Set theory, Foundations, Physics and Causal Structure

[bensprott]

Hey,

I would like to start a discussion about the use of set theory in mathematical physics. I myself have done research in categorical physics and have seen the debates on how it can be an alternate foundation for mathematics. We can discuss here a few things, but try to stick to these things:
1. Positivism is good/bad/ugly in that maybe it doesn't matter what foundation we use in math because math has nothing to do with ultimate reality
2. The use of set theory in mathematics has, without our knowing, provided a false background and false ontology for physics. We take the prototypical example of the set of all pure states of a quantum system and the statement "there exists a vector x in Hilb(n) such that..."...really? the vector exists? That's not how I saw it...discuss
3. Category theory could serve as a better foundation for math in which to do physics because it is based on morphisms. We have structure in the relationships between transformations and we further see the morphisms as part of a realist causal structure. The prototypical example is HILB because we have all the structure in the morphisms and we don't even need to consider the objects as structured sets.
4. If we present the theory of categories IN SET, then we will end up with the same problems of ontology for physics that we had when just did everything in SET.

This is going to be epic...

 

[apeiron]

I would like to start a discussion about the use of set theory in mathematical physics. I myself have done research in categorical physics and have seen the debates on how it can be an alternate foundation for mathematics.

This is indeed very interesting. And I can talk about categorical biology too (Robert Rosen's work). So what are the actual questions?

1. Positivism is good/bad/ugly in that maybe it doesn't matter what foundation we use in math because math has nothing to do with ultimate reality

I think the answer here is that maths has to be "logical" and reality also has to be "logical". So while one is clearly epistemology (a model of reality) and the other is ontological (it is reality), ultimately the two should mirror each other in some fundamental way. Maths may seem arbitrary, but examining its foundational concepts will reveal "there was no choice" as there is only a particular way that causality can operate, worlds develop into existence.

So that makes a search for a basis to maths an intelligible project.

2. The use of set theory in mathematics has, without our knowing, provided a false background and false ontology for physics. We take the prototypical example of the set of all pure states of a quantum system and the statement "there exists a vector x in Hilb(n) such that..."...really? the vector exists? That's not how I saw it...discuss

I agree that sets introduce the notion of the contained, the constrained, without dealing properly with the question of the container, the constrainer. Whereas it is more foundational to have a view based on a functional dichotomy. To have particulars, you must also develop the universals. So set theory gives you only ever half the story (taking the other half for granted).

3. Category theory could serve as a better foundation for math in which to do physics because it is based on morphisms. We have structure in the relationships between transformations and we further see the morphisms as part of a realist causal structure. The prototypical example is HILB because we have all the structure in the morphisms and we don't even need to consider the objects as structured sets.

It is based on the dichotomy of structures and morphisms. It recognises that existence depends on a fundamental duality or division. And it is then the exact nature of this division which needs to be the focus of inquiry.

What do you mean by HILB though? Hilbert space?

And it sounds as though you want to treat morphisms as more fundamental than structures here. I think that would be a false step myself if so.

4. If we present the theory of categories IN SET, then we will end up with the same problems of ontology for physics that we had when just did everything in SET.

Categories have to incorporate sets, but would be the larger description. If you are saying the philosophical argument has to be made at that higher level - so not as a construction of set theory, but a broader justification - I would agree.

It would be useful here to perhaps supply some references that are crucial to your own position. Is there some author or illustrative paper that set you down this path?

 

[bensprott]

Excellent reply. I will need time to respond. I will let others come in too. Your post proves that this will be epic.

 

[Deveno]

as i understand it, all the action in category theory is in the arrows (and the definition of arrow goes beyond "morphism", there are categories in which arrows have no relation to structure-preserving maps), objects are by-and-large irrelevant.

to draw an analogy with group theory, it is a well-known theorem that every group can be seen as a permutation group. a permutation group is a set of bijections on a set, the actual elements being permuted don't matter, the action is the only relevant information.

in categorical terms, one doesn't need a "sub-thing", one only needs a monic arrow, and one doesn't need a "quotient thing" (made out of "co-things"), just an epic arrow (nice pun, eh?). the notions of "inclusion" and "equivalence" can be recovered from this way of looking at it, but why bother?

we live in a world that is changing. a categorical approach adopts the position that "how is it going to change?" is a more worth-while question than "what is it going to change into?". it is interesting to note that first-order logical systems (on which ultimately set theory rests) can themselves be seen as categories in their own right.

there is a sea-change at work, here. questions of what something "is" are supplanted by what something "does". it doesn't matter what a system is made out of, it matters how it behaves.

yes, it seems odd to abandon the objects of a hilbert space, in favor of unitary operators (i think those are the proper morphisms, although perhaps some formulations would use bounded linear operators, depends on whether you want to preseve the inner products, or the topology), until one realizes the identity operator always qualifies, and we may regard that as a "stand-in" for an object.

i don't think "reality" has to be logical, i think "we" have to be logical. it is conceivable that some things in the universe are so sensitive upon initial conditions that they will resist any attempt at ever being "determined". but we need to understand, as best we can, how to use the dynamics of this world, if for no other reason (although there are others) than to continue our survival. and there ARE definite dynamics: the sun shines, and we reap the manifold benfits of the physical consequence of that. we also know that the sun won't do this forever, so if we don't figure out how to get at least some of us from here to some other star that isn't going to die as quickly, we're done for. and unless we get a handle on what this "gravity-stuff" really is, that's not going to happen (thank you, mr. einstein, for taking all the fun out of intersteller travel. you'll get yours.).

but i digress. set theory DID serve a useful purpose, it showed that trying to define a basic structure to unify math (and thus give coherence to this language we call mathematics) could be a fruitful endeavor. it is a relief, after all, that Set turns out to be a category, and that functors to Set are central to the theory.