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
Norman Levitt
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 resonate
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 lesson that he reads 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 far-fetched,
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 have no hesitation in labeling 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 mathematicians,
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 by far 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 mathematics 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 fact 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. Furthr 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
that 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 easily 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 tactics. 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 anything like
the way Cantor did (and aside from a gratuitously insulting reference,
a paragraph on, to Galileo's premonitory work on 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 is irrational-but that dates from ca. 1500 A.D.) 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 celebrate fervently 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 internationalization 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-come-lately. 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 unrivalled, 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 forbears 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 co-authored 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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