The Anti-Defilement Covenant

In Garry Wills on Abortion Rights and Is the Bible Anti-Abortion?, I talked about a convincing argument I found in a recent book, Head and Heart: American Christianities by liberal Catholic historian Garry Wills, to the effect that an abortion in the first two trimesters of pregnancy does not amount to the murder of a human person.

A key part of Wills' line of reasoning was that the Bible doesn't really prohibit abortion, not specifically. In fact, the Old Testament book of Numbers (verses 5.11-27) has God telling Moses:

If any man's wife go aside, and commit a trespass against him,
And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and there be no witness against her, neither she be taken with the manner; And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:
Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth part of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it is an offering of jealousy, an offering of memorial, bringing iniquity to remembrance ... and the priest shall have in his hand the bitter water that causeth the curse: And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness with another instead of thy husband, be thou free from this bitter water that causeth the curse: But if thou hast gone aside to another instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband ... Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell; And this water that causeth the curse shall go into thy bowels, to make thy belly to swell, and thy thigh to rot: And the woman shall say, Amen, amen ... And when he hath made her to drink the water, then it shall come to pass, that, if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.

Today, we would say the priest is coercing this hypothetically "defiled" woman to use a herbal abortifacient ("bitter water") to terminate her pregnancy! All because the woman's "defilement" has been cast as a "trespass" that has been committed against ... her husband!

In other words, the woman's sexuality, fertility, and reproductive capacity here belong to her husband, not to her. When she is "defiled" (actually, or according to her husband's suspicions) it is as if her (suspected) illicit lover has stolen property ... from the husband. This, the worst possible sort of transgression against the husband, demands that the woman (not the man she has supposedly "lain with") be punished in a most extreme way: "the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people."


The society in question was a patriarchal one in which a female's reproductive capacities are originally owned, when she is still a child, by her father. We still say, do we not, that when a woman is married, her father "gives her away" to her new husband.

In such a view of womanhood, the menfolk of the tribe or society participate in a sort of communal "anti-defilement covenant." Together, the men control each and every woman's sexuality from the cradle to the grave.

According to such a covenant, a woman's sexual chastity is a precious possession, to be guarded at all costs by its possessors: the men. The men collaborate to set up the rules of conduct and official ceremony by which this can reliably be done. In the case of the ancient Hebrews, the rules were said to have come to Moses from God.

The underlying notion is, again, that a woman's sexual and reproductive identity belongs to her husband, her father, her brothers (whose job it is to avenge any adultery or rape committed by or against her), and the tribe's menfolk in general (who are adjured, for the sake of domestic tranquility, to rein in their natural tendency to want to "lie with" this woman or that, and with the other as well).

Looked at in one way, this kind of anti-defilement covenant puts every woman on a pedestal. That doesn't sound so bad. But looked at another way, it makes her into a mere object, a container of fertility by means of which her legitimate husband can reliably secure his own posterity.

This patriarchal idea of womanhood is so crucial to the mores of the Bible that womanhood's "defilement" becomes perhaps the only imaginable God-sanctioned rationale for abortion. On the other hand, an undefiled woman who is pregnant by her husband presumably must never have an abortion, for in so doing she is robbing her husband of his posterity.


An interesting aspect of this patriarchal setup is that it may also explain why the Old Testament comes down so hard against homosexuality.

On the one hand, it might be thought that any man who prefers other men to women would be, as one who is not at all inclined to "defile" the womenfolk, welcomed.

But, as a moment's consideration shows, such a man cannot really be part of the covenant, for the covenant is all about the sacrificial obligations men undertake vis-à-vis other men. A man bound by the covenant is giving up something big: his chance to sow his wild oats. He's making a huge sacrifice. (Perhaps this is why circumcision, the painful sacrificing of foreskin tissue at the tip of the penis, betokens membership in the covenant.)

So a shared sacrifice is what binds the covenanters indissolubly together. A homosexual man is making no real sacrifice by agreeing not to "lie" indiscriminately with the womenfolk. That's not very strong glue, by any means.


We who are Christians or Jews today inherit the ideas about womanhood that were inherent in the original, patriarchal covenant of anti-defilement, and transmitted to us via our Bible and our religious doctrines. But these ideas aren't a good match with our own up-to-date, feminist-inspired notions of womanhood. Feminism and women's liberation have taught us that a woman's sexual and reproductive capacities are her own, to do with as she sees fit.

In a society such as ours, where abortion-on-demand has recently (1973) been made legal, is it any wonder that religious conservatives are absolutely opposed to Roe v. Wade? The idea that a woman can, on her own, choose to have an abortion — or not — is diametrically opposed to the ancient understanding of womanhood implicit in the anti-defilement covenant. According to that understanding, a woman's fertility belongs to the menfolk in her tribe. Only they can adjudicate when an abortion is necessary.

And it is likewise no surprise that religious conservatives are equally adamant that homosexuality will not be tolerated — much less blessed by allowing gays to marry.

It all goes back to ancient ideas about how a tribe or society ought to treat the generative capacity of women: as something belonging to, and in turn demanding costly sacrifice from, men.

Is the Bible Anti-Abortion?

As I detailed in Garry Wills on Abortion Rights, the recent book Head and Heart: American Christianities by liberal Catholic historian Garry Wills argues, in one important section, against the "pro-life" stance of today's Evangelical Protestants and conservative Catholics and Jews. After asserting that there is no authoritative text in the Bible in which God specifically forbids abortions, Wills makes a case that an abortion in the first two trimesters of pregnancy does not amount to the murder of a human person.

Absent definite scriptural guidance, we must fall back on philosophical reasoning about "natural law," Wills says. Given what we now know about fetal development, we can be certain a fetus possesses no high, cognitive brain function until neural synapses form in the brain, at a point twenty-five to thirty-two weeks into pregnancy. Before that juncture — which coincides with the onset of the ability of the fetus to survive outside the womb — the fetus is manifestly not "capable of thought, or of speech, of recognizing itself as a person, of assuming the responsibilities of a person" (p. 528).

Hence, a fetus during the first six months of pregnancy does not have the "intellectual soul" which Thomas Aquinas felt, writes Wills, "is directly created by God 'at the end of human generation' (in fine generationis humanae)" (p. 526).

In modern language, "personhood" as Aquinas defined it doesn't begin until late in pregnancy. Prior to the emergence of personhood, the fetus does not have a soul "directly created by God."


It's a convincing argument — one that I imagine (based on an endnote citation) comes from a book by Daniel A. Dombrowski and Robert Deltete called A Brief, Liberal, Catholic Defense of Abortion, which I intend to read.

But the argument depends on there not being a specific proscription of abortion in the Bible, which is a topic Wills expounds in what is perhaps his lengthiest endnote. I'm going to replicate that endnote here:

Though Jewish tradition did not draw any teaching on abortion from Scripture, modern Christians who believe in what they call the Old Testament try to dredge up something on the subject. Here are some of the desperate expedients.

Psalm 139.13-16 [actually, the verses cited are 13-15]:

For thou hast possessed my reins;
thou hast covered me in my mother's womb.
I will praise thee; for I am fearfully and wonderfully made:
marvellous are thy works;
and that my soul knoweth right well.
My substance was not hid from thee,
when I was made in secret,
and curiously wrought in the lowest parts of the earth.

This says only that God foreknows everything, every stage of a thing's coming into existence — even its primordial patterning in the lowest parts of the earth (whatever that means). That this is not a statement of when a person actually comes into being is seen from Jeremiah 1.5: "before I formed thee in the belly I knew thee."

Exodus 21.22: "If men strive, and hurt a woman with child, so that her fruit depart from her, and yet no mischief follow: he shall be surely punished, according as the woman's husband will lay upon him; and he shall pay as the judges determine."

The penalty is not for murder (which calls for the death penalty), but a mere fine to the husband for the possibility of losing an heir. This says nothing of the present status of the fetus, only of the future profit to the male. The woman is not treated as having any stake in the matter. Only the man.

Genesis 38.24-26:

[Wills does not actually give the quote, which is: "And it came to pass about three months after, that it was told Judah, saying, Tamar thy daughter in law hath played the harlot; and also, behold, she is with child by whoredom. And Judah said, Bring her forth, and let her be burnt. When she was brought forth, she sent to her father in law, saying, By the man, whose these are, am I with child: and she said, Discern, I pray thee, whose are these, the signet, and bracelets, and staff. And Judah acknowledged them, and said, She hath been more righteous than I; because that I gave her not to Shelah my son. And he knew her again no more."]

