Do dice play God?
A discussion of Irreligion

A discussion 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)

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Posted Nov 9, 2010. Minor revision posted Sept. 7, 2012.
A previous version of this discussion is found on Angelfire. 

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 below), 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."

He writes: "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). 

Matchmaking
It must be acknowledged that in microbiological matters, probabilities need not always follow a routine independence multiplication rule. In cases where random matching is important, we have the number 0.63 turning up quite often.

For example, if one has n addressed envelopes and n identically addressed letters are randomly shuffled and then put in the envelopes, what is the probability that at least one letter arrives at the correct destination? The surprising answer is that it is the sum 1 - 1/2! + 1/3! ... up to n. For n greater than 10 the probability converges near 63%.

That is, we don't calculate, say 11^-11 (3.5x10^-15), but we have that our series approximates very closely 1 - e^-1 = 0.63.

Similarly, suppose one has eight distinct pairs of socks randomly strewn in a drawer and thoughtlessly pulls out six one by one. What is the probability of at least one matching pair?

The first sock has no match. The probability the second will fail to match the first is 14/15. The probability for the third failing to match is 12/14 and so on until the sixth sock. Multiplying all these probabilities to get the probability of no match at all yields 32/143. Hence the probability of at least one match is 1 - 32/143 or about 78%.

These are minor points, perhaps, but they should be acknowledged when considering probabilities in an evolutionary context.

And so
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.(7)

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 n_1 and n_2, each at least equal to 8
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.

7. Paulos, interestingly, tells of how he lost a great deal of money by an ill-advised enthusiasm
for WorldCom stock in A Mathematician Plays the Stock Market (Basic Books, 2003). The expert
probabalist and statistician found himself under a delusion which his own background should have
fortified him against. (The book, by the way, is full of penetrating insights about the
probability and the market.) One wonders whether Paulos might also be suffering from another
delusion: that probabilities favor atheism.

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

The knowledge delusion

First published Thursday, November 3, 2011

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



Essay by PAUL CONANT

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.

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 (11).

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.

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

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Footnotes

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.

11. Colin J. Humphreys, a Cambridge professor, takes issue with one of Dawkins's barbs. Humphreys, a materials science professor with an interest in biblical mysteries, quotes Dawkins as saying, "The only difference between the Da Vinci Code and the gospels is that the gospels are ancient fiction while The Da Vinci Code is modern fiction."

Humphreys responds that in his book The Mystery of the Last Supper: Reconstructing the Final Days of Jesus (Cambridge University Press, 2011), he has "taken what the biblical scholar F.F. Bruce called 'the thorniest problem in the New Testament,' the date and nature of the last supper" and shown that despite the complexity of the problem "the gospels are in substantial agreement."

Humphreys did extensive research and proposes that Jesus and his disciples ate the last supper on the Wednesday before the crucifixion, not the Thursday before.

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Draft 04</ br> Minor editing on Oct. 1, 2013</ br>

Plato and Cantor v. Wittgenstein and Brouwer

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.

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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_Fn 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_{Fn+1 is true but not provable in Fn+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 PFn is not defined.}

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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_io 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:

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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...

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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...

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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...

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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]

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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...

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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