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Pious Fantasies

A Review of Lost Discoveries: The Ancient Roots of Modern Science — from the Babylonians to the Maya, by Dick Teresi, Simon and Schuster, 2002. 453 pages. This review appeared in Skeptic magazine volume 10, number 2 (2003).

DICK TERESI’S LOST DISCOVERIES will probably flourish commercially because educators eager to augment the canon of multicultural pieties by “decentering” Western Civilization with respect to science — as it has already been dethroned in so many other ways — will embrace it rapturously as the inside story of how Eurocentrists purloined undeserved credit for science and mathematics. It won’t matter much that this book is an intellectual disaster and, worse, a moral disaster. The homiletics of officially sanctioned “diversity” are glibly indifferent to the soundness of the intellectual currency and resound with an exceedingly narrow notion of morality.

Teresi’s book is bad because it is dishonest. Page after page it concocts excuses for belittling the foundational achievements of Western science (that is, the science of the Greeks in classical times and the Western Europeans in early modern times) and for exaggerating the depth and importance of the proto-science done in non-Western cultures. Teresi is here to preach, not to analyze, and the sermon he delivers is that we Westerners have been very remiss in not recognizing that these purported non-European achievements were not only earlier in time, but were crucial to subsequent European developments.

In some cases, this is obviously true, though rather an old story. The key concept of decimal notation for numbers, as well as the idea of zero as a distinct quantity, came to the West from India, via the Arabs. Greek geometry was almost certainly inspired by the mensural geometry of the Egyptians. But Teresi is far from satisfied to repeat these obvious truths. He must puff up these achievements of earlier cultures into foundations of things they did not, in fact, found. When this is too farfetched, even by his standards, he contrives to minimize or disparage unambiguously Western inventions for which no non-Western antecedent (or even parallel) can be found.

At the same time, he denounces failure to recognize his idiosyncratic genealogy of ideas as willful suppression rooted in Western chauvinism. Characteristically, he accepts, without the smallest hint of skepticism, every hypothesis of Martin Bernal’s Black Athena, a book that, by this point, it is only fair to call notorious. He simply ignores the meticulous, exhaustive criticism of Bernal to be found in Mary Lefkowitz and Guy Rogers’s compendium Black Athena Revisited. (He does however list Lefkowitz’s Not Out of Africa in his bibliography, accompanied by the observation, “This book is noteworthy in the way it portrays the anger of some scholars toward non-Western history,” a remark that one should not hesitate to label slanderous.)

Maligned Mathematics

Teresi’s whoppers are not hit-or-miss; practically every page contains one or two. Because I am a mathematician by trade and pretty touchy about the honor of my subject, I shall confine myself to his wobbly presentation of the history of mathematics. I begin with his explicit proclamation: “[N]owhere is non-Western science stronger than in math” (p. 27). Alas, despite his consultation with a number of academic mathe maticians, including Harvard’s eminent Barry Mazur, Teresi simply has no feeling for the subject, no sense of what was crucial to its development, nor any discernment in regard to what is really difficult and deep and what is not. His introductory section, intended to flaunt a slam-dunk case where a pre-eminent achievement of Western science is supposedly revealed as a thief-in-the-night expropriation of non-Western genius, provides instead a telling instance of his lack of insight.

The case in question is that of Copernicus’s heliocentric system — specifically a couple of geometry results Copernicus used to justify his theory. Teresi alleges that these were lifted, uncredited, from earlier Arab mathematicians and that the notion that modern planetary astronomy is a uniquely Western invention is therefore a deception. That “the new math in the Copernican revolution arose first in Islamic, not European minds” is, in his view, “too damaging [for Western chauvinists] to accept” (p. 5).

Now it might well be that these supposedly profound results reached both Copernicus and his Arab predecessors from a common Greek source, since lost (as we have lost the bulk of the achievements of the Greek world in all fields). But this is a side issue; let’s stick to the weighty theorems themselves. The first, sometimes called “Copernicus’s Lemma” (unjustly; but then, many such mathematical labels are unjust), when stripped of some astronomical window dressing, amounts to this:

Let a circle of radius R roll around the inside of a circle of radius 2R so that it is always tangent and does not slip. Then, for a given point x on the small circle, there is a diameter D of the large circle such that x moves back and forth along D.

