Non-Classical Mathematics 2009 (introductory remarks)

Finally, I can write something about what's going on now. I'm in Hejnice, lodged in a cell (it's quite comfortable though) in a monastery pretty much in the middle of Czech mountains. I was here two years ago, but this time I have wireless internet access. It's pretty cool.

So, what's going on? Well, it seems, Non-classical Mathematics (surprisingly??) attracted more mathematicians than philosophers and logicians. In fact, most of the people present here are rather mathematically-minded. This, of course, is not a complaint. For a philosopher (or a philosophically-minded logician, for that matter), dealing with mathematicians is a bit of a challenge though. They usually spend less time looking for philosophical motivations for their work and more time doing real mathematics. This means, if you're a philosopher, listening to mathematical talks will require more effort. You have to overcome the first impression that people sometimes get into extremally complicated technical issues without explaining why we should be interested in them. I mean, I'm pretty sure these people know what they're doing and why they're doing that, but the standards they employ for motivation of technical work are a bit different. Also, this stuff is often more complicated than most of more philosophically motivated work, so it's a bit more difficult to follow (which means, it's easy for me to feel slightly retarded when faced with all those technical results).

There are, however, also certain clearly positive aspects to this experience. For instance, a few brief conversations I had here confirm my view that doing mathematics doesn't require one to have a clear philosophical position about what mathematics is about (I mean, this is not deeply surprising, I've talked with mathematicians before). For instance, a guy who works on weak set theories, when asked about his view on what set theory is about, said cheerfully something like "I don't care, you know, it's a theory, I play around with it, prove stuff and that's it - what else would I need to know?". It's refreshing. This also means that philosophers of mathematics can still claim there's something for them to do.

This reminds me of P. F. Strawson's remark about analysis of concepts used by specific theorists:
The scientific specialist [...] is perfectly capable of explaining what he is doing with the special terms of his specialism. He has an explicit mastery, within the terms of his theory, of the special concepts of his theory [...] the specialist may know perfectly well how to handle these concepts inside his discipline, i.e. be able to use them perfectly correctly there, without being able to say, in general, how he does it. Just as we, in our ordinary relations with things, have mastered a pre-theoretical practice without being necessarily able to state the principles of the practice, so he, the scientific specialist, may have mastered what we may call a theoretical practice without being able to state the principles [...] a mathematician may discover and prove new mathematical truths without being able to say what are the distinctive characteristics of mathematical truth or of mathematical proof [...] even operating within his own specialism, a specialist was bound to employ concepts [...] from the fact that he there employs them quite correctly, it by no means follows that he can give a clear and general account or explanation of what is characteristic of their employment in his specialism. [Analysis and Metaphysics]
There is a downside to it. If you discuss philosophical aspects of mathematical concepts, mathematicians quite likely won't give a rat's ass about it. I mean, it's to be expected, just like you don't expect a competent kettle user to be interested in someone's specification of sufficient and necessary conditions for something to be a kettle (or an ink-spiller to be interested in philosophically interesting ways of spilling ink). Just like a mathematician might have a hard time convincing philosophers that the complex questions he's trying to answer actually matter, a philosopher might have a hard time convincing mathematicians that philosophical considerations about mathematics have some relevance.

Of course, there is no clear-cut distinction between the mathematicians and the philosophers. I presented a very simplified sketch of some aspects of the extremes of a very interesting and often fruitful tension.

Anyway, all this seems to have some bearing on the Burgess-Rosen critique of nominalist reconstructions of mathematical theories. The gist of the critique is this: when you give a reconstruction, you either give something different from what mathematicians actually have in mind, and thus, you put forward a revolutionary view of mathematics (which is highly impractical, because you're suggesting new textbooks have to be written, mathematics in schools should be changed, etc.), or you are claiming that your theory is an actual analysis of what they're doing, and then you have to show that this really is what they have in mind. Now, it seems to me that mathematicians usually don't have anything philosophical in mind at all when they're doing mathematics, just like we don't have a correct analysis of our every-day concepts when we use them. Thus, giving a nominalistic reconstruction is neither a suggestion that mathematics should be revised (in fact, I believe, a correct nominalistic story about mathematics should rather suggest that everything's okay with mathematics and it shouldn't be changed), nor a theory of what mathematicians have in mind. It's rather a proposal as to how a philosopher can make sense of mathematical activity and mathematical truth without being commited to abstract objects. And what would making sense consist in? Well, telling a nominalistically acceptable story which would be consistent with one's philosophical views and which would allow one to understand on the philosophical level how mathematics can be true and yet applicable. Sort of.

Having said all this, I will switch back to the reporting mode now and post some more detailed remarks on the content of the talks some time soon.


Tuomas said…
Hejnice was good last time, perhaps I should consider coming again next year... Hope you've enjoyed it!
Rafal Urbaniak said…
Yeah, it certainly was fun.:) I still have to post some comments about the talks. For now I just need a break from thinking about logic and mathematics.
Rafal Urbaniak said…
Regarding the mathematicians' approach, here's a bit from Tourlakis' second volume of "Lectures in Logic and Set Theory" (p. 101):

...We do not need to attempt to search for the ``real, definitive ontology'' of sets (whatever that may mean) in order to do set theory, andy more than we bother to search for the real ontology of ``number'' or ``point'' before we allow ourselves to do number theory or geometry, respectively.

From the mathematical point of view we are content to have tools (axioms and rules of logic) that tell us how sets behave rather than what sets are -- entirely analogously with our attitude towards points and lines when we do axiomatic geometry, or towards numbers when we do axiomatic arithmetic.