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March 16, 2010

Philosophy in the EMS Newsletter

Posted by David Corfield

The March 2010 edition of the Newsletter of the European Mathematical Society is out, and it includes a short piece by me on realism in mathematics. I was invited to contribute to a series on Platonism by Urs Persson, a series which includes Brian Davies’ ‘Let Platonism Die’ (June 2007), Reuben Hersh’s ‘On Platonism’ and Barry Mazur’s ‘Mathematical Platonism and its Opposites’ (June 2008), and David Mumford’s ‘Why I am a Platonist’ and Philip Davis’ ‘Why I Am A (Moderate) Social Constructivist’ (December 2008). Ulf has himself a riposte to Davies – Let Platonism Live!. I decided to drop the term ‘Platonism’ since I prefer to focus on the issue of the nature of the constraints acting on mathematicians, and don’t wish to tie this wholly to the question of the existence of abstract entities.

Posted at March 16, 2010 12:16 PM UTC

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Re: Philosophy in the EMS Newsletter

Thanks! My ideas on the background of “platonism” in mathematics, with some art and poetry. Barry Mazur remarked on that: “somehow I would have balanced that list (of “platonic” ideas in math, TR) with “things sought that turned out not to exist” (e.g., the squaring of the circle, trisection of a general angle by Euclidean construction methods) as further hints that a platonic mentality seems to be very energetically at work—in the minds of many mathematicians.” Maybe one could wonder about “platonic mirages” like Hilberts proof theory program.

Posted by: Thomas on March 16, 2010 5:29 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Nice piece, David! Anyone who wants to go straight to it should click here.

Ulf needs a more intelligent spell-checker. He speaks of “rightest political views”, and the “a priory”.

Posted by: John Baez on March 16, 2010 11:41 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

I was poking around old issues at the EMS website and discovered an interview with Alain Connes. I remember Connes because John Baez described his “Noncommutative Geometry” book in such glowing terms that I bought it, much to my later chagrin. I found a Platonic article by Mazur which seemed very fair. Connes holds an adamant view in favor of mathematical realism. He had an interesting discussion with Jean-Pierre Changeux in the earlier book “Conversations on Mind, Matter, and Mathematics”.

March 2008, Interview with Alain Connes (2) –C. Goldstein and G. Skandalis page 29

“whether mathematical reality is something created or something pre-existing is much easier to discuss if one uses the distinction which appears in Godel’s theorem between
“truth” and “provability” of a mathematical statement. I discussed this in details in my book “Triangle of thoughts”…

Posted by: Stephen Harris on March 23, 2010 4:55 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

The link to the EMS newsletter was a good idea!

S Y M M E T RY A Journey into the Patterns of Nature by Marcus du Sautoy
“The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.” by Paul Valery

Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication.

plato.stanford.edu/entries/pythagoras/
“Some of these earlier sources were heavily contaminated by the Neopythagorean view of Pythagoras as the source of all true philosophy, whose ideas Plato, Aristotle and all later Greek philosophers plagiarized. Nicomachus (ca. 50 – ca. 150 CE) assigns Pythagoras a metaphysics that is patently Platonic and Aristotelian and that employs distinctive Platonic and Aristotelian terminology … In the Pythagorean Memoirs, Pythagoras is said to have adopted the Monad and the Indefinite Dyad as incorporeal
principles, from which arise first the numbers, then plane and solid figures and finally the bodies of the sensible world”

“The tendency of people to see purpose in symmetry suggests at least one reason why symmetries are often an integral part of the symbols of world religions.”

SH: The ancient Greeks perceived patterns and attributed them to a Divine provenance, as also in the case of the divine proportion. Symmetry in nature and mathematics accounts for patterns ranging from the hexagonal honeybee cells to the way in which “Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory.”

Can you hear the sound of Platonism’s drumbeat vanishing just as the mythological gods have perished when scientific knowledge has replaced primitive belief?!

