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Absolute Truth in Math

Lawrence D'Anna 2021-09-06

People love to hold up mathematics as a realm where absolute truth is available.

This is not only wrong, it’s hilariously, profligately wrong. There is such a fascinating diversity of ways in which this claim is wrong that an entire subfield of math is mostly dedicated to studying them. It’s wrong in simple and obvious ways, and it’s wrong in stunningly subtle ways with amazing connections to seemingly unrelated fields like geometry.

Every mathematical formalism that exists can be contextualized in ways that make it’s claims contingent or relative. They can be placed into more and more powerful formalisms, forming amazing a transfinite hierarchy. Or formalisms of vast power can be seen as mere word games within “basic” ones at the bottom of the hierarchy like PA – and PA can prove profound theorems about them by doing so.

Even seemingly basic concepts like boolean true and false are undermined by logics that see them as special cases.

Formalisms that at first appear to be talking about one specific thing thing inevitably wind up being consistent with other, more surprising and exotic interpretations. Take non-euclidian geometry, uncountable models of PA, countable models of ZFC, or nonstandard calculus as examples.

Math is the last place anyone should look for absolute truths.

Intuitions and hasty conclusions about truth

Matt C. Wilson 2021-09-14

I’m not mathematical enough to confidently understand what you’re saying about ZFC and PA, but I’ve read enough of Godël, Escher, Bach to recognize what you’re pointing at.

I think another way people stumble into “absolutes” is by a kind of extrapolated certainty from applying deductive reasoning against atomic mathematical facts.

Something like - I understand the idea of three. I see three trees. When I wake up tomorrow, three will still represent the same concept, and I’ll reliably be able to use three to count things like trees. I can add with three, and multiply with it, and … all of mathematics is super reliable and definite and complete, in this way, even if I hit a limit mentally where I don’t attach concepts or meanings to the more complicated parts of it.

So, yeah, Godël shoots down that sort of certainty. But there’s still something there, from a layperson’s perspective. Math is a system in which certain formalisms do yield consistent, reliable fact. That isn’t absolute truth, but it isn’t nothing, either.

I think it’s easy for Joe the plumber to mistake math for absolute truth more because it’s the closest thing they’ve previously experienced to what that might be. They haven’t wrestled with the Incompleteness Theorem. There’s an analogy here to losing faith in eternalism, for sure.

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