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October 1, 2008

Mathematical Reality

Posted by David Corfield

Here is the draft of paper I’m contributing to a book on the work of Albert Lautman – Lautman and the Reality of Mathematics. I’m going to be talking about it this Saturday at a workshop I’m jointly organising. Comments are welcome.

Choosing the title for this post reminded me of a post with the same title written for my old blog, which is no longer hosted anywhere. I wrote back in December 2005:

Contemporary differential geometry is dramatically broadening its horizons. For a taste see Non Abelian Differential Gerbes, “We develop a differential geometry theory of non-abelian differential gerbes over stacks using Lie groupoids”, and Higher Gauge Theory “We describe a theory of 2-connections on principal 2-bundles and explain how this is related to Breen and Messing’s theory of connections on nonabelian gerbes”.

What is very noticeable is the number of routes which seem to be leading in the same direction. One shouldn’t underestimate, however, the work of reconciling different viewpoints. Great rewards are due for work which, although it proves nothing new, performs this reconciliation well. In his Racah-Wigner quantum 6j Symbols, Ocneanu Cells for AN diagrams and quantum groupoids, Robert Coquereaux claims:

Our purpose in this paper is very modest. Indeed, all the objects that we shall manipulate have been already introduced and studied in the past, sometimes long ago: 6J symbols, quantum or classical, are considered to be standard material, cells and “double triangle algebras” have been invented in [28], [31] and analyzed for instance in [5], [37], [14] or [39], finally, quantum groupoids are studied in several other places like [7], [25] or [26]. However, it is so that many ideas and results presented in these quoted references are not easy to compare, not only at the level of conventions, but more importantly, at the level of concepts, despite of the existence of the same underlying mathematical “reality”.

Now, why the scare quotes? There are two types of philosophical position that require them. One is a form of idealism which would want scare quotes to be used at the mention of any form of reality. Even the reality of chairs and tables needs putting into question. This is presumably not what Coquereaux believes. What I take it that he is implying is that just as there is a physical world which places severe constraints on what we can and can’t do - we can swim in a river, we can’t walk through trees, we can’t jump up 10 metres, etc. - there is something not so very different which forces mathematicians to work along similar lines, even if this is not always obvious, and this something is not merely logic. In this quotation of Connes, again we see ‘mathematical reality’ in scare quotes. Again, mathematicians often meet each other in the same places:

whatever the origin of one’s itinerary, one day or another if one walks long enough, one is bound to reach a well known town i.e. for instance to meet elliptic functions, modular forms, zeta functions

There is a danger in confusing this mathematicians’ realism (remember not all mathematicians are convinced that this convergence is so important - Zeilberger’s Opinion 49, Ruelle’s Is Our Mathematics Natural?, Bull. AMS 19, 259-268, 1988), with what is at stake when analytic philosophers of mathematics take up realism. Here there is no interest in specific concepts like 6j symbols or elliptic functions. Where the mathematicians will be able to point to concepts that although consistent are not a part of their reality, philosophers generally argue for or against realism across the set theoretic board.

And John replied:

There is a danger in confusing this mathematicians’ realism […]

This is a very important point. Presumably this is because most philosophers don’t actually do mathematics (or even carefully watch people who do).

When you do mathematics, you keep “bumping up against” certain concepts. When you bump up against something - make contact with it unexpectedly, not by design - you start treating it as real. It’s like Boswell’s old anecdote, dating back to 1763:

After we came out of the church, we stood talking for some time together of Bishop Berkeley’s ingenious sophistry to prove the non-existence of matter, and that every thing in the universe is merely ideal. I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered, striking his foot with mighty force against a large stone, till he rebounded from it, ‘I refute it thus.’

Are any philosophers of mathematics willing to grant more “reality” to “natural” concepts than “artificial” ones? A “natural” concept is one you bump up against in the course of doing mathematics, while an “artificial” concept is something you could make up just for fun, but wouldn’t ever feel the need to study.

It’s not clear that “reality” is the right word for what the former concepts have that the latter don’t - but whatever you call it, it’s very important to most mathematicians, and any philosophy that neglects it is not a philosophy of real mathematics.

Posted at October 1, 2008 12:26 PM UTC

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5 Comments & 1 Trackback

Re: Mathematical Reality

“A “natural” concept is one you bump up against in the course of doing mathematics, while an “artificial”; concept is something you could make up just for fun, but wouldn’t ever feel the need to study.”

What about the concept of “useful” in mathematics? Are “natural” cancepts also generally useful (to maths, of course)?

Posted by: Alessandro on October 1, 2008 3:37 PM | Permalink | Reply to this

Re: Mathematical Reality

When I looked at debate surrounding whether groupoids are a good thing for chapter 9 of my book it seemed like there was a trichotomy at play. Some saw them as worthless, some as useful and some as allowing us to get nearer to some truth (about symmetry). It’s very reminiscent of the way we may take a class to fail to pick out anything worthwhile (e.g., things whose English names have ‘f’ or ‘p’ as third letter), or to pick out something useful but not essential (the tree/bush distinction may be of use to a gardener, but is not recognised by a botanist), or to pick out a real class (mammals, emeralds, etc.).

As to whether the latter type, which may be thought to correspond to the mathematically ‘natural’ concepts, must also be useful, well if we follow Polanyi’s position, any such class/universal should reveal its potency by an indefinite range of discoveries revealing similarities between class members.

These themes are discussed in the thread Mathematical Kinds.

Posted by: David Corfield on October 2, 2008 8:48 AM | Permalink | Reply to this

Re: Mathematical Reality

The questions which are raised concerning the reality of mathematical truths and the relationship with the real world can be answered within the platonist approach:
the mathematical ideas and the corresponding mathematical structures have an independent existence outside of the spacetime. Our brains have a capability to see these ideas, i.e. we discover new mathematics, as opposed to the constructivists position of constructing mathematics.
Goedel was a platonist, and that inspired him to formulate his famous theorems in logic. As far as the real world is concerned, within the platonist approach, there is a proposal by a cosmologist Max Tegmark, where our Universe and everything in it, including ourselves, is one mathematical structure. This naturally leads to a concept of a Multiverse. I partially agree with Tegmark´s picture, i.e. I beleive that only a part of our Universe is a mathematical structure, while the new elements which I would add is time and non-mathematical ideas.

Posted by: Aleksandar Mikovic on October 2, 2008 1:04 PM | Permalink | Reply to this

Re: Mathematical Reality

Compare opening sections of Roger Penrose:
The Road to Reality

Posted by: jim stasheff on October 3, 2008 1:33 PM | Permalink | Reply to this

Re: Mathematical Reality

Alas, just because one has a concept, it does not mean there exists anything satisfying the concept. Otherwise, one is just committing the ontological fallacy, exhibited most famously in the proof of God’s existence from the mere existence of the concept of God.

Posted by: abo on October 5, 2008 9:50 AM | Permalink | Reply to this
Read the post Mathematical Robustness
Weblog: The n-Category Café
Excerpt: Does the 'robust' appearance of mathematical entities indicate reality?
Tracked: November 20, 2008 10:03 AM

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