Comments by "Jeff Huffman" (@tejing2001) on "This Is the Calculus They Won't Teach You" video.

On the subject of infinity: Strictly speaking, none of rigorous mathematics deals with actually infinite constructs. You fundamentally can't. You can, at most, allow infinitely many possibilities with a finite description of which possibilities are allowed. It's just a matter of how honest you are about what you're really saying. Coming at things from a more constructive perspective often illuminates a little better what supposedly infinite constructs are really about. And playing at the edge of them leads to interesting conclusions which are often misunderstood to say more than they really do, like how observations about the logical contradictions inherent in supposing you can "pair up" possibilities from 2 finitely-describable spaces leads to the declaration that there are "multiple levels of infinity"... which is a much bigger statement, and doesn't really make any sense. The error was in identifying the philosophical notion of infinity with those equivalence classes of "pairable" spaces in the first place. The same goes for non-standard analysis with its "actual infinitesimals". They're no more "actually infinitesimal" than any other "infinite" construct in mathematics. What makes them different from limits is that they're *actually numbers*, not that they're "actually infinitesimal". "What if we could make limits fully behave like numbers, instead of only mostly? And maybe some other stuff like dirac deltas while we're at it?" I'm all in favor of that, and I think it can definitely lead to more intuitive ways of teaching and understanding calculus.
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@volbla "Although one might ask how different a philosophical construct of infinity is from a mathematical one" Heh, I suppose that's fair. And I don't mean to suggest math can't inform philosophy. It definitely informs mine quite a bit. It's really easy and really dangerous to believe you know more than you do, though.
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@IIIRobIII True, though it may still be an interesting fact that ANY rules can make the concept work.
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@volbla We seem to be mostly on the same page here, I'm just applying the same caution you mentioned about the boundary between math and physics to also put a boundary between math and philosophy. As long as we're clear that we're only talking about a mathematical construct, it's perfectly fine to call those infinite numbers and to say one is bigger than the other (Though in constructive mathematics, for example, you may find that cardinalities can't be proven to be totally ordered, and other wierdness. It's worth some caution even then). It's also worth remembering that there are a lot of different ideas about how to extend the concept of a number or a quantity into the transfinite, and they don't agree about a lot of things. What people often do with the cardinality thing is claim that mathematics has discovered something about the nature of infinity (the philosophical concept), but to understand what really has and hasn't been proven, you need to unpack the stack of definitions you used and be conservative about how you map them to anything real.
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