[Wills paraphrases:] Tamar is brought to be executed, but she is pregnant and is spared. But she is not spared because she is pregnant — that was known when she was condemned to death. She is spared because she produces proof that the father is the head of the tribe, Jonah.

[Then Wills adds:] Actually there is one passage in the Jewish texts that shows God himself inducing an abortion, Numbers 5.11-26 (the Lord is speaking):

[Again, Wills omits the actual text, which (including verse 27) is:]

[And the LORD spake unto Moses, saying,
Speak unto the children of Israel, and say unto them, If any man's wife go aside, and commit a trespass against him,
And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and there be no witness against her, neither she be taken with the manner;
And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:
Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth part of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it is an offering of jealousy, an offering of memorial, bringing iniquity to remembrance.
And the priest shall bring her near, and set her before the LORD:
And the priest shall take holy water in an earthen vessel; and of the dust that is in the floor of the tabernacle the priest shall take, and put it into the water:
And the priest shall set the woman before the LORD, and uncover the woman's head, and put the offering of memorial in her hands, which is the jealousy offering: and the priest shall have in his hand the bitter water that causeth the curse:
And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness with another instead of thy husband, be thou free from this bitter water that causeth the curse:
But if thou hast gone aside to another instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband:
Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell;
And this water that causeth the curse shall go into thy bowels, to make thy belly to swell, and thy thigh to rot: And the woman shall say, Amen, amen.
And the priest shall write these curses in a book, and he shall blot them out with the bitter water:
And he shall cause the woman to drink the bitter water that causeth the curse: and the water that causeth the curse shall enter into her, and become bitter.
Then the priest shall take the jealousy offering out of the woman's hand, and shall wave the offering before the LORD, and offer it upon the altar:
And the priest shall take an handful of the offering, even the memorial thereof, and burn it upon the altar, and afterward shall cause the woman to drink the water.
And when he hath made her to drink the water, then it shall come to pass, that, if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.]

[Wills does provide this interpretation:]

If a woman is suspected of infidelity, her husband must bring her before a priest, who will make her drink "bitter water" into which he has mingled cursed (23). "If she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people" (27).

I don't know whether to call this last example "God himself inducing an abortion," so much as God calling for the priest to induce one, using "bitter water" as an abortifacient. But never mind. It does seem to show that God in that period of history and among those people ordered an illicit fetus of a "defiled" woman to be aborted, does it not?

Garry Wills on Abortion Rights

Garry Wills' Head and Heart: American Christianities is a new book about the many strands of (mostly Protestant) religion in American history. Wills is a liberal Catholic historian with many popular books to his credit, including Why I Am a Catholic.

Head and Heart posits that there are two main strands of American Protestantism: Evangelical and Enlightened. Though at times they have overlapped and cross-pollinated, the former is characterized by a personal, passionate piety, and the latter by a reasoned, philosophical approach.

The two strands didn't really start to separate until the 18th century, during the pre-Revolutionary and then the Revolutionary eras. Going all the way back to the early 1600s, the New England colonies, especially Massachusetts, were home mostly to Puritans, who were Congregationalists officially and Calvinists theologically. Though many of their beliefs and practices sowed the seeds of Evangelical outlooks yet to come, the Puritans were by and large also highly educated and committed to an intellectual approach to religion.

So were the mainly Anglican (Church of England) denizens of Virginia, another leader among the original thirteen colonies. Though these Virginian forerunners of today's American Episcopalians had a different religious outlook from that of the Puritans, they too combined personal piety with reasoned discourse.

The Virginia Anglicans were, however, notably more "liberal" in their outlook than the original Puritans, who imagined that only certain human individuals were predestined to be saved — or "regenerated" — and thus find a heavenly reward. The Puritans feared that even the regenerate among them could be deceived by Satan's wiles and consequently fall away, so they spent a lot of time worrying about the Quakers, Baptists, and supposed "witches" who were their close neighbors and might be agents of the devil.

The Puritans' corporate piety was thus highly normative; dissenters were dealt with harshly. Though there was officially what we would today call "separation of church and state" in the original Puritan colonies, the good offices of local government more often than not functioned to allay the Puritans' religious fears. Unorthodox fellow Puritans and others who had radically different ideas were often silence, exiled, or executed.


Fast forward to the early 21st century. We now have Evangelical Christians pillorying not just atheists and "secular humanists" but also liberal Christians, whom Wills sees as the heirs of the Enlightenment Protestants of the America of the mid-19th century: the Transcendentalists, as represented in Wills' book mainly by Ralph Waldo Emerson.

The Transcendentalists equated their version of Protestant Christianity with a sort of "nature mysticism." Their God was a wonder-of-it-all Creator who did not need to reveal himself through Scripture, or to work miracles. To Emerson, the traditional, Biblical idea of "miracle" was "not one with the blowing clover and the falling rain," and so was a false notion. "Emerson had claimed that all the universe is the only real miracle ... " (p. 264).

Nor was the God of Emerson triune. The Transcendentalists jettisoned the doctrine of the Trinity — as did their intellectual forebears among the Founding Fathers: Jefferson, Madison, and John Adams, among others. The great minds who conceived America were mainly Deists and Unitarians, no matter their nominal (Protestant) denominations.

Those two terms, Deist and Unitarian, were originally synonyms, says Wills. They both referred to a Protestant outlook that was wedded to the notion that our One God has to be uniquely divine ... and so it is wrong to say that He is Three Persons, Father, Son, and Holy Ghost.

Jefferson, Madison, Adams, and contemporary Unitarian/Deist children of the Enlightenment were responsible for that touchstone of religious freedom in America, the portion of the First Amendment that bars the government from "establishing" any one particular religion or denomination as supreme over all others, or from interfering with citizens' "free exercise" of personal conscience in choosing which religion, if any, to follow.

Jefferson in particular, Wills shows, felt that if the government carefully avoided channeling the religiosity of its citizenry, citizens would naturally gravitate to the sort of Unitarian Deism he espoused. But Jefferson was wrong about that. There had already been in America a "Great Awakening," which reversed some of the liberalizing trends of the early 1700s, and a "Second Great Awakening" would occur in the wake of the Emersonian liberalizing of religion in the 1800s. There were Culture Wars even then.


The Culture Wars today are, Wills says, about a lot of things: school prayer, stem cell research, homosexual marriage, sex education, Darwinism, pornography, and on and on. But the centerpiece of the Evangelicals' fight against — well, against the rest of the culture — concerns abortion.

A woman's right to choose an abortion during the first two trimesters of pregnancy was established by the Supreme Court in 1973, in the case of Roe v. Wade. The Court held that the states' erstwhile laws prohibiting abortion were unconstitutional because they violated a woman's "right to privacy," a right said to be latent in the language of the U.S. Constitution as duly amended.

Ever since the Roe decision made abortion legal, Evangelical Christians and many Catholics — together, they form the heart of the "pro-life" movement — have sought to roll Roe back, and to make it possible for the states to once again ban most or all abortions.

Wills shows that the political success of our current president, George W. Bush, was engineered by a close adviser, Karl Rove, by means of uniting Evangelical Protestants with pro-life Catholics, and also equally pro-life Jews, and getting them to vote for the candidate most likely, as president, to appoint Supreme Court justices who would reverse Roe.

As such, says Wills opposition to abortion rights has become "the ecumenical issue" that today suffuses American politics with one particular style of religion, the Evangelical style, writ large to include Catholics and Jews (see pp. 523ff.) Except, Wills demonstrates, abortion is not really a "religious" issue at all.

Wills' argument to this effect is a deft one — and one with which I mostly concur — and I would like to spend some time dissecting it in this post and posts to come.


One of the main points Wills makes in his argument in favor of legal abortion rights is that there is no authoritative passage either in Jewish Scripture (the Old Testament) or in the New Testament that conveys God's opposition to abortion. Though one of the Ten Commandments forbids killing, or the doing of murder, there is no purely scriptural reason to assume that abortion qualifies, in God's eyes, as murder.

