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Monday, April 15, 2013

Where is Zion?

First published ca. 2005
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By PAUL CONANT
Many wonder about the emergence of the modern state of Israel. What is the divine purpose? Is this reconstituted Jewish state a sign of the end times? How should Christians deal with this situation?

The believer is admonished by scripture to beware false prophets, perhaps appearing as born again 'angels of light,' teaching demonic theories intended to hinder the work of the gospel of salvation through Jesus Christ [2 Cor 11.13]. Peter also warns that unspiritual people can falsely interpret the word of God. [2 Pet 1.20] Therefore, it is important that the believer, with the aid of the Spirit, examine the basis for various apocalyptic theories and timetables that are popular these days, rightly and diligently dividing scripture [2Tim 2.15]
 

An appendix (scroll down) contains interpretations of scriptures that contain the word 'Zion.'

Because many of those who adhere to such theories are known as Christian Zionists, it seems appropriate to examine the concept of Zionism from the point of view of gospel-centered, born-again Christianity, in accord with Christian precepts that anything can be tested against the Bible.

As Paul wrote: 'All scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness; so that the man of God may be adequate, equipped for every good work.' [2 Tim 3.16,17]. However, when the flesh interprets scripture, the result is dead wrong, as we know from Satan's temptation of Jesus. Peter, in his struggle against false prophets, said that 'no prophecy of scripture is a matter of personal opinion,' not being an act of human will but rather of the Holy Spirit motivating the speaker [2 Pet 1.20,21]. 

EARLY IDEAS OF 'ZION'
The first biblical mention of 'Zion' is Deut. 4.48 and is another name for Mount Hermon in Lebanon. This 'Zion' derives from 'Sirion,' a name given to the mountain by the Sidonians. This Zion is referred to again in Psalm 133.3 when the poet compares brotherly love to 'the dew of Hermon coming down on the mountains of Zion.'

'Zion' is next cited in 2Sam 5.7, where it is reported that 'David took the stronghold of Zion (that is, the City of David).' This was a fortified enclave held by the Jebusites on a site later known as Jerusalem. David built his palaces in this fort and laid the groundwork for the first temple, which was built by his son Solomon, on a nearby hill. The term 'Zion' then came to refer to the entire palace and temple compound. It was often used specifically for the temple or the temple and its hill. Scholars think 'zion' was originally a word for 'hill.'
'Zion' was often extended in the Bible to mean the entire city of Jerusalem, where the general populace lived adjacent to the palace-temple compound. In fact, sometimes the Old Testament Bible extends the definition of 'Zion' to include the people of Israel or a remnant of that people. 

THE CONCEPT OF TIME
Some Bible commentators seem to unconsciously believe that, when it comes to time, God is not as subtle as he is in other matters. They would have the anticipated fulfillments of prophecies fit nicely onto a 'railroad timetable.'

However, this illusion is refuted directly and indirectly by scripture.
Josh 10.13, describing the halting of the sun and moon, may be read literally or figuratively. The author may have meant that the victory was so astonishing that it was tantamount to the sun and moon standing still (and this may have been a jibe against a monarch whose authority supposedly included the sun and the moon).

Or, he may have been recounting that the sun, along with a pale daylight moon, stood still during the battle. If literally interpreted, then we see that God may extend a day's duration at will. If figuratively interpreted, then the author says that God is so powerful he could, if he wished, make the sun stand still, meaning in God's hands time is truly flexible, a point made by the prophet Daniel, who lauded God as one who 'changes times and the seasons' [Dan 2.21].

God's control of time is demonstrated in Is 38.8, when God told Hezekiah that the sun's shadow would go back ten steps on the stairway of Ahaz, which is what then happened. God showed that he erased a period of history that led up to the point in time of Hezekiah's death. He gave Hezekiah a new lease on life by bypassing a sin-laden time in this man's life.

The disciples found that their concepts of space gave way to the power of God when they were with Jesus in a boat in the middle of the Sea of Gallilee and suddenly found themselves at the shore. The time it takes to cross that space was voided. So, we can accept that God can and does compress time. Likewise, he can expand it, as we know from Hezekiah's experience.

Of a prophecy given to Habbakkuk, God said, 'For the vision is yet for the appointed time; it hastens toward the goal and will not fail. Though it tarries, wait for it; for it will certainly come, it will not delay'[Hab 2.3]. Note that the appointed time may seem, to human observers, to be close at hand or even overdue. So we can see that if the prophecy isn't fulfilled at an expected time, then it must be fulfilled in an unexpected way.
This point is driven home by the chronologically inconsistent narratives found in Genesis and elsewhere. For example, ages of Abraham and his family at key points don't jibe -- if we use modern western definitions. The number for age represented a benchmark, such as '40' representing the period of mature manhood, rather than a specific year.

But even if we 'solve' this difficulty, others remain: for example, when were the people of earth who married into the family of Adam and Eve born?

What happened to the giants (the nephilim or 'fallen') when the flood hit? These demon-people appear again after the flood. Could it be that time is not really representable as a real number line? Should we accept that perhaps the Bible contains serious errors (we certainly know that different versions contain minor errors that Bible scholars are continually attempting to correct), or might we conclude that there is simply much that we don't fully understand?

So then, if the Bible itself demonstrates the impossibility of drawing up a superficially logically consistent timeline of events of old, what makes us think we can use biblical references to map a 'logically consistent' history of the future?

Often in scripture, a time is said to arrive when the measure of something (such as sin or wrath) is full. The Greeks had two words for time: 'kronos' and 'kairos.' Kronos is the time of clocks and astronomical cycles. Kairos implies the propitous time. It is more spiritual in nature. A kairos time may be thought of as a balloon that fills up, perhaps quickly, perhaps slowly, with a gas until the point of bursting. That is when the consummation of the kairos time has arrived. We might consider the second coming as a the consummation of a kairos time.

Even atheists who seek the truth in modern physics have found that time is supple and relative, and not at all like the typical person's view of it. At extremely short intervals, time can be said not to exist. On larger scales, as Einstein proved, time is relative to the observer. Here's an idea of what that means: If you left earth on a starship traveling near the speed of light and traveled deep into space and then back, you would find that your twin brother was substantially older than you upon your return. In fact, you could conceivably return to an earth where centuries had elapsed, though for you only months had elapsed.

When Martha was grieving for Lazarus, dead for four days, she told Jesus she believed Jesus could get any favor from God. Jesus then told her that her brother would rise again. 'I know that he will rise again in the resurrection on the Last Day,' she answered. 'I am the resurrection and the life; he who believes in me will live even if he dies, and everyone who lives and believes in me will never perish. Do you believe this?'

You see, Martha had the traditional Jewish understanding of the Old Testament prophecies, but she was hoping Jesus would find a way to revive her brother anyway! Her faith was honored when Jesus said, in effect: Don't stand on tradition! Just BELIEVE! This is GOD talking! [John 11.22-26]

The great and awesome day of the Lord had arrived for Lazarus and was witnessed by Martha.
It wasn't the sidereal day that counted, but rather belief when the word came. The awesome day of the Lord is on a spiritual (which does not mean 'unreal') plane: 'SEE IT NOW! ETERNAL LIFE!'

In general, we may say that the launch of Jesus' salvation mission was indeed the dawn of the great day.
The malleability of time and the physical world is indicated in the variant accounts of the resurrection. Some skeptics see these seemingly irreconcilable versions as proof of biblical error. And, it is conceivable that God permits minor structural or form errors on the basis that it's the spirit that counts, just as when one hears a spirit-filled person give the word, even though he makes some grammatical or even factual slips. The slips are not important; it's the spirit, the word, that counts.

Even so, there is a serious school of thought among physicists known as the 'many worlds theory' -- reflecting Jesus' word, 'In my father's house are many mansions [possibilities]' [Jn 14.2] -- in which alternate realities can coexist. In some related areas of scientific conjecture, such alternate realities are thought to interact strongly rather than weakly.

Mathematicians are accustomed to dealing with 'projection,' in which a more complex form is projected as a set of simpler forms. For example, if you shine a flashlight above a globe, you will see an elliptical or circular shadow below.

Suppose we think of a unified event occurring in a purely spiritual dimension, such as heaven or hell, as being projected into the earthly dimension where man lives. It is quite reasonable to suppose that its projections, or reflections, will be fragmentary, meaning that, from a human perspective, the one 'higher dimension' event is seen in various ways at different times and places on the earthly plane.

You needn't believe that this idea is correct to see the point that the commonplace idea of linear time may not suffice for prophecy.

Earthbound people can be misled by a spirit of deception that puts falsehoods into the mouths of wicked prophets [1 Kings 22.22] and thus blinded by a 'strong delusion' [2 Thes 2.11]. While it is true that the born-again believer is fit to understand scripture, as illuminated by God's Spirit, it is still possible for the believer to be influenced by the delusive nature of this world. 

DEAD LITERALISM
As Christians, we are strongly advised to study scripture [2 Tim 2.15]. We cannot however understand what we read without the help of the Holy Spirit. Though scripture is written in Hebrew and Greek, its origin is divine, and divine language is beyond human ken [Rom 8.26]. Scripture means what God says it means.
Who has plumbed the mind of God? Who has exhausted the deep well of the wisdom of God? [Rom 11.33,34]

God's thoughts are far above our thoughts [Is 55.9]. We must beware assuming that we can know the full meaning of a scriptural passage or group of scriptural passages.

Dead literalism is effectively refuted by Jesus in a number of places, such as when he healed on the sabbath day and when he upbraided the scribes and pharisees for focusing on technicalities rather than on the point of God's teachings: love of God and one another.