I invite readers with even a little mathematical skill to take a crack at this one. It is an easy result of elementary Euclidean geometry and it would not have given a competent geometer of Classical — or High Renaissance — times the slightest difficulty. It is, at best, a minor observation in its own right. Proving it from scratch took me about 20 seconds — without any help from modern mathematical tricks and without benefit of pencil and paper. It is simply not a big deal. It is nothing more, in fact, than a mildly interesting high-school geometry exercise. Even more, I would bet that it was known, empirically, to many horologists, millwrights, and other Medieval folk who worked with cogs, gears, sprockets and the like.

The other result can be stated thus:

Let AB and CD be line segments of equal length that meet line L at points A and C respectively. Let the non-acute angle formed by L and AB equal that formed by L and CD. [To help visualize, AB and CD needn’t be parallel, and one possible case is that this non-acute angle is actually a right angle.] Then the straight line M determined by points B and D is parallel to L.

This proposition, it must be said, is almost totally trivial; it is pretentious to call it a “result” at all. Classical geometry is full of propositions that are enormously deeper.

To intimate that this stuff in the context of late 16th century Europe constitutes powerful “new math,” the way calculus did at the end of the 17th century, betrays either mathematical incompetence or a willingness to let a predetermined agenda override everything else. Teresi may well be guilty on both counts.

The Axiomatization of Geometry and the Mathematical Shallowness of Teresi

Mathematicians are, of course, apt to be picayune and snotty. We are, perhaps, too dismissive when those outside the fraternity try to explain or recount serious mathematical work. But Teresi has chosen to propound a case that is by his own account radical and challenging, offhandedly dismissing the judgment of historians of mathe matics like Lancelot Hogben and W.W. Rouse Ball who were themselves competent at a professional level. It would have behooved him, then, to have made sure of his footing before treading this tricky ground. But political urgency overrides caution, and we frequently encounter oracular judgments that turn out to be blatantly wrongheaded.

Consider, for example, Tiresi’s treatment of the Pythagorean Theorem and its role in the mathematics of sundry ancient cultures. He cites the substantial evidence that Indian mathematicians knew this result at about the same time that the Greeks discovered it and that the Babylonians knew it well before. But he dismisses the significance of the fact that “knowledge” of a mathematical principle like this may have several different senses and that these differences are crucial. There is no evidence that either the Babylonians or the Indians knew the theorem as anything beyond an empirical fact of practical geometry or that they looked for demonstration beyond a large ensemble of examples. The crucial difference, of course, is that in Greek mathematics, the theorem is embedded in a vast, rigorous, and comprehensive deductive system replete with equally powerful results. Teresi, though aware of this distinction, minimizes its importance and, in fact, tries to convert it into a pretext for downgrading the Greek achievement. On this view, the Greeks’ obsession with axiomatics and logic merely displays their head-in-the-clouds taste for sterile abstraction and their effete disdain for the practical use of mathematical knowledge. This is bizarre in view of the fact that the greatest of Greek “pure” mathematicians, Archimedes, was at the same time the greatest engineer and inventor of Classical times, possibly the greatest who ever lived. But aside from that blunder (hardly surprising, since Archimedes’s work, either mathematical or practical, wholly escapes Teresi’s consideration), the slighting treatment of the axiomatization of geometry reveals deplorable mathematical shallowness. The fruits of this great innovation are astonishing beyond measure.

For one thing, the axiomatic approach requires systematic organization and presentation of developed results in a way that brings to the forefront their logical interrelationships and avoids circularity. It illuminates the very deep fact that the corpus of knowledge reposes, ultimately, on a minimal set of intuitive assumptions. It makes starkly clear the vital distinction between a conjecture or surmise supported by example and a validly established result. We have no evidence that any other ancient culture ever came close to this insight.