Posted by: Stephen Harris on March 20, 2010 3:11 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Thomas wrote: “The sketch of the “platonic” mindset seems to fit esp. that of number theorists. An interesting suggestion I received is about Wilfried Owen’s poem “The naming of the parts” as illustration of emotions related to “naming” and the short stories of Tim OBrien “The things they carried” on that and a complex concept of “reality” similar to those of the ancient greeks.”

The idea of the Muses has been hear since at least the days of the ancient Greeks. The eye receives something like two million bits of information per second. Only about two thousand bits of information are passed onto consciousness as a basis for decisions. Perhaps the unconscious can process other conscious ideas and provide unusual creativity. Some people report creativity being passed on in dreams. I think this Muse idea (unconscious) makes a lot more sense than creative input received from an eternal realm of ideas existing beyond space and time. Muses don’t have to be actual personas, just functions of the unconscious. So perhaps everyone has that potential. But I don’t think one has to buy into the notion that everyone is networked into a Jungian Collective Unconscious. If there are archetypes, it is because they are part of the normal evolved brain information that babies are born with that allows them to learn without needing to be taught to learn, early, not language. And because someone writes a historical perspective doesn’t mean that they don’t reveal their own belief inclinations within that primarily historical account. No matter whether they had the intention to do so or not.

Posted by: Stephen Harris on March 22, 2010 5:53 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

I am more interested in e.g. how “platonic” ideas work even when they seem not to work. There I wonder if the story of p-adic Birch Swinnerton-Dyer, where the p-adification of a classical theory run into unexpected troubles, were an interesting example. An other interesting case may be K. Kato’s use of nonexistent “fundamental functions” a guide in ch. III, §1 here. There could be a lot of interesting stuff aside the wellknown cases like F_1, primes”=”knots, numbers”=”function fields,”mysterious functor”,Fontaine-Mazur conjecture, etc.

A history-of-ideas of “platonism” makes sense simply because much nonsense is often associated with that term. There is a funny story by an austrian physicist, “Einsteins Erben”, sketching a possible similar degeneration of the idea of science-mentality, as obviously happened there. That one has to distinguish things shows e.g. Gromov’s essay: On the one hand it fits perfectly into the chain of “platonistic” theorising and he stresses the “platonic” mindset among his colleagues, on the other hand, his questioning of speaking of really huge numbers, or his use of Zellig’s semantics do not. His examples of how hidden symmetries in very simple situations turn into interesting theories and how “good” definitions differ from “unproductive” and “boring” ones are very interesting, independent from ontological beliefs, and it would be great to see if and how other cases of that happen.

Posted by: Thomas on March 22, 2010 1:17 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Thomas wrote: “I am more interested in e.g. how “platonic” ideas work even when they seem not to work.”

I cannot tell what you mean by “platonic”. I think some definition of platonic is necessary because how can you match the concept of platonic with some situation? How do you identify or associate programs in mathematics or number theory that are governed or controlled or fall under the jurisdiction of a platonic event, unless one knows what the word platonic means. Do you mean when some area of mathematics unexpectedly, or it comes as a subjective surprise, is related to another apparently distant area of mathematics.

I don’t think you can give me an example of a mathematical result which required any concept called platonism or any mathematician who required such ideas. David avoided going into the source of a platonic idea, and just focused on whether the idea was considered: discovered or invented.

But if a person wants to say an idea had a platonic inspiration, that means there has to be some (abstract perhaps) place where the idea was discovered. Some place that was available to hold the ideas so that some mathematician could discover them.

So there is a religious/mythological set of beliefs which Pythagoras dreamed up. And Plato adopted many of these ideas. The eternal platonic realm held all mathematical truths. [See Tom Silverman’s post.]
Another idea was Jung’s Archetypes. Jung, Eliade, and Joseph Campbell were all mythologists.

Take the idea of the unicorn which can dip its horn in contaminated waters and purify them. The unicorn is imaginary, with no physical presence unless one can answer,
Where it comes from, what it is, and what it applies to and how it came into existence.

The same criteria apply to platonic inspirations for the mathematician. If one can’t provide concrete answers then platonism is just another religious/mythological/imaginary belief.

