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Great article, very well written and actually understandable without hunting down mathematical definitions. I think it is quite cool that people are still hunt
by auggierose 10y ago
Great article, very well written and actually understandable without hunting down mathematical definitions.
I think it is quite cool that people are still hunting down the various different incarnations of infinity and are able to prove deep results like this.
On the other hand, I don't think that these questions are as essential as they are made out to be. How do we know that everything finite is on unshakable foundations? Doesn't even thinking about what that statement means involve reasoning about infinities? Let's just accept that nothing ever is really unshakable and let's use the tools that will get the job at hand done.
- Smaug123 10y agoBut we can be more sure of things that are less shakeable. Additionally, these techniques sometimes give us more effective proofs: for example, if we know that A is true, that is less useful than knowing that A is true and that we can find a finite certificate of its truth. We can make computers verify finite certificates, for instance. A useful distinction to have in mind is: I know that the number 91 must have prime factors, because I can prove that all numbers have prime factors. But that's much less useful than knowing that its prime factors are 7 and 13. Similarly, if I could prove A using infinitistic methods, that's in some sense "a bit less useful" than a proof which uses finitistic and/or constructive methods.
- auggierose 10y agoMy point is that if you prove that 91 has as its prime factors 7 and 13, I don't care if you did that proof using infinite methods or not. In general, I am only proving theorems that are useful to me, so "a bit less useful" doesn't apply to these situations. If I needed that "bit more usefulness", I would go ahead and prove it (if I could). Your reasoning mostly only applies if you prove theorems for their own sake, and not because you already know what you want them for.
- Smaug123 10y agoIt's only an analogy. The analogy is meant to be "prove that 91 has prime factors" using infinitistic methods, and "prove that 91's prime factors are 7 and 13" using finitistic ones.
- auggierose 10y agoI am aware of that. That's why analogies are usually not very useful.
- Smaug123 10y agoI have no idea how you learn, but analogies are basically the only way I use of absorbing new concepts.
- auggierose 10y agoInteresting. I learn by playing around with the concept until I gain some intuition about it. No analogies required.
- ZeroFries 10y agoI think you're on to something here. We often assume a playing field of finite, seperate, individual objects interacting with other finite, seperate, individual objects, but I feel this is more a relic of how our brains break down reality than what reality actually is.
- Manishearth 10y agoYour recasting of the problem reminds me of the Axiom of Choice, which leads to similar questions if not included.
- jsprogrammer 10y agoAxiom of choice is just an ad hoc theorem taken as axiom. Check out HoTT.