Prisca Sapientia

It's ironic that most of the men who participated in the "scientific revolution", whose contributions seem (to us) so original and innovative, were themselves convinced that they were merely re- discovering the vast body of pristine knowledge (prisca sapientia) that had been possessed by the ancients, but somehow lost and forgotten during the centuries that came to be called the "dark ages" of western civilization. This was not an entirely unreason- able belief, because the great works, both material and intellectual, of the classical civilizations were (and to some extent still are) very imposing. The intellectual culture of Western Europe really did decline during the fall of Rome, and the institutions for preserving and passing along knowledge, as well as the inclination to do so, were severely diminished. Then, after so long an absence, when the ancient texts were re-discovered, the scholars of the Renaissance and later periods were acutely aware of their intellectual inferiority vis-à-vis "the ancients". Also, the fact that many of the ancient texts were now available only in fragmentary form, often in third- hand translations, and many of the references were to works totally unknown and presumably lost, contributed to the impression that the ancients had known far more, if we could only find it out.

This attitude toward the past is, in some ways, the exact opposite of our usual view today, which is of a totally ordered sequence of eras progressing from less knowledge in the past to more knowledge in the future. It's hard for us to imagine, today, the intellectual climate among people who believed (knew) they were scientifically and mathematically inferior to their ancestors in the distant past. Interestingly, this peculiar historical circumstance undoubtedly contributed to the unique flourishing of intellectual affairs in Western civilization that occurred soon after the ancients had been re-discovered. Part of the psychological impetus came from the great appreciation they felt for recorded knowledge, and the esteem they had for the great thinkers of antiquity. Also, the enduring value of the recorded knowledge (if it was preserved), and the kind of immortality it gave to the authors, surviving a millennium of neglect only to be more wondered at when finally re-discovered, was a source of immense fascination, and inevitably tempted men to participate in the process, even if only (at first) by translating and copying the great works.

The early 16th century discovery of the general solution of cubic polynomials is regarded by some people as a significant turning point in scientific history, because this was the first time a "modern" man made a significant discovery that went beyond the ancient knowledge. (Needless to say, there were acrimonious disputes between Cardano, Ferro, Tartaglia about who deserved to be credited with this discovery.) The tantalizing prospect of "bettering" the ancients was thus raised, and was an incredibly powerful incentive for making intellectual discoveries. Of course, far more important for convincing Europeans that it was possible to know more than the ancients was the discovery of The New World, beginning with Columbus's voyage in 1492, a world of which the ancients had not even dreamed.

Nevertheless, the belief that the ancients had possessed a vast body of knowledge, of which we have only fragments and scattered hints, persisted. As late as the 1600's men like Fermat were developing their original ideas in the form of speculative "reconstructions" of lost works from antiquity. For example, Fermat completed a re-construction of Appolonius' lost work on "Plane Loci", and Fermat himself said that this effort led directly to his development of what we now call analytic geometry. (Needless to say, there was an acrimonious dispute about whether Fermat or Descartes deserves credit for this discovery.)

Newton was convinced that "the ancients" had used analytical methods to arrive at their results, and then consciously concealed their methods by expressing the results in synthetic form. Regarding the solution of the locus problem, Newton remarked

 ...they [the ancient geometers] accomplished it by certain simple proportions, judging that nothing written in a different style was worthy to read, and in consequence concealing the analysis by which they found their constructions.

Even with regards to the calculus, Christianson's biography of Newton tells us that

 ...in May 1694...Newton had recently completed his brilliant mathematical treatise 'De quadratura' which introduced the now familiar dot notation for fluxions, and he expressed the belief [to David Gregory] that its contents were known to the Greeks, who had destroyed all evidence of algebraic analysis in favor of more elegant geometrical proofs.

In discussing the question of why Newton, the inventor of the calculus, avoided any explicit use of it in his Principia, Christianson says

Had he been more forthright, he would have simply admitted to his preference for classical geometry on the grounds that it was more elegant than the analytic algorithms of the fluxional calculus, and to his belief that it had enabled the ancients to discover what he was only rediscovering some two millennia later.

Of course, Heiberg's 1906 discovery of "The Method", in which Archimedes described (in a private letter) analytical techniques - including what we would call an intuitive form of fluxional calculus - that he had used to discover his most important theorems, showed that Newton's suspicion had some validity.

Newton was clearly influenced by the hermetic tradition, which attributed all kinds of wisdom and secret knowledge to "the ancients", not just mathematical. For example, he told Fatio in 1692 that the ancients knew the law of gravitation, and David Gregory noted that "Newton believed his natural philosophy was most consistent with the teaching of Thales, while the Egyptian Thoth 'was a believer in the Copernican system'."

Returning to mathematics, not everyone shared Newton's generous view of the motives of the ancient sages in presenting their results in the more elegant (i.e., synthetic) form. Wallis (never having seen "The Method", of course) commented on the distinctly cryptical progression of many of Archimedes' presentations that they seemed to him

...as it were of a set purpose to have covered up the traces of his investigation, as if he had grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results… Not only Archimedes, but nearly all the ancients so hid from posterity their method of Analysis (though it is clear that they had one) that more modern mathematicians found it easier to invent a new Analysis than to seek out the old.

Boyer says that Torricelli expressed similar sentiments. These men evidently believed that the motive of the ancients in presenting their work in synthetic form was not the striving for elegance but just a selfish effort to keep their methods secret. (It's amusing to speculate that the synthetic form of presentation so closely associated with rigorous mathematics, and so widely used in education for so many centuries (e.g., Euclid) may have been originally just a cynical strategy to conceal the actual thought processes, like zero-information proofs!)

Going even further in impugning the not-so-prisca motives of the ancients, not to mention disparaging the fullness of their sapientia, Descartes wrote (Regulae, Rule IV, quoted in Calinger) that

We have sufficient evidence that the ancient geometers made use of a certain "analysis" which they applied in the resolution of their problems, although, as we find, they grudged to their successors knowledge of this method...

I could not but suspect they were acquainted with a mathematics very different from that which is commonly cultivated in our day. Not that I imagined that they had full knowledge of it; their extravagant exultations, and the sacrifices they offered, for what are minor discoveries suffices to show how rudimentary their knowledge must have been... [!]

 ...[but] I am convinced that certain primary seeds of truth implanted by nature in our human minds - seeds which in us are stifled owing to our reading and hearing, day by day, so many diverse errors - had such vitality in that rude and un-sophisticated ancient world that the mental light ... enabled them to recognize true ideas in philosophy and mathematics, although they were not yet able to obtain true mastery of them... These writers, I am inclined to believe, by a certain baneful craftiness, kept the secrets of this mathematics to themselves.

 Acting as many inventors are known to have done in the case of their discoveries, they have perhaps feared that their method being so very easy and simple, would if made public, diminish, not increase public esteem. Instead they have chosen to propound, as being the fruits of their skill, a number of sterile truths, deductively demonstrated with great show of logical subtlety, with a view to winning an amazing admiration, thus dwelling indeed on the results obtained by way of their method, but without disclosing the method itself - a disclosure which would have completely undermined that amazement.

Here Descartes is claiming that the ancients not only kept their true methods secret, but did so for the basest of reasons, to cover up the fundamental simplicity of these results when approached analytically. Essentially Descartes accuses the ancient sages of perpetrating a conscious fraud on the uninitiated - and on posterity. This shows what a distance the western intellectual community had come from the wonder and awe that they once felt toward the ancient writers.

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