an inference is a philosophical (which does not mean trival) difficulty.

2011-11-10T14:01:00.001-08:00

First published Thursday, November 02, 2006

Pseudorandom thoughts on complexity

Draft 2

This post supplements the previous post "Does math back 'intelligent design'?"

With respect to the general concept of evolution, or simply change over time, what do we mean by complexity?

Consider Stephen Wolfram's cellular automata graphs. We might think of complexity as a measure of the entropy of the graph, which evolves row by row from an initial rule whereby change occurs only locally, in minimal sets of contiguous cells. Taken in totality, or after some row n, the graphs register different quantities of entropy. That is, "more complex" graphs convey higher average information than "less complex" ones. Some graphs become all black or all white after some row n, corresponding to 0 information after that row. There exists a significant set of graphs that achieve neither maximum nor minimum entropy, of course.

How would we define information in a Wolfram cellular automaton graph? We can use several criteria. A row would have maximum entropy if the probability of the pattern of sequential cell colors is indistinguishable from random coloring. [To be fussy, we might use a double-slit single photon detector to create a random sequence whereby a color chosen for a cell is a function of the number of the quadrant where a photon is detected at time t.]

Similarly for a column.

Obviously, we can consider both column and row. And, we might also consider sets of rows and-or columns that occur as a simple period. Another possibility is to determine whether such sets recur in "smooth curve quasi-periods" such as every n^2. We may also want to know whether such sets zero out at some finite row.

Another consideration is the appearance of "structures" over a two-dimensional region. This effectively means the visual perception of at least one border, whether closed or open. The border can display various levels of fuzziness. A linear feature implies at least one coloration period (cycle) appearing in every mth row or every nth column. The brain, in a Gestalt effect, collates the information in these periods as a "noteworthy structure." Such a structure may be defined geometrically or topologically (with constraints). That is, the periodic behavior may yield a sequence of congruent forms (that proliferate either symmetrically or asymmetrically) or of similar forms (as in "nested structures"), or of a set of forms each of which differs from the next incrementally by interior angle, creating the illusion of morphological change, as in cartoon animation.

At this juncture we should point out that there are only 254 elementary cellular automata. However, the number of CA goes up exponentially with another color or two and when all possible initial conditions are considered.

So what we are describing, with the aid of Wolfram's graphs, is deterministic complexity, which differs from the concept of chaos more on a philosophical plane than a mathematical one.

We see that, depending on criteria chosen, CA graphs, after an evolution of n steps, differ in their maximum entropy and also differ at the infinite limit in their maximum entropy. Each graph is asymptotic toward some entropy quantity. By no means does every graph converge toward maximum entropy as defined by a truly random pattern.

So we may conclude that, as Wolfram argues, simple instructions can yield highly complex fields. The measure of complexity is simply the quantity of information in a a graph or subgraph defined by our basic criteria. And what do we mean in this context by information? If we went through all n steps of the rule and examined the sequence of colors in, for example, row n, the information content would be 0 because we have eliminated the uncertainty.

If, however, we don't examine how row n's sequence was formed, then we can check the probability of such a sequence with the resulting information value. At this point we must beware: Complete aperiodicity of cell colors in row n is NOT identical with maximum entropy of row n. Think of asking a high school student to simulate flipping of a coin by haphazardly writing down 0 or 1 in 100 steps. If one then submits the sequence to an analyst, he or she is very likely to discover that the sequence was not produced randomly because most people avoid typical sub-sequences such as 0 recurring six times consecutively.

So then, true randomness (again, we can use our quantum measuring device), which corresponds to maximum entropy, is very likely to differ significantly from computed chaos. This fact is easily seen if one realizes that the set of aperiodic computable irrational numbers is of a lower cardinality than the set of random digit sequences. Still, it must be said that the foregoing lemma doesn't mean there is always available a practical test to distinguish a pseudorandom sequence from a random sequence.

We might also think of deterministic complexity via curves over standard axes, with any number of orthogonal axes we like. Suppose we have a curve y = x. Because there is no difference between x and y, there is effectively no information in curve f(x). No work is required to determine f(x) from x. The information in y = 2x is low because minimal work (as counted by number of simple steps in the most efficient algorithm known) is required to determine g(x) from x. Somewhat more information is found for values of h(x) = x^2 because the computation is slightly slower.

A curve whose values hold maximum information -- implying the most work to arrive at an arbitrary value -- would be one whereby the best method of determining f(x+k) requires knowledge of the value f(x). Many recursive functions fit this category. In that case, we would say that a computed value whose computational work cannot be reduced from n steps of the recursive function or iterative algorithm holds maximum information (if we don't do the work).

