### Philosophy of physics

During my year long teaching post at Oxford (2002-3), I spent most of my time with the philosophers of physics there. Philosophers of physics tend to be much more au fait with contemporary physics than philosophers of mathematics are with mathematics. Worringly for us philosophers of mathematics they tend to know more mathematics too. One piece of evidence for these last two assertions is the 202 page paper Algebraic Quantum Field Theory by Princeton philosopher Hans Halvorson (including a 61 page appendix on reconstructing groups from their categories of representations type results by Michael MÃ¼ger). This is to appear in Handbook of the Philosophy of Physics edited by Jeremy Butterfield and John Earman. In this paper Halvorson identifies himself with analytic philosophy:

Against Halvorson, we cannot separate a theory from the socially-embodied arguments in which its adherents and opponents engage. To the extent that Halvorson has done something worthwhile in writing this long paper, it will modify the thinking of people working in and around quantum field theory. Once we realise this, mathematics seems strikingly similar, full of extended conversations and arguments. I shall end with the moving remarks of Ross Street, an exponent of my favourite mathematical research program, from his talk at the IMA n-categories workshop:

In philosophy of science in the analytic tradition, studying the foundations of a theory T has been thought to presuppose some minimal level of clarity about the referent of T. (Moreover, to distinguish philosophy from sociology and history, T is not taken to refer to the activities of some group of people.) In the early twentieth century, it was thought that the referent of T must be a set of axioms of some formal, preferably first-order, language. It was quickly realized that not many interesting physical theories can be formalized in this way. But in any case, we are no longer in the grip of axiomania, as Feyerabend called it. So, the standards were loosened somewhatâ€”but only to the extent that the standards were simultaneously loosened within the community of professional mathematicians. (pp. 3-4)So philosophers should study AQFT because:

AQFT is our best story about where QFT lives in the mathematical universe, and so is a natural starting point for foundational inquiries.That parenthesised sentence in the first of the quotations above worries me, since it seems to me to express a sentiment that will forever condemn philosophy of mathematics to avoid serious engagement with the activities of practicing mathematicians. On pp. 4-5 of my book I raise the question as to why the philosophies of physics and mathematics have such differing relationships to the discipline each studies, and suggest that much of the difference is due to events in the history of analytic philosophy and the role it assigned to logic in philosophical analysis. I expanded on this in a review of Martin Krieger's

*Doing Mathematics.*There I am critical of some parts of philosophy of physics for not realising the extent to which they have diverged from analytic philosophy. Most philosophers of physics avoid confrontation with that section of the analytic heartland know as metaphysics, but it is striking that when they do, they often reveal very large problems, such as when Jeremy Butterfield confronts David Lewis's metaphysics with the fact that it is incompatible even with classical mechanics. Personally, I feel the best way to frame what philosophers of physics are doing when they think hard about the presuppositions of physical theories is what R. G. Collingwood described as engaging in metaphysics. In an earlier post, I outline this in terms of what I call the Historical Stance with reference to the Empirical Stance of Bas van Fraassen, another Princeton Philosopher.Against Halvorson, we cannot separate a theory from the socially-embodied arguments in which its adherents and opponents engage. To the extent that Halvorson has done something worthwhile in writing this long paper, it will modify the thinking of people working in and around quantum field theory. Once we realise this, mathematics seems strikingly similar, full of extended conversations and arguments. I shall end with the moving remarks of Ross Street, an exponent of my favourite mathematical research program, from his talk at the IMA n-categories workshop:

For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established.

## 2 Comments:

When I saw that parenthesized sentence I paused as well. I mentally decided it should be rewritten as "T is not taken to refer

merelyto the activities of some group of people". While mathematics might in fact be the activities of some group of people, this doesn't seem entirely constitutive of what it is, as a historian or sociologist might say. The activities are relevant for finding out what counts as mathematics, but in a sense they are merely illustrative. I think Maddy has some nice suggestions for her naturalistic philosopher, who should use the practice of mathematicians as a guide to see what the real goals and methods of mathematics are. The philosopher can then use these goals and methods to show what the mathematician really should be doing, in case she isn't engaged in totally productive mathematics. There's not much room for criticism, but there is some.I think this room for criticism is why we want it not to be merely the activities of some people.

Kenny,

I appreciate Maddy's work very much but I think it leaves out an important dimension, namely, arguments internal to mathematics. It is all to easy to imagine that consensus reigns within mathematics, aside from some fringe constructivist or finitist programs. I noted somewhere in my book, which I don't have to hand, that she tends to take the goals of a branch of mathematics as already in place, and thus unchallengeable. But there are serious disagreements about the worth of programs from outside, and from within programs about the road they should take and the ways they should develop. Our role as philosophers is not to take sides, although to understand a particular program we may have to devote such considerable resources to it that we grow to love it. Rather one of our main roles, according to my MacIntyrean conception of rationality, is to foster more open critical discussion. For a small attempt to do this see here.

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