Equally important, an axiomatic context for mathematics highlights the possibility of continued expansion, generalization, and synthesis of existing knowledge. Indeed, it makes clear that the real function of a mathematician is to expand, generalize, and synthesize. Further it makes the teaching and preservation of mathematics immensely more efficient, which is precisely why Euclid’s Elements was the central foundational text for mathematical learning in several cultures over the course of two millennia (a role it is still, in some sense, fitted to play). This stunning achievement of Hellenic (and Hellenistic) civilization is precisely what kept its mathematical insights from dispersing into unrelated fragments, each of limited value. By contrast, that dismal fate has overtaken the mathematics of all the other ancient cultures we know of. It is not that Greek mathematicians were, as individuals, somehow brighter than their counterparts in the Nile Delta, the Fertile Crescent, or the Indus Valley. Rather, what triggered their stupendous achievement was the fact that the axiomatic organization of the subject made it possible for one genius to climb upon the shoulders of another and thus to see the subject whole.

This realization, of course, is precisely what Teresi wants to make disappear, since it undercuts his political agenda. Rather than dispassionately comparing Greek mathematics with distinct ancient traditions, he singlemindedly unearths whatever grounds he can — specious grounds, in most cases — for demeaning the Greek achievement. A case in point:

The concept of infinite sets of rational numbers was grasped by Jaina (Indian) thinkers in the sixth century B.C. and by Alhazen in the tenth century A.D. It entered Europe nearly a thousand years later, when the nineteenth-century German mathematician Georg Cantor refined and categorized infinite sets. (p. 22)

Aside from the fact that early Indian and Arab thinkers did not grasp the notion of infinite sets in the sense that Cantor did (and aside from a gratuitously insulting reference to Galileo’s premonitory workon infinite cardinality), this ignores the fact that Euclid gives a prominent place to notions of the infinite in his very well-known exposition of the result that the set of prime numbers is infinite. (Speaking of prime numbers — which Teresi never does — the Fundamental Theorem of Arithmetic (that is, all integers greater than 1 are uniquely factorable into prime factors) is also proved, to all intents and purposes, by Euclid, without precedent or parallel in other ancient cultures, so far as I am aware.)

A similar bit of badmouthing occurs with respect to irrational numbers: “The obsession with purity kept the Greeks from embracing irrational numbers” (p. 55). What he seems to mean is that other cultures attempted to approximate some useful irrational quantities numerically, without realizing that they were, in fact, irrational. (Teresi does cite, in a note, the great Indian mathematician Nilakantha who surmised, without proof, that π is irrational — but that dates from ca. 1500 C.E.) It was the Greeks who attacked the specific problem of “incommensurability” and, in fact, Euclid devotes considerable attention to the existence of such quantities and the geometric situations in which they arise. It is the Greeks, if anyone, who “embraced” irrational numbers.

Aside from the Copernicus episode mentioned above, Teresi does not seem terribly interested in the early modern period, a time when Western mathematics began its great acceleration. To a certain extent, he harms his own case by neglecting this era. Around 1600, very roughly speaking, there seem to have been three cultures, including Western Europe, with an ongoing tradition of serious mathematics. India was probably the equal of Europe in its understanding of geometry and seems to have been somewhat more advanced in its grasp of infinite series and the foundations of analysis. Japan was developing its own unique school of synthetic geometry centered around the so-called “Temple Problems,” a strain of mathematical thought that was highly original, beautiful, deep, and difficult.

Teresi doesn’t even consider these stunning non-Western achievements. Perhaps this is merely an oversight. But perhaps the reason lies in the subsequent course of mathematical history. During the next few centuries, Indian and Japanese thought remained confined to narrow channels. In the West, however, mathematics developed explosively, a kind of intellectual “hyper-inflation.” By the turn of the 20th century, Western mathematicians had progressed from more or less classical geometry and some understanding of the theory of algebraic equations to the full development of calculus, differential equations, power series, complex variables and analytic functions, Fourier series, probability theory, number theory, differential geometry, non-Euclidean geometry, topology, algebraic geometry, and abstract set theory. At that juncture, the mathematics libraries of Europe held a vast and continually expanding record of deep, broad, powerful mathematical accomplishments, a huge reservoir of ideas from which all sorts of scientists, from physicists to psychologists, have drunk deeply.