Jesus, obeying the Spirit, told his hearers that a scripture from Isaiah that he had just read had been 'fulfilled in your hearing.' [Luke 4.21] Jesus halted his reading at 'to proclaim the favorable year of the Lord,' leaving unread the other part of the couplet 'and the day of vengeance of our God.' [Is 61.1-2]

Does this mean the unquoted prophecy is untrue? No, but we can assume that the fulfillment of the entire passage from Isaiah does not necessarily follow a simple human schedule.

After the long-lost Book of the Law was recovered, it was read to the people and interpreted for them by the Levite priests [Neh 8.7]. Language and custom had evolved, so that strict literalism would have left the hearers perplexed. The Bible here shows that, without inspired interpretation, we have the dead letter of the law.

The word of God is 'alive and active and sharper than any two-edged sword, and piercing as far as the division of soul and spirit, of both joints and marrow, and able to judge the thoughts and intentions of the heart,' says the writer of Hebrews [Heb 4.12], after quoting several Old Testament passages.
So if a man is to understand the word, he cannot rely on human wisdom [1 Cor 1-9]. Just as the spirit within a man knows his thoughts, the Spirit of God knows God's thoughts [1 Cor 2.11]. In fact, the Spirit reveals scriptural meaning to believers 'for the Spirit searches all things, even the depths of God.' [1 Cor 2.10]

Or, as Paul wrote elsewhere, the new arrangement for Christians is 'not of the letter but of the spirit; for the letter kills, but the spirit gives life.' [2 Cor 3.6]. So the assertion that 'the Bible is literally true' should perhaps be changed to read that 'the Bible contains no error.' Since God 'makes all things work together for the good' for his chosen [Rom 8.28], he is able to make human missteps into truth. We know this from Caiaphas, who, as high priest, spoke a truth, though he had no idea of the divine profoundness of his words [Jn 18.14].

An example of literalness is the Mosaic command for Jews to put symbols of God's word on their foreheads and hands [Deut 6.8]. Some Jews to this day observe this custom. But, isn't it plain that the real point of this thought is that the Jews were to think about God's word and practice it? Why would God want people to honor this custom but not honor God's will in the more important area of behaving on the square with one's neighbors?

People who want to be excessively literal may not be aware that Jewish and Christian editors have over the centuries sifted ancient writings in efforts to obtain God's word at its purest. The fact that the Bible is such an authoritative source of truth and wisdom shows that many of these editors were, like Paul who rightly divided the word of truth [2 Tim 2.15], divinely inspired.

Many modern Bible texts identify words attributed to Jesus with red type or quotation marks. However, the Greek texts have no such graphical devices. This means that there are some passages sometimes attributed directly to Jesus that might actually be divinely inspired editorial comments. Also, there are variant literal versions of some of the words attributed to Jesus, though they are close in substance. With the aid of the Spirit, it is possible for the contemporary mind to understand the meaning of these words.

And the sayings of Jesus clearly were compiled early on and later woven into the synoptic gospel accounts. Does this make the gospel accounts wrong? Only if one insists on literally precise chronological narratives. Otherwise, they are rather like the modern newspaper report that lays out the relevant details in a way that 'tells a story.'

If a newspaper reporter extracts a couple of relevant sentences from some document and includes a one-sentence paraphrase in his story, he could not fairly be accused of improperly quoting the document. We should give Bible writers the same leeway. To wit, Paul, whose eyesight was poor and who probably relied on memory, in several places conflates scriptural passages in order to make a point. For example, Rom 9.33 apparently splices Is 28.16 and Is 8.14. This splice is easily defended as summarizing later prophetic fulfillments. So Paul is not in error, but we would be foolish to be excessively literal: there is no single such passage in the Septuagint (the Greek-language Bible in use at the time).

In Rom 11.26, Paul quotes scripture thus: 'The deliverer will come from Zion, he will remove ungodliness from Jacob. This is my covenant with them, when I take away their sins.'
The quotation is more a paraphrase than an exact quote. In fact, some might argue that the paraphrase misquotes Is 59.20,21 to give it a meaning it does not have. However, the import of the paraphrase is justified elsewhere, such as Psalm 14.7 and Psalm 15.

I sympathize with the apostle. I had in my mind that Jesus had, in assuring believers of his steadfastness, asked rhetorically whether a mother could forsake her child. It turns out that the relevant verse is Is 49.14, where I hear the voice of the Messiah.

Consider Mt. 21.5, which tells of Jesus' triumphal donkey ride into Jerusalem. Two old testament scriptures, Is 62.11 and Zech 9.9, are spliced together here, illustrating that different prophetic utterances can be fulfilled in one action.

Or consider John 12.15 which quotes Zech 9.9 but omits 'shout, O daughter of Jerusalem.' Yet, the omission is inconsequential. The meaning of Zech 9.9 is faithfully preserved. Writers of the period did not follow rules of quotation used today. So we cannot easily ascribe error here.

When we are told that 'the scripture cannot be broken' [Jn 10.35], the Old Testament was meant (specifically, Ps 82.6). So then, what do we make of Jude's citation from the Book of Enoch, which is regarded by Jewish and Christian scholars as largely a group of entertaining tall tales that on the whole lack divine inspiration?

Accepting that the Book of Jude is scripture, then is the scripture broken if Jude cites a non-scriptural source? [Jude 1.9]

Firstly, Jude was using an example from a book familiar to his contemporaries in order to make the moral point: If even Michael, in his dispute with Satan over Moses' body, wouldn't revile the devil himself, who are we to revile anyone at all? Now, regardless of the scriptural accuracy of this story, the moral point is substantiated by the words of Jesus, who occasionally alluded to the Book of Enoch.

And, we cannot rule out the possibility that the Book of Enoch contains fragments of real scripture and that Jude cites such a fragment.

New Testament writers cite numerous passages from the Old Testament in order to make their case about Jesus. However, if one were to apply the railroad timetable technique to the Old Testament writings, one would doubtless be able to argue that the writers were taking scripture out of context in order to justify their pet theories. But, the reality is the reverse: the Spirit illuminated the authors' minds as to how the written word was being fulfilled in their day. But the human wisdom -- railroad timetable analysis -- was to be scorned. As Paul said to the Corinthians, he decided to put Jesus first and then see what the Spirit revealed.

He always respected scripture, but insisted that the Spirit should lead in its interpretation.
So it is that sometimes a person apprehends a spiritual reality to such an extent that, for him, prophecy is being fulfilled. One wakes up to a truth, which was there all along, previously unnoticed. It's like suddenly seeing the solution to a math problem. The solution always existed, independently of the observer, but now that solution has become a potent part of the observer's reality.

Paul, when caught up to the third heaven, could not tell whether he was in the body or out of the body [2 Cor 12.2-4]. That is, the spirit world was so intense that the material world faded out, a non-issue. In paradise, Paul heard inexpressible words, which a man is not permitted to speak. That is, the language of heaven far exceeds the capacity of the earthly vessels. So we should be careful about trying to interpret spirit reality according to earthly standards.

Now Paul's experience here is echoed elsewhere in scripture: Sky (heaven) and earth had passed away! The sky was torn asunder and rolled up like a scroll! The stars fell from their places and sun and moon became nothing in comparison with God's glory [Is 34.4]! He was in paradise and his world was gone in the blink of an eye! Yet, we can expect that such scriptures will be fulfilled again in other ways.

Consider the believer content in Christ and filled with the Spirit not even noticing supposed bodily hardships. Sky and earth have begun to pass away, as with Stephen during his martyrdom. Or think of Jesus in the desert being comforted by angels following his victory over Satan. That desert was no desert! It was gushing with refreshing streams and blooming with beautiful flowers!

So we can see that fulfillment of scripture is not naively chronological but comes in surprising ways. Often a prophecy had already been in part fulfilled in the shadowy pre-gospel sense but awaited complete fulfillment in Jesus. Also, even when a prophecy is being fulfilled at the dawn of the Christian era, that does not mean there isn't more to come. For example, take Acts 2.17,18 when the spirit-filled Peter quotes Joel's prophecy that in the last days God will pour out his spirit on all flesh (1).

And indeed, since that time that prophecy has been fulfilled by God granting his Spirit to people of every race and background. And its fulfillment is not yet complete.

As Paul said, human wisdom is insufficient for understanding God. So we can appreciate that when God speaks to man through prophetic scripture, he is infusing man's tribal viewpoint and his local legends with the Spirit in order to lift the hearer into an expanded consciousness. He is using man's language and man's way of thinking to reflect God's thoughts.

Because of this, the word of God may seem to some to be ambiguous or inconsistent. Also, linear history seems to matter to modern humans far more than it does to God, who is focused on spiritual growth.

Our ability, even with the Spirit, to comprehend prophecy 'through a glass darkly' [1 Cor 13.12] (as a dim reflection) has a counterpart in modern astronomy. Before powerful telescopes made most of the physical cosmos observable, one could only make educated guesses as to most of what was out there, even though scientists had Einstein's highly accurate general theory of relativity as their 'bible.'

Though the theory's general predictions are repeatedly verified as astronomy progresses, no one in the early years of the 20th century anticipated the fantastic view of the cosmos that eventually unfolded.
Previously, scientists had the physical truth, and yet they still perceived the cosmos 'through a glass darkly' before having their consciousness expanded to a much bigger reality (and this reality is nothing in comparison to what God has in store).

The Book of Revelation can be thought of as the Big Picture of spiritual reality. It gives a panoramic sweep that includes events of the past and of the future. We can think of it as a completed picture of a jigsaw puzzle. But, try as we might, our human mental strength does not suffice to perfectly match puzzle pieces found elsewhere in scripture and in contemporary events to that Big Picture.

However, that does not mean we should ignore various 'end times' events unfolding about us. We should indeed use these events to toughen up in our Christian walk. Why are we being so worldly? Can't we see the Bible is right? 

