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May 28, 2008

Manin on Foundations

Posted by David Corfield

Out of the series of observations made by Yuri Manin in Truth as value and duty: lessons of mathematics most relevant to us here is:

For a working mathematician, when he/she is concerned at all, “foundations” is simply a general term for the historically variable set of rules and principles of organization of the body of mathematical knowledge, both existing and being created. From this viewpoint, the most influential foundational achievement in the 20th century was an ambitious project of the Bourbaki group, building all mathematics, including logic, around set-theoretical “structures” and making Cantor’s language of sets a common vernacular of algebraists, geometers, probabilists and all other practitioners of our trade. These days, this vernacular, with all its vocabulary and ingrained mental habits, is being slowly replaced by the languages of category theory and homotopy theory and their higher extensions. Respectively, the basic “left-brain” intuition of sets, composed of distinguishable elements, is giving way to a new, more “right brain” basic intuition dealing with space-like and continuous primary images, both deformable and deforming.

Has mathematics learned better to employ its corpus callosum?

Posted at May 28, 2008 6:29 PM UTC

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Re: Manin on Foundations

Is that an emoticon on page 5 of Manin’s paper? I don’t think I’ve ever before seen an emoticon in a philosophy paper, though it’s bound to become common eventually.

Posted by: John Baez on May 29, 2008 3:40 AM | Permalink | Reply to this

Re: Manin on Foundations

I agree with his distinction between the styles, but I like to call it holistic/reductionistic rather than right/left brain. And I usually hate the words holistic and reductionistic, but here they seem to do exactly the right job.

Posted by: James on May 29, 2008 4:43 AM | Permalink | Reply to this

Re: Manin on Foundations

Yes, you would have thought we’d got beyond simple left/right brain dichotomies. Still, I suppose it works as a kind of shorthand. For example, Arnold declares:

In the middle of the twentieth century a strong mafia of left-brained mathematicians succeeded in eliminating all geometry from the mathematical education (first in France and later in most other countries), replacing the study of all content in mathematics by the training in formal proofs and the manipulation of abstract notions. Of course, all the geometry, and, consequently, all relations with the real world and other sciences have been eliminated from the mathematics teaching.

All the same, it might be interesting to do some brain imaging of real mathematicians thinking about real mathematics. A couple of years ago I was discussing the idea with a brain scientist and even put out a call for participation. Existing studies seem rather limited, if indicative.

For example, in ‘Interhemispheric Interaction During Global-Local Processing in Mathematically Gifted Adolescents, Average-Ability Youth, and College Students’, Singh, Harnam; W. O’Boyle, Michael; Neuropsychology, Vol 18(2), Apr 2004. pp. 371-377, the authors argue that the ‘mathematically gifted’ have greater interhemispherical activity.

Abstract:

Interhemispheric interaction in mathematically gifted (MG) adolescents, average-ability (AA) youth, and college students (CS) was examined by presenting hierarchical letter pairs in 3 viewing conditions: (a) unilaterally to the right hemisphere (RH), (b) unilaterally to the left hemisphere (LH), or (c) bilaterally, with 1 member of the pair presented to each hemisphere simultaneously. Participants made global-local, match-no-match judgments. For the AA and CS, the LH was faster for local matches and the RH for global matches. The MG showed no hemispheric differences. Also, AA and CS were slower on cooperative compared with unilateral trials, whereas the MG showed the opposite pattern. These results suggest that enhanced interhemispheric interaction is a unique functional characteristic of the MG brain.

Atiyah speculates here about a much more ambitious project to search for the neural underpinnings of mathematicians’ aesthetic sense:

Despite all the difficulties associated with understanding or defining beauty we can still ask what mechanisms in the brain are involved in its appreciation. This can be asked about beauty in the various arts or in mathematics. It is a fascinating question whether there is any commonality across all areas. Are we just misled by the inadequacies of language and the misleading power of metaphor?

As I argued at the beginning, mathematics is a pure form of thought and so it may provide an easier field for physiological study. It is still a daunting task. We have to identify many instances of what a mathematician finds beautiful and see, by experiment, if there is any region of the brain that is common to them. We can for example compare beauty, as illustrated in geometric form, with the more formal beauty of an elegant algebraic formula or of a subtle abstract argument.

Posted by: David Corfield on May 29, 2008 10:01 AM | Permalink | Reply to this

Re: Manin on Foundations

DC wrote: We can for example compare beauty, as illustrated in geometric form, with the more formal beauty of an elegant algebraic formula or of a subtle abstract argument.