After making this argument about Scripture, which I will go further into in a future post, Wills goes on to demonstrate that St. Augustine, Thomas Aquinas, and the Christian Church in general have traditionally addressed abortion as a matter of "natural law," not revealed theology ... and so we today, to settle our disputes about the morality and legality of abortion, "must turn to reason and science, the realm of Enlightened religion" (p. 527).

Because the question is one of natural law, it is a "misconception" to say "that this is a religious issue, that the pro-life advocates are acting out of religious conviction" (p. 526; italics in the original).

Once Wills establishes that the question of abortion must be addressed as a matter of reason rather than scriptural revelation, he does just that. First, he shows it to be a misnomer to call the anti-abortion position "pro-life," since there is no principled way to show that the "life" of an embryo is any more sacred than the "life" of growing human hair or fingernails, semen, egg cells ... or, tellingly, the vast majority of fertilized eggs/human embryos! Most embryos naturally fail to get embedded in a womb wall, or, if they do, result in early miscarriages that are perceived to be no more significant than heavy menstrual flow:

Are these millions of embryos that fail to be embedded [in the wall of a womb] human persons? Then "intelligent design" [the doctrine about God's role in evolution supported by many pro-life Evangelicals] aborts far more persons than nay human abortioners can. God is responsible for this silent holocaust. (p. 528)

Then Wills addresses what he says the true question ought to be: not when human "life" begins, but when during the gestation period the human person emerges:

Is it when it is capable of thought, or of speech, of recognizing itself as a person, of assuming the responsibilities of a person? Is it when it has a functioning brain? Thomas Aquinas said that the fetus did not become a person until God infused the intellectual soul. A functioning brain is not present in the fetus until the end of the sixth month at the earliest (what Roe called the beginning of the third trimester). Only then can the cerebral cortex process information from the various senses. (p. 528)

Wills cites widespread medical opinion that it is only at about the point in gestation at which the fetus becomes viable outside the womb — roughly, the start of the third trimester of pregnancy — that the already abundant nerve cells present in the fetus' cerebral cortex become linked together through the formation of synapses: the gaps between nerve cells that impulses travel across during higher brain functioning.

Before that juncture, Wills argues, the life of the fetus — though the fetus does respond to external stimuli — is merely "vegetative," in the sense that Terri Schiavo was in a "persistent vegetative state" at the time she was taken off life support after fifteen years of never waking up.

Aquinas, intimates Wills, would have held that a soul in such a state was one of the "nutritive (vegetable)" variety, and not even of a "sensing (animal)" sort — much less an "intellectual soul" of the type God manifestly cherishes (see p. 526).

Even plants react to stimuli such as sunlight, etc. Animals have their own well-developed sensory lives. But only human persons have the full-fledged "intellectual soul" that was identified in Christian thought by Thomas Aquinas. And human fetuses, prior to the establishing of full brain function and the consequent onset of fetal viability outside the womb, presumably lack an "intellectual soul."

Hence, says Wills, abortion prior to the third trimester does not equal the killing of a human person.

Michael Shermer: "The Mind of the Market" | Scientific American

Michael Shermer writes "The Skeptic," a regular column in Scientific American. In February 2008's "The Mind of the Market," Shermer discusses why the seeming irrationality of our financial decision-making — we are anything but rational in any purely abstract sense — is really much more sensible than it is often given credit for being.

A scientist puts you in a game with another individual in which the scientist offers you alone $100 and tells you to offer any part of it to your counterpart in the game. If he or she accepts your offered amount, you will receive the $100 and dole out the agreed portion of it to your counterpart. But if he or she refuses your offer, neither of you gets a penny.

How do you decide how much to offer your counterpart?

You might think this way: my counterpart is surely a rational individual who should accordingly be satisfied with any pittance of a sum I'm willing to part with. After all, he (or she) would rather have a tiny amount of free money than no free money at all, right?

But, no. It turns out that most players in the counterpart role won't settle for less than, say, $30 of the original $100.

Why not? Because it offends their (and our own) built-in notion of fairness that the split should be any less than 70-30.

Why does it make sense to be fair, and to expect fairness in return? Because (a) we know we'll be dealing again and again with our various societal counterparts, not just on a one-shot basis, but continually, and also (b) a conscious awareness of the fairness imperative has been programmed into us by evolution.


The key thing here is that we humans actually have a built-in idea of fairness. It's not just a learned-through-culture add-on. Our primate relatives, experiments show, have one too! So we and our hairier cousins share a capacity for fairness, because evolution gave it to us. Shermer:

Just as it is a myth that evolution is driven solely by “selfish genes” and that organisms are exclusively greedy, selfish and competitive, it is a myth that the economy is driven by people who are exclusively greedy, selfish and competitive. The fact is, we are ... selfish and selfless, cooperative and competitive. There exists in both life and economies mutual struggle and mutual aid. In the main, however, the balance in our nature is heavily on the side of good over evil. Markets are moral, and modern economies are founded on our virtuous nature. The Gordon Gekko “Greed Is Good” model of business is the exception, and the Google Guys “Don’t Be Evil” model of business is the rule. If this were not the case, market capitalism would have imploded long ago. 

Here is an area, then, where science and religion chime together to challenge philosophical attitudes of the past which held that human beings are, at base, selfish and venal. Religion has long claimed that we are destined to have angels' wings. Now, evolution science can confirm that a sense of fairness and morality is our birthright. But much in the modern history of philosophy has claimed the opposite.

The 18th-century Scots philosopher David Hume, according to Arthur Herman's How the Scots Invented the Modern World: The True Story of How Western Europe's Poorest Nation Created Our World & Everything in It, was a religious skeptic who insisted that:

... self-interest is all there is ... [so] a secular Golden Rule: I won't disturb your self-interest, if you don't disturb mine [is] the best we can hope for ... [since] morality is largely a matter of convention and ingrained habit [,] the laws of nature offer nothing to help, and appeals to reason fall on deaf ears ... . (pp. 200-202)

Hume made no allowance for Darwin's theory of evolution, which was more than a century off when he published A Treatise on Human Nature in 1734, or for how that theory might be extended in the 21st century to show how the laws of nature are perfectly capable of generating a species (or more than one) with an inbuilt moral sense.

So much so, in fact, that our species can be criticized for not being perfectly rational when it comes to money matters. Our innate sense of fairness sometimes trumps our "reason."

(Another, similar article by Michael Shermer can be read here.)

What I.Q. Doesn't Tell You About Race

My title for this piece is the subtitle of "None of the Above," Malcolm Gladwell's review of What Is Intelligence?, a new book by James Flynn, a social scientist at the University of Otago, New Zealand. The book and the Gladwell review deal with "the Flynn effect," discovered over 20 years ago by Flynn himself. Flynn noticed in 1984 that, according to Gladwell, "I.Q.s around the world appeared to be rising by 0.3 points per year, or three points per decade, for as far back as [intelligence] tests had been administered."

Many social scientists have long thought I.Q. ("intelligence quotient") reveals something innate and largely genetic in each individual human being's intrinsic mental capacity, something "real" which I.Q. tests supposedly show is larger in some of us than in others of us. True, the nurture we get as infants and the environment we are brought up in contribute heavily to our person-to-person I.Q. differences, for better or for worse. Yet we each supposedly start out life with an immutable mental potential that varies with our heredity. If we have the right genes, we can be smart. If not, we're average — or, worse, we're morons, idiots, imbeciles.

Thus, when intelligence tests in the first half of the 20th century showed significant I.Q. differences among various races and ethnic groups, those groups who scored lower on average were deemed genetically inferior. The people Gladwell calls "I.Q. fundamentalists" believed — as many still do — that the genes we inherit by virtue of our race put brackets around our potential for mental achievement, as measured by I.Q. tests.


I.Q. fundamentalism is something like religious fundamentalism as it applies to Darwin's theory of evolution and the origin of species. Creationists counter Darwin by asserting that natural selection might be able to account for some degree of variation within fixed species established by the divine creator, but it isn't powerful enough to make new species. I.Q. fundamentalists say the influence of environmental factors might be able to swing I.Q. up or down around a point fixed either high or low by heredity or race, but it can't turn a dope into a mental giant.

But that's what the Flynn effect seems to show: the passage of time and the succession of generations can turn below-average I.Q.s into above-average I.Q.s, and above-average scores into genius-level scores. That's why the I.Q. tests have to be constantly "renormed" (made harder) to hold the average score at 100.