THE TRUE ZION
The issue of Judaism's primacy was addressed by Jesus and the apostles with such sayings as 'Salvation comes through the Jews' [Jn 4.22] 'to the Jew first, and then to the Gentile' [Rom 2.9,10], 'the first shall be last and the last shall be first' [Mt 20.16].

The Samaritan woman was puzzled by religious differences. 'Our fathers worshiped in this mountain,' she said, 'and your people say that in Jerusalem [on Mount Zion] is the place where men ought to worship.' Jesus replied, 'Woman, believe me, an hour is coming when neither in this mountain nor in Jerusalem will you worship the Father. You worship what you do not know; we worship what we know, for salvation is from the Jews. But an hour is coming, and now is, when the true worshipers will worship the Father in spirit and in truth.' [John 4.20-23].

In other words, God was creating a new world, or covenant, where the heart of the believer was the issue, as opposed to a particular geographic location.
In that same chapter, Jesus -- who was breaking the Middle Eastern taboo of conversing with a woman without her male folk present -- tells her that he would be happy to fill her up with 'living water' from a spring that won't run dry. 'Living water' (what in English is called 'running water') is a poetic description of the Holy Spirit. Jesus was bringing salvation to a hapless, roughly handled Palestinian woman.

The Lord was not terribly interested in geography or property rights [Luke 12.14]. His aim was and is to confer the gift of salvation and eternal life. Recall his assertions that 'the kingdom of heaven is within you' [Luke 17.21] and 'my kingdom is not of this world' [Jn 18.36].

The writer of Hebrews says that Jewish religious customs are a 'copy and a shadow of heavenly things.' [Heb 8.5]. Likewise, Paul wrote that the Jewish law, including the ten commandments, was intended to school the Jews as to the meaning of right and wrong so that they would become aware of how much they needed the unmerited favor of salvation [Gal 3.24].

A new order replaces the old order, as Heb. 8.8-13 says, quoting prophecy [Jer 31.31]:

'"Behold, days are coming, says the Lord, when I will effect a new covenant with the House of Israel and with the House of Judah; not like the covenant which I made with their fathers on the day when I took them by the hand to lead them out of the land of Egypt; for they did not continue in my covenant, and I did not care for them, says the Lord. For this is the covenant that I will make with the House of Israel after those days, says the Lord: I will put my laws into their minds, and I will write them on their hearts. And I will be their God. And they shall be my people..." When he said, "a new covenant," he has made the first obsolete.
But whatever is becoming obsolete and growing old is ready to disappear.'

And when does the new covenant appear? It is made evident at the Feast of Pentecost, when, as Peter said, quoting from Zechariah, Jerusalem became 'the city of truth.' That passage [Zech 8.2-17] also says that the Lord is 'exceedingly jealous for Zion.' The covenant dawned at Pentecost and is still in process of fulfillment.
Or, as Heb 12.22-24 points out, the terrifying experience of Mount Sinai is no more, but rather, 'you have come to Mount Zion and the City of the Living God, the heavenly Jerusalem, and to myriads of angels, to the general assembly and church of the firstborn who are enrolled in heaven, and to God, the judge of all, and to the spirits of the righteous made perfect, and to Jesus, the mediator of a new covenant...' Plainly, Zion here is a spiritual place, the realm of salvation. The old world of the old covenant will be shaken and only what is unshakable will remain, but 'we receive a kingdom which cannot be shaken' [Heb 12.27,28].

Paul says that eventually 'all Israel will be saved; just as it is written, "The deliverer will come from Zion, he will remove ungodliness from Jacob." "This is my covenant with them, when I take away their sins" [Rom 11.26,27]. Paul adds, 'From the standpoint of the gospel, they [the Jews] are enemies for your sake, but from the standpoint of God's choice they are beloved for the sake of the fathers; for the gifts and calling of God are irrevocable' [Rom 11.28,29].

He is explaining that the Jews have been partially hardened in order to permit the 'fulness of the Gentiles to be gathered into the body of Christ' (indicating that many who perished without seeing Jesus must still arise to meet him as their Savior).

A recurrent theme of Paul is the unity of Jews and Gentiles in Christ. In God's eyes, there is no distinction. 'For by one Spirit we were all baptized into one body, whether Jews or Greeks [Gentiles], whether slaves or free, and we were all made to drink of one spirit.' [1 Cor 12.13].

So then the new Zion is filled with the new Israel, the redeemed Jews and Gentiles. Under the new covenant, 'the virgin daughter of Zion' [2 Kings 19.21] has become the bride of Christ [2 Cor 11.2, Rev 21.9,24]. 'For out of Jerusalem will go forth a remnant, and out of Mount Zion, survivors' [2 Kings 19.31]. This prophecy was fulfilled by the fact that the Jewish identity survived Nebuchadnezzar's destruction of Jerusalem, but that prophecy is fulfilled in an expanded sense by the fact that born-gain Jewish Christians, the ultimate survivors, fanned out from Jerusalem. 

A MAJOR SIGN
Though interpretation of end-times prophecies requires a special gift, there is one prophecy that stands out as a beacon in our time. Luke 21.24 quotes Jesus thus: 'and they will fall by the edge of the sword, and will be led captive into all nations; and Jerusalem will be trampled underfoot by the Gentiles until the times of the Gentiles are fulfilled.'

After the Romans led the Jews away captive, Jerusalem remained under Gentile domination for nearly 2,000 years. Upon Israel's founding in 1948, Jerusalem became the Jewish state's capital, though this status has long been a diplomatic sore point. After a fight, Jordan occupied east Jerusalem, but then lost that sector during the 1967 war.

So then, it is fair to say that the times of the Gentiles are about over. However, we might also note that Arabs predominate in East Jerusalem, possibly indicating that the Gentile era is not quite over. Certainly, the Israeli government cannot end the Gentile era by trying to complete this prophecy.

For one thing, the Gentile era is the period when the message of salvation is going out to the Gentiles. Once the full complement of Gentiles destined for salvation has been reached, then the Gentile era ends, according to Paul, writing in Romans 11.11-32 -- in particular Rom 11.25. Once this occurs, Paul says, Jews who had been unable to receive the gospel will be saved.

Of course, that implies that a powerful evangelical campaign must come in Israel, accompanied by great pressure to thwart the gospel.

Further light on the Gentile era is shed in the Book of Revelation (Rev 11.2) where John is told not to measure the outer court of the temple, which is given over to the Gentiles, who will tread Jerusalem underfoot for 42 months (three-and-a-half years).

Two witnesses will prophecy for 1,260 days (three-and-half years), clothed in sackcloth. They have tremendous power, and can kill with the fire of their mouths. (Rev 11.3-6)

In Rev 12.14, we are told that the woman flies from the serpent's presence to the wilderness to be nourished for three-and-a-half times.

This echoes Daniel 12.7, where it is prophesied that wonders would cease -- the world would end -- after three-and-a-half times. A general resurrection occurs after the holy ones are overcome, a theme found in Rev 11 and in Jesus' end-times discourse.

In scripture, a 'time' can be taken to be a solar year. And it is interesting that the Roman-Jewish war lasted about three-and-a-half years, culminating in the destruction of Jerusalem.

However, if we keep in mind the Greek idea of kairos, we might think of 'a time' as meaning a period in which certain trends accumulate until reaching the critical point, which ends that period. So then, we might interpret the three-and-a-half years of Revelation as three-and-a-half kairos times, meaning that the era of the Gentiles ends amidst some other era -- perhaps the age of grace, which must still be extended to the Jews.

Here we have another argument against eschatological timetables, since kairos eras can overlap. On the other hand, it should be acknowledged that some accept that Jeremiah's prophecy -- that Jerusalem would be resettled by the Jews after 70 years of desolations by Gentiles -- was fulfilled after about 70 solar years (Jer 25.11,12, Jer 29.10). However, Daniel, in Babylon, seems to have been perplexed that the 70 years was up and the prophecy in need of fulfillment, if solar years were meant. The angel Gabriel responded by giving Daniel a vision of the Messianic era to come, and explained that this era would begin after '70 weeks.'
Interpreting a week as seven years, the kronos time of 490 years seems to accord with the Jewish resettlement of Jerusalem in 538 BCE and the ascension of Jesus around 34 CE (about 496 years), though the Messiah was also supposed to be cut off after 62 weeks -- in kronos time, 434 years, which does not jibe.]

Realizing that the Spirit can cause believers to see a scripture or set of scriptures as being fulfilled in many ways, we offer this scenario as a possibility:

In Rev 11.2, the Gentiles are in the outer court of the temple. That is, these are the Gentiles being brought to salvation; meanwhile, the Gentiles are walking all over the holy city. Now, the 'holy city' may signify the kingdom of God -- which they, like the prostitutes, are storming into -- and-or it may signify earthly Jerusalem, which seems likely, since Jesus in Luke 21.24 certainly was talking about earthly Jerusalem. So then, we would assert that once earthly Jerusalem came under Jewish rule, the times of the Gentiles begin drawing to a close. Now which times are meant? One time evidently is the era for the gospel to the Gentiles; another time apparently is the era of Gentile rule over earthly affairs.

Now the two witnesses of Rev 11 may mean the Jewish Christian and the Gentile Christian components of the body of Christ. True Christians have tremendous power, as the gospel accounts and the Acts of the Apostles, show. But they are at their best when they are humble, eschewing earthly wealth. They are 'in sackcloth' during the age of witness, but eventually they will wear divine apparel. However, I am aware that such an interpretation might constitute a partial fulfillment of prophecy, with a complete fulfillment to come.

We are now in a position to suspect that the three-and-a-half times of Daniel may correspond with the three-and-a-half years of Rev 11. Similarly, we can interpret the woman of Rev 12 as the church -- the body of born-again believers -- who, harshly persecuted in Judea, escaped to the 'wilderness' of Gentile lands, where she has been nourished for three-and-a-half times.

