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April 16, 2011

Report on Peter Freyd’s Lecture

Posted by David Corfield

Guest post by Aaron Smith on Freyd’s lecture – An Anti-Philosophy of Mathematics

What follows is a quick-and-dirty summary of Peter Freyd’s talk “An Anti-philosophy of Mathematics” delivered April 11th at UPenn.

Professor Freyd’s talk, although interwoven with several anecdotes about his interactions with many bright lights of the field of logic/mathematical foundations, was mainly concerned with discussing characteristics of mathematics with a view toward giving some account of what mathematics is. (With regard to the provocative title, I was unable to make a lot of sense of the explanation he gave, or at least it’s relevance to the rest of the discussion. He pointed out that he wouldn’t be surprised if we eventually saw a Godel-like theorem stating that any sufficiently rich language will produce questions which are inherently unanswerable and perhaps even nonsensical within the language –perhaps to say that the question at hand “What is mathematics” is such a question?]

In any case, he brought out the main theme by focusing on the challenge of giving a definition of mathematics to an uninitiated person. He first drew a distinction between mathematics [what mathematicians do] and the mathematical [mathematical techniques], the former being what he was describing as mathematics –and the latter not. The existence of this distinction seemed to imply that mathematics was about something or concerned with something rather than merely being a methodology. But what is this something?

In order to suggest an answer to this question Freyd reproduced a comment that had once been made to him concerning what mathematics should be about: the man said something to the effect of “I don’t know if mathematics is about something, but I do know that it’s not arbitrary.” Freyd then claimed that perhaps mathematics could be seen as concerned with the non-arbitrary itself. There was a lot of discussion attempting to flesh out his notion of the non-arbitrary. This notion seemed to first encompass matters of information content and complexity: for example the game of chess is far more arbitrary than go in the sense that chess requires a rather significant amount of data to specify its starting positions, whereas go involves a very few simple rules with few “special rules” to control endless looping and to prescribe scoring. Somehow mathematics would concern itself with the structural features of a game like go or chess abstracting away from the arbitrariness of the multitude of starting positions. But Freyd’s non-arbitrariness seemed to span much more, being specified by intuitions that humans have [for instance the empirical-psychological fact that young humans assume Euclidean geometry – or have Kantian forms of intuition thereabout]. And going even further still Freyd pressed the notion of non-arbitrariness to something specified by features of all possible worlds [going beyond the arbitrariness of our own human predicament]: in one example he claimed that in some sense Euclidean geometry should be regarded as less arbitrary than curved geometry, suggesting that intelligent beings living in a curved universe would still construct flat blackboards in order to understand calculus (which is flat geometric analysis) before studying curved geometry. [To me this claim seemed like a gross lack of imagination, or at least an underestimation of incommensurability between distinct forms of life.]

Given that non-arbitrariness incorporates the intuitions of human groups and sub-societies, it is not surprising that Freyd singled out aesthetics and practical usefulness as determining forces of non-arbitrariness. He emphasized that “theorems” rather than merely necessary conclusions are the output of mathematics, understanding theorems as a technical term connoting mathematically relevant necessary conclusions – non-arbitrary conclusions of reason. Ultimately these are facts that mathematicians find useful or interesting for various reasons. This is all folded into the idea of non-arbitrariness as suggested by the maxim (posed by Murray Gerstenhaber in the audience): mathematicians study the questions that pose themselves.

Professor Freyd’s coda involved the suggestion that mathematics should go about understanding itself by articulating its interests/motives/aesthetics more fully.

[My own commentary based on the bit of background I have along these lines goes as follows: Freyd is dealing with themes that have been part of this discussion for quite some time. For instance a major feature of Wittgenstein’s Investigations (and Kripke’s analysis) is concerned with (non-)arbitrariness and rule-following in mathematics. The ultimate skeptical question concerns whether or not mathematical statements/theorems even possess life-form-independent features/meaning or whether or not they would better be thought of as moves in a big socio-linguistic game. As moves which follow rules which are universal across the particular form of life in question, they are not arbitrary, but the non-arbitrariness is not a result of some consideration concerning possible worlds. Non-arbitrariness would be a realization that there are in fact rules – and a task of understanding what mathematics is would involve articulating these rules. Freyd’s concept seems much different, something more like a mathematical natural kind. There also seems to be something very Heideggerian about the whole thing, especially this issue of concern, and the “questions that pose themselves”, and of course in the final suggestion that mathematics should articulate itself. Anyway, I’d love to hear what others have to say about this issue – assuming my presentation is coherent!]

Posted at April 16, 2011 1:28 PM UTC

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Re: Report on Peter Freyd’s Lecture

Thanks very much for this, Aaron. Given that Freyd is raising what I take to be very interesting philosophical questions, I do find his title bizarre. The issue of the non-arbitrariness of our mathematics is one I’ve been trying to steer people towards for a long time. A recent effort appeared in Philosophia Mathematica last year.

Posted by: David Corfield on April 16, 2011 1:45 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

You question if your presentation is coherent. I tried parsing it, but I get a compile-error in the first paragraph due to incorrect bracketing. (…]
;-)

That aside, a 20th century philosopher whos name I’ve forgotten, argued that sets of rules could not themselves specify how to use them correctly. In teaching you therefore have to rely on practical examples showing how the rules are applied. For instance, we have learned E = mc^2 as a textbook example, and some of us have memorized the whole of special and general relativity, but that E = mc^2 should _not_ be used when calculating the energy content of a bar of chocolate, that is not a theorem in relativity theory. Sure, you can make a theorem stating it, but that’s not how most of us got clued-in on it.

It is my opinion that when you discuss the relation between mathematics and reality as we experience it, learning and teaching cannot be ignored. They are what performs the translation between math and subjective world. They and therefore the translation is not itself a pure set of rules.

