Mathematics and René Descartes

Question

Why are axioms of mathematics not recognized by René Descartes?

How can they be untrue? Can we not treat them as basics which are absolute and build upon them?

Answer 1

At Descartes' time, only Geometry (see Euclid's Elements) had axioms.

See Descartes' Mathematics: Descartes' new (analytical) Géométrie was not axiomatized.

But Descartes clearly conceived mathematics as the paradigm of certainty; see the Rules:

"In the Rules, he sought to generalize the methods of mathematics so as to provide a route to clear knowledge of everything that human beings can know. His methodological advice included a suggestion that is familiar to every student of elementary geometry: break your work up into small steps that you can understand completely and about which you have utter certainty, and check your work often."

The Cogito's argument "applies" also to mathematical statements:

"[Regarding] those matters which I think I see utterly clearly with my mind's eye... when I turn to the things themselves which I think I perceive very clearly, I am so convinced by them that I spontaneously declare: let whoever can do so deceive me, he will never bring it about that I am nothing, so long as I continue to think I am something; or make it true at some future time that I have never existed, since it is now true that I exist; or bring it about that two and three added together are more or less than five, or anything of this kind in which I see a manifest contradiction. (Med. 3, AT 7:36)"

In conclusion: mathematical truth are "evident" (i.e. their denial implies a contradiction) and they are guaranteed by God.

Answer 2

It is an irony of history that Aristotle's First Philosophy came to be seen as coming last and was dubbed Meta-physics; for Descartes it is still the First and its answers determine the rest. And first of all are the questions about what exists. So in the Discourse part IV Descartes wrote:

I was disposed straightway to search for other truths and when I had represented to myself the object of the geometers... I went over some of their simplest demonstrations. And, in the first place, I observed, that the great certitude which by common consent is accorded to these demonstrations, is founded solely upon this, that they are clearly conceived in accordance with the rules I have already laid down In the next place, I perceived that there was nothing at all in these demonstrations which could assure me of the existence of their object: thus, for example, supposing a triangle to be given, I distinctly perceived that its three angles were necessarily equal to two right angles, but I did not on that account perceive anything which could assure me that any triangle existed: while, on the contrary, recurring to the examination of the idea of a Perfect Being, I found that the existence of the Being was comprised in the idea in the same way that the equality of its three angles to two right angles is comprised in the idea of a triangle...

One should note that Descartes had no qualms about admitting "degrees of reality" and, as most of his contemporaries, thought 'the real' to be somehow superior to 'the possible'. (Leibniz said that god gives reality to the best of possible worlds.) It was only after the discovery of non-Euclidean geometries that people came to accept 'the real' as a particular case of 'the possible'. Eschewing ontology early in the XX.c. was called "ifthenism"