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Really interesting to read your post, despite half of it being a bit over my head. That is quite cool you got to have a long conversation with Penrose! As to
by VirusNewbie 5y ago
Really interesting to read your post, despite half of it being a bit over my head. That is quite cool you got to have a long conversation with Penrose!
As to your statement about platonism or mathematical realism, I'm curious about something. Surely most axiomatic systems are a choice, so I can understand why we would say they're man made (euclidean geometry, ZFC, etc).
However we also know about 'meta axioms', which seem to go beyond being a choice. There are simply some axioms that don't seem optional. Do you hold those to be also man made or is that a separate phenomenon?
- stared 5y agoI am not sure what is a "meta axiom". Could you give an example for one?
- VirusNewbie 5y agoI'm thinking Godel's theorems or AIT.
- stared 5y agoGödel's theorem is a theorem, not an axiom.
- VirusNewbie 5y agooops in my original post I meant to say meta axioms and meta theorems, how embarrassing. As to your reply above, regardless of what you call axioms, semantically speaking there is an isomorphic subset in common for all useful axiomatic system is what I'm postulating. Can you think of an axiomatic system that doesn't have something iso to identity and equivalence for instance?
- stared 5y agoI never went deep into formal logic, so I can misname a few things as well. 1. When it comes to the Gödel's incompleteness theorem - for non-recursive systems you may you can get without that, vide https://math.stackexchange.com/questions/3359878/what-is-some-easy-illustrative-example-of-an-non-recursive-formal-system-and-rec https://math.stackexchange.com/questions/3359878/what-is-som... 2. In general, you can do a lot without equivalence. For example, in set theory (ZF), you get only trivial equivalence (sets having the same elements are the same set). But you can still do a lot with "contains". A similar thing works with partially ordered sets, in which you have the "<=" operator.
- cevi 5y agoThe rules of propositional logic (or first-order logic) would seem to fit: i.e. that fact that "A and B" has the same truth-value as "B and A", or the use of modus ponens as a deduction rule. These rules are usually left out of axiomatic descriptions of mathematical fields.
- stared 5y agoIn formal mathematics, everything needs to be stated explicitly. For a moment, forget about "and" and use "zxhds" instead. You need to specify how it works. There is nothing magic, intrinsic or natural about "truth" or "implies". It is all rules of symbol rearrangement.