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Thursday, April 27, 2006

Thinking loudly

Blogs are great places to think aloud, or to think loudly, as Imre Lakatos used to say. I heard one of his former students John Worrall once lament that there was so much more of great substance that Lakatos had said than appears in his surviving writings. It will be interesting to see whether the keeping of a blog effects one's 'proper' writing, whether more of those small but important insights, which the brevity of a blog post can support, sharpened by readers' comments, will be preserved when one writes more formal pieces.

After our discussion following this post, Denis Lomas has written to me to say:

Set theory universalism is a great overarching story of the past century of mathematics. (It's likely part myth.) Of course, it shouldn't override or erase other stories.
This is a useful corrective if I seemed a little dismissive of set theory. I couldn't agree more with this comment, but time stops still for no-one, and now that the set theoretic impetus has led it far from the centre of mathematical concerns, we should keep our eyes open for other grand stories. These we can help articulate by pushing them to extremes, or by forcing them to confront rival stories. The previous blog post was an exercise in the former. If you take the categorification program seriously, basic theory like Euclidean geometry ought to be categorifiable (not the prettiest word). I wish the noncommutative geometry program had a resident philosopher to probe it, and help us understand the tensions between rival accounts (besides Connes, try this blog). Then we might link up to see how Grothendieck's and Connes's conceptions of space face up to each other.

Other fields allow philosophers to do this. In research for the illness book (publication date now pushed back to March 2007), I came across an interesting paper by Kenneth F. Schaffner, 'Assisting immunologists to examine the philosophical foundations and implications of the new theories of tolerance', in Singular selves, Historical issues and contemporary debates in immunology, AM Moulin, A Cambrosio (eds.), Elsevier. pp. 86-93, 2001. As the title suggests, exponents of rival conceptions of immunity, such as self/nonself and danger theorists, were brought together in a debate run by Schaffner. One of the advantages of having a philosopher well-informed in the history of immunology to host the discussions is that they can pick up on a loose use of a term of, say, Popper or Kuhn. Any philosopher who has studied earlier scientific debates will understand what can be disappointing about them, and so try to steer contemporary ones in productive directions. What might have been achieved if someone had been there to mediate relations between Hilbert and Frege or between Hilbert and Brouwer?


Pete said...

Funny, I had the thought as I randomly dropped in here: "just how much maths do philosophers really need to know to do philosophy of maths?" and I read your statement:

I wish the noncommutative geometry program had a resident philosopher to probe it, and help us understand the tensions between rival accounts [...] Then we might link up to see how Grothendieck's and Connes's conceptions of space face up to each other.

Well, I find myself wondering if the "tensions between rival accounts" are mathematical or philosophical tensions, and why a philosopher, rather than a mathematician, would be needed to expose them. Isn't the arena in which G and C's conceptions of space square off a mathematical arena?

(BTW on "categorifiable" - google, that fount of wisdom, advises "categorizable" ;-)

Thanks for the ever-fascinating bloggage!

April 28, 2006 10:35 PM  
david said...


I'm not expecting philosophers to be the ones to reconcile tensions between rival accounts, but I would like to see them help draw them out, if only by provoking their emergence by questioning mathematicians. I have found mathematicians to be generally generous with their time.

I'm not expecting all philosophers of mathematics to engage with contemporary developments. And I'm not expecting every contemporary development to be investigated. I have the sense that noncommutative geometry should be interesting to look at. I'm sure higher-dimensional algebra's reconceptualisation of sameness/difference is important.

In many ways I'm expecting less than the philosophy of physics expects of its workers. Just follow up the paper mentioned here. Indeed, I would say that I understand mathematics less fully than many philosophers of physics understand physics. There are several philosophers who understand second-order logic or set theory very well (just read Penelope Maddy's Naturalism in Mathematics. If they'd spend a small but reasonable portion of their time on, say, noncommutative geometry, they'd soon reach a level that would suffice for the kind of work I'd like to see.

The 'arena in which G and C's conceptions of space square off' is indeed a mathematical arena, but to gain a sense of the nature of that arena is to learn something very important of mathematics functioning at its highest level. It's like a chance to visit the Coliseum, or better the Senate, if you wished to understand Ancient Rome. This doesn't mean you should wade in on the side of one of the gladiators, or take them all on, nor try to defeat Cicero in an argument.

I think what the philosophy of mathematics most lacks at the current time is any thoroughgoing conception of what mathematicians are trying to do.

May 01, 2006 6:15 PM  
Anonymous said...

Re-reading your post and the comments, David, made me wonder if the problem you identify in current philosophy of mathematics (of a failure to engage sufficiently with actual mathematical practice) is in fact part of the broader problem of western philosophy, and indeed western intellectual activity in general, identified by philosopher Stephen Toulmin in his book, "Cosmopolis: The Hidden Agenda of Modernity" (U. Chicago Press, 1990). Toulmin argued that, since Descartes, western society has been obsessed with formality and abstraction at the expense of gritty, specific reality. We favour the abstract over the concrete, the general over the particular, universal laws over particular instantiations and exceptions, text over image, the written over the spoken or performed, and the recorded over the ephemeral. This bias permeates most of our contemporary intellectual endeavours, at least in the west. I wish you success in your struggle, but I think your enemy is larger and more deeply entrenched than you may realize.

-- Peter

May 03, 2006 11:55 AM  
david said...

We don't seem very good at treating human concerns at the right level of generality. This is especially so in psychology. For instance, Anglophone research on the mind and illness have largely become expression of theses of the type: people of type X are more likely to have higher bodily parameter Y. The thorough, sensitive case studies of a handful of individuals with a certain condition, common 40 years ago, have all but disappeared.

As Collingwood put it,
'They neglect their proper task of penetrating to the thought of the agents whose acts they are studying', who merely 'content themselves with determining the externals of these acts, the kinds of things which can be studied statistically.'

Similarly, we can learn much more about the mathematics of a given era, and so about mathematics in general from a well-chosen allusive case history than we can from some ahistorical assertion about the supposed nature of mathematical entities.

However, I detect a danger in making general pronouncements about how we moderns favour the general over the particular. Different disciplines seem to go through different phases. History of Science, for example, appears currently to be swinging towards a preference for the particular.

May 04, 2006 12:26 PM  
Dennis said...

You quoted my email above: “Set theory universalism is the great overarching story of the last century of mathematics. (It’s likely part myth.) Of course, it shouldn’t override or erase other stories.” This might sound a little too grand. What I was leaning toward is something like this: If narration takes centre stage, then set theory universalism must be part of the narration. (I say set theory universalism is part myth in the sense that it is incomplete, questioned, etc.)
Often today, the shortcomings of this universalism are pointed to as evidence that mathematics is not really rigorous, but a subjective enterprise, etc. But acceptance of this universalism, at least for a time, despite its problems (its part-myth character, as I put it) probably had the opposite effect of promoting rigour and systematization – of helping to usher in modern mathematics.
This observation suggests another: the blessings set theory universalism – without another universalism to replace it – may compensate for its problems, maybe even now.

May 17, 2006 7:29 PM  
John Baez said...

David writes:

However, I detect a danger in making general pronouncements about how we moderns favour the general over the particular.

Yes... though it would be thoroughly modern to do so. Everybody generalizes these days. :-)

July 16, 2006 6:08 AM  

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