Descartes' foundationalism [closed]

Question

Is the cogito an axiom from which we can reason axioms of mathematics? Was Descartes' aim to make mathematics (and other fields of knowledge) reducible to the cogito?

Answer 1

The cogito has to serve as the bedrock, the foundation, the keystone, for all epistemology. It's the only knowledge that's certain. In other words, if math isn't based on the cogito, it should be. However, math is an axiomatic system and so can do without cogito support.

Answer 2

Descartes was not just a philosopher, he was a genuine mathematician who made genuine contributions to the field. If his beliefs about the cogito cohered in his mind with his beliefs about his mathematical assertions, then he would have to have either found a relevant "derivation" somewhere down the line, in his system, or at least have shown that his fundamental mental distinctions (e.g. between adventitious and innate concepts) were reflected in the substance of mathematics (mathematics-as-an-instance, an instance of innate knowledge).

I don't remember the details, but Descartes brings up a 1000-sided regular shape, a "chiliagon" I think (I can't help but always imagine a polygon whose sides are made of chili), to argue about the difference between higher/purer and lower/weaker knowledge, where the visualizable knowledge of a 1000-sided figure is less than the abstracted knowledge of the same. Depending on how strong his representation of algebra, especially algebra-as-logic (or logic as algebraic), was, this all might have played into deriving the foundations of mathematics from the cogito.

It is frequently seen in the presentation of the cogito, even in its more strictly axiomatic form, a case where circular knowledge is implicitly supposed or at least there are still unfairly unfounded claims being made in the background. If it is a theorem, what is it a theorem of? If in some sense it is a theorem, then the reasoning involved has a form, however basic and unramifying it might be. The form of this reasoning might be styled, in a Cartesian system, as the form of mathematical, as algebraic, logic.

Descartes does go so far as to say that even arithmetic is subject to abstract doubt. I don't recall what Descartes' use of the "chiliagon" concept was, I think it was prearithmetical maybe. But so far as an act of doubt is the initial node to which his assertoric regress goes, there is an erotetic moment in the system,⁺ so that Descartes could have adapted other erotetic structures as erotetic premises in his philosophy of mathematics: the use of variables in algebra is the use of erotetic terms (not full erotetic sentences), so that inputting such a term into an equation yields an erotetic sentence. Which forms/parts of algebra Descartes would have described as indubitable, as encoded into the premises or the pattern of inference involved in Descartes' theory of the cogito, I have no real idea of my own about (I guess one would look into which family of algebra/geometry his system has been associated with under modern taxonomies of algebras and geometries).

⁺In a more modern-sounding way, you could say that there is a doubt operator whose major functions include taking a question and reforming it as an assertion: "Is X true?" goes to, "I doubt that X," and vice versa. By virtue of, "What is X? you get wh-terms, and then wh-terms are algebraic terms par excellence.

Answer 3

No.

Descartes was a dualist par excellence. Mind and body, finite and infinite, thought and extension exist side by side. He was an amazing writer, his prose is crystal clear, he solved many long standing philosophical problems, and provided a new foundation for philosophy.

However, his dualism created just as many new problems, and philosophers will argue about them for centuries to come. For example, sure, thought and extension exist side by side, but we as humans definitely have a body and a mind, and they clearly interact, but how and in what way remains very problematic.

Same thing goes for his mathematical and philosophical theories. They are separate, they exists side by side, and they have nothing to do with one another. Once again, they seem to overlap and that again gives birth to a whole bunch of new problems.

I have read his challenging "Geometry", and there is no cogito business in that book as far as i can remember. Cogito is discussed in his philosophical text "Mediations". Cogito is an epistemological concept needed for some areas of knowledge, but not pure mathematical knowledge. Axioms that lie at the foundation of pure mathematics require no cogito.