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March 2, 2009

Two Streams in the Philosophy of Mathematics

Posted by David Corfield

A conference with this title will be held on 1-3 July 2009 at the University of Hertfordshire, Hatfield, UK. A description of the rationale for the conference is here. Mathematicians are more than welcome to participate. The deadline for abstracts is 30 April.

Update: I forgot to include a link to the conference homepage.

Posted at March 2, 2009 10:09 AM UTC

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17 Comments & 1 Trackback

Re: Two Streams in the Philosophy of Mathematics

That looks right up your alley. I guess you’ll be attending?

It’s a bit sad that philosophers of mathematics who note that ‘formal logic cannot justify the choice of concepts’ in mathematics are called ‘mavericks’. But at least these mavericks are joining up to challenge the ‘mainstream’ philosophers who think logic is the be-all and end-all of mathematics.

The conference description says ‘seminars combining the history and philosophy of mathematics are a regular part of the Parisian university scene’. Does someone here know the people involved in that scene?

My wife Lisa Raphals talks a lot to Karine Chemla, an expert on ancient and medieval Chinese mathematics. She has interesting ideas on what substituted for the concept of ‘proof’ in this stream of mathematics. Anyone wanting to rethink the role of logic in mathematics could do well to ponder that.

But I wonder which seminars are being alluded to above.

Posted by: John Baez on March 2, 2009 4:45 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

Not just attending. I’m jointly organising and speaking.

One good seminar in Paris is this one - you’ll see I was there on 16 April 2008.

There’s also this one, continued this year. And, generally, a feast of meetings.

There’s nothing approaching this in the UK.

Posted by: David Corfield on March 2, 2009 4:57 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

David wrote:

Not just attending. I’m jointly organising and speaking.

Oh, so it’s your British modesty that explains why your name isn’t mentioned in the description of the conference. I assumed it was being organized by some people who either didn’t know your work or didn’t think it was very important! I was wondering who those jerks were.

Posted by: John Baez on March 3, 2009 2:41 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

Whoops. I forgot to link to the conference homepage.

Posted by: David Corfield on March 3, 2009 2:47 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

I’d love to come to this, but unfortunately I’ll most likely be at the Australasian Association of Philosophy conference at the time. I wonder though if the discussion in this call for papers is a bit too inflammatory to encourage people from the traditional side to come.

Posted by: Kenny Easwaran on March 3, 2009 12:36 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

Shame you can’t make it, Kenny.

‘Too inflammatory’? Gosh. We thought we were being as open and inviting as possible. But I’m sure I’m not nearly sensitive enough to pick this up. Was there a passage in particular, or is it the whole document?

Posted by: David Corfield on March 3, 2009 11:41 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

I re-read it and now I can’t tell exactly what part gave me this impression. I don’t know if it was the terms “mainstream” and “maverick” themselves that somehow seemed that way. I think it was in part due to the fact that there’s a solid argument in there for the “maverick” position, while the only argument that’s given in favor of the “mainstream” position seems to be based on an interpretation of the maverick position as “an attempt to replace validity with credibility.” I’m not sure if this is what gave me that feeling the first time I read it, but it’s probably part of it.

Posted by: Kenny Easwaran on March 4, 2009 1:34 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

I wrote the conference blurb. It seems I couldn’t help revealing who I think are the goodies and baddies, in spite of my efforts at even-handedness. Actually, I’m doubtful about the whole two-stream picture. We’ve inherited it from Aspray and Kitcher, but I wonder whether it still describes the situation. I adopted it as a fund-raising tactic: say there is a debate going on and we want to have it out. I would have been quite happy to call the conference “Proof, eh?”

In the event, the British Academy declined to fund this meeting (which is why we have a long list of other sponsors).

So if anyone reads this and thinks, well I have something to say about proof, but I’m not sure it fits into this frame, I would say, send us an abstract anyway, we’re not hung up on our title.

Posted by: Brendan Larvor on March 4, 2009 6:42 PM | Permalink | Reply to this

Proof, eh?; Re: Two Streams in the Philosophy of Mathematics

“To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.”
[Feynman, The Character of Physical Law (1965) Ch. 2]

“Each piece, or part, of the whole nature is always an approximation to the complete truth, or the complete truth so far as we know it. In fact, everything we know is only some kind of approximation, because we know that we do not know all the laws as yet. Therefore, things must be learned only to be unlearned again or, more likely, to be corrected…….The test of all knowledge is experiment. Experiment is the sole judge of scientific ‘truth.’”
[The Feynman Lectures on Physics (1964), Volume I, 1-1, Introduction]

“I don’t know anything, but I do know that everything is interesting if you go into it deeply enough.”
[Feynman, from Omni interview, The Smartest Man in the World (chapter 9)]

Richard Feynman and The Connection Machine
by W. Daniel Hillis for Physics Today

One day when I was having lunch with Richard Feynman, I mentioned to him that I was planning to start a company to build a parallel computer with a million processors. His reaction was unequivocal, “That is positively the dopiest idea I ever heard.” For Richard a crazy idea was an opportunity to either prove it wrong or prove it right. Either way, he was interested….

He distrusted abstractions that could not be directly related to the facts…. He asked, “How is anyone supposed to know that this isn’t just a bunch of crap?”

“When you get as old as I am, you start to realize that you’ve told most of the good stuff you know to other people anyway.”

Posted by: Jonathan Vos Post on March 4, 2009 9:32 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

What do philosophers think about possible “robot mathematicians”? (…and mathematicians about robot philosophers?) Having heard repeatedly about interactive proof-checking projects, which could some day explain a proof to students, I wonder what philosophers prognose for that too.

