Remarks in honor of Gian-Carlo Rota on the occasion of his 60th birthday.
by Jack Schwartz
I need a hero and a text for the brief sermon that I want to preach in Gianco's honor. As my text, I take an observation made in September 1583 by the celebrated Jesuit Matteo Ricci, when he arrived in China carrying a copy of Euclid's Elements: "In China, the penalty for studying mathematics without the Emperor's authorization is death. However, this law has fallen into disuse." (An interesting character this Father Ricci: born Ancona, 1552; a law student at Rome from his 16th year and a member of the Jesuit order at 19; graduate student of mathematics and astronomy from 20th to 24, 31 when he entered China, 49 when, 18 years later, he was finally allowed to enter Beijing. He remained there for 9 years till his death.)
As my hero I take Aristarchus of Samos, who in about 280 BC determined the sizes of the moon and sun (showing that the sun was much larger than the earth, even though his estimate of the sun's size was off by a factor of 15: Aristarchus made the sun out to be 6 1/2 times the linear size of the earth, instead of the true 109 times.) I ask: why did the Chinese, though ahead for a millennium in some relevant branches of science, never come close to this accomplishment?
Since it calibrates the intellectual tools at Aristarchus' disposal so nicely, let me quote a heading from his monograph "On the sizes and distances of the sun and moon": Proposition 15: The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6. Geometry is here in all its rigor and system; but tools adequate for routine numerical computation plainly are not.
Until the birth of European algebra with Leonardo da Pisa in 1202 and its first steps to leadership with Cardano and Tartaglia in 1545, the Chinese stood well ahead of Europe in numerical technique. Their arithmetic was decimal (and implicitly place based) from the earliest period. Chinese numerical solutions of systems of any number of linear equations go back to the second century AD. Many techniques for arithmetic calculation were well understood by the fifth century; for example, in 480 AD, in his Art of Mending, Chu Chongchi used the inscribed polygons method to determine Pi accurate to 6 decimal places. This held the record for nearly a thousand years. The Pascal triangle of binomial coefficients is clearly seen in Jia Xian's Solutions to Mathematical Problems of Huangdi, published around 1030 AD, a full 6 centuries before Pascal. By the 14th century, numerical solution of systems of n-th degree equations in four variables had been described.
The form in which Chinese algebra existed is particularly interesting, in that they had nothing approaching an algebraic notation. What they had instead were abacus-like (but two-dimensional) 'counting boards' configured into rectangles in which decimal digits could be represented by deposited sticks. Such a board is exactly equivalent to a computer memory of a hundred bytes or so. So Chinese algebra existed in the form of systematically described algorithms for the manipulation of these digits to accomplish a wide variety of algebraic calculations. In this 'computer algebra', which antedates symbolic algebra by several centuries, the Chinese had a significant potential advantage. A second advantage lay in their sophisticated map-making, which could draw on a fully systematic imperial geodetic survey. But none of this led to so much as a good theoretical discussion of the size of the earth, a matter familiar to Greek science from 300 BC.
Thus Aristarchus' accomplishment reveals the depth of the Euclidean influence on early science. Like Einstein's leap to Special and General Relativity 23 centuries later, Aristarchus' work, though not without intellectual fore-runners and subsequent improvements, is particularly striking as an act of nearly pure intellect. Like Einstein's work, it rested on just a few, already well established, observations. In Aristarchus' case these were the size of the earth's shadow as observed during eclipses, the (almost equal) angular sizes of the sun and moon, and (most sensitively) the observed moon-earth-sun angle at the precise instant of half-moon. In this his work contrasts, say, with Galileo's discovery of the moons of Jupiter, which like the many of greatest scientific discoveries belonging to the second rank intellectually, followed instead from the improvement of instrumentation.
Why was nothing of Aristarchus' height achieved in China? Though this question is far too deep and obscure for after-dinner remarks to answer, I would like to say just a few things about it. First of all, Euclid had established a model for fully systematic, precise, and indisputably reliable reasoning, much transcending the merely verbal techniques of Greek political, and even philosophical, reasoning. As Aristarchus' concern for the precise angular position of the sun relative to the moon at half-moon reveals, the scientific role of mathematical rigor is to provide a framework within which the salience of otherwise indifferent or minor details stands out sharply. In scientific history the sharp highlights thrown on these quantitative details have continually encouraged development of 'magnifiers' of all sorts, of microscopes, telescopes, spectrometers, diffusion chromatography, and so forth, thereby encouraging creation of science's characteristic instrumentation .
