The Mathematicians' Misconception

[@Abhinav-Bhat]

You are so underrated

[@phulcq6716]

I always find it curious when people refer to axioms as "true" - the implication is that mathematical axioms are true in some cosmological sense. But axioms are abstract and we can choose them however we like! Our choices may not always obviously map onto the physical world, but they aren't required to. To suppose that axioms that readily map onto our intuitions are true in the same way as a physical theory seems mistaken to me. The way I think about it is that statements in mathematics are always contingent, so rather than saying things like "Q is true because we can prove it from P", it would be more precise to say "Assuming my axiom P, then Q". Of course this might be unsatisfying for some people because they like to think of Q as being true in a more physical sense. "1 + 1 *is* 2" sounds better than "assuming [everything I need to prove 1+1=2], then 1+1=2". This becomes especially clear when you consider that to be really precise, you would have to account for the rules of inference to get from P to Q in P! So we would have to say something like "If P, and (P -> Q), then Q". It kind of takes the profundity out of any Q when you really show all your working this way. And of course, even though reasoning this tightly appears to be error free, we can never fully eliminate the possibility of making a computational or semantic error. If my rule (P -> Q) is subtly different and actually represents (P -> Q'), then I will have made a mistake.

[@MikolajKonarski]

Of course that mathematics has no notion of an axiom being true (as opposed to a set of axioms being not contradictory, etc.). Every logician knows that, every theoretical computer scientist does, every mathematician born in the second half of XX century and not steeped too much in physics or applied mathematics knows that. Some philosophers do.

[@danwylie-sears1134]

A proof is an abstraction. A particular proof remains the same proof, regardless of whether it's written on paper, carved in stone, encoded in the flows of ions across membranes, or stored as charge on transistors.

[@anatomicallymodernhuman5175]

But physics has to be interpreted. It requires a prior interpretive framework. Concepts are processed in our brains, but they themselves are immaterial, as are the imaginations we grasp them with.