However, I am uncertain as to what three eras are meant, though I would guess that the 'half-era' refers to the age of grace.

At various points, the Bible says that the coming of the last days is accompanied by birth pangs. Severe stress presages the birth of a new world era.

I suggest that the closing days of the age of the Gentiles mean great suffering on the part of the Gentiles. The plague of AIDS appears poised to sharply reduce the earth's population, possibly by 2028 (2). Poor nations are ill equipped to combat this disease, which has already devastated sub-Saharan Africa -- leaving 11 million AIDS orphans -- and which is on the verge of bringing about a similar catastophe across Eastern Europe, Russia and Asia. In addition, the rapid increase in global warming is likely to have numerous calamitous effects, including triggering devastating malarial and ebola epidemics.

We also cannot rule out the possibility that weapons of mass destruction may yet be unleashed on a genocidal scale. But, even if such arms are contained, pandemics and related famines are likely to promote civil disorder and war. The developed nations can no more escape the judgment than the Titanic could escape sinking. The great distress will cross borders, despite all efforts, and then spiral out of control in the supposedly safe areas.

We cannot say that the state of Israel will be unaffected. However, a remnant of Jews will remain, perhaps still in control of Jerusalem. Yet, the scepter of world rule will not be passed to earthly Israel. If so, why would the Bible repeatedly point to a Messianic era as the culmination of Jewish aspirations? My guess is that, while the distress of nations is under way, Israel will undergo a harsh internal fight as Christian Jews begin to make headway.

At some point, the Christian Jews will be eradicated. BUT, we cannot assume that this will occur at any particular time.

The scenario I have outlined is plausible, but I do not claim that I am correct in every jot and tittle. 

A NOTE ON THE ABRAHAMIC BLESSING
An important topic of end-times discussions is the blessing given to Abraham, which was routed through the Jews.

'And I will bless those who bless you, and curse those who curse you: and in you shall all the families of the earth be blessed' [Gen 12.3].

Abraham's real descendents are those who heed God, not merely people with certain DNA characteristics who have been circumsized [Jn 8.39]. Obviously, the blessing points to Jesus, who blesses all the nations of the earth. Those who bless Abraham will be blessed and those who curse him will be cursed. This certainly applies to Abraham's spiritual progeny, the born-again Christian.

But supposing that God's pledge also applies to Abraham's descendents in the flesh, a question must be posed: does the pledge apply to the state of Israel, the Israeli government, Jewish residents of Israel, or Jews in general, wherever they live? I raise that question because some seem to equate any criticism of Israeli government policies as tantamount to invoking a curse against onself. I don't wish to attempt to neutralize the potency of that scripture, but I am concerned that some are deploying it for propaganda purposes. 

THOUGHTS ON 'ELOHIM'
A verse in the Song of Moses, Deut. 32:8, reads:

'When the Most High apportioned the nations,
when he divided humankind,
he fixed the boundaries of the peoples
according to the number of the sons of God.'

The phrase 'sons of God' was rendered 'sons of Israel' in the traditional Jewish Bible, but a fragment from the Dead Sea scrolls attests to the former.

Who were these sons of God? Were they related to the sons of God who intermarried with earth women and spawned demonic offspring? What is known is that various non-canonical writings, along with some clues in the Bible, indicate that these were the 70 (probably a symbolic number) angels who led the nations surrounding ancient Israel. The highest of these divine beings, who came to be known as Elohim and also YHWH (roughly: 'Yahweh'), chose the nation of Israel for himself.

Scholars have found that, though 'Elohim' is used in the singular in the Hebrew scripture that we have now, the word itself is a plural form of 'El,' which was the name of the Canaanite's chief god. So 'Elohim' simply means 'the gods.'

But the Hebrews were forbidden to worship any God but Elohim -- which evolved into a a generic name for God -- who eventually revealed his special name to them as YHWH, just as he has revealed to Christians his special name of his son Jesus. Elohim brooked no competition from his 'sons.'

So it is apparent that the Hebrews knew of the Canaanite pantheon -- which is similar to other pantheons, in which one god is chief of the others.

However, as Hebrew religion and language evolved, we find that the Canaanite 'sons of El' -- which is equivalent to 'the gods' -- becomes 'sons of Elohim,' which is later rendered as 'angels,' which are directly created divine beings (note that this makes Jesus a hybrid of son of man -- because born of a woman -- and angel, or son of God, because directly created, which is also what born-again believers are). Angels could have god-like power but were not to be worshiped. The theological issue bothered the Pharisees, who took 'son of God' to mean a god who must not be worshiped.

Elsewhere we learn that Michael is Israel's special protecting angel (whose name means 'Who is like God?'). So then he is, possibly, identified as a projection (message) of Elohim. In addition, we learn in Daniel and Revelation (and in extra-biblical texts) that Michael is waging war with the devil and his angels. In particular, we find that the prince of Persia is a demonic angel. So we can guess that the angels that lead the nations are demonic, or at any rate inferior, angels, perhaps those who intermarried with earth girls.

We might think of the pantheon of the gods as a muddled (reprobate) intuition of a situation in which the human soul is a projection of a particular angel. So all those who are a projection of a particular angel are in the 'nation' ruled by this angel. But Jesus taught that those who are not ruled by the Father must be ruled by Satan, implying that the unregenerate are either reflections of fallen angels, or, are tares that have been planted by an enemy.

Be this as it may, clearly some will suspect that the Jewish religion is simply a variant of the older Canaanite religion with which it clearly has similarities. Scholars would add that as Jewish theology evolved, it took on some of the imagery and concepts of Zoroastrianism, a Persian religion of great antiquity. (A number of Zoroastrian concepts make sense in terms of Judaeo-Christian traditions. Take, for example, 'the Lie,' which can be thought of as a place where many are headed. It is also the pull to proceed into such a place. The Lie is the door opening to one of the 'many mansions' in God's universe, as well as the pull toward that door. The Lie is the pull toward Death, death itself being a lie.)

So my thinking is that God was using the imagery with which the Hebrew tribesmen were familiar in order to gradually unveil the higher things of God. Recall Paul on Mars Hill, when he preached to the Athenians on the 'unknown God.' The monument to which he referred was put there in the event that the polite and cosmopolitan Athenians had neglected to honor some visitor's god. Yet Paul used the concept of 'unknown God' as a jumping off point to preach the good news that the way to know God was via Jesus Christ. Similarly, we can see that God could have used old stories, imbuing them with his spirit, in order to reveal some of the depth and magnitude of the true God.

Think of the transfiguration:

Jesus had previously quoted a scripture that said 'you are gods' (or possibly 'you are as gods') and had held that if God IS the God of Abraham and Isaac, then God is the God of the living.

On the mountain, his disciples saw him with Moses and Elijah, all blazing with unearthly light, and then, after closing their eyes in fear, saw only Jesus.

That is, these divine beings were united by a single Spirit.

As Paul says, born-again believers are heirs with Jesus of the kingdom [Eph 3.6], sons of God that is. In fact, they are as gods, indeed they are gods. Nevertheless, they, who are junior partners of Christ, are certainly not to be worshiped.

The promise to the believer is that, as he becomes more Christ-like, he will become one with God -- though he actually is already one with God, just as Jesus is one with the Father.
In any event, we can see that the Hebrew word 'Elohim,' though it stemmed from a pagan theory, fits well with God's revelation to the Jews, who he did indeed choose as his special people, though their heory of God is past its prime. 

APPENDIX: 'Zion' prophecies interpreted
Following are my interpretations of a number of scriptures concerning Zion:

Ps 9.11
'Sing praises to the Lord, who dwells in Zion.'

Under the old covenant, the invisible God dwelt in the ark of the covenant. Within that ark were the stone tablets on which God had personally inscribed the 10 commandments. The two stones can be viewed as a token of God's covenant with the children of Israel. But also we see that the Lord himself was the true covenant -- the word and the guarantor of that word.

Under the new covenant, the invisible God -- the Spirit of God -- dwells in the mind of the believer, making the believer's body, 'the temple of the Holy Spirit' [1 Cor 6.19]. So then, would God go back to the old covenant in order to take up residence in another stone temple?

Ps. 14.7
'O that the salvation of Israel would come out of Zion, when the Lord restores his captive people, Jacob will rejoice, Israel will be glad.'

Jesus rose to heaven out of Jerusalem, where he could then send the Spirit to the Jews -- and Gentiles -- who were to be set free. 'Whom the son sets free is free in deed'[John 8.36].

Ps. 65.1
'silence before you and praise in Zion'

Or, 'be still and know I AM' [Ps 46.10]. This intimacy with God requires the presence of Jesus in the believer, who then can enjoy God.

Ps. 69.35,36
'For God will save Zion and build the cities of Judah, that they may dwell there and possess it. The descendants of his servants will inherit it, and those who love his name will dwell in it.'

Zion and the cities of Judah are reserved for the descendants of his servants and those who love God's name. A true descendant of God's servant is also God's servant. Otherwise the devil is his father [Jn 8.44]. So is earthly Jerusalem filled with such people?

The 'cities of Judah' needn't refer to old Middle Eastern towns; instead what is indicated is a land where the redeemed, who love the name Jesus, can enjoy life.

Ps. 78.67,68,70
'He also rejected the tent of Joseph, and did not choose the tribe of Ephraim, but chose the tribe of Judah, Mount Zion which he loved.'... 'He also chose David his servant and took him from the sheepfolds.'

Though this refers to the preservation of David's tribe, Judah, after the destruction of the other 11 tribes, these words are fulfilled by Jesus, the good shepherd descendant of David, who established spiritual Zion by his conquest of evil at the cross.

Ps. 84.5,7
'How blessed is the man whose strength is in you, in whose heart are the highways (to Zion)... every one of them appears before God in Zion.'