Obviously, this stretch of time coincides pretty exactly with the era during which Western hegemony over virtually all other regions and peoples of the Earth reached its zenith. This might account for Teresi’s reluctance to enter these waters. To sing the praises of the Western mathematical genius displayed during this period, or merely to let it be seen how brightly it outshone the best efforts of what had once seemed equally proficient cultures, can be read as extenuating, or even justifying, the arrogance and cruelty of European expansionism and colonialism. Clearly, that is something Teresi’s political agenda will not allow. His book, after all, is part of an intense movement among American pedagogues to browbeat students (especially males) of European descent into diffidence, even shame, in regard to the accomplishments of their ancestral culture. Simultaneously, non-Eurogenic youths are urged to fervently celebrate the moral, artistic, and intellectual achievements of their forbears.

But to speak of mathematics in Greek antiquity or in early modern Europe without conceding that a kind of collective cultural genius must have been at work is to assert, when you get down to it, that the brightest minds on the Western rim of Eurasia must simply have been individually brighter than their counterparts on its Southern or Eastern rims. This is palpably silly. The internation al iza tion of mathematics over the past century demonstrates just how silly it is.

In speaking of a collective cultural genius, I don’t mean to be mystical or to embrace any “blood and soil” notion of national and ethnic destiny. I am merely trying to give a name to the coming-together of ideas, ideology, economics, and modes of prestige seeking that catalyzed the achievements of the Greeks and of my own (speaking as a mathematician) recent intellectual ancestors. I profess utter ignorance of what the mechanisms might actually have been, yet I insist that it is perverse to doubt that they existed. But that’s pretty much what Teresi wants to do: demolish the very concept of Western culture’s singularity and establish that all of its geniuses must have stood on the shoulders of non-Westerners.

Defenders of Teresi might claim that even if his most radical contentions are insupportable, at least his work has the virtue of bringing attention to the heretofore neglected and disparaged achievements of non-Western and pre- Western cultures. After all, hasn’t he diligently unearthed and organized an important story that might otherwise have escaped notice indefinitely? I don’t think so. Let me tell you about another book that once passed through my hands, one that systematically recounted the achievements of Egypt in mensural geometry and in the invention of algebra, the ingenuity of Babylonian mathematicians along with the story of how traces of their work endure in our common systems for measuring time and angle, the calendrical virtuosity of the Maya and, as well, their invention of zero, the independent Indian invention of the same, the Arab transmission of this new arithmetic along with their own germinal algebraic ideas to the backward West. This covers the greater part of what Teresi has to say in this regard.

The volume in question is, as it happens, the book from which I learned ninth grade algebra eons ago (when the Giants were still playing at the Polo Grounds and the Dodgers at Ebbett’s Field). The facts in question were presented in occasional sidebars, included, I suppose, to provide mild distractions from the rigors of solving simultaneous equations, factoring polynomials, and mastering Cartesian coordinates. They were matter-of-factly stated, without any sense that they revealed state secrets, and without any “multicultural” heavy breathing — it was far too early for any of that! My point is that Teresi’s “lost” discoveries never have been lost, in the sense he implies, and that serious historians of mathematics have never neglected them. To the extent that Teresi tells a true story, it’s an old one.

The Whole of Science

Teresi’s treatment of mathematics does not depart, in tone at least, from the rest of his book. True, with respect to most of the empirical sciences, he doesn’t have to contend with the Greeks to such a great degree (though his studious avoidance of Archimedes’s achievements, along with those of Eratosthenes and Aristarchus, also spares him some embarrassment from that quarter). But the central question in the history of science and civilizations goes unasked. That question, which I have already posed for mathematics, can be formulated for science in general: What combination of cultural and historical factors allowed Europe to bring forth the enormous and enormously powerful engine of knowledge creation we call science?

This question has to be seen whole. It cannot be answered — nor, for that matter, can its premise be challenged — simply by pointing to one or a dozen or a hundred isolated innovations in ideas or technology. But Lost Discoveries is an exercise in avoiding seeing it whole. Teresi’s technique is to remain constantly on the lookout for an “anticipation” of something in modern science and then having found it, at least to his own satisfaction, to use it to reprove the West as a Johnny-comelately. Sometimes, he is obliged to contort matters to the point of absurdity in order to find one of those precious foreshadowings.

