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by zogomoox 1059 days ago
If you consider that mathematical ideas are discovered rather than invented, then the whole point becomes moot. Sure, we may not be smart enough to discover all of it, but we can be pretty certain there is still mathematics beyond our ability to understand it. As an analogy, we'll never be able to look at or visit beyond our cosmic horizon, but it's certain there are stars and galaxies out there.
2 comments
DecayingOrganic 1059 days ago
Math as we know it, due to Gödel's Incompleteness Theorems, is not fully consistent - we can't even prove everything that is true with our current mathematical framework. This means that our understanding of math is indeed limited, not just by our intellectual capabilities, but by the very structure of the math we currently use also.

This makes me wonder, will we be able to develop new mathematical frameworks that bypass these issues? And if so, what will they look like?

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kbrkbr 1059 days ago
> is not fully consistent

Not being able to prove the consistency of a system within the system does not entail that the system is inconsistent (or not fully consistent).

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pixl97 1059 days ago
I think it means we could only ever prove it is inconsistent, and never prove that it is consistent.
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tux3 1059 days ago
If we can only prove that an inconsistent system is inconsistent, and we can never prove our (presumably) consistent system is consistent, then it is incomplete.

We can't say it's inconsistent just because we can't prove otherwise. We can say it may be incomplete (or inconsistent, and we haven't noticed yet)

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zmgsabst 1059 days ago
The insight is that mathematics is not complete — which is the property that a system can prove every true theorem.

Consistency is there as a technical detail: an inconsistent system can prove every true theorem, by virtue of being able to prove every theorem.

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arketyp 1059 days ago
But it does become relevant if you consider the mathematics we can understand expanded to some ontological boundedness, for instance perhaps something that relates to the validity of the Church-Turing thesis. Then talking about mathematics beyond such a scope becomes moot too. With this I'm not saying we can't reason about Gödel's results by the way.
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