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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 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 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 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 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 at 21:58

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