Last Friday, Feb. 15, I had the opportunity to host a Friday Forum discussion at the Honors College on whether Mathematics is created or discovered.

One can address the question from a technical metaphysical point of view, but currently I do not find this approach too illuminating or interesting. This was the path followed by Kit Fine in a talk he gave here about two years ago (April 15, 2011). I commented briefly on Fine’s talk on Twitter: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11:

http://news.boisestate.edu/update/2011/03/23/kit-fine/

I attended yesterday a public lecture by Professor Fine, entitled “Mathematics: Invented or discovered”. The auditorium was packed.

I didn’t like some of the points Fine made, and the direction in which he took the discussion, but there were some interesting highlights.

His conclusion: The heart of mathematics is not axioms but procedures for extending the domain of discourse.

For example, we extend the concept of “number” from “natural” to “integer”, “rational”, “real”, …

Fine introduced a calculus based on dynamic logic for “extension procedures”.

This was the core of his talk, one of the parts I mostly disagreed with. Another: Fine seems to think there “is”, e.g., a unique “number 1”.

(As opposed to: this makes no sense, but there are many essentially equivalent representations.)

A cute detail was his portrayal of constructivism, equating it with writers creating fictional characters.

(It made me think all I do is write fan fiction, which made me smile (snicker?).)

I guess Fine’s conclusion is that mathematics is both invented and discovered as they are different parts of his “extension procedures”.

The Friday Forum was a very nice experience. The problem is complex and has a long history. One of the questions it leads to is how to explain the applicability of mathematics. I consulted several references while preparing for the forum, and I think someone else may find at least some of them useful. Let me list a few. Books:

- The applicability of mathematics as a philosophical problem. Mark Steiner. Harvard University Press, 1998.
- Is God a mathematician? Mario Livio. Simon & Schuster, 2009.
- Conversations on mind, matter, and mathematics. Jean Pierre Changeux, and Alain Connes. Edited and translated by M. B. DeBevoise. Princeton University Press, 1995.

Papers:

- Philosophy of mathematics. Jeremy Avigad. In Constantin Boundas, editor,
*The Edinburgh Companion to Twentieth-Century Philosophies,*Edinburgh University Press, 234-251, 2007, also published as*The Columbia Companion to Twentieth-Century Philosophies,*Columbia University Press, 2007. - Does mathematics need new axioms? Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel. The Bulletin of Symbolic Logic,
**6 (4)**, (2000), 401-446.

Related: http://philosophyinprogress.blogspot.com/2010/07/invention-vs-discovery.html