Just as a photon is either a wave or particle depending on how it is measured, it seems like these difference in math philosophy may all be neither wrong nor right - it depends on how the topic is approached. — EricH
For example, certain principles or properties involving collections of sets may be more naturally expressed in SOL.
Second-order logic plays a significant role in model theory, which is the branch of mathematical logic that deals with the study of mathematical structures (such as sets) using formal languages.
Pretty much. So mathematical expressions are true only if there is a proof-path that shows it to be true. There are, one concludes, mathematical expressions that are neither true nor false. — Banno
This is opposed to Platonism — Banno
But then again, prima facie there is nothing necessary about the idea of cats, protons, or communism. It could be that numbers are innate ideas, being then "world-independent".
But then the claim "it is not the case that this proof-path pre-exists our construction of it", the syntax being the proof-path, and in our case being the FOL that we see in things such as ZFC, did we really construe relations such as ∧ and →? If so, it would then bring up "how did we"? — Lionino
I'm not following what you say here. — Banno
it's the relation of abstract entities, nature, and mind from which knowledge emerges — Count Timothy von Icarus
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