Medieval-Islamic meditation on metamorphosis and transformation, for example, is touted as a precursor of Darwinian evolution. Mining various cultures for odd bits of metaphysical fluff that can be manipulated to vaguely resemble quips and punch lines from modern physicists, he solemnly pronounces that they anticipate quantum mechanics (which he doesn’t seem to grasp well in any case). Old creation myths are portentously placed alongside contemporary cosmology on the basis of flimsy metaphorical similarities. The fact that contemporary cosmology is still in all probability a long way from a final, stable model is cited as a pretext for dignifying this ancient folklore as worthy of the comparison. None of this special pleading is helpful when we try to reflect on what mature science might really be and what enables it to take shape historically.

There is some interesting historical material in Lost Discoveries, the most extensive and thorough of which is a brief survey of Chinese technology. Teresi reminds us that many of the technological innovations seminal to the efflorescence of modern Europe were imported from China. I say “remind” advisedly, because no one who looks at history at all seriously has ever doubted this. There are no “lost discoveries” here. But again, Teresi avoids ever asking the really serious question evoked by contemplating the splendors of Chinese inventiveness: Why is it that an encompassing tradition of intellectual inquiry based on general theory, continually modified and extended by purposeful experiment and observation, and enriched, where appropriate, by suitably powerful mathematics, never grew up around the astounding Chinese virtuosity in creating practical devices? In short, why didn’t China develop a science worthy of its technology?

Perhaps Teresi, like many others, feels that the question is condescending and demeaning to an incomparably vast and deep culture. But I think that it is demeaning not to ask it. It is precisely the brilliance of Chinese civilization that provokes the question in the first place! It is also possible that Teresi fails to bring it up because, inevitably, it points up the fact that it was modern Europe, after all, that devised the now-universal ethos of science.

As I have already hinted, it is difficult for some people to contemplate the soaring triumphs of European culture without brooding on its great cruelties and crimes. Many of them react with bitter irony, brooding on the crimes and dismissing the triumphs as illusory or meretricious. Lost Discoveries doesn’t quite take that tack — there are no extensive tirades against Conquistadors or gunboat diplomacy — but the intent of the book is clearly deflationary as regards Western pride in Western science. I suggest that a degree of immaturity is at work here, a refusal to face head on history’s grim habit of interweaving the glories of a civilization and its villainies.

But we have to take what Elizabeth Janeway aptly called “the Goddamn human race” as we find it. Glory and villainy are everywhere intertwined in all great cultures, living and dead, and it is not our task to keep either out of our field of view. The best we can do is to deplore what must be deplored and to celebrate what is to be celebrated, without kidding ourselves on either front. Teresi seems, indeed, to be celebrating something — the very real cleverness and insight of a spectrum of non-European peoples — but this is a sly way of refusing to celebrate something unquestionably more important: the emergence, from a distinctly European cultural substrate, of an unrivaled, incomparably powerful framework for investigating the material world.

At heart there is something immoral about Teresi’s effort because it strengthens a myth that is already far too powerful in the current culture, ridden as it is with “identity politics.” The myth, in a nutshell, says that your genetic ancestors do your thinking for you. If you aren’t European (and male), you are impotent as a thinking, creating being unless it can be shown that your forbearers were equal to the Europeans as begetters of knowledge. You cannot hope to be a successful scientist (or poet or philosopher or historian) unless you convince yourself that, twenty generations back, your particular ethnic group was replete with world-class scientists. As a corollary, you learn that the only way to cut into European predominance is to prove to the Europeans amongst us that their ancestors really weren’t what they’re cracked up to be.

The trouble is, however, that, at least as far as the genesis of mature science goes, the Europeans were everything the earlier myths (the ones Teresi tries so frantically to refute) say they were. That won’t stop Lost Discoveries from finding a secure place in the hearts and book purchases of the multicultural priesthood. So far as those folks are concerned, it fills the bill perfectly. After all, Teresi has coauthored a book with the justly celebrated physicist Leon Lederman. He really comes across as a very respectable fellow, and, as multiculturalists reckon virtue, a very virtuous one. He preaches exactly the lesson they wish to hear preached, and does so under a seemingly impeccable canopy of respectability.

This, alas, is the wrong lesson. The true lesson is that your ancestors can’t and won’t do your thinking for you. You have to do that all by yourself. But you’re free — this is a Western innovation too, by the way — to climb the shoulders of whatever giant you choose, regardless of race, color, or national origin.

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