Those who love God are approaching Zion.

Ps. 87.2
'The Lord loves the gates of Zion more than all the other dwelling places of Jacob.'
English-language translations of the Bible customarily italicize words not found in the Hebrew or Greek scriptures but that seemed necessary. In this case, if we omit 'other,' we can interpret this passage to say that 'Jacob' refers to the unregenerate Jew. He cannot enter Zion until he has been transformed, by the renewing of his mind [Rom 12.2] into 'Israel.' The name 'Israel' means 'wrestler with god' and the name 'Jacob' means 'he who grabs by the heel' or more generally 'trickster.'

Recall the encounter between Jesus and Nathanael, who knew salvation was to proceed from Jerusalem. Nathanael's question 'Can any good thing come out of Nazareth?' was followed by Jesus' retort: 'Look, a real Israeli, in whom there is no guile.' Or, that is: 'Look, a real contender with God, in whom there is no Jacob.'

Ps. 87.6
'The Lord will count when he registers the peoples, "This one was born there [in Zion]."

As Jesus said, the birth that counts is the spiritual birth. Those written in the Lamb's Book of Life [Rev 20.12] have been born of the Spirit, not of flesh and blood [Jn 3.5,6]. So there is no use having been born in Jerusalem if one hasn't been born again, saved by the intervention of Jesus Christ.

Ps. 97.7
'Let all those be ashamed who serve graven images, who boast themselves of idols ... Zion heard this and was glad.'

The real Zionists are glad when God shames the money-worshipers and their type.

Ps. 99.2
'The Lord is great in Zion'

Or, Zion is a place where the Lord's glory shines brightly. And isn't that in the hearts of those who worship God 'in spirit and in truth'?

Ps. 102.21
'that men may tell the name of the Lord in Zion'

Utterance of that name -- as represented in scripture by the Hebrew letters YHWH -- came to be forbidden to Jews of old and such utterance is still considered impious for Jewish residents of modern Jerusalem, assuming anyone could be sure of the pronunciation.

The name of the Lord, under the old covenant, is not, under the new covenant, fully revealed since the name of 'Jesus' suffices as the name of the Lord.

It has become possible to tell the name of the Lord openly, even if with difficulty.

'Jesus' or 'Yah shua' means 'Yaweh saves.' Knowing this name suffices just as seeing Jesus sufficed for seeing the Father [Jn 14.8].

Ps 110.1,2
'The LORD says to my Lord: sit at my right hand until I make your enemies a footstool for your feet. The LORD will stretch forth your strong scepter from Zion...'

Jesus cites this passage [Mt 22.44] to show that David calls his son, the messiah, his lord. So then we can be sure that Jesus is one with YHWH (the name of God using only consonants, as translated from Hebrew) whose great power over heaven and earth extends from the kingdom of God, which is Zion.

The fourth verse of this psalm says that God has determined that this savior is 'a priest forever according to the order of Melchizedek,' a passage that Heb. 7.21 says refers to Jesus, the consummate priest.

Ps. 129.5
'May all who hate Zion be put to shame and turned backward.'

Those who love their sin more than the light remain under judgment and are continually injured by their own wilfulness [Jn 3.18], since they reap what they sow [Gal 6.7].

A specific instance of this scripture being fulfilled occurred when an armed troop came to arrest Jesus. Jesus asked whom they sought. When told, he replied, 'I am he.' On hearing 'I am,' which is a name for God, the men stumbled backward in confusion and were then only able to proceed with God's permission [Jn 18.4-6].

Ps 135.21
'Blessed be the LORD from Zion, who dwells in Jerusalem.'

The writer was likely thinking of the Temple in Jerusalem. Yet, when Jesus died on the cross, the veil of the Temple was torn apart, indicating the departure of God's Spirit from the veiled ark of the covenant. Where does the Lord dwell today? Within those who put their trust in Jesus.

Is 1.27
'Zion will be redeemed with justice and her repentant ones with righteousness.'

Justice was served by the sacrifice of Jesus, the son of God, whose crucifixion paid in full the price of all sin.

Those who humbly turn to him will be made right with God.

Is 8.18
'Behold, I and the children whom the Lord has given me are for signs and wonders from the LORD of hosts, who dwells on Mount Zion.'

Heb 2.13 takes this passage to refer to Jesus.

Is 24.23
'The LORD of hosts will reign on Mount Zion and in Jerusalem.'

We know that Mount Zion is a spiritual place and the Book of Revelation tells us to expect a new, heavenly Jerusalem [Rev 3.12].

Is 28.16,18
'Behold, I am laying in Zion a stone, a tested stone, a costly cornerstone for the foundation, firmly placed. He who believes in it will not be disturbed [moved].'

'Your pact with Sheol will not stand.'

Paul in Rom 9.33 and Peter in 1 Pet 2.6 say Is 28.16 refers to Jesus. Now the old, earthly Zion was overthrown by the Romans in 70 A.D. So the new Zion is the kingdom of God that Jesus is yet bringing forth.

Jesus will cancel the sin-sick soul's contract with death.

Is 29.8
'It will be as when a hungry man dreams--and behold, he is eating; but when he awakens, his hunger is not satisfied, or when a thirsty man dreams--and behold, he is drinking, but when he awakens, behold, he is faint and his thirst is not quenched. Thus the multitude of all the nations will be who wage war against Mount Zion.'

Anybody who fights God is deluded. Any 'victory' over the LORD is hollow.

Though this passage specifically refers to Gentiles, certainly it also holds for Jews.

A nearby passage [Is 29.10] says that God 'has poured over you a spirit of deep sleep [or 'strong delusion']' blinding the prophets and the seers. Paul [Rom 11.8] says this scripture applies to the bulk of the Jews, who 'were hardened.'

That is, those who resist God, and his works, are turned away from his kingdom by a strong delusion, reminiscent of the angels guarding the gates of Eden.

Is 30.19-21
'O people in Zion, inhabitant in Jerusalem, you will weep no longer. He will surely be gracious to you at the sound of your cry; when he hears it, he will answer you. Although the Lord has given you bread of privation and water of oppression, your Teacher will no longer hide himself, but your eyes will behold your Teacher.

Your ears will hear a word behind you, "This is the way, walk in it," whenever you turn to the right or the left.'

God will comfort his people. Yet, who can see God? Yet his people will see the Teacher, who will guide their minds. Those who walk with Jesus recognize the Teacher's voice [Jn 10.27], which can only be heard via the Holy Spirit. He is the road and he makes sure that every follower will learn more and more about God.

Is 31.4,9
Like a lion with its prey unworried by the shepherds, so will 'the LORD of hosts come down to wage war on Mount Zion and on its hill.'

'whose fire is in Zion and whose furnace is in Jerusalem.'

Have you ever noticed that on occasion titanic contests occur with little recognition of what is going on -- hardly anyone is paying attention?

Jesus was sent by the Father to wage spiritual warfare against the devil and his hordes. A scene in that struggle occurred when he picked up a whip and drove the businessmen from the Temple.

Certainly Jesus could not be deterred from achieving his victory over hell.

That victory was exemplified when, during the feast of Pentecost, tongues of fire descended upon the apostles in Jerusalem to launch the true church.

Is 33.5
'He has filled Zion with justice and righteousness.'

Not much of that in wordly Jerusalem. However, Jesus embodies the phrase 'justice and righteousness.'

Is 33.20
'Look upon Zion... Your eyes shall see Jerusalem ... a tent which will not be folded; its stakes will never be pulled up. No oarboat or warship will pass on its great streams.'

What great streams? These must be streams flowing in the heavenly Jerusalem of Rev 3.12, where commerce and war are no more.

Is 35.5,6,10
'Then the eyes of the blind will be opened and the ears of the deaf will be unstopped. Then the lame will leap like a deer, and the tongue of the mute will shout for joy.'

'And the ransomed of the LORD will return and come with joyful shouting to Zion, with everlasting joy upon their heads.'

When John the Baptist queried Jesus as to whether he was the Messiah, Jesus replied: 'Go and report to John what you have seen and heard: the blind receive sight, the lame walk, the lepers are cleansed and the deaf hear, the dead are raised up, the poor have the good news preached to them.'

All of Isaiah 35 foretells, in rhapsodic imagery, Zion's wonderful future. From Jesus' response, we can see that he was a sign of the true Zion. And a person who has truly received Jesus as Lord has entered that Zion, the land of eternal bliss, and, despite hardships, will shout for joy at the wonder of this event.

Is 37.32
'For out of Jerusalem will go forth a remnant and out of Mount Zion, survivors.'
Those who have been born by the Spirit.

Is 40.9
'Get yourself up on a high mountain, O Zion, bearer of good news ... say to the cities of Judah "Here is your God."'

Jesus was lifted up on Mount Calvary with the inscription: 'King of the Jews.'

Is 49.14
'But Zion said, the LORD has forsaken me...'

Jesus cried out on the cross: 'My God, my God, why have you forsaken me?' [Mk 15.34]

Is 51.3
'Indeed, the LORD will comfort Zion...and her wilderness he will make like Eden.'

How sweet it is to follow Jesus.

Is 51.11
'So the ransomed of the LORD will return and come with joyful shouting to Zion.'

How wonderful to have your sins paid for by Jesus.

Is 52.3
Zion 'will be redeemed without money.'

Those destined for eternal life have had their sin paid for by the blood of Jesus.

Is 52.7
'How lovely on the mountains are the feet of him who brings good news, who announces peace and brings good news of happiness, who announces salvation, and says to Zion, "Your God reigns!"'

Jesus announced the good news of salvation, and so do his followers.

Is 52.8
'For they will see with their own eyes when the LORD restores Zion.'

