> Even if you did that, you should show somehwere this finitist foundation disagrees with the results obtained by the standard foundation, otherwise there's no reason to think the standard foundation is in error.

Well these are probably easy to find even now? E.g the Banach-Tarsky paradox is unlikely to be provable in finitist math which is somewhat of an improvement.

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griffzhowl 3 days ago | parent [-]

I was thinking more about applications in physics where calculus and irrational quantities are used all the time.

At more advanced levels the theories are based on differential geometry and operators on Hilbert space. I'm not sure if fully worked out finitist versions of these even exist. Where finitist versions do exist, they're often technically more difficult to use than the standard versions, which is the opposite of an improvement in my view.

Whether it's undesirable for your mathematical foundation to prove the Banach-Tarski paradox is debatable. It's counter-intuitive, but doesn't lead to contradictions, as far as is known. It doesn't apply to physics because the construction uses non-measurable sets.

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alexey-salmin 2 days ago | parent [-]

I'm not a finitist myself but my understanding is that it has to do as much with physics as does ZFC, which is very little. The math used in physics works on practice and did work long before the question of foundations even came up.

The problem that bothers some mathematicians is that despite working well math still lacks a solid foundation. Furthermore it's basically proven that these foundations can't even exist, or at least for the mainstream version of math. This is where non-mainstream versions pop up. The denial of uncountable sets does help you resolve some of the paradoxes. Not all unfortunately, even the countable sets already lead to things like incompleteness theorems. Well, one can dream.

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griffzhowl 12 hours ago | parent [-]

> Furthermore it's basically proven that these foundations can't even exist,

What are you referring to? The current working foundation is ZFC but there are equivalent type theoretical foundations like what Lean and other proof-checking software uses. I guess you know that, but that's why I don't know what you mean by saying this

	▲	alexey-salmin 3 hours ago \| parent [-]
		I'm referring to the failure of Hilbert's program. All the incompleteness, undefinability and undecidability results arise when and only when some sort of infinite objects are present so I can definitely see the allure of finitism. ZFC is a working foundation of math but it's unknown whether it's consistent or arithmetically sound and important statements like CH are independent from it. It's a "working foundation" but not a "true foundation" which alas cannot exist. As mentioned above I'm personally not a finitist and think that math without infinite and uncountable sets is intellectually poorer. I don't mind however developing further a finitist subset of math and see what's provable (and describable) in it, much like there's value in proving theorems in ZF instead of ZFC whenever possible.