### 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!]

## 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.