Witnesses saw the works of Jesus, including his ascension into heaven. However, many of those suffering spiritual blindness were unable to see that Zion was being restored. Isaiah begins his description of the suffering servant with 'who has believed our report?' [Is 53.1]

Is 61.2,3
'...to comfort all who mourn, to grant all those who mourn in Zion...'
Jesus tells us to fear not, to be cheerful, because he has overcome the world [Jn 16.33].

Jesus told his hearers that Is 61.1,2 had been fulfilled in their hearing.

Is 62.1,2
'For Zion's sake will I not hold my peace, and for Jerusalem's sake, I will not rest' until the 'nations [Gentiles] will see your righteousness...and you will be called by a new name, which the mouth of the LORD will designate.'

The name of Jesus is lifted up among the Gentiles.

Is 66.8
'Can a land be born in one day? Can a nation be brought forth all at once? As soon as Zion travailed, she also brought forth her sons.'

The new Zion was brought forth by the sacrifice of Jesus and began in power on the day of the feast of Pentecost, when immediately thousands turned to Jesus for salvation [Acts 2.41].

Jer 3.14
'...return O faithless sons...and I will take you one from a city, and two from a family, and I will bring you to Zion.'

Many are called, but few are chosen for Christ's kingdom.

Joel 2.32
'And it will come about that whoever calls upon the name of the LORD will be delivered; for on Mount Zion and in Jerusalem there will be those who escape.'

Paul in Rom 10.13 applies this scripture to anyone who calls on Jesus, who will then escape hellfire. Joel 2.28 -- 'I will pour out my spirit on all mankind' -- is cited by Peter during the Pentecost speech.

Had Jesus' warning been remembered [Lk 21.20], those in Jerusalem during the Roman-Jewish war would have known it was time to flee when they saw the Temple profaned by Jewish rebels.

In the case of the prophecies of Joel and of Jesus, we can see that they may be specifically fulfilled and yet we cannot rule out their being fulfilled again. However, Paul is less interested in 'future history' than in getting people to see the main point: the salvation provided by Jesus more than fulfills scripture.

Obadiah 17
'...on Mount Zion will be those who escape...'

Those who escape the wrath to come [Mt 3.7] are those who put their trust in Jesus.

Mic 4.2
'Many nations will come and say, Come let us go up to the mountain of the LORD and to the house of the

God of Jacob, that he may teach us about his ways and that we may walk in his paths. For from Zion will go forth the law, even the word of the LORD from Jerusalem.'

Some may read this passage literally, but I see it as fulfilled by non-Jews who are able to get into the kingdom of God through the mercy of Jesus, who is the word of the LORD personified and who is the fulfillment of the law of the LORD. As Jesus taught at age 12, the House of God is the work of the Lord [Lk 2.49] and for humanity the greatest work is salvation.

Zeph 3.12-15
'But I will leave among you a humble and lowly people, and they will take refuge in the name of the LORD.

The remnant of Israel will do no wrong and tell no lies, nor will a deceitful tongue be found in their mouths; for they will feed and lie down with no one to make them tremble.

'Shout for joy, O daughter of Zion! Shout in triumph, O Israel!...The LORD has taken away his judgments against you, he has cleared away your enemies.'

Jesus told us to learn of him, who is humble and lowly [Mt 11.29]. Those who trust in Jesus, who paid for all crimes against God, no longer need fear anything at all, and as they are perfected, they become less and less prone to wrongdoing. And God does not look upon sin because the believer has been pardoned by Jesus. So those put right with God through Jesus have every reason to shout for joy.

Zec 9.9
'Rejoice greatly, O daughter of Zion...Behold your king is coming to you; he is just and endowed with salvation, humble and mounted on a donkey...'

Fulfilled by Jesus' triumphal entry into Jerusalem, where he was greeted as the son of David -- the Messiah.

Zec 9.10
'I will cut off the [war] chariot from Ephraim and the [war] horse from Jerusalem; and the war bow will be cut off. And he will speak peace to the nations; and his dominion will be from sea to sea, and from the River to the ends of the earth.'

Jesus, who has been granted all power in heaven and earth, gives his followers the 'peace of God, that passes understanding'[Phlp 4.7]. Once his kingdom has spread everywhere, warfare will cease.


(1) Colin J. Humphreys, a Cambridge scientist and Bible investigator, has analyzed Peter's interpretation of the prophecy of Joel as related to historical events that can be corroborated. His keen mind is revealed in his book The Mystery of the Last Supper: Reconstructing the Final Days of Jesus (Cambridge University Press, 2011). He is also the author of The Miracles of Exodus: a Scientist Reveals the Extraordinary Natural Causes Underlying the Biblical Miracles (Harper Collins, 2003).

His web site: http://www.csap.cam.ac.uk/network/colin-humphreys/

(2)  When that was written, such a global catastrophe seemed plausible. However, education and distribution of anti-AIDS medicines seems to have blunted the threat, though we cannot be sure it won't return.
______________________________

Sept. 2013. This article has had minor editing on several occasions between 2005 and Sept. 2013.
Note on Wolfram's 'principle of computational equivalence'

Stephen Wolfram has discussed his "principle of computational equivalence" extensively in his book A New Kind of Science and elsewhere. Herewith is this writer's understanding of the reasoning behind the PCE:

1. At least one of Wolfram's cellular automata is known to be Turing complete. That is, given the proper input string, such a system can emulate an arbitrary Turing machine. Hence, such a system emulates a universal Turing machine and is called "universal."

2. One very simple algorithm is Wolfram's CA Rule 110, which Matthew Cook has proved to be Turing complete. Wolfram also asserts that another simple cellular automaton algorithm has been shown to be universal or Turing complete.

3. In general, there is no means of checking to see whether an arbitrary algorithm is Turing complete. This follows from Turing's proof that there is no general way to see whether a Turing machine will halt.

4. Hence, it can be argued that very simple algorithms are quite likely to be Turing complete, but because there is no way to determine this in general, the position taken isn't really a conjecture. Only checking one particular case after another would give any indication of the probability that a simple algorithm is universal.

5. Wolfram's principle of computational equivalence appears to reduce to the intuition that the probability is reasonable -- thinking in terms of geochrons -- that simple algorithms yield high information outputs.

Herewith the writer's comments concerning this principle:

1. Universality of a system does not imply that high information outputs are common (recalling that a bona fide Turing computation's tape halts at a finite number of steps). The normal distribution would seem to cover the situation here. One universal system is some algorithm (perhaps a Turing machine) which produces the function f(n) = n+1. We may regard this as universal in the sense that it prints out every Turing machine description number, which could then, notionally, be executed as a subroutine. Nevertheless, as n approaches infinity, the probability of happening on a description number goes to 0. It may be possible to get better efficiency, but even if one does so, many description numbers are for machines that get stuck or do low information outputs.

2. The notion that two systems in nature might both be universal, or "computationally equivalent," must be balanced against the point that no natural system can be in fact universal, being limited by energy resources and the entropy of the systems. So it is conceptually possible to have two identical systems, one of which has computation power A, based on energy resource x, and the other of which has computation power B, based on energy resource y. Just think of two clone mainframes, one of which must make do with half the electrical power of the other. The point here is that "computational equivalence" may turn out not to be terribly meaningful in nature. The probability of a high information output may be mildly improved if high computation power is fairly common in nature, but it is not easy to see that such outputs would be rather common.

A mathematician friend commented:

I'd only add that we have very reasonable ideas about "most numbers," but these intuitions depend crucially on ordering of an infinite set.  For example, if I say, "Most integers are not divisible by 100", you would probably agree that is a reasonable statement.  But in fact it's meaningless.  For every number you show me that's not divisible by 100, I'll show you 10 numbers that are divisible by 100.  I can write an algorithm for a random number generator that yields a lot more numbers that are divisible by 100 than otherwise.  "But," you protest, "not every integer output is equally likely under your random number generator."  And I'd have to agree, but I'd add that the same is true for any random number generator.  They are all infinitely biased in favor of "small" numbers (where "small" may have a different meaning for each random number generator).

Given an ordering of the integers, it is possible to make sense of statements about the probability of a random integer being thus-and-so.  And given an ordering of the cellular automata, it's possible to make sense of the statement that "a large fraction of cellular automata are Turing complete."

My reply:

There are 256 cellular automata in NKS. The most obvious way to order each of these is by input bit string, which expresses an integer. That is, the rule operates on a bit-string stacked in a pyramid of m rows. It is my thought that one would have to churn an awfully long time before hitting on a "universal." Matthew Cook's proof of the universalism of CA110 is a proof of principle, and gives no specific case.

As far as I know, there exist few strong clues that could be used to improve the probability that a specific CA is universal. Wolfram argues that those automata that show a pseudorandom string against a background "ether" can be expected to show universality (if one only knew the correct input string). However, let us remember that it is routine for functions to approach chaos via initial values yielding periodic outputs.

So one might need to prove that a set of CA members can only yield periodic outputs before proceeding to assess probabilities of universalism.

Perhaps there is a relatively efficient means of forming CA input values that imply high probability of universalism, but I am unaware of it.

Another thought: Suppose we have the set of successive integers in the interval [1,10]. Then the probability that a randomly chosen set member is even is 1/2. However, if we want to talk about an infinite set of integers, in line with my friend's point, the probability of a randomly selected number being even is meaningless (or, actually, 0, unless we invoke the axiom of choice). Suppose we order the set of natural numbers thus: {1,3,5,7,9,2,11,13,15,17,4...}. So we see that the probability of a specific property depends not only on the ordering, but on an agreement that an observation can only take place for a finite subset.

As my friend points out, perhaps the probability of hitting on a description number doesn't go to 0 with infinity; it depends on the ordering. But, we have not encountered a clever ordering and Wolfram has not presented one.

