### Purity of Method

#### Posted by David Corfield

Last week I participated in the inaugural conference of the Association for the Philosophy of Mathematical Practice in Brussels. I decided to sketch some of my thinking on coalgebra, that Café people helped me formulate back here and here. This has turned into an article which will appear next year in *Studies in History and Philosophy of Science*.

Given twenty-five minutes to speak, there was only time to gesture at the connections with computer science, algebraic set theory, and analysis. It’s very tricky choosing a rich and interesting case study which is philosophically salient. To encourage the reader or listener to follow up the mathematics to understand what you’re saying, there must be a decent pay-off. An intricate twentieth century case study had better pack plenty of meta-mathematical punch. The trouble is that mathematics has become so enormously interconnected that when you pull on one strand, it’s easy to find yourself dragging along the whole edifice.

One of the talks I found most interesting was by Andy Arana, which managed to combine simple but deep mathematics, with a historical meta-mathematical debate of possibly current significance. He told us about work on projective geometry in the late nineteenth century, and in particular about reactions to the proof of Desargues’s theorem in real projective planar geometry. One reaction was to think it a theorem that ‘really’ belonged to projective space, rather than the plane. Certainly there’s a very straightforward three-dimensional proof. Others considered this an importation of foreign concepts into the proof – a statement about the plane should not use concepts from three-dimensional geometry.

One group who did not wish to keep planar and spatial geometry separate were the *fusionists*. Here’s Felix Klein from the Preface to *Elementary Mathematics from an Advanced Standpoint*

It has long been the custom in the schools as well as the university, first to study the plane and then, entirely separated from it, the geometry of space. On this account space perception, which we possess originally, is stunted. In contrast to this the ‘fusionists’ wish to treat the plane and space together, in order not to restrict our thinking artificially to two dimensions…

Andy has a strong interest in the notion of *purity* of proof, where a proof only uses concepts which belong to the theorem. This was a live issue for as important a mathematician as David Hilbert. In particular he discusses Desargues’s theorem in terms of the “Reinheit der Methode”, and shows by constructing a non-Desarguesian plane that an extra axiom is necessary.

One modern response, due to Michael Hallett, is to say that implicit within Desargues’s theorem, there are spatial notions. Therefore to invoke spatial concepts is not impure. Andy’s thought is that by the time we cash out this ‘implicit’ we’ll find so much of mathematics involved that just about any proof will count as pure, and the notion won’t do any work. This makes sense when we come to think of all the decategorified shadows of richer structure we like to discuss here.

Say we avoid the ‘implicit content’ approach, and stick with an idea of purity in terms of a tight reading of what is contained in the statement of the theorem. So, we have that no pure proof of Desargues’s theorem is possible, and presumably this is the case also for a whole raft of theorems. Then either we can lose interest in purity, or continue to think it has a mathematical or philosophical salience. I wonder how many readers would say that we need to follow wherever the mathematical structure leads us, whether pure in the strict sense or not.

Hilbert himself writes:

In many cases, our understanding is not satisfied when, in a proof of a proposition of arithmetic, we appeal to geometry, or in proving a

geometrical truthwe draw onfunction theory. Nevertheless, drawing on differently constituted means has frequently adeeper and justifiedground, and this has uncovered beautiful andfruitful relations; e.g. the prime number problem and the $\zeta(x)$ function, potential theory and analytic functions, etc. (p. 6 of Ivahn Smadja, Local Axioms in Disguise: Did Hilbert Really Dismiss Diagrams?)

By the way, John has written some fascinating material about Desarguesian planes. See especially week 145 and week 173, where we find out why the associativity of the coordinate algebra is necessary to extend a plane to a space, and so why there is an octonion plane (non-Desarguesian), but not an octonion space. It seems it comes down to the transitivity of the action of the group of collinearities on quadrangles within the plane.

## Re: Purity of Method

Interesting! Many examples of “impure proofs” that I can think of are now regarded as displaying a deep underlying connection between seemingly disparate fields of mathematics. If this were always the case, then I might be inclined to say that a desire for “purity” in proofs is misguided. Can anyone think of examples of “impure proofs” which you would

notregard as displaying any deep underlying connection?On the other hand, I can probably think of some cases where a pure proof gives more, or at least

different, insight into “why” a theorem is true than an impure one does. But is it better to regard that through the lens of pure vs. impure, or is should it rather be subsumed under the general principle of “the more proofs we have, the better, since each one gives different insight”?(Those are real questions, not rhetorical ones.)