How could time and generational succession, occurring over far too few years for genetic improvements to take root and make a difference, make people smarter? What sort of cultural evolution accounts for it?


Flynn and Gladwell say the question itself is flawed. We are not getting smarter in any fundamental sense. Rather, we are getting more modern. Gladwell: "An I.Q. ... measures not so much how smart we are as how modern we are."

I.Q. tests don't really measure intelligence, then. They measure the extent to which our cultural milieu has fashioned what Gladwell calls "scientific spectacles" for us and taught us to put them on.

When the Flynn effect is broken down into subcategories of mental functioning, it is the category of I.Q. test questions called "similarities" in which we beat the pants off our fathers, and especially our grandfathers. Questions in this category ask us to say what goes with what, and why — as in "In what way are dogs and rabbits alike?" The "right" answer is supposedly the "taxonomic" one: dogs and rabbits are both mammals. "A nineteenth-century American," though, says Gladwell, "would have said that 'you use dogs to hunt rabbits.' "

For a like reason, when I.Q. tests were given to members of the Kpelle tribe in Liberia, the self-styled "wise" of the tribe would — wrongly, per the official answer book — sort potatoes (a food) with knives (a tool), because you use the latter to cut the former. Only when asked how a "fool" would answer the same questions did the Kpelle sort potatoes with other foods and knives with other tools.

The Kpelle, as members of a traditional society, did not wear "scientific spectacles" and automatically assume that a "taxonomic," category-based answer was best. Nor did our nineteenth-century American ancestors, unless they were in the minority who were highly educated. Today, college degrees and "scientific spectacles" are more prevalent ... and people generally have higher I.Q. scores (unless, that is, the tests are renormed).


But not all Americans have benefited to an equal extent from the Flynn effect, or at an equal pace. In the last century, various immigrant groups (Italians, Eastern Europeans, Chinese, Hispanics, and others) were called inferior for failing to score as high on intelligence tests, on average, as people of Northern and Western European descent. The gaps were erased — and then some, in the case of Asians — in succeeding generations.

Except, sadly, in the case of African Americans. Though blacks narrowed the I.Q. gap during the 25 years following World War II, "that trend stalled" over the subsequent quarter century, Gladwell notes. Though, Gladwell adds, "more recent data showed that the race gap had begun to close again," the gap still perplexes social scientists and encourages I.Q. fundamentalists to deem blacks inferior.

But, Flynn shows, blacks start out life with I.Q. parity, according to the (admittedly crude) tests measuring infant cognitive functions. By age four, blacks' I.Q.s are on average just a tad behind whites', at 95.4 to whites' 100. But by age 24, the black shortfall has widened to 83.4.

It is a slew of adverse environmental factors that account for this, Flynn shows, by keeping African Americans from learning to put on the "scientific spectacles" of the modern American mind. There's no reason to believe blacks aren't just as smart, but they don't necessarily end up filtering the world through the same cognitive categories as the supposed mental giants among us do.

Miracles and Mini-Miracles

The End of Certainty by Ilya Prigogine

In A God of Beckoning? I embroidered upon the science laid out in Ilya Prigogine's book The End of Certainty: Time, Chaos, and the New Laws of Nature a theological theory, if you will. It was a theory of mini-miracles. Where Prigogine saw occasions of pure random chance sparking evolutionary changes in living systems in the natural world, I saw opportunities for God to work little miracles.

Every time one of Prigogine's so-called dissipative systems — all living systems are dissipative systems — reaches a crossroads in the course of its development, a little miracle can occur. Instead of blind luck being responsible for which future path the system is going to take out of its present instability, God can beckon and the system can freely follow ... or not. God does not coerce compliance.

If the system indeed responds positively to God's call to it, it is as if it is echoing Mary, the mother of Jesus, when she answered the angel Gabriel at the Annunciation told of in Luke's Gospel. Mary assented by saying, "Behold, I am the handmaid of the Lord; be it to me according to your word." No coercion there.

Prigogine's pure science contains no such theological speculations. And it applies equally well to other types of systems subject to motion and change, including the moving, roiling particles that make up atoms, as understood today by quantum physics. Quantum objects, too, are supposed to be probabilistic with respect to the odds of whether certain events occur or not. I'd say they, too, are candidates for mini-annunciations and mini-miracles.

Likewise, the systems studied in the branches of physics called thermodynamics and chaos theory — not to mention in the theory of classical dynamics inherited from Newton — are aptly described by Prigogine's mathematics. These as well can receive and assent to mini-annunciations, in my theory, and be the occasions of mini-miracles.

All these types of systems have in common the fact that they change irreversibly over time. This means they evolve. Evolution is a word we typically associate with the origin of biological species, à la Darwin. Some believers in God feel the theory of evolution works against religion. In my view, there is ample reason to believe that the evolutionary history of living systems and of the several other kinds of systems subject to Prigogine's mathematical descriptions amount to a skein of mini-annunciations assented to, making for mini-miracles without number. All that is necessary to bridge theology to science is an image of every so-called "chance event" as being an opportunity for a mini-annunciation and a mini-miracle to occur.


It seems to me all of a sudden — I don't think I really believed this before — that a belief in miracles is indispensable to a belief in God. Paul Greenberg, editorial page editor of the Arkansas Democrat-Gazette, confirms this in his op-ed piece "Much revealed in telling of the Hanukkah story."

The Jewish festival of Hanukkah which began last Tuesday night, and which Greenberg discusses, might be seen as a memorial of a military victory of the Jews over Seleucid Greek hegemony some 2,200 years ago, but Greenberg brings out the fact that what it really celebrates is a miracle. After the Jews' victory, there didn't seem to be enough oil to light the lamps of their Temple in Jerusalem, and keep them lit until a new oil supply could be prepared. But, no; a seeming one-day supply lasted fully eight days: the "miracle of the lights."

Greenberg explains it thus:

In the glow of the candles, the heroic feats of the Maccabees [in defeating their non-Jewish enemies] have become transmuted into acts of divine intervention. The blessing over the candles recited each night of the holiday goes: "Blessed art Thou, O Lord our God, King of the universe, who wrought miracles for our fathers in days of old." Miracles, not victories.

... The victory is to be celebrated not for its own sake but for what it reveals.

One more violent confrontation has been lifted out of history and enters the realm of the sacred. A messy little guerrilla war in the dim past of a forgotten empire has become something else, something that partakes of the eternal. ... [This] may be the greatest miracle of Hanukkah: the transformation of the oldest and darkest of human activities, war, into a feast of illumination.

... If there is one, unchanging message associated with this minor holiday magnified by time, it can be found in the unchanging portion of the Prophets designated to be read for the Sabbath of Hanukkah. It is Zechariah 4:1-7, with its penultimate verse:

Not by might, nor by power, but by My spirit, saith the Lord of Hosts.

The Spirit of God is revealed not by might, nor by power ... but by miracles. The candlelight which by ordinary reckoning would last only a day instead lasts a week and a day. A world which by ordinary reckoning hasn't a prayer of evolving animals with God-blessed souls does so anyway. Who is to say that all such miracles are not skeins of mini-miracles? Such occasions of surprise would not defy the laws of nature at all, except that an event which by ordinary reckoning is the product of blind chance would actually be in a long series of nature's numberless assents to God's ceaseless mini-annunciations?

A God of Beckoning?

The End of Certainty by Ilya Prigogine

The late Nobel laureate Ilya Prigogine's 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature reformulates the mathematics of physics to show an irreversible arrow of time. Prigogine focuses on systems made of objects in motion — objects as large as planets, stars, and galaxies and as small as molecules, atoms, and the subatomic particles studied in quantum physics. He shows that the objects' changing locations — their movements — are describable as the shifting relationships of mathematical waves.

When waves are superposed atop one another, they may cancel each other out, or they may reinforce one another. When waves of closely related wavelengths reinforce one another in just the right way, they create resonances. Prigogine shows how resonances in nature make for instability, first, and then new order.

When a violinist bows a string, the body of the violin resonates in sympathy with the frequencies at which the string vibrates in air — the fundamental frequency of the note being played along with all its harmonic overtones — to make music. When resonances occur in natural systems, they provide opportunities for new and more complex order to arise. I discussed these ideas about resonances in more detail in Chance ... or God? and in The World as Music.