So let's say we have the best-arranged sieve of Eratosthenes to produce the sequence of primes. On an xyz grid, we map this discrete curve z = f(y) over y = x^2, using only integer values of x. Now suppose we perceived this system in some other way. We might conclude that a chaotic system shows some underlying symmetry.

It is also possible to conceive of two maximally difficult functions mapped onto each other. But, there's a catch! There is no overall increase in complexity. That is, if f(x) is at maximum complexity, g(f(x)) cannot be more complex -- though it could conceivably be less so.

This conforms to Wolfram's observation that adding complexity to rules does little to increase the complexity of a CA.

Now what about the idea of "phase transitions" whereby order suddenly emerges from disorder? Various experiments with computer models of nonlinear differential equations seem to affirm such possibilities.

Wolfram's New Kind of Science shows several, as I call them, catastrophic phase transitions, whereby high entropy rapidly follows a "tipping point" as defined by a small number of rows. Obviously one's perspective is important. A (notional) graph with 10^100 iterations could have a "tipping point" composed of millions of rows.

Wolfram points out that minor aymmetries in a high entropy graph up to row n are very likely to amplify incrementally -- though the rate of change (which can be defined in several ways) can be quite rapid -- into a "complex" graph after row n. I estimate that these are low entropy graphs, again bearing in mind the difference between true randomness and deterministic chaos or complexity: the entropies in most cases differ.

What we arrive at is the strong suggestion -- that I have not completely verified -- that a high information content in a particular graph could easily be indicative of a simple local rule and does not necessarily imply an externally imposed design [or substitution by another rule] inserted at some row n.

However, as would be expected, the vast majority of Wolfram's graphs are high-entropy affairs -- no matter what criteria are used -- and this fact conforms to the anthropomorphic observation that the cosmos is en toto a low entropy configuration, in that most sets of the constants of physical law yield dull, lifeless universes.

I should note that New Kind of Science also analyzes the entropy issue, but with a different focus. In his discussion of entropy, Wolfram deploys graphs that are "reversible." That is, the rules are tweaked so that the graph mimics the behavior of reversible physical processes. He says that CA 37R shows that the trend of increasing entropy is not universal because the graph oscillates between higher and lower entropy eternally. However, one must be specific as to what information is being measured. If the entropy of the entire graph up to row n is measured, then the quantity can change with n. But the limiting value as n goes to infinity is a single number. It is true, of course, that this number can differ substantially from the limiting value entropy of another graph.

Also, even though the graphs display entropy, the entropy displayed by physical systems assumes energy conservation. But Wolfram's graphs do not model energy conservation, though I have toyed with ways in which they might.

The discussion above is all about classical models arranged discretely, an approach that appeals to the computer science crowd and to those who argue that quantum physics militates against continuous phenomena. However, I have deliberately avoided deep issues posed by the quantum measurement/interpretation problem that might raise questions as to the adequacy of any scientific theory for apprehending the deepest riddles of existence.

It should be noted that there is a wide range of literature on what the Santa Fe Institute calls "complexity science" and others sometimes call "emergence of order." I have not reviewed much of this material, though I am aware of some of the principle ideas.

A big hope for the spontaneous order faction is network theory, which shows some surprising features as to how orderly systems come about. However, I think that Wolfram graphs suffice to help elucidate important ideas, even though I have not concerned concerned myself here with New Kind of Science's points about networks and cellular automata.

2011-11-10T13:19:00.001-08:00

This first appeared on Sunday, June 24, 2007

The Kalin cipher

Note: The Kalin cipher described below can of course be used in tandem with a public key system, or it can be done by hand with calculators. An appropriate software program for doing the specified operations would be helpful.

The idea of the cipher is to counteract frequency analysis via matrix methods.

Choose a message of n characters and divide n by some integer square. The remainder can be padded out with dummy numbers.

Arrange the message into mxm matrices, as shown,

H I H x
O W A y
R E U z

We then augment the matrix with a fourth column. This column is the first key, which is a random number with no zeros. A second 4x3 matrix is filled with random integers. This is the second key. The keys needn't be random. Easily remembered combinations might do for some types of work because one would have to know the cipher method in order to use guessed keys.

We then matrix multiply the two as MK and put MK into row canonical form.
This results in the nine numbers in M being reduced to three in [I|b], where b is the final column.

8  9  8 x   7 9 2      1 0 0 a
8  15 1 y   5 5 4      0 1 0 b
18 5 21 z   3 2 3  =   0 0 1 c
           5 2 3

where (a, b, c) is a set of three rationals.