Posted by: Robert on April 17, 2011 8:35 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

You’re most-surely referring to Wittgenstein. For example see section 3.5 of

http://plato.stanford.edu/entries/wittgenstein/

After reading your comment I realized that I confused the description of Wittgenstein’s skeptical argument and certainly shouldn’t have used the word “rule” because this connotes an abstract, Platonic kind of specification. And that’s precisely the picture he is undermining, namely that we can identify abstract rules which are grasped/understood and which specify which mathematical statements can be made and in which ways –and furthermore, whether what we do as mathematicians could be described as following such abstract rules. So articulation of mathematical practice on this view would have to be a softer subject: somehow explaining and teaching people what it is we do. Likely the best way to teach someone about what mathematicians do (if you have a lot of time) is to teach them mathematics…

Posted by: Aaron Smith on April 18, 2011 12:50 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

Without elaborating some case studies there’s little to be done in this kind of debate other than exchange intuitions.

…in one example he claimed that in some sense Euclidean geometry should be regarded as less arbitrary than curved geometry, suggesting that intelligent beings living in a curved universe would still construct flat blackboards in order to understand calculus (which is flat geometric analysis) before studying curved geometry.

Was there an argument for this claim?

Posted by: David Corfield on April 18, 2011 2:09 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

We are intelligent beings living in a curved universe. As long as we’re talking about a universe where spacetime is described by a Riemannian or Lorentzian manifold, the curvature becomes small at short distance scales and flat geometry becomes a useful approximation. The Earth is not flat, either, but plane geometry was developed a bit before spherical geometry (with the latter being developed mainly to study the spherical sky at first, and only later applied to the Earth). So, it’s easy for me to imagine creatures starting with flat geometry in a wide variety of circumstances, and then moving on to curved geometries.

Of course there could also be curvature at distance scales too small to easily see—like the ‘curled-up extra dimensions’ of string theory.

Posted by: John Baez on April 21, 2011 4:47 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

By the way, anyone interested in this question should read Greg Egan’s novel Incandescence, which explores the development of science on a world where spacetime is severely curved. You can read his discussion of the underlying physics here.

Egan is now writing a trilogy called Orthogonal, about a universe that’s a Riemannian rather than Lorentzian manifold. In his discussion of the underlying physics, he writes:

For the past year or so I’ve been spending most of my waking hours in a place where light, matter, energy and time obey different laws of physics than those that rule our own universe. Studying the way things move and interact under these alternative laws reveals some familiar behaviour, some strange and eerily beautiful phenomena, and some terrifying risks.

To reach what I will call the Riemannian universe involves nothing more than changing a minus sign to a plus sign in a simple equation that governs the geometry of space-time. And curiously enough, although the consequences sometimes seem bizarre, the basic laws here can be understood more easily and intuitively than those that apply in the real world.

“I’ve been spending most of my waking hours in a place where light, matter, energy and time obey different laws of physics than those that rule our own universe”—this explains why we haven’t been seeing him here lately. The first volume of Orthogonal is due out in September.

Posted by: John Baez on April 21, 2011 5:01 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

That’s a brave enterprise. I could imagine spending years on such a project only to have someone point out to them something which would make the physics of the imagined universe rather boring.

If our universe is supposed to be fine-tuned for life, changing the signature of the metric seems rather radical.

Posted by: David Corfield on April 21, 2011 11:45 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

Without getting into the merits of the idea of fine-tuning itself, I think that one reasonable interpretation of it is that the parameters of physical laws are only locally optimal for life, and that other — possibly radically different — versions of physics may also be fine-tuned for life.

Posted by: Mark Meckes on April 21, 2011 2:19 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

What’s the Riemannian equivalent of the classification of Unitary Representations of the Poincaré Group? I see there was a document – The Wigner Classification for Galilei, Poincaré and Euclid – referred to there, but the link no longer works.

Posted by: David Corfield on April 21, 2011 12:57 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

David wrote:

What’s the Riemannian equivalent of the classification of Unitary Representations of the Poincaré Group?

In brief, there are massive particles with the same kind of spins as you’d find in our universe, arising from SO(3) again appearing as the isotropy subgroup. There are no luxons, and there’s no tardyon/tachyon distinction. There’s no difference between parity and time reversal.

Lots of gory details can be found here.

Posted by: Greg Egan on April 27, 2011 2:18 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

Wow. It’s an amazing project to integrate all this in a fictional work.

Posted by: David Corfield on April 28, 2011 11:18 AM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

I won’t be mentioning induced bundle representations in the novels. But by the end of the second volume hopefully every reader will have a good intuitive grasp of the fermion-antifermion annihilation cross-section in a universe where opposite charges repel at short distances.

Posted by: Greg Egan on April 29, 2011 1:03 PM | Permalink | Reply to this

Re: Report on Peter Freyd’s Lecture

I must sluggishly hasten to interject (I had mixed reactions to the claim itself, whether Freyd was making it himself or merely mentioning it, some time ago) that while Euclidean geometry is simpler to define, and while there is an obvious ammount of arbitrariness in any particular Riemannian space, still there are more Riemannian spaces than there are Euclidean spaces, meaning that Riemannian geometry has fewer axioms in it than Euclidean geometry has (and hence fewer things that one can prove about all Riemannian spaces…).

ON THE OTHER HAND: it would be an amusing challenge (amusing for me to challenge someone else, that is) to define Riemannian geometry — still more challenging: to motivate it — without reference to any particular homogeneous isotropic geometry.

(John essentially levels the same challenge with his observation “the curvature becomes small at short distances”; but my way was fun to write!)

Posted by: some guy on the street on April 30, 2011 6:36 PM | Permalink | Reply to this

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