Resonances in nature's dynamical systems — systems made of moving objects and thus undergoing change over time — are associated with instabilities. In the "phase space" representing all the possible states a system may visit during the course of its journey in time, at those points where resonances occur there is chaotic unpredictability. Certain of the systems which interest Prigogine most are "dissipative" ones; imbibing energy and exporting waste, they hold themselves far from thermodynamic equilibrium. These dissipative systems include all living organisms. When these far-from-equilibrium systems experience the instabilities associated with resonances, they "choose" between two or more new arcs through their phase space. These new paths represent a higher and more complex order than the order which existed before the instability. Accordingly, new order emerges spontaneously out of chaos.

What Prigogine's mathematical descriptions do not tell us is why the system chooses one new path instead of another. In the simplest case, that of a "pitchfork bifurcation," the system has two equally likely choices and picks one of them. Why that one? Why not the other? How is the choice made?

The choice is probabilistic. Descriptively, the same laws of chance apply to a pitchfork bifurcation as to the flip of a coin. Statistically, over a long enough series of trials, half the time the coin will come up heads and half the time tails. Likewise, at a pitchfork bifurcation the dissipative system seemingly has a 50% chance of picking path A and a 50% chance of picking path B.

In Chance ... or God? I suggested that the supposedly random decision made by a dissipative system is somehow influenced by a God who "stills the seas" of chaos to allow new order to emerge. This is an idea which does not appear in Prigogine's discussion, since it lies beyond science.


It may be that at any given branching point, God is equally happy with path A or path B. But it seems likely — since God is thought by many believers to prefer the specific evolutionary pathways that allowed intelligent creatures such as ourselves to arise on earth — that in at least some bifurcations God has a definite preference. What happens then?

Setting aside the question of how God does whatever he does, what does he do? Does he, for example, somehow coerce a dissipative system at a crossroads of instability to take path A rather than path B?

My own inclination is to believe that God never forces such divinely favored choices on nature. If we think about what such a policy of coercion would imply, we have little choice but to imagine God as canceling whatever freedom nature-per-se was originally granted to make its own choices. But why would God create nature in such a way as to have nominal freedom of choice but no real freedom, in that God himself preempts nature's freedom at the drop of a hat?

My feeling is also that the imaginable divine methodology that lies at the opposite conceptual extreme is not the case either: the one in which God takes no purposive action whatever. True, it is not unthinkable that God somehow designed the world in such a way that, left entirely to its own devices, it might (albeit gradually) evolve creatures like us. But that would imply that God is blithely unconcerned about the possibility of "wrong" choices made by nature along the way, not really caring whether a free Mother Nature errs and evolves a soulless world by accident.

We are taught by our religions, however, that God rues our choices when they don't correspond to his own desires. Why wouldn't that apply to nature writ large?


If God neither coerces nature nor is content with whatever probabilistic choices nature happens to make, "right" or "wrong," is there some middle conceptual ground which allows God to call forth from nature the evolutionary pathways he prefers? I would like to believe that there is.

Specifically, I would like to imagine that God in effect "beckons" to the physical world. The world can then freely "choose" whether or not to follow.

If the world does not respond to God's beckoning, then the "godless" picture drawn by Prigogine says it all. When an evolving dissipative system arrives at a pitchfork bifurcation — to take the simplest case — it chooses blindly which of two possible paths to follow. The two paths, in the case of the pitchfork bifurcation, are equiprobable. One is as likely to be taken as the other, and the question of how the actual choice is made is unanswerable.

But if the world indeed responds positively to God's beckoning, then the system in question "defies the odds" and follows the path God intends it to follow.


It is for several reasons that I like this image of God beckoning and the world freely responding by defying the odds and following the evolutionary pathways God favors.

One reason is that, as far as I can see, it comports perfectly with Prigogine's mathematics and physics. According to his model, various types of physical systems, including those which he terms dissipative, have certain characteristics in common: they are best described by probability distributions; they can be modeled abstract waves stacked atop one another; they have irreversible arrows of time; they are prone to points of instability that mimic chaos; from these chaotic instabilities new order arises probabilistically. Nothing in my image of a God "beckoning" to a natural world, which then freely responds by "defying the odds" and picking from supposedly equiprobable options in accordance with God's preferences, contradicts that.

Rather, when the metaphorical coin of a probabilistic choice made by nature happens to come up "heads" — as presumably desired by God, inasmuch as it furthers his plans for an evolving world — then for all we know it happened by blind luck.

Every metaphorical or actual coin flip represents what students of probability and statistics call a "trial." There is no way to learn whether or not an ordinary, non-metaphorical coin-flip, seen as an individual trial, has a truly random result ... as opposed to being a response to some unseen factor that has influenced the coin to come up heads.

True, when one makes a long series of trial flips, one expects half the flips to come up heads and half tails. But if it just so happens that there are a lot more heads than tails in any series of trials, then, well, so be it. It is not totally impossible for a truly fair coin to come up heads 100 times in a row, or 1,000, or even 1,000,000. So if the results of 10,000 flips are 7,500 heads and 2,500 tails, the coin could still be fair.

Likewise, if the results of 10,000 choices made by evolving dissipative systems in the natural world go in favor of what we assume God would prefer fully 7,500 times out of 10,000 — or 9,000, or 9,999, or 10,000 — we still have no proof that there really is a God. Prigogine's "godless" picture of chance alone making choices in the natural world could still be all that's going on.


Prigogine's systems cannot help but evolve. This is one meaning of the arrow of time which his equations demonstrate. Just as Darwin showed, as biological systems evolve, new species originate. Presumably, evolutionary novelty enters the picture at bifurcation points where dissipative systems' behavior patterns grow unstable. At these points, resonances make for chaotic behavior ... and then new order unfailingly arises.

My suggestion is that Darwinian evolution is real, but that God "beckons" to systems at crossroads of instability. Then they may (or may not) freely choose to follow one particularly "blessed" arc through phase space instead of any of the other possibilities. For all science can detect, this choice is a matter of blind chance.

The late Harvard paleontologist Stephen Jay Gould, one of the best known popular writers on evolution science, had it that our species appeared on earth by blind luck. If we were to rewind the tape of evolutionary history and play it again, he famously said, Homo sapiens would surely never appear. The odds against our ever having evolved are astronomical. That we did evolve is a matter of, again, blind luck.

Gould may have been right about the astronomical odds against our appearance, if evolutionary changes happen at random. Other theorists offer better odds, ones that hinge on the tendency of the systems which Prigogine calls dissipative to "self-organize." Self-organizing systems are biased in favor of ever-increasing complexity, which makes it easier to imagine ourselves as (in the words of theoretical biologist Stuart Kauffman) "we the expected."

Yet the views of Gould, Kauffman, and Prigogine are all basically "godless" — as they should be, since science has no way to detect the influence of God on evolutionary processes. But it should be kept in mind that there still may be a God who beckons to nature and elicits behavior which science is powerless to disentangle from the workings of chance.


Beckoning-and-following is one useful metaphor for what happens when a superintending God modifies the workings of chance. Another might involve an image of calling-and-responding: God calls to nature and nature answers back by doing his bidding. Yet another analogy is my favorite: annunciation-and-assent.

Chapter 1 of the Gospel of Luke tells the story of the angel Gabriel coming unexpectedly to a not-yet-married Mary and announcing God's intent to give her a son, to be named Jesus, without benefit of having had the usual marital relations. "And Mary said, 'Behold, I am the handmaid of the Lord; be it to me according to your word.' "

When God beckons to a dynamical system undergoing a crisis of instability, I take what happens to be a sort of mini-annunciation. If the system responds by, in effect, saying "be it to me according to your word," a mini-miracle occurs. It is by a series of such mini-miracles that evolution defies the odds and produces creatures like us that find favor with God.

Chance ... or God?

The End of Certainty by Ilya Prigogine

In The World as Music I outlined why Ilya Prigogine's 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature leads me to the fanciful speculation that the world is basically made of music. Now I would like to show how such a view can be extended to include a God who calls the tune.

