- #1
bensprott
- 4
- 0
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...
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...
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