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Assume for the sake of this question that mathematics is reducible to set theory in such a way that the only mathematical objects there really are, are sets.

Suppose further that the Indispensability Argument is successful and shows us that we need to believe in sets.

My question is would this show us that impure set theory is indispensable? Does physics require impure sets? In other words, are there primitive relations of physics (where a primitive relation of physics is something unique to physics--- not something like equality/identity) that hold between mathematical entities (like numbers) and physical entities?

Are there any good papers that discuss indispensability and impure set theory?

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What would it mean for a relation to be "unique to physics", especially if one believes in the indispensibility of mathematical objects? –  Niel de Beaudrap Jan 3 '13 at 11:58
I'm not sure what you mean here. How does the indispensability of mathematical objects have anything to do with relations being "unique to physics"? Are you assuming that all relations are mathematical objects? If you're making that assumption then my point in making that remark is that I wish to focus on those relations distinctive of physics. My example of what I was not looking for, identity, was meant to illustrate that identity is a relation assumed by most (if not all) theories and plays no interestingly different role in physics. –  Dennis Jan 3 '13 at 19:31
Well, I was asking precisely what you meant by a relation being "unique to physics". How is a supposedly-purely-physical relationship prevented from arising in mathematical theories in some other way than empirical motivation? Could you present an example of a physical relationship that is in some way characteristically "physical" like this? As to the other part of my remark, I was simply observing that if you believe that mathematical objects are features of reality, it might be harder still to describe what it might mean for a relationship to be purely physical. –  Niel de Beaudrap Jan 3 '13 at 19:38
Ok, now I understand a bit better. I don't have a clean definition of this notion of "unique to physics". It was meant to be vague (but hopefully somewhat intuitive) and I introduced it simply to avoid answers like "equality is a physical relation since force is equal to mass times acceleration"--- I didn't want anything crucial to hang on it. As for an example, if you look at this paper, an example of what I had in mind is the primitive s, where s(p,t) may be glossed as a vector which gives the position of a particle p at a time t. –  Dennis Jan 3 '13 at 21:50
@NieldeBeaudrap I'm really not sure that is a great example, but it is the best I can come up with at the moment, especially given my lack of knowledge of physics. Finding putative examples of such relations, and seeing whether they needed to relate particles to numbers, for instance, was a big part of the reason I asked this question. –  Dennis Jan 3 '13 at 21:58

1 Answer 1

I'm afraid I don't know of any papers that discuss indispensability and impure set theory directly. In a sense there's a reason to think that impure set theory is nice to have for physics. As you know, it's natural to represent functions in set theory as sets of ordered pairs, hence if you want to talk about functions that map spacetime points to other spacetime points, you'll need impure sets (if those functions don't float your boat, pick your favourite physical objects that you want to talk about functions from/to).

However, to say that impure set theory is indispensable is a much stronger claim. It's also going to be tough to establish, given (as I'm sure you're aware) that the universe(s) of pure sets is pretty big, far bigger than the physical universe (on any plausible theory of the physical universe; the biggest I think you can reasonably get it is if you allow mereological sums of spacetime points then you might be able to argue for the universe having cardinality $2^{2^{\aleph_0}}$, which is diddly squat in set theoretic terms). Given then the sheer abundance of sets, we can always represent new an interesting physical phenomena by pure sets, in such a way that the impure set theory is dispensable.

However, one should be mindful of the dialectic into which an indispensability argument is often inserted. Usually there is some sort of Quinean holism in the background providing the necessary oomph to think that indispensability to science matters. Given this, one's question really should not be "what is indispensable to science?", but rather, "what is indispensable to our best theory of the world?".

If the latter is the question, and if one thinks that categoricity is important for a mathematical theory (say for worries about first-order theories being unable to pin down their intended model up to isomorphism), one might be interested in the following paper by McGee:

McGee, Vann; `How We Learn Mathematical Language', The Philosophical Review Vol. 106, No. 1 (Jan., 1997), pp. 35-68

There he gives a full categoricity proof for $ZFC$ (on the assumption of unrestricted first-order quantification), by first adding urelemente and proving the categoricity of the pure sets from the impure universe. Thus if it turned out that urelemente were indispensable for this task, one might think that that impure set theory is an indispensable part of our best theory of the world after all.

[As a footnote, it should be noted that there are plenty of other ways to get categoricity given unrestricted first-order quantification. See, for example:

Martin, Donald A. (2001). Multiple universes of sets and indeterminate truth values. Topoi 20 (1)

who argues that any two universes of sets (satisfying certain criteria) can be combined,


McGee, Vann (1992) Two Problems with Tarski's Theory of Consequence Proceedings of the Aristotelian Society New Series, Vol. 92, (1992), pp. 273-292

where he argues for categoricity through the introduction of a satisfaction predicate.]

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