Monday, September 24, 2012

Freaky facts about 9 and 11
Scroll down to Force 1089 for a fun psychic power trick

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

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

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

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

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


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

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

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

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

Example:

27, 36, 45, 81 all divide by 9

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

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

Example:

819/9 = 91

891/9 = 99

198/9 = 22

189/9 =21

918/9 = 102

981/9 = 109

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

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

Example a.:

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

Example b.:

Number chosen by random number generator:

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

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

372969/9 = 4144

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

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

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

Let's check 5863.

We sum 5 - 8 + 6 - 3 = 0


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

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

6358 = 11*578

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

253 = 11*23, 143 = 11*13

143253 = 11*13023

Now let's test this pseudorandom number:

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

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

7051770517/11 = 641070047

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

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

Sometimes another trick works.

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

Check: 589402/11 = 53582

Let's play with 157311.

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

Positive permutations are

117, 711, 171

Negative permutations are

531, 513, 315, 351, 153, 135

So integers divisible by 11 are, for example:

137115 = 11*12465

711315 = 11*64665

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

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

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

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

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

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

Force 1089
An amusing property of the numbers 9 and 11 in base 10 arithmetic is demonstrated by Force 1089, the name of a trick you can use to show off your "mentalist" powers.

"It is important that you clear your mind," you say to your target. "Psychic researchers have found that a bit of simple arithmetic will help your subconscious mind to focus, and create a paranormal atmosphere."

You then instruct the person to choose any three-digit number where the first and last digit differ by at least 2. "Please do not show me the number or your calculations," you insruct, handing the person a calculator to help assure they don't flub the arithmetic.

The target is then told to reverse the order and subtract the smaller from the larger number. Take that difference and do a reversal of order on that. Then add those two numbers.

You toss a couple of books over to the target and say, "Be careful to conceal your number. Now use the first three digits to find a page in one of the books you choose." Once the page is found, you instruct the person to use the last digit to count words along the top line and stop once reaching that number.

"Now, PLEASE, help me, and concentrate hard on the word you read."

After a few moments, you announce -- of course -- the exact word your target is looking at.

Your secret is that the algorithm always yields the constant 1089, meaning you only have two words to memorize.

These symmetries always, in base 10, produce numbers divisible by 9 and 11 (or 99), even though the first number may not be so factorized.

Consider 854-458 = 396. 396+693 = 1089.
Further, 396 = 4*99; 693 = 7*99; 1089 = 9*11^2.

If we differ the first and last digit by 0, of course, the algorithm zeros out on the next step. If we differ them by 1, the subtraction yields 99, as in 100 - 001 = 99, which turns out to be a constant at this step.

Consider 433-334 = 99.

In the case of a number with 2n digits, we discover that the algorithm always yields numbers divisible by 9. However, for 2n+1 digits, the algorithm always yields a set of numbers divisible by 99. BUT, it does not yield a constant.

(I have not taken the trouble to prove rigorously the material on Force 1089.)

Ah, wonderful symmetry.

Tuesday, July 24, 2012

Mathematics of evolution: a note on Chaitin's model

In his book Proving Darwin: Making Biology Mathematical (Knopf Doubleday 2012), Gregory Chaitin offers a "toy model" to demonstrate that progressive evolution works in principle. The idea is that DNA behavior is very similar to that of cyber software.

http://pantheon.knopfdoubleday.com/2012/05/08/proving-darwin-by-gregory-chaitin/

Computation of numbers then becomes the work of his evolution system. A mathematician, he finds, can "intelligently design" numbers that get very close to a Busy Beaver number with work growing at n. An exhaustive search of all possible computable numbers under some BB(n) requires exponential work. But, he found, his system of a climbing random walk arrived at numbers close to BB(n) on the order of n^2.

Chaitin posits a Turing-style oracle to sieve out the "less fit" (lower) numbers and the dud algorithms (those that get stuck without producing a number). The oracle represents the filtering of natural selection.

Caveat:

His system requires random mutations of relatively high information value. These "algorithmic mutations" alter multi-node sets. Point mutations, he found, were unproductive. Hence, he has not succeeded in demonstrating how the DNA system itself might have evolved.

Chaitin says he was concerned that there existed no mathematical justification for evolution. But, this assertion gives pause. The existence of the universal Turing machine would seem to demonstrate that progressive evolution is possible, though not necessarily highly probable. But granting that Chaitin was focused on probability, we can agree that if a system is of a high enough order, the probability of progressive evolution is strong. So in that respect, one may agree that Darwin has been proved right.

However, there's an old saying among IT people: "Garbage in, garbage out." The probability that random inputs or alterations will yield increased functionality is remote. One cannot say that Darwin has been proved right about the origin of life.

Remark
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), or some routine binomial combinatorial multiple, 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.





Saturday, June 2, 2012

Periodicity 'versus' randomness
A few observations, with no claim to originality

Please also see a more recent post on the probabilities of periodicity
http://kryptograff5.blogspot.com/2013/08/draft-1-please-let-me-know-of-errors.html

We do have a baseline notion of randomness, I suggest. Within constraints, the clicks of a Geiger counter occur, physicists believe, at truly random intervals. An idealized Geiger counter hooked up to an idealized printer might -- assuming some sort of time compression -- print out a noncomputable transcendental. The string's destiny as a noncomputable has probability 1, though, from a human vantage point, once could never be sure the string wasn't destined to be a computable number.

Stephen Wolfram's New Kind of Science, on the other hand, tends to see randomness in terms of pseudorandomness or effective randomness. In my view, an effectively random digit string is one that passes various statistical tests for randomness (most of which are based on the normal approximation of binomial probabilities, which are natural to use with binary strings).

Now, if presented with a finite segment of, say, a binary string, we have no way of knowing, prima facie, whether the string represents a substring of some random string (which we usually expect also to be of finite length). Perhaps the longer string was published by a quantum noise generator. Even so, inspection of the substring may well yield the suspicion that a nonrandom, deterministic process is at work.

Now if one encounters, absent a Turing machine (TM) executor or equivalent, the isolated run 0 0 0 0 0 0 0 0 0 0, one is likely to suspect nonrandomness. Why so? We are doubtless assuming that truly random binary digits occur with a probability of 1/2 and so our belief is that a run of ten zeros is too far from the expectation value of five zeros and five ones. Even so, such a run by itself may simply imply a strong bias for 0, but an otherwise nondeterministic process and so we need to filter out the effect of bias to see whether a process is largely deterministic.

We are much less likely to see the string 0101010101 as anything other than almost surely deterministic, regarding it as a strong candidate for nonrandomness. If we use independence of event (digit on space m) as a criterion, the probability of such a string is 2^(-10), or one chance in 1024 -- the same probability as for any other permutation. Still, such a quantification doesn't tell the whole story. If you predict the sequence 10011110101, you have one chance in 1024 of being right, but if instead you predict some sequence that contains six 0s and four 1s, then you'll find that the relevant set contains 210 strings, yielding a probability that you are correct of 210x2^(-10), or better than 20 percent.

So why do we regard 0101010101 as having a strong probability of nonrandom generation? Because it is part of a small subset of permutations with what I call symmetry. In this case, the symmetry accords with periodicity, or 10 ≡ 0 (mod 2), to be precise.

Is the sequence 100111010 likely to be the result of a random process. A quick inspection leaves one uncertain. This is because this particular string lacks any obvious symmetry. The first sequence is a member of a small subset of length 10 strings that are periodic or "nearly periodic" (having a small non-zero remainder) or that have other symmetries. Many strings, of course, are the result of deterministic processes and yet display no easily detected symmetry. (We might even consider "pseudo-symmetries" in which "periods" are not constant but are polynomial or exponential; some such strings might escape statistical detection of nonrandomness.)

Consider 01001001001

In this case we may regard the string as a candidate for nonrandomness based on its near periodicity of 10 ≡ 1 (mod 3), which includes three full periods.

Consider

0110101101011

The sequence is too short for a runs test, but we may suspect nonrandomness, because we have the period 01101, giving 13 ≡ 3 (mod 5).

We see here that strings of prime length have no periods of form a ≡ 0 (mod c), and hence the subset of "symmetrical" substrings is lower than for a nearby composite. So prime length strings are in general somewhat more likely to look nonrandom.

To get an idea of the smallness of the set of exactly periodic strings, observe that the binary string 101,XXX,XXX has 15 permutations of the last six digits, only one of which yields the periodic string 101101. By similar reasoning, we see that subsets of near-periodic strings are relatively small, as long as the remainder is small with respect to length n. It might be handy to find some particular ratio m/n -- remainder n over string length n -- that one uses to distinguish a union of a periodic and near-periodic subsets, but I have not bothered to do so.

Aside from periodicity and near-periodicity, we have what I call "flanking symmetry," which occurs for any string length. To wit:

0001000

or

0111110

And then we have mirror symmetry (comma included for clarity):

01101,10110

which is equivalent to two complete periods (no remainder) but with the right sequence reversing the order of the left.

We might try to increase the symmetries by, let's say, alternating mirror periods. But note that

0101,1010,0101 is equivalent to 010110,100101

and so there is no gain in what might loosely be called complexity.

Speaking of complexity, what of patterns such that g(n) is assigned to digit 0 and f(n) to digit 1 as a means of determining run lengths? In that case, as is generally true for decryption attacks, length of sample is important in successful detection of nonrandomness.

We may also note the possibility of a climbing function g(0,m) = run length x alternating with a substring of constant length y, each of which is composed ofpsuedorandom (or even random) digits, as in

0,101101,00,111001,000...

In this case, we have required that the pseudorandom strings be bracketed by the digit 1, thus reducing the statistical randomness, of course. And, again, sample size is important.

That is, though 010010001 has a whiff of nonrandomness, when one sees 010010000100001000001, one strongly suspects two functions. To wit, f(x) = k = 1 for runs of 1s and g(x) = next positive integer for runs of 0s. Though a human observer swiftly cognizes the pattern on the second string, a runs test would reveal a z score pointing to nonrandmoness.