This influence is technical and intellectual. There is a social-intellectual factor of even greater importance: openness to surprise. Any radical resolve to protect the details of an ideology, whether religious or secular, breeds a hostility to science which can be crippling. This is seen in all fundamentalisms, alive and dead, from the creationists and radical deconstructionists of our own day to the official Hegelians of the former Soviet Union. As a classic, and characteristically ferocious, statement of this anti-scientific view I quote the Emperor Diocletian's Edict Against Manichaeism of about 300 A.D.: 'It is the height of criminality to reexamine doctrines once and for all settled and fixed by the ancients, doctrines which hold and possess their recognized place and course.' Till after 1630, the official statutes of Cambridge University fined Bachelors and Masters of Arts five shillings for each point in which they diverged from Aristotle's Organon. Though secular, China's relentless official insistence on social relevance as the goal of all study had similar stunting effects. To Diocletian I counterpose Francis' Bacon's 'There are four classes of idols which beset men's minds. To these for distinction's sake I have assigned names, calling the first class Idols of the Tribe; the second Idols of the Cave; the third Idols of the Market-place; the fourth Idols of the Theatre. .. There are idols which have immigrated into men's minds from the various dogmas of philosophies, and also from wrong laws of demonstration. These I call Idols of the Theatre, because in my judgment all the received systems are but so many stage-plays, representing worlds of their own creation after an unreal and scenic fashion. .. Neither again do I mean this only of entire systems, but also of many principles and axioms in science, which by tradition, credulity, and negligence have come to be received.'
(Diocletian's AD 296 Edict on Maximum Prices is also worth quoting: Maximum legal Elementary School Tuition, per boy, per month 50 cents; maximum legal tuition for Arithmetic teaching per boy, per month 75 cents; for Geometry 200 cents; for Rhetoric and Public Speaking 250 cents. So even in the third century mathematics teachers outdid their unspecialized colleagues;-- but law school professors did even better.)
In their continuing concern for the integrity of the ideological apple-cart one may see an important reason for the Chinese failure. Even the names of the Chinese Emperors' palaces and cities: 'Gate of Heavenly Peace' - 'Hall of Abiding Tranquillity' - 'Pacified West' proclaim a distaste for surprise. Imperial hostility to the unauthorized study of mathematics had the same root. Mathematics could all too easily draw unauthorized students into the far more forbidden subject of astronomy. To the Chinese bureaucracy, astronomy's proper role was preparation of official responses to celestial phenomena, like comets, novae, and eclipses, which could threaten a change of dynasty. Astronomy also underlay empire-wide promulgation of the official calendar, a carefully guarded Imperial prerogative, which the founders of rebellious dynasties often revised to announce new regimes. For this reason, astronomical observations were guarded in the way that critical nuclear cross-sections are held today, and made available only within a narrow circle of government officials with a need to know.
We can understand something of the weight of this continuing control by contrasting the consequences of Columbus' 1492 voyage with that of the Chinese Admiral Cheng Ho's expedition to East Africa in 1405. Columbus traveled 3,600 miles with three ships crewed by about 300 men; Cheng Ho 5,000 miles with 62 ships and roughly 30,000. But Columbus' voyage triggered an immediate fever of exploration and conquest: Cortez to Mexico 19 years later; Pizarro in Peru 39 years after Columbus; Coronado to Kansas in 1541. In China, the Emperor saw no sufficient reason to follow up actively on Cheng Ho's voyages. The expedition records were simply filed in the Imperial archives, and scrapped some time later by an administrator who deemed them an unfortunate wast of government money. Perhaps our own failure, after 27 years, to follow up on the 1969 moon landings reflects influences of the same sort.