Posted by: Thomas on March 3, 2009 11:13 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

The real test is to produce robots that can serve as examiners of maths PhDs, or robots who can edit maths journals. When machines can articulate judgments about the novelty and significance of a piece of mathematics, I’ll be impressed. Mathematicians make such judgments about their own work as they go along, so if you don’t have this element, you don’t have a mathematician.

On the way to that, one would want to see a robot that could ask and answer Polya-style heuristic questions: do I know how to solve a related problem? This would require a specification of the relation of heuristic promise between a solved problem and an as yet unsolved problem. Is it possible to give this specification in a general form, i.e. without reference to any particular domain of mathematics?

One hesitates to judge serious research on the basis of breathless journalism, but Sloman’s investigations of children seem perilously close to all that work on language-acquisition that treats toddlers as little scientists. “As toddlers, we soon translate our experiences into general theorems which we use to make predictions.” Show me a toddler with a general theorem! In fairness, I think this loose talk is by the journalist rather than Sloman.

Posted by: Brendan Larvor on March 6, 2009 11:23 AM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

I very much agree.

You should see (and can by turning to page 43 of my book) the trouble machines have in producing a proof of the two-dimensional version of the Heine-Borel theorem, given the one-dimensional proof. The difficulty is that you can’t just mechanically translate a proof that eventually members of a sequence of nested intervals are contained in an interval centred on the point contained in the intersection. You must make sure that rectangles are small enough in both directions.

Posted by: David Corfield on March 6, 2009 12:29 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

Thanks to google, I saw your note. Yes, the journalist (or the Newscientist editorial team) did use wording I would not have used. If you wish to take a closer look at what I am investigating there’s a pdf presentation on the main ideas here: http://www.cs.bham.ac.uk/research/projects/cogaff/talks/#toddlers

The key observation (which I think comes originally from Immanuel Kant over 200 years ago) is that something initially discovered empirically can later be understood as more than a well confirmed generalisation, e.g. that counting your fingers from left to right gives the same result as counting from right to left, or that spatial containment is transitive.

The fact that children can make that transition appears not to have been noticed by developmental psychologists.

The information-processing architectures and mechanisms required to support that ability are far from obvious.

The biological benefits have to do with being able to solve problems in novel situations, and I think humans are not the only animals that can do that, though only humans subsequently codify and formalise the discoveries made (after appropriate cultural evolution).

The currently popular attempts to make machines learn by using statistical techniques do not seem to me to be capable of supporting these processes.

There are some partial suggestions on my web site, but much more work to be done.

Posted by: Aaron Sloman on March 7, 2009 11:05 PM | Permalink | Reply to this

Sloman’s right; Re: Two Streams in the Philosophy of Mathematics

I agree with Aaron Sloman. I have 75 days of Directed Teaching to do before I get my California Single Subject Secondary School teaching certificate in Math – notwithsting having been an adjunct professor of math. The Colleges of Education are blissfully unaware of breakthroughs in Cognative Science and of Chaos Theory and so much more. They teach teachers by ad hoc combinations of Pop Psych, Pop Sociology and ideology.

I did manage to petition out of the required “Introduction to Educational Psychology” on the basis that some of my publications in that are already used in curricula in several countries.

Hence I plan to start my Ed.D. dissertation late this year in Educational Leadership. Working title: “Students on the Edge of Chaos: Neurological Pedagogy.”

Posted by: Jonathan Vos Post on March 8, 2009 7:20 PM | Permalink | Reply to this

Re: Two Streams in the Philosophy of Mathematics

The observation you’re talking about sounds to me very much like Alison Gopnik et al.’s
“How Babies Think: The Science of Childhood” and “The Scientis in the Crib.”

Posted by: Eugene Lerman on March 9, 2009 6:06 PM | Permalink | Reply to this

Darwinian meta-concepts, developmental psychology, math curriculum; Re: Two Streams in the Philosophy of Mathematics

The stochasticity component of the chaotic evolution of concepts has long been proposed in terms of Natural Selection in the plane of ldeas.

Thank you, Eugene Lerman. I’ve not read either book, but have seen interesting reviews of both.

Yes, I agree that something is known about how children acquire adult knowledge and reasoning, and my mother’s Master’s degree was on the epistemology and developmental psychology of Piaget and Montessori, before my Mom taught 3rd grade Math with Cuisinaire rods and other manipulatives.

There’s some of this trickling into Math education.

Public Schools Outperform Private Schools in Math Instruction

ScienceDaily (Mar. 3, 2009) — In another “Freakonomics”-style study that turns conventional wisdom about public- versus private-school education on its head, a team of University of Illinois education professors has found that public-school students outperform their private-school classmates on standardized math tests, thanks to two key factors: certified math teachers, and a modern, reform-oriented math curriculum.

Posted by: Jonathan Vos Post on March 9, 2009 6:33 PM | Permalink | Reply to this

Re: Darwinian meta-concepts, developmental psychology, math curriculum; Re: Two Streams in the Philosophy of Mathematics

I should have been more clear. It’s one book with two titles: one British, one North American. For a more technical account have a look at “Words, thoughts and theories” by Alison Gopnik and Andrew Meltzoff), MIT Press, 1997.

Posted by: Eugene Lerman on March 10, 2009 1:33 AM | Permalink | Reply to this
Read the post A River and a Trickle
Weblog: The n-Category Café
Excerpt: On the brief appearance of Lakatos in a Handbook of philosophy of mathematics
Tracked: April 3, 2009 4:43 PM

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