A reply like, well so and so believes in Platonism is no evidence. It reminds me of asking a sadhu if he is enlightened.
The sadhu says no, but that his guru is enlightened. Question: How do you know that your guru is enlightened if you are not?
Answer: Because, look at all the other unenlightened aspirants who belong to my order, they all think that the guru is enlightened.

So please provide an example of what your term “platonic insights” means, something you can say for sure is a result of a mathematician receiving platonic inspiration. So where does a “platonic insight” come from if not from a “platonic reality” Manin’s term?
Speaking of computers, Manin said:
“So if he doesn’t have the imagination to distinguish some features of this Platonic reality, he can experiment.” [he = mathematician]

I think it is logical to define an instance where “platonic ideas work” before attempting to figure out where platonic ideas don’t work, or can’t work.

Posted by: Stephen Harris on March 22, 2010 5:13 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

“what you mean by “platonic””: I think the examples make that clear. I refer to what among mathematicians is usually regarded as such, e.g. the idea behind class field theory and Kronecker’s “Jugendtraum”. Gromov’s “h-principle” may be one, but I know too little about that. I don’t think that there is a need to define it. Aside that, to repeat, what could be called “platonic” is a midset and mentality, not some ontological proposition (which however can result in entertaining pieces of literature, like some Borges’ stories). People can believe a lot of things, change their beliefs, or do better things than norishing beliefs with their time. But it is interesting to see which kinds of theories come out of it and how that develops.

Posted by: Thomas on March 22, 2010 6:57 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Here’s what Alain Connes says about mathematical realism. I find it quite congenial to my own way of thinking.

Q: In your conversations with Changeux you discussed mathematics and reality. Have you advanced in your thinking about this?

I have no doubt that mathematical reality is something which exists, that it exists independently of my own brain trying to see it, and has exactly the same properties of resistance as external reality. When you want to prove something, or when you examine if a proof is correct or not, you feel the same anguish, the same external resistance as you do with external reality. Some people will tell you that this reality does not exist because it is not “localized” somewhere in space and time. I just find this absurd and I adopt a diametrically opposed point of view: for me even a human being is better described by an abstract scheme than by a material collection of cells – which in any case are totally renewed and replaced over a relatively short period of time and hence possess less meaning or permanence than the scheme itself, which might eventually be reproduced in several identical copies……

If one wants to reduce everything to “matter localized somewhere” one soon meets a wall which comes from quantum mechanics and one finds that this reduction of the outer reality to matter is an illusion that only makes sense at intermediate scales but by no means at a fundamental level. Thus I have no doubt on the subtleness and existence of a reality which can be neither reduced to “matter” nor “localized”.

Posted by: John Baez on March 25, 2010 4:44 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Good, but I always get a little nervous when people reach straight for proof when they want to illustrate mathematical resistance. This lends itself to being interpreted as merely logical resistance. I know Connes elsewhere (see bottom of TWF218) explains how reality is revealed by one’s repeated encountering of the same entities: elliptic functions, modular forms, zeta functions.

Looking to Lautman’s distinction between entities, facts, theories, and ideas, we can see mathematical resistance in all quarters. What’s intriguing me at the moment is the possibility of mathematical accounts of mathematical resistance.

For example, you can run a debate on the inevitability of the quaternions appearing once mathematicians had happily accepted the complex numbers.

The contingentists tell you about Hamilton’s idiosyncratic use of the Coleridgean metaphysical understanding of space and time in his quest to move beyond the 2 dimensions of \mathbb{C}.

The inevitabilists point to the classification of normed devision algebras, allowing a 4-dimensional but not 3-dimensional analogue of \mathbb{C}.

The contingentists say that the notion of a normed division algebra was invented to make sense of the quaternions.

The inevitabilists point to how the triple ,,\langle \mathbb{R}, \mathbb{C}, \mathbb{H} \rangle, or quadruple ,,,𝕆\langle \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O} \rangle, manifests itself over a range of fields.

