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Anyway, I would argue that as 'our understanding' expands, what we understand also changes, and it can hardly be claimed that we are understanding the same thing pre and post expansion. If you make a claim about some elaborated kind of proof, you cant infer anything about a proof was prior to your elaboration. Or better said, you are now using the word 'proof' in a different way than how you had been before you choose to elaborate into what a proof is.

And I would say the same about what Math is. We define the word math, not math itself, and no definition of the word will fully capture all the ways the word is used.

As a side note, I have since studied a bit of proof theory in the last 4 months, and I find it interesting.