It seems to me that any modern philosophical investigation of infinity reduces to looking at the mathematical concept of infinity (as introduced by Georg Cantor) and then delving into philosophy of mathematics (where I include questions regarding applied mathematics, like if mathematical infinity is “instantiated” in physical reality).

Have there been any important works about, or fruitful approaches to infinity which were independent of the mathematical concept / philosophy of mathematics? One certainly would assume that pre-Cantor there must have been attempts to give a satisfying philosophical definition of infinity.

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    Even for Cantor, the philosophical notion of the infinite extended beyond the purely mathematical. See Absolute Infinity. – Nick Feb 21 '19 at 5:36
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    Infinity.doesn't exist in reality. And no matter how much people complain about that statement... it's just a fact. It's like unicorns.. we have a noun for it.. we know what it looks like... but you're not going to see one in.a zoo enclosure. – Richard Feb 21 '19 at 11:11
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    This is an astute observation. Infinity outside of mathematics has always been inexplicable and ineffable, hence the run to Cantorian concepts that "tame" it and can be talked about. A recent attempt to talk about something like metaphysical or absolute infinity, as distinct from mathematical infinity, is Meillassoux's "hyper-chaos" in After Finitude. – Conifold Feb 21 '19 at 18:21

I am unable to draw a line between the foundations of mathematics and metaphysics, and am not sure one can be understood without the other. I'd recommend the writings of Hermann Weyl on this topic, As I read him he doesn't just deny infinity but denies extension, a view partly justified by the infinities that arise for the idea of extension.

It seems to me the foundation of mathematics simply is metaphysics, such that infinity in philosophy is no different from infinity in mathematics. The various 'philosophies of (mind, maths, science, whatever)' would all reduce to metaphysics.

A suggestive example would be Spencer Brown's metaphysical solution for Russell's mathematical paradox.

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  • Question. Can you draw a line between the rules of chess and metaphysics? I mean, does the knight "really" move that way? Of course not. Chess is a formal game. Why can't you see math the same way? A set is whatever behaves in accordance with the rules of set theory; just as a knight is whatever piece (of any shape or design) moves according to the laws of knights in the game of chess. That's not metaphysics, it's just formalism. Yes? – user4894 Feb 23 '19 at 1:48
  • @user4894 - Quite so. Your examples are not metaphysics. But when you try to axiomatise mathematics you arrive at exactly the same problem that arises when you try to axiomatise the world. This was what I was getting at by mentioning Spencer Brown and Hermann Weyl, who both wander happily between foundations of mathematics and metaphysics. I often use chess as an analogy for metaphysics and find it nearly perfect. Big topic. . . . . – user20253 Feb 23 '19 at 12:15

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