# Does the mathematics of physics require impure set theory?

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

Now, there is a common distinction between pure and impure set theory. Pure set theory is built up from the empty set and only involves "pure" sets--- sets that only contain other sets as members. Impure set theory, by contrast, involves urelements--- non-set individuals which can be members of sets.

My question is whether the mathematics needed for physics (assuming it all gets cashed out in terms of set theory) is of the pure or impure variety? 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? If so, could you provide examples?

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If further clarification is needed, let me know. Also, I hope this question is appropriate for this forum. –  Dennis Jan 3 '13 at 5:05
Further clarification needed: I don't understand what the non-equality/identity primitive relations of physics are supposed to be. Physical theories are written down in mathematical terms where some of these term get associated with a rule where to read off some value from a measurement tool. The formalism is arbitrary, it must only be readable to the scientist. I can't see how the question comes up, as all the formalisms of mathematics, which can be used as foundation for mathematics will obviously suffice for this task. If formalism A can model formalism B, then formalism B isn't "required". –  NikolajK Jan 5 '13 at 22:35
Your question is essentially an answer to my question. I was curious as to whether impure set theory was indispensable. You are saying that any theory that could plausibly be a foundations for mathematics would suffice for physics. So, no one theory is strictly required and no impure set theory in particular is required. –  Dennis Jan 5 '13 at 22:42
It's essentially what the Timelike Cat said, with "interpreted" written out as modeled (understood in the rigorous set theoretic sense). The question called to mind all the threads on MathSE and even PhilosophySE where people discuss how second order theories can really be formalized in first order logic. - Anyway, my question still stands, what do you mean by "primitive relations of physics", how are these beyond mathematical formulations of some ideas for theories of physicist. –  NikolajK Jan 5 '13 at 23:03
@NickKidman I'll admit that I don't have a clear conception of what the "primitive relations of physics" are. I was hoping to get some guidance and plausible examples here. I suppose a rough characterization of what I have in mind are the primitive (undefined) relations of fundamental physics. Obviously this is somewhat cagey, but hopefully it is clear enough to elicit some helpful suggestions? Should this be a separate question ("What are the primitive relations of fundamental physics?")? –  Dennis Jan 7 '13 at 4:03

Impure set theory can be interpreted in terms of pure set theory, so the question is moot.

You may ask which interpretation is "more natural". But embedding arithmetic, analysis, and calculus in terms of set theory is already a fairly convoluted and unnatural thing.

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Impure set theory cannot be interpreted in terms of pure set theory. Well, if by that you mean "there's no real difference" then you're wrong. They have different primitives and impure set theory can only be "interpreted" within pure set theory by adding an additional theoretical primitive, the predicate "is a urelement". But once you do that, you haven't interpreted impure set theory within pure set theory, you've simply turned pure set theory into impure set theory! –  Dennis Jan 5 '13 at 22:02
I agree with your second point, and do not favor this reductionist attitude. I'm just trying to approach this from the standpoint of someone who favors reduction for reasons of simplicity of theory, since that is who I am attempting to respond to. –  Dennis Jan 5 '13 at 22:04
That isn't the only way to interpret impure set theory in pure set theory. You can map the urelements into pure sets such that nothing collides. The predicate "is an urelement" can then be expressed in terms of pure sets. See e.g. section 3.1 of this –  Retarded Potential Jan 5 '13 at 22:22
Hmm, very interesting. I stand corrected! Has there been any discussion of this paper in the literature that you know of? –  Dennis Jan 5 '13 at 22:38
Also, do you have any thoughts on differences like these between pure and impure set theories? If the theories really are "synonymous" then shouldn't the metatheoretic results coincide? –  Dennis Jan 5 '13 at 22:44

I'm not entirely sure this question is answerable in terms of physics.

The reason is that all physics theories are reducible to combinations of data structures (e.g. scalars, vectors, matrices, tensors) and associated behaviors. Data structures and associated behaviors are in turn (and pretty much by definition) entities that can be represented within any formal system that has enough richness and complexity to build a Turing machine within it. Even odd physics issues such as quantum entanglement can be modeled in this fashion, just very inefficiently.

So, the problem is that pure and impure set theories are both rich enough (easily!) to build Turing machines. Either one could be used represent any formal physics theory. Since pure sets are a bit simpler, that would (I guess) be the winner between the two options.

However, your real question may be more along the lines of asking if there are any unique, atomic, indivisible things or properties in physics that match better with urelements than with pure sets. Asked that way, I'd say there is a pretty good chance that there are. Physics hasn't quite found them yet, but structure does get simpler as you get smaller. For example, nothing in physics seems to get below $1/3$ of an electron charge or less than $1/2$ of a unit of spin, so something firmly "atomic" or "indivisible" may be going on with those units.

However, I would also point out that such smallest-units in physics tend to come in mutually annihilating pairs, not as simple additions. The idea of mutually annihilating pairs is not typically seen in the all-positive-additions way in which sets are typically constructed, or at least not something I've seen in set theory.

So, if you want to broaden the question a bit and say "what kinds of mathematical systems best seem to capture the natural structure of physics as we observe it?", I would say "something that is based on the creation and annihilation, at many different levels, of mutually annihilating pairs of properties or entities." It would be akin perhaps to a more discrete version of the framework found in quantum field theory.

I am not specifically aware of any specific formal system of that type in mathematics, at least not as part of the search for more fundamental mathematical building blocks like sets.

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The second and third paragraphs ("The...options.") provide exactly the sort of information I am looking for. Do you know of any papers that discuss what you describe in those paragraphs? The reason I'm interested in this is that there is a common argument in the Philosophy of Mathematics (the area I work in), the "Indispensability Argument" which argues that we must accept mathematical entities because they play an indispensable role in our best theory of the world (assumed to be something like fundamental physics). -> see next comment –  Dennis Jan 3 '13 at 6:12
Tomorrow, sorry! (1:14 AM here) –  Terry Bollinger Jan 3 '13 at 6:14
I'm currently reading a paper where the author assumes that physics requires set theory, and in particular impure set theory. He then proceeds to argue for a set-theoretic construal (as opposed to mereological) of physical geometry and defends it on the grounds that it is simpler (in just the sense you describe in the second paragraph) than its mereological counterpart. I was skeptical of his claim that impure set theory was indispensable to physics and it seems that you think my suspicions are right. –  Dennis Jan 3 '13 at 6:17
Oh no worries! Thanks for any help at all that you can provide. I am a philosopher/logician by trade and am woefully under-informed with respect to physics. So I appreciate any assistance in better understanding this topic. –  Dennis Jan 3 '13 at 6:18
Dennis, sorry, busy days. Alas, I don't have any specific references for that, at least not that I can recall. The Turing argument often used to be brought up in the context of whether human sentience could be modeled accurately using computers, and I think such arguments just sort of settled deeply into my bones. I would completely concur, however, that an argument that only impure sets can model physics is invalid, for the same reasons I gave: If you can make a Turing machine with it, you can model physics with it -- and pure sets certainly have sufficient power to create Turing machines. –  Terry Bollinger Jan 4 '13 at 5:23