Briefly put, Prigogine asserts that many systems which are subject to change in the physical world naturally exhibit resonances of a mathematical variety originally discovered by Henri Poincaré. Poincaré resonances exist at points on a graph where equations describing dynamical systems — systems in the physical world made of objects in motion — intersect one another. At these points of intersection, crucial denominators become vanishingly small, such that algebraic divisions have undefined results. The customary approach to modeling dynamical systems, based on the idea that particles trace out hard-and-fast trajectories, breaks down.

Prigogine saves the day by substituting for the classical model a model based on probability distributions. His probability distributions substitute a "cloud" of system states for each single point ostensibly designating the state with certainty. They replace every simple, hard-and-fast trajectory with an expanding vapor trail of system pathways.

One of the desirable results of Prigogine's strategy is that the "instabilities" typically encountered by dynamical systems under particular sets of circumstances turn out to be related to and engendered by Poincaré resonances.

Basically, resonances are where waves — sound waves, light waves, mathematical waves — reinforce one another. When waveforms are coupled together, the amplitude (i.e., the height) of the combined waves can under the right circumstances be the (possibly very large) sum of the amplitudes of the individual waveforms. By envisioning certain mathematical constructs — "plane waves" and related "wave vectors" — as the basic stuff of which systems are built when they undergo motion and change, Prigogine is able to show that what we think of as particles occupying specific points in space and moving along specific trajectories in time are really made of these wavelike mathematical entities.

Such a wave-based mathematical model of the physical world achieves Prigogine's prime objective of introducing a directional "arrow" of time into the basic equations of physics, such that the clock of the universe cannot run backward after all.

Furthermore, the Prigogine model spreads out from the description of classical dynamical systems to include thermodynamic systems, chaotic systems, "dissipative" systems, and subatomic systems subject to the laws of quantum mechanics.


In traditional interpretations of quantum theory, subatomic particles such as electrons are never assigned trajectories per se. Instead, they have "wave functions." Different from the similarly named "plane waves" Prigogine alludes to, these attributes of the tiniest bits of matter are usually modeled using an equation named after its discoverer, Erwin Schrödinger.

Schrödinger's equation, notably time-reversible or time-symmetrical, does not allow for a directional flow of time. The irreversibility of time which we attribute to events (quantum, or at macroscopic scales larger than the subatomic) occurs, in traditional theoretical interpretations of quantum mechanics, as a result of external observation of those quantum events. Under observation, and only then, the supposed wave functions of quantum particles are said to "break down," creating a measure of scientific certainty about such things as the particles' velocities and positions where there had only been a fuzzy set of probabilities.

That reduction of the quantum wave function only under conditions of external observation is called the "quantum paradox" — for why should external observation radically affect the basic attributes of subatomic stuff?

Prigogine succeeds in sidestepping the quantum paradox by replacing specific wave functions à la Erwin Schrödinger with "ensembles" of possible states of a quantum entity, where an ensemble is represented mathematically by a probability distribution. His strategy simultaneously brings into the picture an irreversible arrow of time and removes from the quantum picture "the mysterious intervention" of an "observer" (p. 131).


When it comes to systems larger than the innards of an atom, Prigogine's selfsame model works equally well for any "nonintegrable" system — any system undergoing motion and change, whose overall behavior cannot be reduced to the sum of the behaviors of the individual pieces. And, for systems that are simple enough to be integrable, Prigogine's model can easily be transformed into the description given by classical Newtonian dynamics, whose equations turn out to be but a special case of Prigogine's more general ones.

Among the nonintegrable systems Prigogine discusses are systems in "deterministic chaos," a hot topic in physics over the last few decades. Also covered are thermodynamic systems, in which entropy and disorder increase over time. Meanwhile, one of the centerpieces of Prigogine's view concerns "dissipative" systems. Such systems hold themselves far from thermodynamic equilibrium by taking in and dispelling energy as well as matter over time. They can be merely physical or chemical systems, but all organisms are also dissipative systems.

Dissipative systems are wont arrive at crossroads of instability when they have been pushed far enough away from equilibrium by their characteristic energy flows. Prigogine calls these crossroads "bifurcation points" because the system will resolve each temporary instability by branching over onto one of two (or more) new pathways exhibiting increasingly complex order and stability. Which branch is "chosen" by the system at any particular bifurcation point is, according to Prigogine, a matter of irreducible probability — a coin flip, as it were.


Although Prigogine does not make this point explicitly, it looks to me as though he feels Poincaré resonances engender the "fluctuations" or "perturbations" which he says are associated with an evolving dissipative system's movement over onto one new branch or another at any given crossroads. These resonances seem to be intimately associated with the bifurcation points' fundamental character as instabilities. But Prigogine does not say how the resonances "decide" which of two possible paths the system will actually take.

Specifically, he does not say how a system undergoing characteristic junctures of instability makes its "choice" among alternate pathways in response to the minute "fluctuations" or "perturbations" that occur at a critical moment and that the reader must assume are associated with Poincaré resonances.

So, the reader is entitled to ask, if the resonances themselves do not "decide" which of two possible paths the system will actually take, what does? Prigogine seems to be content to allow that question to remain unanswered, except by reference to the influence of his "fluctuations" or "perturbations."

As for me, I consider "fluctuations" or "resonances," as an answer to the "what decides" question, to be just as mysterious as Prigogine calls the deciding influence of an outside observer, in discussing the paradoxes of quantum theory.


It looks to me as if there are two principled responses to this ambiguity. The first is the approach which Prigogine himself appears to take. He categorizes his probabilistic description of dynamical systems at various crossroads of mathematical instability as "primitive" and "irreducible."

You can't account for the primitive or irreducible elements that are necessarily found in every scientific theory. For example, the constant speed of light is a primitive, irreducible element of Einstein's theory of relativity. There is no way to explain why light alone should have an absolute velocity that is not relative to that of other moving entities. But when you predicate relativity theory on such an unexplained assumption, the theory then goes on to provide you with excellent explanations for, and reliable predictions about, many other phenomena.

Prigogine takes a like attitude toward the probabilistic-but-not-otherwise-predictable behavior of systems subject to Poincaré resonances. His main reason: only by taking the probabilistic uncertainty of the system as a given can you introduce the directional flow of time into a mathematical model of the physical world.

Put a different way, Prigogine settles for a theory that is stunningly powerful from a descriptive point of view, in that a single mathematical model adequately describes the behavior of systems from multiple disciplines: classical dynamics, thermodynamics, chaos theory, quantum mechanics, and the theory of dissipative, self-organizing systems.

Yet the model which describes the behavior of these systems does not explain it. It does not explain, in effect, why the probabilistic coin flip produces tails, not heads.

In fact, it undermines one of the assumptions classical physics traditionally made about coin flips and the like: if we could only ascertain the forces acting on the coin down to the nth degree of accuracy, we would be able to know in advance which coin flips would produce heads and which tails. The assumption was that coin flips and all other seemingly random events in the physical world were actually deterministic. Randomness was thought to be an illusion based on ignorance — as was the apparently unidirectional flow of time.

In undermining that assumption about the arrow of time being a bogus perception on our part, Prigogine likewise explodes the parallel assumption that coin flips are deterministic events. According to him, crucial events in the evolution of "nonintegrable" systems subject to Poincaré resonances are truly inexplicable except by reference to the laws of chance.


The other principled way to treat the "what decides" question is to decline to settle for the supposition that the choices made by systems at key crossroads of instability that stem from to Poincaré resonances have no further explanation outside the laws of chance.

What if there is a God behind these chance events?

According to Prigogine, if I read him aright, bifurcation points of dynamical instability match up with regions of "phase space" wherein chaos prevails (see p. 41). "Phase space" is an abstract realm made up of all the possible points on a hypothetical graph plotting the conceivable states of a system. These conceivable states in phase space are defined in terms of the instantaneous locations and movements of its constituent particles.

Under classical assumptions, at every individual moment the system as a whole occupies a single point in phase space. Since the system is "dynamical," it is subject to change over time. As it changes, in the classical view it traces out a "nice" (i.e., deterministic) trajectory, which amounts to a single, crisp line or curve through phase space.

In Prigogine's revision of the classical view, that single point in phase space has to be replaced by a "cloud" of possible points describing the system state probabilistically. The classical trajectory becomes an "ensemble" of possible trajectories through phase space. The "ensemble" can under some circumstances be viewed as a "random" trajectory, rather than a "nice," deterministic one. Under these circumstances, the system is chaotic. "Random" trajectories "wander erratically through regions of phase space" (p. 41). The future states of the system become exponentially less predictable as time marches on.

