Descartes and Viete

Viete was a French mathematician whose `The Analytic Art' of 1591 was a standard text on algebra. Viete starts the book with a description of mathematical analysis, breaking it into three parts: zetetics, poristics, and rhetics, with the following definitions:

1. Zetetics: setting up an equation or proportion between the unknown and the givens
   
2. Poristics: testing the truth of a theorem by means of an equation or proportion
   
3. Rhetics: determining the value of an unknown term in an equation or proportion
   

As an example, suppose I know that 3 doughnuts cost 2 dollars; how much would 12 doughnuts cost? Saying `2 is to 3 as x is to 12' or '2/3 = x/12' is zetetics, while solving for x to find x = 8 is rhetics. I can generalize this to find a general theorem: N doughnuts cost (2/3)N dollars; plugging in values for N to verify this theorem is poristics (although this is a lame example of poristics; I think it's more along the lines of trying to figure out the truth value of a proposed equation, not a derived one).

One would think that Descartes was familiar with this book, and with this sort of analysis. The sixteenth century was a time of re-definition for mathematics, in which mathematicians included these little prefaces about how to practice mathematics; in a sense, they were establishing professional methodology. Descartes and other early moderns would inherit these explicit statements of methodology; would then they seek to carry mathematical methodology over to other fields? Note in particular that mathematicians of the time claimed that their field led to truths whose certainty was unchallenged (it's annoying that `certain' has two meanings; `certain truths' means either `some, but not all, truths' or 'undoubtable truths', and I want the latter). How was the early modern search for axioms and a deductive system (Descartes wanted to start with just `I think, therefore I am' and deduced from there; Hobbes started with a state of nature and deduced from there) influenced by the professionalization program of 16th century mathematics?

And thanks have to go out to Jeff Mullins at this point; this sort of a question wouldn't have occurred to me without having gone through his History 610 course.