These diametrically opposed outcomes hint at another, probably very significant influence: the force of competition. The Chinese Emperors, unquestioned masters of their own continental world, could ignore, at least for a while, any development in the outer 'barbarian' zone with which they chose not to be concerned. The violently contending 16th century European powers (like the U.S. during the Cold War's 'Sputnik' episode) were driven inexorably forward by the fear that one of their competitors would get the better of any situation which they neglected to exploit fully. One sees this clearly enough in the sponsorship of Cabot's New World explorations (nearly contemporaneous with Columbus) by the generally parsimonious Henry VII of England. One sees it even more nakedly in a very interesting 1493 letter of Queen Isabella to Columbus, quoted in Felipe Fernandez-Armesto's excellent biography of the Admiral: "Don Cristobal Colon, my Admiral of the Ocean Sea, Viceroy and Governor of the islands newly discovered in the Indies. With this messenger I send you a copy of the book which you left here, which has been so long delayed because it has been made secretly so that the Portuguese emissaries here should not know of it, nor anyone else; and for the same reason it has been done in two hands, as you will see, for the sake of speed. Certainly, according to what has been said and seen in the present negotiations here [ed. note: 1493, under Papal auspices; concerning division of the continents between Spain and Portugal], we know from day to day the importance, greatness, and substantial nature of the business... The sea-chart which you have to make you will send me when it is finished; and to serve me you will make great speed in your departure, so that, if the Lord is gracious, the chart may be commenced without delay, for you must see how important it is to the progress of the negotiations. And of all that happens at your destination you will write and always let us know. In the Portuguese negotiations nothing has been decided with the envoys who are here, although I believe their King will come to see reason in the matter..."
A closely related factor, which dependence on, or control by, a central authority tends to limit, is the size of the population of research workers able to contribute, if not major monuments, then at least crucial heaps of brick. As with today's Internet fever, flourishing application links make initially esoteric scientific ideas familiar to a very large community, and this can eventually illuminate their further possibilities even to professionals directly involved in developing new ideas. It is often recognized that Europe's first renaissance steps in algebra were tied to the demand of expanding commerce for improved techniques of business arithmetic. Today we can note that industry has been a more important source of new programming languages and of such basic ideas as transaction-based systems than has the academic research community. Speaking here, I am happy to say that these are influences with which MIT has engaged much more successfully than Harvard.
Finally there is luck. The Greek love affair with lines, circles, and spheres was pregnant with much science: just the tools need to advance astronomy, and then optics. Even the initially esoteric mathematics of conic sections developed by Archimedes and his Alexandrian contemporaries proved to be just what Kepler needed to describe planetary orbits 19 centuries later. (This step from pure aesthetic to scientific application shows how truly the mathematician's work can count on being amortized over eternity.) Algebra, the zone of Chinese advantage, offered much less until Newton elevated it to calculus.
The net result of all these influences was beautifully summarized by another China-based Jesuit, J. B. Du Halde. In 1735 he wrote: 'When we cast our eyes on the great number of libraries in China magnificently built, finely adorned, and enrich'd with a prodigious Collection of Books; when we consider the vast number of their Doctors and Colleges established in all the Cities of the Empire, their Observatories, and their constant Application to watch the course of the Stars, and when we further reflect that by Study alone the highest Dignities are attained, and that Men are generally prefer'd in proportion to their Abilities; that according to the Laws of the Empire the Learned only have, for above four thousand years [sic], been Governors of Cities and Provinces, and have enjoy'd all the Offices about the Court, one would be tempted to believe, that of all the Nations in the world China must be the most knowing and most learned.
However a small acquaintance with them will soon undeceive one; for tho'
it must be acknowledg'd that the Chinese have a great deal of Wit, yet it is not an inventive, searching, penetrating Wit, nor have they brought to perfection any of the speculative Sciences which require Subtilty and Penetration.
Yet I am not willing to find fault with their Capacity, since it is very plain that they succeed in other things which require as great a Genius and as deep a Penetration as thc speculative Sciences; but there are two principal Obstacles which hinder their Progress in these kinds of sciences: 1. There is nothing within or without the Empire to stir up their Emulation; 2. Those who are able to distinguish themselves therein have no Reward to expect for their Labour.
The chief and only way that leads to Riches, Honors, and Offices, is the study of the Canonical Books, History, the Laws, and Morality; it is to learn to write in a polite manner, in Terms suitable to the Subject treated upon; by this means the Degree of Doctor is obtained, and when that is over they are possessed of such Honour and Credit that the Conveniences of Life follow soon after, because then they are sure to have a Government Post in a short time; even those who wait for this Post, when they return into their Provinces, are greatly respected by the Mandarin of the Place, their Family is protected from vexatious Molestations, and they there enjoy a great many Privileges.
But as there is nothing like this to hope for by those who apply themselves to the speculative Sciences, and as the Study of them is not the Road to Affluence and Honours, it is no wonder that these sort of abstracted Sciences should be neglected by the Chinese.'
So wrote Father DuHalde in the 18th century. I invite those of you who administer science or guide its development to reflect on his remarks, and on this history.
Jacob T. Schwartz is a professor of Mathematics and Computer Science at the Courant Institute of Mathematical Sciences at New York University.