Viète's Enigmatic Species
What were "species" in the algebra of Francois Viète? The 16th-century mathematician Viète is rightly recognized as the "founder of modern algebra," although, curiously, he regarded himself not so much an innovator as a restorer of a lost art of the Greek mathematicians. The formal structure that Viète imposed on algebraic reasoning has remained in effect even to this day. Yet his mathematical innovations (such as the introduction of literal symbolism and the first true theory of equations) were all motivated and coordinated by a central concept he called "species," which for Viète constituted the proper object of algebra. Isn't it a sorry state of affairs that no one knows exactly what Viète's new algebraic entities were, or the exact role they played in his overhauling of algebra?
It has long been realized that Kepler's astronomical researches were motivated by deep mystical and Pythagorean convictions about the numerical harmonies in the cosmos. What has never been recognized is that the algebraic art of his near contemporary François Viète, arguably the "father of modern algebra," was a metaphysical arithmetic and logistic, a deliberate attempt to reconceptualize the numerical algebra of his day in terms of the mysterious "eidetic" or "ideal" numbers of Plato - hence Viète's choice of the term "species" to describe the proper object of algebra. This may come as a surprise to modern mathematicians, who for the most part balk at anything smacking of metaphysics in mathematical inquiry.
Viète made only a few enigmatical statements about his species in his published writings and died just as his writings were being edited and collected together. For this reason, his successors did not know exactly what he meant by this central concept and attempted to again reconceptualize the species concept in terms of mathematical number, which, ironically, the species were intended to replace. This observation has led us to several metamathematical questions: Does algebra have a "proper object?" How could Viète's species have been reconceptualized as number without leaving serious fault lines in algebra itself? Are Viète's species still invisibly at work whenever we do algebra, masquerading as numbers, or were they lost through reconceptualization?
Again, Viète regarded himself as a restorer of a lost art that was deliberately hidden by the Greek mathematicians, traces of which Viète thought he discerned in the algebra of Diophantus, in the Greek geometrical art of analysis and synthesis, and in other classical sources. Viète's profound rehabilitation of algebra, then, was not simply the work of a mathematician working on mathematical problems. Accordingly, we must ask ourselves whether Viète would ever have undertaken his bold rehabilitation of algebra and entertained such "innovative" concepts, if he had not thought that he was restoring the lost art of a golden age.