SH: Providing an opportunity to mention the last two books I’ve read,
“Why Beauty Is Truth: The History of Symmetry”
by Ian Stewart
and
“The Equation That Couldn’t Be Solved”
how mathematical genius discovered the language of Symmetry by Mario Livio

Both authors mention symmetric faces being more attractive and bees are attracted to flowers whose petals are more symmetric.
They both mention the connection to group theory but I think Livio’s book is a bit broader and deeper.

Posted by: Stephen Harris on May 29, 2008 12:52 PM | Permalink | Reply to this

Re: Manin on Foundations

DC wrote…

Not DC, but Sir MA, FRS, OM.

Posted by: David Corfield on May 29, 2008 1:17 PM | Permalink | Reply to this

Re: Manin on Foundations

I’m interested in how some of the cognitive findings in “Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being” by Lakoff and Nunez can be related to mathematical aesthetics. I really think that their “mathematical idea analysis” gives a good method.

Posted by: Matt on May 29, 2008 3:41 PM | Permalink | Reply to this

Re: Manin on Foundations

The handful of mathematicians I’ve heard talking about ‘Where Mathematics Comes From’ seemed united in thinking that while it points in an interesting direction, something isn’t quite convincing about their position.

Perhaps something in our discussion is more promising.

Posted by: David Corfield on May 29, 2008 10:19 PM | Permalink | Reply to this

Re: Manin on Foundations

Could I trouble you to give some sort of summary of their position? I have been rather curious, although the casual glance at the bookstore did not result in a very positive impression.

Posted by: Minhyong Kim on May 30, 2008 6:51 PM | Permalink | Reply to this

Re: Manin on Foundations

I’ll have to come clean and admit that I have only browsed it too. I did read earlier work of Lakoff with Mark Johnson. I wrote a little on what they called ‘image schemas’ in my PhD thesis. I was struck at the time by the relevance of their analysis of ‘over’ in Metaphors We Live By for the mathematical notions of cover and sheaf.

Examples like these might have been better choices than the run-of-the-mill ‘infinity’, ‘continuity’, ‘set’, ‘number’ examples of Where mathematics comes from. A synopsis of the book in the Wikipedia article seems a reasonable starting point.

Posted by: David Corfield on June 2, 2008 1:19 PM | Permalink | Reply to this

Re: Manin on Foundations

There is one remark I think of often that illustrates rather well the difficulty of capturing what geometric intuition really is in any simple terms, coming as it does from Sir MA’s student Graeme Segal, known mainly for his work in geometry and topology:

`A good sentence is worth a thousand pictures.’

We were discussing knots and QFT’s in that conversation, and I think John was present. Segal has a very non-trivial appreciation of high-brow literature (as far as I’m able to judge, of course).

Posted by: Minhyong Kim on May 29, 2008 2:53 PM | Permalink | Reply to this

Re: Manin on Foundations

Perhaps JB will recall an incident at a conference in which the host of the conference badgered the speaker to not demonstrate any more pictures. The speaker was prepared with a slide that contained the arcane formulas that were restatements of the pictures. The identity of neither should be made public. Those who were present may also recall.

Yes, thoughtful and careful writing can create imagery, and the mathematical author who insists upon not using pictures has an obligation to create the unambiguous image that (s)he is experiencing.
As a practitioner who illustrates with drawings and with words, I don’t think either task is particularly easy. By using both, one creates redundancy in exposition, but the redundancy facilitates the flow of information across the channel and between the lobes.

The 4-dimensional imagery of a mathematical mind at work would indeed be interesting. Perhaps tenure and promotion decisions could be determined by how much of the brain is firing during the production of the candidates magnus opus ;-)

Posted by: Scott Carter on May 29, 2008 4:10 PM | Permalink | Reply to this

Re: Manin on Foundations

I do indeed recall both these incidents. Thanks for reminding me!

Posted by: John Baez on May 29, 2008 4:49 PM | Permalink | Reply to this

Re: Manin on Foundations

> we can still ask what mechanisms in the brain are involved in its appreciation.

I expect this to be a difficult task. Appreciation of mathematical beauty isn’t necessarily something that takes place while doing mathematics. When doing mathematics, you’re often way too busy doing mathematics to stop and think about whether or not what you’re doing is beautiful.

