### Is Mathematics Special?

#### Posted by David Corfield

There’s to be a conference held in Vienna next May which asks Is Mathematics Special?

A bringing together of researchers from different fields interested in the question what makes mathematics special (if anything). Even if mathematics presents itself or is presented as a (quasi-)empirical matter, the status of an epistemic exception that mathematics forms among the sciences asks for explanations.

So, is mathematics special? Well, of course, any discipline is special. No other discipline than particle physics builds 27 km. long pieces of experimental apparatus. But one very noticeable feature of the Anglo-American philosophical treatment of mathematics is that it takes it to be a discipline very unlike the natural sciences. This divergence has developed out of the Russellian conviction that mathematics is a (dressed-up) form of logic, and so true in all possible worlds, while the sciences deal with particular empirical events in our universe.

There’s a quotation on the conference website from Felix Klein which suggests some will be looking to play down differences:

Quite often you may hear non-mathematicians, especially philosophers, say that mathematics need only draw conclusions from clearly given premisses and that it is irrelevant whether those premisses are true or false – provided they don’t contradict themselves. Anybody who works productively in mathematics, however, will talk in a completely different manner, In fact, those people base their judgements on the crystallized form in which mathematical theories are presented once they’ve been worked out. The research scientist, like any other scientist, does not work in a strictly deductive way but essentially makes use of his imagination and moves forward inductively with the help of heuristic aids.

Personally, given where we are today in philosophy, I should prefer people to side with Klein and attend to the similarities with other disciplines. In particular, we need to focus on what has been terribly neglected – mathematics’ higher-level aspirations. In its pursuit of phantoms, dreams, or vistas, my sense is that it is not so very different from other forms of intellectual enquiry.

John recommended a short article by Minhyong Kim – Mathematical Vistas – in which after a passage by Grothendieck the author says

It is hoped … that Grothendieck’s candid expressions might convey even to the casual reader that curious sense of the unknown attached to any process of deep learning and thinking, and the urgent conviction at the core of worthwhile endeavor.

If mathematics is special it’s because its practitioners have been deeply learning and thinking for such a very long time.

Posted at December 17, 2007 11:42 AM UTC
## Re: Is Mathematics Special?

It seems that the distincion between

empirical natural scienceand

pure logicis indeed a subtle one, or at least one that deserves further attention.

Math is only the extreme case of various sorts of “intellectual enquiries” that are more or less rooted in phenomenology and more or less in pure logic.

To start on the far side, there is exobiology: people are reasoning about which kind of organisms are thinkable in principle, which might exist somewhere, or, if not existing already somewhere or having existed at some point, might be forced into existence by suitable experimental setups.

Similar thoughts are being thought in chemistry, clearly. I guess here it can already happen that reputable practitioners spend the better part on their career investigating properties of compounds, that possibly nobody has ever, so far, created in the lab. Or maybe at least not probed in detail in the lab.

Then of course physics. I don’t have to remind anyone here about the huge discussion about to what extent it is healthy for theoretical physics to move into territory completely detached from experiment.

Viewed from this perspective, math seems to be not singular but just extreme. I think.