So let us use the periodicity test to estimate probability of a deterministic process thus: For n = 20, we have the four aliquot factors, 2, 4, 5, 10. The permutations of length 2 are 00, 01 and their mirrors 11, 10, for 4 strings of period 2. For factor 4, we have 4C2 = 6, yielding, with mirrors, 12 strings of period 4. For factor 5, we have 5C2 =10, yielding, with mirrors, 20 strings of period 5. For factor 10, we have 10C2 = 45, yielding, with mirrors, 90 strings of period 10. So we arrive at 132 periodic strings, which gives a probability of one chance in 138,412,032 if we consider every period to be equiprobable. This concession however may sometimes be doubtful. And, the probability of nonrandomness is also affected by other elements of the set of symmetries discussed above. And of course there are standard tests, such as the runs and chi square tests that must be given due consideration.

Now consider

00000000001111111111

Why does this string strike one as nonrandom? For one thing it is "mirror periodic," with a period of 2. However, one can also discern its apparent nonrandomness using a runs test, which yields a high z score. The advantage of a runs test is that it is often effective on aperiodic strings (though this string doesn't so qualify). Similarly, a goodness of fit test can be used on aperiodic strings to detect the likeliness of human tweaking. And one might, depending on context, apply Benford's law (see http://mathworld.wolfram.com/BenfordsLaw.html ) to certain aperiodic strings.

So it is important to realize that though a small set of symmetrical strings of length n exists whose members are often construed as indicative of nonrandom action, there exists another small set of aperiodic strings of the same length whose members are considered to reveal traits of nonrandom action.

For example, a runs test of sufficiently large n would disclose the nonrandom behavior of Liouville's transcendental number.

Worthy of note is yet another conception of randomness, which is encapsulated by the question: how often does the number 31 appear in an infinite random digit string? That is, if an infinite digit string is formed by a truly random process, then a substring that might occur once in a billion digit spaces, would have probability 1 of recurring infinitely many times.

That is, in base 10, "31" reaches a 95% probability of a first occurrence at the 17,974,385th space. In base 2, "31" is expressed "111111" and a 95% probability of a first occurrence is reached at the 1441th space. Similarly, in an infinite string, we have probability 1 that a run of 10^(100^100) zeros will recur not just one, but infinitely often. Yet if one encountered a relatively short run of, say, 10^10 zeros, one would be convinced of bias and argue that the string doesn't pass statistical randomness tests.

The idea that such strings could occur randomly offends our intuition, which is to say the memory of our frequency ratios based on everyday empirical experience. However, if you were to envisions an infinity of generators of truly random binary strings lined up in parallel with strings stretching from where you stand to the horizon, there is prbability 0 that you happen upon such a stupendous run as you wander along the row of generators. (Obviously, probability 1 or 0 doesn't mean absolute certainty. There is probability 1 that Goldbach's conjecture is true, and yet perhaps there is some case over infinity where it doesn't hold.

Concerning whether "31" recurs infinitely often, one mathematician wrote: "The answer lies in an explicit definition of the term 'infinite random digit string.' Any algorithm for producing the digit string will at once tell you your answer and also raise the objection that the digit string isn't really random. A physical system for producing the digits (say, from output of a device that measures quantum noise) turns this from a mathematical into an experimental question."

Or, perhaps, a philosophical question. We know, according to Cantorian set theory -- as modified by Zermelo, Frankel and Von Neumann (and including the axiom of choice) -- that there is a nondenumerable infinity of noncomputable numbers. So one of these noncomputables, R0 -- that could only be "reached" by some eternal randomization process -- does contain a (countable) infinity of 31s. But this means there is also a number Ra that contains the same infinite string as R0 except for lacking all instances of the substring denoted 31. Of course, one doesn't know with certainty that a randomization process yields a noncomputable number. Ro might be a noncomputable and R1 a computable. Even so, such a procedure might yield a noncomputable R1. So we see that there is some noncomputable number where the string "31" never shows up.

If we regard Zermelo-Frankel set theory as our foundation, it is pretty hard to regard an element of the noncomputables as anything but a random number.

The next question to ponder is whether some pseudorandom aperiodic string mimics randomness well enough so that we could assume probability 1 for an infinity of 31s. Plenty of people believe that the pi decimal extension will eventually yield virtually every finite substring an infinity of times. And yet it is not even known whether this belief falls into the category of undecidable theorems. And even if pi could be shown to be "universal," there is no general way of determining whether an arbitrary program is universal.

No discussion of randomness can readily ignore the topic of Bayesian inference, which, despite a history of controversy, is used by philosophers of science to justify the empirical approach to establishment of scientific truth. Recurrence of well defined events is viewed as weight of evidence, with probability modified as evidence accumulates. Such thinking is the basis of "rules of succession, so that a sequence of, say, five 1s would imply a probability of (m+1)/(m+2) -- 86% -- that the next digit will also be a 1.

In this case, a uniform a priori distribution is assumed, as is dependence. So here, Bayesian inference is conjecturing some deterministic process that yields a high degree of bias, assumptions which impose strong constraints on "background" randomness. (Interestingly, Alan Turing used Bayesian methods in his code-breaking work.)


A longer discussion of Bayesian ideas is found in the appendix of my essay, The Knowledge delusion, found at http://kryptograff5.blogspot.com/2011/11/draft-03-knowledge-delusion-essay-by.html

Monday, May 7, 2012

In death's borderland

By PAUL CONANT
--------------------------------------------------------------------------------
Time: 12:30 a.m. to 6:30 p.m. Sept. 12, 2001.
Place: In the Manhattan buffer zone uptown from the twin towers catastrophe.
---------------------------------------------------------------------------------
When I first arrived in the buffer zone between 14th and Houston streets, surrealistic scenes greeted me.

Police cars in motion covered with ash and dust; a convoy of giant earth movers filled with skyscraper rubble; emergency rescue vehicles on unspecified missions.

No one was afoot except for me and a few drunks, addicts and homeless persons.

At the key intersection of Houston and 6th Av. (also known as the Avenue of the Americas) I shared a bench with a homeless woman, watching as emergency vehicles came and went, convoys of dump trucks were deployed and city buses ferried police, firefighters, volunteers and construction workers in and out of the death zone.

I wandered up and down East Houston, noting the trucks laden with scaffolding parked and ready to roll. I stood on a footbridge over FDR Drive watching streams of emergency vehicles, some marked, some not, some with lights flashing and sirens blaring, some not, streaming in and out of Houston Street or heading around the FDR curve to approach the disaster from the toe of this forever-changed island.

In my wanderings, I frequently came across homeless men sleeping fitfully on sidewalks and loading docks, in jarring contrast to the more than 10,000 dead buried a few blocks away. Yet, I noticed around 4 a.m. that most residences in the buffer zone had all lights out, so I presumed that many New Yorkers must have simply gone to bed.

All I could think when watching the emergency activities was that New York should be glad of such efficiency and cool-headedness in response to this outrage.

Once dawn came, I saw groups of professionals hustling off toward West Street (aka the West Side Highway), apparently on their way to work.

Overheard snatches of conversation:

"We all know somebody who is dead," said a woman striding along with two men.

"We had a very, very close friend who was on the 92d floor," says a bearded man into a cellphone.

Cellphones of course were ubiquitous. People at Houston and 6th were using them to report details of what they were seeing. One man sitting on the by-now packed bench was reading his notes in French, most probably to an editor at the other end of his cellphone.

Channel 3 News from Hartford encamped at the intersection at about 5 a.m. and all day long conducted TV interviews with New Yorkers who had been at or near the catastrophe site.
By 10 a.m., the pace was picking up, as more and more New Yorkers ventured out, looking for newspapers (none delivered in the buffer zone), visiting neighbors and just plain looking around.

But it was eery. A perfect summery day. The residents of the buffer zone were perhaps defiantly nonchalant. Those in the streets showed no trace of fear, spoke animatedly to one another and played with their children, doing their best to enjoy a very bad day. And they somehow were succeeding.

If the point was to terrorize the New Yorkers, I can tell you they were not at all terrorized. New Yorkers were well behaved. A few onlookers were a bit of a pain at the key intersections, but when one considers the number of people in New York, things went very well. And the harried police handled the onlookers good naturedly.

Among the contrasts:

Throngs of curious Manhattanites near the death zone acting as if it was a nice day off (and that feeling of confidence was quite clearly contagious); yet every once in a while silent rescue workers, individually and in small groups, would trudge past the police checkpoint and walk uptown. You knew who they were even if they were not in uniform. Their footgear was covered in ash. They trudged, stonefaced, staring straight ahead, overcome with exhaustion, both physical and emotional.

As a Vietnam veteran, I could identify with them somewhat. Once past the checkpoint, many in the crowds failed to notice them, but of course that didn't really matter.

Over on West Street, crowds made a point of cheering and applauding the rescue workers as they drove to and from the death zone. Many news trucks lined West Street. It gave one of the clearest views in Manhattan of the 'hole' in the skyline.

I couldn't help but wonder: is it wise to put so many people in one place? Do we really need skyscrapers anyway in this new economic era of computer teleconferencing?

I recall seeing a family walking their children up the bikepath, their little girl racing along gaily, clutching her dolly -- completely oblivious to the tower of smoke billowing up behind her.
Similar scenes played out in Washington Square Park where parents supervised toddlers laughing an giggling under the turtle sprinkler.

Looking downtown, three crosses atop churches abutting the park stood out in stark relief against the heavy pall of smoke.

I heard a helicopter whipping high overhead and was watching it for a while before I realized: I had heard it because there was no traffic making the usual Manhattan cacophony around the park.

There just wasn't enough noise in the air.