According to Prigogine, chaos, even when it is not "fully developed," can be associated with points in phase space that are "characterized by resonances" (p. 41). It is not clear to me exactly how closely the behavior of dissipative systems at resonance-engendered bifurcation points matches the "randomness" of chaos, but I assume the path the dissipative system "chooses" at such points of instability equates to a reemergence of order out of chaos.

Metaphorically, then, the choice could represent what naturally happens when God "stills the seas" of chaos.

The World as Music

The End of Certainty by Ilya Prigogine

Ilya Prigogine's 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature, which I last discussed in Seeds of Surprise, Part 2, leads me to the fanciful speculation that the world is basically made of music.

Consider this admittedly nearly impenetrable passage from pages 122-124:

We can now introduce the effect of Poincaré resonances into the statistical description of dynamics. These resonances couple dynamical processes exactly as they couple harmonics in music. In our description, they couple creation and destruction fragments ... which lead to new dynamical processes that start from a given state of correlations ... and eventually return to exactly the same state. ... [T]hese dynamical processes are depicted as bubbles ... .

These bubbles correspond to events that must be considered as a whole. They introduce non-Newtonian elements [into the theory of dynamical systems in physics] in that no analogue of such processes exists in trajectory theory. Such new processes have a dramatic effect on dynamics because they break time symmetry. Indeed, they lead to the type of diffusion that had always been postulated in phenomenological theories of irreversible processes ... .

I'll try to translate. The book talks about how Prigogine, the now-deceased Nobel laureate, had spent many years in search of ways to revise the basic theoretical understandings of physics inherited from Newton and, more recently, from the original investigators of the quantum theory of subatomic particles in the early 20th century. Their explanations found time's seeming directional flow, past to future, to be an illusion not borne out by the mathematical equations at the heart of their theories.

Prigogine, resisting that, revises Newton et al. by representing the motion of physical particles with "probability distributions" rather than hard-and-fast "trajectories." In the branch of physics known as thermodynamics — the study of energy as it relates to heat — one of the principal concerns is figuring out how heat and other forms of energy derive from the motion of molecules, particularly in a gas such as the earth's atmosphere in which the moving molecules are continually bumping into one another. The higher the temperature of the gas, the faster they speed about, and the more often they collide.

Colliding molecules in the real world, Prigogine says, do more than respond to heat sources. They establish persistent "correlations" with one another. This is a fact which scientific models based on individual trajectories lose sight of, and so it is generally ignored in the equations of standard theory.

I'll go over that again. Once two particles have collided, what happens to them next happens in part because they are now at least temporarily something of an "item," no longer totally independent of one another. If one them has yet another collision with a third particle, the motion of all three particles is thenceforth correlated. This is a fact models based on individual trajectories lose sight of, and so it is generally ignored in the equations of standard theory.

Correlations of particle motion can also be destroyed by ensuing events as well. Prigogine's "creation fragments" and "destruction fragments" are shorthand ways of referring to such beginnings and endings as they apply to correlations.

So are his "bubbles." They refer to the fact that a correlation's creation event (think of a bubble being blown) is thereafter "coupled" to its eventual destruction (when the bubble pops). The correlational bubble's very ability to exist, in turn, derives from a "Poincaré resonance."


In the mathematics of probability distributions, the location and motion of particles are expressed in terms of waves, and waves tend to have Poincaré resonances when they combine. Light (when it is not considered to be made of particles called photons) is made of waves. A vibrating guitar string produces audible waves in the air. The air molecules move in such a way that their collective density increases or decreases in proportion to the amplitude of the wave at any particular point in space and at any given instant of time.

Any physical thing that oscillates or vibrates, furthermore, will produce a wave whose wavelength, the distance between two adjacent crests of a wave, is inversely proportional to the frequency of vibration.

Prigogine abstracts from the mathematics of acoustical waves in applying the same sorts of equations to Newtonian dynamics and to thermodynamics as well. The equations used by Prigogine feature such constructs as "wave vectors" and "plane waves."

A wave vector is an algebraic term that shows up repeatedly in the equations; it is inversely proportional to wavelength. A plane wave, another mathematical construct, is a periodic function of the wave vector, and also of the spatial coordinate that specifies what position or location in space is being referred to. As the focus of attention moves to different positions in space, or as the wave moves outward from its source, the plane wave wiggles up and down between its maximum and minimum values. These maxima and minima depend on the wave amplitude or height ... which is also a function of the wave vector. (Yes, it gets complicated!)

Now, it's important at this point to realize that mathematical constructs such as wave vectors and plane waves are treated by Prigogine as if they were also physical things. In essence, he is suggesting that we ought to think of the pathways that moving physical particles trace — their trajectories, in old-fashioned physics — as no longer quite as real as the plane waves from which these pathways can be mathematically derived.

When plane waves of various wavelengths are superposed — stacked on top of one another — they wind up either increasing or decreasing the composite wave amplitude at any given coordinate in space, depending in whether their "interference" is constructive or destructive. If constructive interference is set up in just the right way, the otherwise cloudy, probabilistic locations of molecules in motion — making tracks as fuzzy as a vapor trail of a jet airplane — can resolve to distinct localized trajectories.

In a crucial special case, the superposed waves reinforce one another in such a way as to establish a "resonance," much as a violin's body resonates with a vibrating violin string and amplifies its sound. This is where things get really interesting. As physicist-mathematician Henri Poincaré (1854-1912) discovered, when systems resonate, key terms in the Newtonian equations that model them tend to develop "dangerous," verging-on-zero denominators, resulting in the inability to use these equations to calculate real-world trajectories.

Prigogine turns this minus into a plus by using Poincaré resonances as a springboard to a mathematics of dynamical systems which avoids dependence on trajectories and thereby succeeds in "breaking time symmetry." Prigogine's math gives scientists for the first time a principled way to see how dynamical systems can — or, indeed, must — evolve over the course of a time in a mathematically irreversible direction.


But what intrigues me personally is not so much the breaking of time symmetry, in the sense of establishing the flow or "arrow" of time as a valid physical reality. It is the admittedly rather fanciful notion that the world is basically made of music. That is, if Prigogine is right and "resonances couple dynamical processes exactly as they couple harmonics in music," then we are entitled to imagine the stuff of the universe as the same as the stuff of music.

The stuff of music boils down to composite audible waves succeeding one another in an orderly fashion over time. Each instrument or voice produces composite tones with harmonic overtones built atop a fundamental note frequency. When a piano sounds middle C, its struck string also sounds the C above that, and the G above that, and any number of higher harmonic frequencies which superimpose to give the piano its distinctive timbre or tone color. Resonances created by the sympathetic vibrations of the wood of the piano and the strings of other, unstruck notes add to the richness of the sound.

Then the pianist strikes the next note, and the stuff of the music instantly changes. Over time, a melody plays out which, thanks to the craft of the composer, usually ends by returning to the tonic note of whatever key the piece started out in — much as the "destruction fragment" of one of Prigogine's dynamical "bubbles" is coupled to its associated "creation fragment."

In music, resonances are what create timbre or tone color. They can also break glass. In Prigogine's physics, resonances break time symmetry. If there were no Poincaré resonances, the standard equations which seek to define the hard-and-fast trajectories of motion and change would serve scientists just fine ... and those equations show no arrow of time. So they don't really show how dynamical systems that operate far from thermodynamic equilibrium evolve unidirectionally with respect to the flow of time.

All living organisms, as well as certain physical and chemical systems that don't qualify as living, are far-from-equilibrium dynamical systems. These systems, which Prigogine labels "dissipative" because they take in and dissipate energy and matter as they hold themselves far from equilibrium, are prone to crises of instability — "bifurcation points" — at which their trajectories fork. Which fork they take depends on minute fluctuations that are seemingly due to chance. Each time a branch is taken, there is new, more complex order in the system.

Reading between the lines, it looks as if bifurcation points in dissipative systems correspond to Poincaré resonances in the more general theory of dynamical systems.