A while back a researcher friend of mine complained to me about how there was little beauty to be seen in his day-to-day work. He was just getting on with his job. It was only when he wasn’t doing mathematics he could look back and appreciate what he had been working on. Mathematical appreciation isn’t a state of mind that exists at any given instant, it’s a long-term ongoing process.

Posted by: Dan Piponi on May 29, 2008 7:19 PM | Permalink | Reply to this

Re: Manin on Foundations

Dan wrote:

A while back a researcher friend of mine complained to me about how there was little beauty to be seen in his day-to-day work. He was just getting on with his job.

Ugh! It sounds like he’s gotten himself stuck in a trap. Please remind him that he’s going to die soon and he needs to have fun now if ever.

I spend a lot of time enjoying the beauty of mathematics. I’m sure I could be more ‘productive’ if I knuckled down and worked harder… but why, really? It’s not as if the world needs more theorems per year. I actually work too hard already! If I were really smart I’d goof off more.

Posted by: John Baez on May 30, 2008 8:49 AM | Permalink | Reply to this

Re: Manin on Foundations

But it’s not just a question of the search for beauty as provider of fun versus the unenjoyable graft of theorem proving.

We dwell on mathematics and affirm its statements for the sake of its intellectual beauty, which betokens the reality of its conceptions and the truth of its assertions,

says Michael Polanyi. Without this betokening how would you know where to go?

While in the natural sciences the feeling of making contact with reality is an augury of as yet undreamed of future empirical confirmations of an immanent discovery, in mathematics it betokens an indeterminate range of future germinations within mathematics itself.

Posted by: David Corfield on May 30, 2008 10:31 AM | Permalink | Reply to this

Re: Manin on Foundations

Perhaps this remark takes us off-topic, but I’ve never felt much sympathy for a clear division between ‘work’ and ‘fun.’ The reality of such categories may even be problematic in general, but certainly seems artificial in the quotidian existence of a university professor.

Perhaps I’ll go out on a limb and state my suspicion that the rigid psychological dichotomy of leisure vs. labor creates real tension in a broad spectrum of human concerns, including (obviously) education as well as the administration of industrial economies.

Justifying this suspicion, of course, would take too much work!

Posted by: Minhyong Kim on May 30, 2008 4:31 PM | Permalink | Reply to this

Re: Manin on Foundations

Afterthought:

Maybe it wasn’t entirely off-topic. I think I was vaguely trying to sympathize with Dan Piponi’s friend and his last sentence. And then, I did mention `categories’ in the post.

Posted by: Minhyong Kim on May 30, 2008 4:45 PM | Permalink | Reply to this

Re: Manin on Foundations

Minhyong wrote:

Perhaps this remark takes us off-topic, but I’ve never felt much sympathy for a clear division between ‘work’ and ‘fun.’ The reality of such categories may even be problematic in general, but certainly seems artificial in the quotidian existence of a university professor.

I have no sympathy for this distinction at all. I hate it! But for me, at least, the distinction seems real — secretly self-created, but not easy to escape.

I start projects because they’re ‘fun’; when I want to start the next project, the current one becomes ‘work’. However, especially when I’m writing a paper with coauthors, or I’ve promised a paper for a conference proceedings, I feel compelled to finish my projects! So then I’m stuck doing ‘work’ when I want to have ‘fun’ — that is, think about something new. I then start procrastinating… and can make myself really miserable. Last fall I was really depressed about the huge amount of work I’d set up for myself, which stood like a huge wall between me and the new ideas I wanted to think about.

When I’m writing a paper by myself, without any deadline, I don’t seem to fall into this trap quite so badly. The reason is that then I write the paper at approximately the rate at which I figure things out. I figure things out by writing the paper! Figuring things out is fun. Ideally, this means that when I’m done having fun, the paper is done.

Unfortunately I like having coauthors, where the ‘figuring out’ stage consists of sitting around talking. That’s fun. But then the writing up somehow becomes ‘work’.

So, it’s a complicated mind game, which I keep feeling I should be able to escape simply by taking a more enlightened attitude… but I haven’t succeeded yet.

Now I have to get to work.

Posted by: John Baez on May 30, 2008 5:58 PM | Permalink | Reply to this

Re: Manin on Foundations

Aha! I always knew the reason we aren’t able to collaborate is that you work so much harder than I do!

Posted by: Minhyong Kim on May 30, 2008 6:46 PM | Permalink | Reply to this

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