### A River and a Trickle

#### Posted by David Corfield

I was sifting through The Oxford Handbook of Philosophy of Mathematics and Logic to see whether it included any trace of my philosophical heroes. Lautman was always going to be an outside bet, but I reckoned on finding Lakatos there.

Conspicuously absent in the index, I resorted to Google Books to help me search, and there, along with some mentions of Lakatos as editor of a book that Kreisel had contributed to, I finally found in the introduction:

Sometimes developments within mathematics lead to unclarity concerning what a certain concept is. The example developed in Lakatos [1976] is a case in point. A series of “proofs and refutations” left interesting and important questions over what a polyhedron is. For another example, work leading to the foundations of analysis led mathematicians to focus on just what a function is, ultimately yielding the modern notion of function as arbitrary correspondence. The questions are at least partly ontological.

This group of issues underscores the

interpretativenature of philosophy of mathematics. We need to figure out what a given mathematical concept is, and what a stretch of mathematical discourse says. The Lakatos study, for example, begins with a “proof” consisting of a thought experiment in which one removes a face of a given polyhedron, stretches the remainder out on a flat surface, and then draws lines, cuts, and removes the various parts–keeping certain tallies along the way. It is not clear a priori how this blatantly dynamic discourse is to be understood. What is the logical form of the discourse and what is its logic? What is its ontology? Much of the subsequent mathematical/philosophical work addresses just these questions. (Shapiro 2005, p. 10)

What on earth does that final sentence mean? Why bother dealing with Lakatos so briefly in the introduction to a 800 page handbook if he is not to be considered at greater length later?

Where we represent the state of philosophy of mathematics as two streams, it seems as though those working in one of them take theirs to be the Ganges and ours an irrigation ditch.

But the ditch continues to issue forth good, if expensive, work. Jean-Pierre Marquis has recently published From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory.

The main thesis is that Klein’s Erlangen program in geometry is in fact a particular instance of a general and broad phenomenon revealed by category theory. The volume starts with Eilenberg and Mac Lane’s work in the early 1940’s and follows the major developments of the theory from this perspective.

I also have my hands on Ladislav Kvasz’s Patterns of Change, one of which,

…called relativization, is illustrated by the development of synthetic geometry, connecting Euclid’s geometry, projective geometry, non-Euclidean geometry, and Klein’s Erlanger Programm up to Hilbert’s Grundlagen der Geometrie. In this development the notions of space and geometric object underwent deep and radical changes culminating in the liberation of objects from the supremacy of space and so bringing to existence geometric objects which space would never tolerate.

One book I don’t have yet is an explicit attempt to bridge philosophy’s two cultures. It is called The Philosophy of Mathematical Practice.

## Re: A River and a Trickle

`left interesting and important questions over what a polyhedron is/

reading current math literature, I think the term is not yet uniquely defined!

polytope seems doing a bit better