In my earlier posts in this series, I called these bifurcation points "seeds of surprise" because it can't be foretold with certainty which of two or more forking branches a system will follow as it finds its way out of an evolutionary instability. What if each bifurcation point corresponds to a Poincaré resonance? According to the interpretation at this web page, resonances are points on a graph "where the equations [being graphed] intersect in such a way that one of the members is divided by zero." Here, "the equations" could be taken as those which describe a path leading to or away from a bifurcation point. At the bifurcation point itself, there is a mathematical discontinuity such that the various equations have different solutions — another way of saying that crucial terms or "members" in them turn out to be "divided by zero." As a result, the standard equations can no longer be used to predict what the system will do.

Events transpiring at bifurcation points qualify as "divergences due to resonances," at least in my understanding of Prigogine (see p. 40). They also qualify as divergences due to fluctuations (see pp. 68ff.). That would seem to imply that fluctuations are due to resonances. Writ large, that would seem to mean that key events in the evolution of life on this planet, wherein new forms of order arose over time, came about by virtue of the intrinsic behavior of the stuff of music!

Seeds of Surprise, Part 2

The End of Certainty by Ilya Prigogine

I'll take another go at setting forth some of the ideas in Ilya Prigogine's excellent 1997 book The End of Certainty: Time, Chaos, and the New Laws of Nature — which I first discussed in Seeds of Surprise, Part 1 — along with my ideas about those ideas.

One of Prigogine's notions is that, inasmuch as atoms are made up of particles and these particles are in motion, they are dynamical systems. As such, the motion of electrons and other particles in an atom possesses an "arrow of time." So there ought to be a way to mathematically model the quantum behavior of subatomic particles such that time appears in the equations as an irreversible variable.

In standard quantum theory, time is a reversible variable. The equations normally used apply equally well to forward-directional time as to reverse-directional time.

In fact, the forward and reverse directions of the flow of time are not apparent in the standard quantum equations. Accordingly, at least as far as quantum phenomena are concerned, many theorists attribute the arrow of time that we experience, to the effects on a quantum system of its being observed, as scientists study it from the outside.

When scientists try to measure the position of an electron, they can pinpoint that position only if they abandon hope of also knowing the electron's momentum: its velocity in a particular direction, times its mass. Hence measuring the electron's position is incompatible with measuring its velocity, and observing its velocity makes it impossible to know its position precisely.

Think of trying to photograph the end of a horse race. If a flash photo is taken right at the finish line, it freezes the horses' motion. The stop-motion photo thereby pinpoints which horse's nose crossed the line first — i.e., the relative positions of the horses at the finish of the race. On the other hand, if a non-flash, time-exposure photo is taken, using a long exposure time, the lead horse's nose will be but a blurred streak in the picture. The form and length of the streak will be a function of how fast the horse was moving — its speed, which relates to its velocity.

Therefore, how you set up the camera determines whether it captures position or momentum. Either alternative can be measured with certainty only if the other alternative is sacrificed.

The stop-action alternative which pinpoints the horses' relative positions has no time dimension whatever. The blurry-streak alternative which records the horses' speed does not, meanwhile, tell you in which direction the horses really moving. Common sense tells us, it is quite true, that horses run "forward," i.e., with their noses out in front of the rest of them. But strictly speaking, the camera does not know this. The blurry streak formed by the moving equine noses would look just the same if the horses were being transported backward!

So either act of photographic commemoration, stop-action or blurry-streak, lacks an arrow of time. We, of course, supply one when we look at and interpret the blurry-streak photo.


Similarly, the standard equations of quantum theory lack an arrow of time. Yet various interpretations of the theory exist in which time irreversibility is imposed by virtue of acts of external observation. I discussed one such interpretation, which exalts the importance in quantum phenomena of outside observers' choices as to what to observe and how, in my earlier post Genesis by Observership.

In a well-known thought experiment in quantum physics, that of Schrödinger's Cat, a live cat is imagined to be put in a closed box with a mechanism that may (or may not) indirectly cause the cat's death by at some moment releasing (or not releasing) a radioactive particle as a trigger to the breaking open of a vial of poison gas. In our imagination, we let the experiment run unobserved for a period of time. Then, after a while, we open the box and see whether the cat is alive or dead.

It is only at the moment we do so, Schrödinger showed mathematically, that the outcome becomes etched in stone ... even though the release of the fateful particle and the resulting death of the cat happened (if at all) earlier in time! Inconceivable as it seems, apparently no other understanding is consistent with the laws of standard quantum mechanics. Our observation of the event alone is what makes that (earlier) event certain.

Thus, in this interpretation of the mathematics of quantum theory there can be no arrow of time, except under conditions of external observation.


Prigogine objects to that approach. He seeks (and finds) an alternative mathematical formulation of quantum theory which builds in the arrow of time from the get-go. In return, it trades in the (limited) certainty which the standard equations can yield and replaces those equations with probabilistic formulas that are unable to pinpoint which event, out of a multitude of possibilities, is actually so.

The standard equations can yield certainty in the limited way a photo of a horse race's finish can: one variable can be pinpointed with certainty by sacrificing another. But those standard equations lack an arrow of time. That leads to, among other paradoxes, the one concerning Schrödinger's imaginary cat in which a seemingly later "photo" — a visual observation, that is — determines a seemingly earlier event.

Prigogine's equations remove that proneness to paradox in quantum theory at the expense of making certainty fundamentally impossible at the level of subatomic phenomena. But his equations do successfully account for the arrow of time.

More than that, they generalize to other types of systems involving physical "stuff" in dynamical motion. Such systems include thermodynamic systems, with their vaunted entropy, and dissipative systems which self-organize and evolve in seemingly the opposite direction. They also include systems whose futures are in thrall to "deterministic chaos."

Dissipative systems are quite important to us. After all, all living things are dissipative systems. So, too, are various kinds of non-living physical and chemical systems, at least under certain circumstances.

Prigogine's equations involving "probability distribution functions" are basically the same for all these kinds of dynamical systems.


My thought about all this is that those other, non-quantum kinds of dynamical systems might, like quantum systems, have alternative mathematical models in which acts of external observation would successfully change probabilities into certainties — just as they do in the study of quantum phenomena, at least according to some interpretations of the relevant theory.

For brevity, I'll refer to such hypothetical alternative mathematical models as AMMs. If an AMM exists for, say, a dissipative system, certain things would likely result. One of them is that the choice of which new stable state a dissipative system enters at a dynamical "bifurcation point" is not a coin flip but reflects an act of external observation. The event which results from the bifurcation in turn can depend on an act of choice made by the outside observer (see Genesis by Observership for more on the astounding consequences of observational choices).

If an AMM exists for a living, evolving, dissipative system, it just might be that the bifurcation events of its evolutionary pathways are determined by choices made by God, who sees all worldly events from outside the context of irreversible time. Hence, such an AMM would be expected to represent time as reversible, just as do the equations of standard quantum theory. Accordingly, such an AMM would violate one of Prigogine's foundational assumptions: his preference for a mathematical model in which time is irreversible.

Moreover, if such an AMM exists for a living/evolving/dissipative system, that AMM would very likely allow the chance-based alternatives intrinsic to a mathematical model such as Prigogine's to be supplanted by descriptions of events which are knowable (at least in limited fashion) as certainties.

At the same time, such an AMM would not be wholly deterministic. There would be types of systems experiencing types of situations in which the future is not set. There would be an element of surprise in these systems' future evolutionary paths that could not even in concept be eliminated by our gaining more thorough knowledge of their present states.


Such an alternative mathematical model would not invalidate Prigogine's, any more than Riemannian geometry invalidates Euclidean. Prigogine's model succeeds in accounting in a very general way for the arrow of time as it affects many different types of systems which are usually treated as separate. If you are interested in a purely physical-world approach, it looks to me (as admittedly a layman) that Prigogine's mathematics are spot on target when it comes to explaining time's irreversible flow.

The alternative approach I am suggesting seems to require that there be an observer outside the world system whose observational choices we experience as the workings of chance.

Prigogine's original description of bifurcation points in dissipative systems is a case in point. It involves chance fluctuations, which "choose" which path the systems will take out of states of instability. His probabilistic, statistical mathematical model replaces this description by one in which chance (another name for probability) is involved at a fundamental level, at all times, not just at discrete points of instability.

The Prigogine model accepts both chance (or probability) and the arrow of time as brute facts or bottom-level realities in our world. The AMM I seek sees them both as artifacts of the choices made by God in his "observation" of our otherwise freely evolving world.