The keys can be supplied by a pseudorandom number generator in step with a decoder program. A key can vary with each message matrix or remain constant for a period, depending on how complex one wishes to get. But, as said, if one is conducting an operation where it is advantageous for people to remember keys, this method might prove useful.

By the way, if a row of the original message matrix repeats or if one row is a multiple of another, a dummy character is inserted to make sure no row is a multiple of another so that we can obtain the canonical form. Likewise, the key matrix has no repeating rows.

In fact, on occasion other square matrices will not reduce to the I form. A case in point:

a+b  b+c  a+b
a    b    c
1    1    1

The determinant of this matrix is 0, meaning the I form can't be obtained.
In general, our software program should check the determinant of the draft message matrix to make sure it is non-zero. If so, a dummy letter should be inserted, or two inconsequential characters should be swapped as long as the meaning isn't lost.

But, if the program doesn't check the determinant it will give a null result for the compression attempt and hence would be instructed to vary the message slightly.

Notice that it is not necessary to transmit I. So the message is condensed into three numbers, tending to defeat frequency analysis.

Here is a simple example, where for my convenience, I have used the smallest square matrix:

We encrypt the word "abba" (Hebrew for "father") thus:

1 2 3
2 1 1

where the first two columns represent "abba" and the third column is the first key.
We now matrix multiply this 2x3 matrix by a 3x2 matrix which contains the second key.

1 2 3  x  2 1  = 10  16
2 1 1     1 3    7   8
         2 3

We then apply key 1 (or another key if we like) and reduce to canonical form:

10  16  3
7   8   1

which is uniquely represented by

1  0  -1/4
0  1  11/32

By reversing the operations on (-1/4, 11/32), we can recover the word "abba." If we encode some other four characters, we can similarly reduce them to two numbers. Make a new matrix

-1/4  x 3
11/32 y 1

which we can fold into another two number set (u, v).

We see here an added countermeasure against frequency analysis, provided the message is long enough: gather the condensed column vectors into new matrices.

We can do this repeatedly. All we need do is divide by, in this case 4, and recombine. If we have 4²ⁿ matrices to begin with, it is possible to transmit the entire message as a set of two numbers, though usually more than one condensed set would be necessary. Of course we can always pad the message so that we have a block of x²ⁿ. Still, the numbers tend to grow as the operations are done and may become unwieldy after too many condensations. On the other hand, frequency analysis faces a tough obstacle, I would guess.

So a third key is the number of enfoldments. Three enfoldments would mean three unfoldments. Suppose we use a 4x4 system on 64 characters with three enfoldments.
We write this on 16 matrices. After the transform, we throw away the I matrices and gather the remaining columns sequentially into a set of 4 matrices. We transform again and are left with a single column of four numbers. So if the adversary doesn't know the number of enfoldments, he must try them all, assuming he knows the method. Of course that number may be varied by some automated procedure linked to a public key system.

Just as it is always possible to put an nxn matrix without a zero determinant [and containing no zeros] into canonical form, it is always possible to recover the original matrix from that form. The methods of linear algebra are used.

Decryption is also hampered by the fact that in matrix multiplication AB does not usually equal BA.

The use of canonical form and the "refolding" of the matrices is what makes the Kalin cipher unique, I suppose. When I say unique, I mean that the Kalin cipher has not appeared in the various popular accounts of ciphers.

An additional possibility: when an nxn matrix is folded into an n column vector, we might use some "dummy" numbers to form a different dimension matrix. For example, suppose we end up with a 3-entry column vector. We add a dummy character to that string and form a 2x2 matrix, which can then be compressed into a 2-entry column vector. Of course, the receiver program would have to know that a 2-vector is compressed from a 3-vector. Also the key for the 2x2 matrix is so small that a decryption program could easily try all combinations.

However, if a 100x100 matrix folds into a 10-entry column vector, six dummy characters can be added and a 4x4 matrix can be constructed, leaving a 4-vector, which can again be folded into a 2-vector. All sorts of such systems can be devised.

Additionally, a "comma code" can be used to string the vectors together into one number. The decipherment program would read this bit string to mean a space between vector entries.

Clearly the method of using bases or representations as keys and then transforming offers all sorts of complexities -- perhaps not all useful -- but the emphasis on matrix condensation seems to offer a practical antidote to frequency analysis.

BTW, I have not bothered to decimalize the rational fractions. But presumably one would convert to the decimal equivalent in order to avoid drawing attention to the likelihood that the numbers represent canonical row vectors. And, of course, if one is using base 2, one would convert each rational to a linear digit string. Not all decimal fractions can be converted exactly into binary. However, supposing enough bits are used, the unfolded (deciphered) number will, with high probability, be very close to the correct integer representing the character.