Yes. It had some adherents in the 1930’s then lost out on mindshare to set theory. It’s making a comeback via approaches like homotopy type theory, and on the philosophical side, neo-intuitionism.
Yes I’m aware of type theory. Maybe I don’t get where you’re coming from. Of course there are alternatives to set theory. The fact that foundations are in a constant state of turmoil, and that there are interesting alternatives, doesn’t invalidate set theory, nor disturb its place as the dominant paradigm up to the first couple of years of grad school. In areas like abstract algebra, differential geometry, and algebraic geometry, set theory has already been supplanted by category theory.
Alternative approaches are perfectly fine. I get the feeling that you think we are having a difference of opinion over whether type theory exists or whether set theory may or may not be supplanted by a better foundation. I agree to all of it.
Can you just explain to me where you’re coming from? The point is that if you, or anyone, out of a sense of what, contrariness or whatever, wants to argue that set theory is some kind of evil plot designed to suppress the Truth, well I just can’t hold up my end. Set theory is a tool. Math is a big toolbox with many tools.
No I disagree with you. There is intuitionist philosophy that corresponds to type-theoretic math. And there’s also philosophy that corresponds to full infinitary math. I do not believe you are correct that all the philosophers are intuitionists or finitists. That’s simply not the case.
That expression is undefined. When I speak of mathematical infinity I speak of no such thing. A set is infinite if it may be bijected with a proper subset of itself. That’s the working definition.
Nonsense. The reification of reciding? What the hell does that mean?
I don’t care if circles don’t exist. Mathematical circles do, and they’re interesting and useful. And by Putnam’s indispensability argument (linked earlier), circles are entitled to abstract existence by virtue of the fact that they are indispensable in understanding the world.
Take it up with Putnam and Quine, not me.
ps Here is the SEP link. These are high fallutin’ philosophers making the case that because abstract mathematics turns out to be indispensible – as I’ve been putting it, interesting and useful, but indispensability seems to be a higher standard – that abstract mathematical objects have a claim on existence.