### Foundations

#### Posted by David Corfield

Something that has long troubled me is the question of why philosophers have shown what to my mind is an unwarranted interest in ‘foundational’ mathematical theories which make little or no contact with mainstream mathematics. Two posts have made me reconsider the problem. Alexandre Borovik discusses Zilber’s work on the Schanuel conjecture, while Kenny Easwaran wonders whether string theorists may use mathematics which depends on conjectured axioms of set theory.

In comments to both I mention the model theorist Angus MacIntyre’s interest in foundational theories which do make contact with the mainstream. He personally wants to relate some of Gropthendieck’s ideas to model theory. To return to the philosophers, is their interest in disconnected theories just a vestige of earlier failed foundationalist projects, or is there a continuing rationale?

For myself, I find Yuri Manin’s notion of foundations very attractive:

I will understand ‘foundations’ neither as the para-philosophical preoccupation with the nature, accessibility, and reliability of mathematical truth, nor as a set of normative prescriptions like those advocated by finitists or formalists. I will use this word in a loose sense as a general term for the historically variable conglomerate of rules and principles used to organize the already existing and always being created anew body of mathematical knowledge of the relevant epoch. At times, it becomes codified in the form of an authoritative mathematical text as exemplified by Euclid’s Elements. In another epoch, it is better expressed by the nervous self-questioning about the meaning of infinitesimals or the precise relationship between real numbers and points of the Euclidean line, or else, the nature of algorithms. In all cases, foundations in this wide sense is something which is relevant to a working mathematician, which refers to some basic principles of his/her trade, but which does not constitute the essence of his/her work. (Georg Cantor and His Heritage: 6)

This notion of foundations is closely related to what I’m driving at in this post about getting the ‘causal’ ordering of concepts right.

## Re: Foundations

I doubt that there is anything in the relation of string theory (as currently understood) to set theory that is not already in the relation of quantum field theory to set theory.

From time to time I hear physicists chat about whether or not Gödel’s incompleteness theorem has any implication on physics. But I have never seen a convincing argument.

I could imagine that parts of physics depend on the axiom of choice. But I don’t know to which degree.