And on we go. The inevitabilist will tend to resort to mathematical explanations, which opens them to the charge that that mathematics itself is the result of certain contingent choices. But I’m wondering whether there can be an overwhelming force of evidence from the snugness of fit especially into areas where the concepts had developed independently. Then again this claimed independence is open to challenge.

Posted by: David Corfield on March 25, 2010 10:27 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Alain Connes:

…you feel the same anguish, the same external resistance as you do with external reality.

These seem to be the keywords that describe why mathematics feels “real”.

I suppose everyone knows about Popper’s three worlds?

Alain Connes:

…for me even a human being is better described by an abstract scheme than by a material collection of cells…

To even speak this words requires the mental picture aka abstract scheme of a human body as an interacting cell complex (pun intended), which is neither universal nor very old. I’m sure that the accompanying paradigm of modern medicine, namely “all diseases are diseases of cells”, will change somewhen in the future, just like the comparison of a human mind with a telephone switchboard, customary a hundred years ago, got replaced by the comparison with a computer (which will seem as inept in a hundred years as the switchboard analogy seems to me now).

If one wants to reduce everything to “matter localized somewhere” one soon meets a wall which comes from quantum mechanics…

I don’t understand the rôle that quantum mechanics plays in the argument.

Posted by: Tim van Beek on March 25, 2010 10:45 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

“the rôle that quantum mechanics plays in the argument”: As far as I remember the trialogue book, it is just a plausibility reason that simple concepts of “reality” don’t work anyway, independent from “platonism”. An example without advanced physics gives this nice text of an archeologist on how to present data by drawings. Aside that, Connes takes noncommutativity as fundamental concept, and seems to have the idea of a “quantum platonism”, i.e. that individual minds act as projectors on “platonic ideas”. I’d be curious what composers he thinks at: “Some composers reached, by an hallucinatory work of precision, a level of perfection close to that of some of Riemann’s work.”?

Posted by: Thomas on March 25, 2010 2:40 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Connes wrote:

If one wants to reduce everything to “matter localized somewhere” one soon meets a wall which comes from quantum mechanics…

Tim wrote:

I don’t understand the rôle that quantum mechanics plays in the argument.

Lacking any secret knowledge of what Connes meant, I’d take what he said quite literally: if we try to pursue a simple-minded materialism where we try to reduce the world to ‘objects located somewhere’, we run into a wall. Namely, quantum mechanics says the world isn’t like that. At best, it’s approximately like that under certain circumstances.

Of course, one could switch to a more sophisticated version of materialism which takes quantum mechanics into account…

Posted by: John Baez on March 27, 2010 11:41 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Here is a very nice article by Laurent Lafforgue sketching the “platonic mindset”. Philosopher’s ideas differ (that link explains Connes’ harsh remarks on that).

Posted by: Thomas on March 29, 2010 10:18 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Thanks for pointing this out to me. You can find more of Lafforgue’s philosophical writings here.

Posted by: David Corfield on March 29, 2010 2:19 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

His description of the relation between mathematics and language is very nice too and complements the other text. His quote of S. Weil in one of the texts (“popularisations of science are in part responsible for a general loss of sense of reality”) makes me wonder how Edge etc. work - do they really increase the complexity of their reader’s way of thinking?

Posted by: Thomas on March 30, 2010 1:52 PM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

In hope not to restart silly disputes, here a nice new article of the New Scientist on language shaping perception etc.

Posted by: Thomas on September 6, 2010 11:03 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Here is a talk by Yves André comparing Poincaré’s and Grothendieck’s writings on their “methods”.

Posted by: Thomas on April 29, 2010 11:51 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

News on the “Plato code”

Posted by: Thomas on June 29, 2010 9:52 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

The press release is full of pretty wild claims!

The hidden codes show that Plato anticipated the Scientific Revolution 2,000 years before Isaac Newton, discovering its most important idea…

Posted by: John Baez on June 30, 2010 6:49 AM | Permalink | Reply to this

Re: Philosophy in the EMS Newsletter

Yes - I wait for the movie

Posted by: T. on June 30, 2010 2:27 PM | Permalink | Reply to this

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