Climbing at last into contemporary thought about the nature of the infinite, Graham Priest’s book is finally losing that weird feeling of frustration. When he was examining the Ancient, Medieval, and Enlightenment periods of philosophy, the tension was difficult between his own mission of exploring the inherent contradictions of infinity and the utterly different philosophical priorities of his analytical subjects. In the context of Ancient Greek ontology, Medieval speculation about the nature of God, and the onto-moral twists and turns of German Idealism, Priest’s concerns jarred a little too much.
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| I'm also interested in Cantor's approach to paradoxes. Priest, dialetheic logician that he is, has no problem with set theoretical mathematics implying inevitable paradoxes. Cantor and other set theory mathematicians desperately wanted to erase or nullify the paradoxes that arose because they took contradictions to be impossible, invalidating an entire mathematical approach. Priest doesn't care, and I think philosophy is better for escaping its fear of paradoxical reasoning. |
You can’t say the same when he starts in on Georg Cantor. I’m still not entirely sure where he’s going (as I’m working on editing a paper, working at my other editing job, and preparing to move apartments at the end of the winter, so I’m not reading the book quite as fast as when I started), but one idea in his first chapter on modern set theory is fascinating to me. This is the switch in status of potential and actual infinity.
For pretty much the entire history of Western philosophy until Cantor et al’s set theory was developed, actual infinity was conceived as a weird, strange, impossible thing. It was spoken of in the hushed and information-free tones of negative theology. We are finite creatures living in a finite world, so we can’t conceive of an actual infinity as anything other than the negation of the finite. As Hegel said, this isn’t even really an infinity, just putting a ‘not’ in front of the word ‘finite.’ The concept doesn’t even have any content.*
* Priest discusses Hegel’s conception of the infinite in terms of the absolute briefly, but the Hegelian metaphysics of thought as dialectical progression is rather outside his philosophical wheelhouse. I myself find the idea that there is a single concept, arrived at through a single determined progression of dialectical logic, which encompasses all that can be, rather repugnant.
But transfinite numbers, which are defined as sets with an infinite number of members, are now pretty easy to do mathematics with, once you conceive of w as an operation performed at the end of enumerating an infinite series. You can conceive of w + 1, 2w, w2, or whatever. Once you can increase an infinite set, things can get crazy. But the upshot of this regarding how philosophers have traditionally conceived infinity is that set theory mathematics clearly define an actual infinite, define multiple such infinities, and you can increase or decrease them.
The reversal appears when you think of the potential infinite, an infinity that isn’t complete, but is growing serially over time. In other words, it’s a process that’s finite now, but will continue to grow indefinitely into an infinite span of future time. It used to be that this was the sensible infinity, because it was the only infinity that a finite human mind could conceive of with any content: an ongoing series that isn’t done yet. Actual infinities were conceived as some body so big that it literally was infinite, and it was thought that understanding such a body meant comprehensively conceiving or experiencing all its parts.
But set theory mathematics gives us a shorthand for conceiving of actual infinities and comparing them. On paper, they look like ordinary bits of algebra that obey idiosyncratic rules of arithmetic, like these operations of belonging, inclusion, and diagonalization. Yet these mathematical operations define the nature of actual infinite quantities. They can be clearly defined and compared.
Potential infinities can’t work in quite the same way. The mathematics of limits come close, but a limit is a point that is infinitely deferred in a mathematical progression. A limit makes what would be a potential infinite a clearly defined quantity: an asymptotic approach to a given point.
Potential infinities are continuing movements, developments, or becomings. Only the philosophies of Henri Bergson (and to my mind, Gilles Deleuze, when he’s thinking in a Bergsonian direction) can manage this. Even then, these durational infinities, movements of infinite growth if you will, aren’t expressible in mathematics. Bergson himself defined it that way, describing mathematics as limited to abstract generalities. Only experience, he says, can give you the potential infinite: a power of development that need never stop. Yet this is exactly the impasse with which Priest ends his chapter on Cantor’s set theory.
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