Comment by j-pb
4 years ago
I often wonder if we simply constructed math the wrong way around.
People tend to mentally construct the natural numbers from set theory, wholes from naturals, rationals from wholes, reals from rationals and so on.
But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
The very act of formalising mathematical concepts into words and paper is a quantisation step after all, because both are symbols. Maybe there are proofs that can be inuitioned about (assuming brains are continuuous in some sense) but neither verbalised nor formalised.
> But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
Presumably this hypothetical universe implements arithmetic, so it’s not complete.
You missed my point.
Incompleteness means that there are true statments for which there are no proofs.
But that doesn't preclude proofs that are beyond the proof system that you proofed incompleteness for.
A non discete/symbolic proof might exist after all.
It would therefore not be the existence of the natural number subset that causes undecidability, but the missing parts of the non-natural superset required to talk about the proofs that cause undecidability.
Doesn’t the incompleteness theorem say that any consistent proof system which implements arithmetic must be incomplete?
Then the system you are proposing, does it implement arithmetic or not?
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Indeed! I suspect/realize that the incompleteness proofs are in fact artifacts of our discrete symbolization - a discrete symbolization that though has been incredibly effective about reasoning about continuous 'phenomena'.
Perhaps that is exactly what mathematics is? and all it can be? Or perhaps there is a 'higher mathematics' that we cannot reach yet? (or ever?)
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