>>12464288

It's not the "humaneness" that we should care about (I'm actually not even sure what that means) but rather how logically sound our theory is. We want to know that what we're saying is meaningful, we don't want to be simply shuffling meaningless formulas around with evidence-free faith that the formula "0=1" cannot be proven from such manipulations, and pretend that the formulas have any semantics behind them.

I do think that some limited amount of infinitist tools can be good to study in mathematics, to investigate what kind of results could be out there that you may later try to verify using finite mathematics. For example, the fundamental theorem of algebra being proven again and again with infinitist tools suggest that there might be something going on here, and we should try to prove it using finite tools. Surprisingly, nobody has been able to do it yet.

Now the problem in my opinion is that mathematicians care too little about the logical soundness of their theories. If you tell them the status of finite fundamental theorem of algebra, they would act confused and ask who cares? It is known that much of the mathematics that was once thought to have require infinities can be actually understood in a finite, and hence much more clearer and logically sound, way. There needs to be a program which brings as many of these results to light as possible. After all, mathematics is a very cool and interesting subject. The state of it now to my mind is something like the state of italian algebraic geometry before they started to be "rigorous", where implicit assumptions and leaps of logic are thrown around carelessly. People recognized that what was done was actually interesting, even though a lot of it was very logically questionable, and tried to recover it with tools they regarded as actually rigorous. The same effort needs to happen to the whole of mathematics, this time with genuinely rigorous finitistic tools.