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I'm not a mathematician, but what you're asking is a valid question in philosophy of mathematics. Generally, mathematical realism sees mathematical objects as e

by Delk 4y ago

I'm not a mathematician, but what you're asking is a valid question in philosophy of mathematics. Generally, mathematical realism sees mathematical objects as existing independently of the human mind while anti-realist schools of thought don't.

still_grokking 4y ago
I came across this. As non-mathematician and non-philosopher I have no strong opinion on that question. It's not important to me whether something exists "only in my mind" (because I could imagine it form some principles) or whether it exists "in general". (In the end "reality" only ever exists in someones mind, because that's how our minds work. But that's not relevant here, imho). The question would be more relevant to the divide between mathematical intuitionism and realism. But the question about reals is imho deeper: I have no strong feelings about, say, the "law of excluded middle". Both sides of the debate have good arguments. You can e.g. "logically claim" that excluded middle must hold, or, equally, needs to be refuted. But I didn't find anything similar regarding the "existence" of reals. There seems to be no logic form which one could derive their "existence". The point is: I don't know any method by which one could even "sense" most reals. It is just assumed that there "are" between any two constructible numbers infinite many (non constructible) reals. But nobody ever seen them, nobody ever will see them, not even if you would have an infinitely powerful computer and know anything that is knowable (which would amount to be like a "god"; so not even a god can "see" most reals). The question around the existence of most reals is like asking "What's 'between' two points in space-time which are less than half a Planck unit away?". A physicist would most likely answer with: "Even asking such a question does not make any sense". (That's once again the mentioned tangent to physical reality). But mathematicians seem to happily answer such a question with "There are just more reals. Never mind that you can't see them". Completely unrealistic things like Banach–Tarski get accepted than as a consequence without much fuss… People are still riddled by things like the continuum hypothesis… But what if there isn't any "continuum" at all? In physical reality there is also nothing like that! (Which was an unexpected and deeply concerning discovery, the consequences of which we still don't understand; but we're quite sure by now that this new picture, a discrete description of reality, is closer to what's actually out there than the "naïve" description that assumes some "continuous smooth space").

by Delk 4y ago

still_grokking 4y ago

I came across this. As non-mathematician and non-philosopher I have no strong opinion on that question. It's not important to me whether something exists "only in my mind" (because I could imagine it form some principles) or whether it exists "in general". (In the end "reality" only ever exists in someones mind, because that's how our minds work. But that's not relevant here, imho). The question would be more relevant to the divide between mathematical intuitionism and realism. But the question about reals is imho deeper: I have no strong feelings about, say, the "law of excluded middle". Both sides of the debate have good arguments. You can e.g. "logically claim" that excluded middle must hold, or, equally, needs to be refuted. But I didn't find anything similar regarding the "existence" of reals. There seems to be no logic form which one could derive their "existence". The point is: I don't know any method by which one could even "sense" most reals. It is just assumed that there "are" between any two constructible numbers infinite many (non constructible) reals. But nobody ever seen them, nobody ever will see them, not even if you would have an infinitely powerful computer and know anything that is knowable (which would amount to be like a "god"; so not even a god can "see" most reals). The question around the existence of most reals is like asking "What's 'between' two points in space-time which are less than half a Planck unit away?". A physicist would most likely answer with: "Even asking such a question does not make any sense". (That's once again the mentioned tangent to physical reality). But mathematicians seem to happily answer such a question with "There are just more reals. Never mind that you can't see them". Completely unrealistic things like Banach–Tarski get accepted than as a consequence without much fuss… People are still riddled by things like the continuum hypothesis… But what if there isn't any "continuum" at all? In physical reality there is also nothing like that! (Which was an unexpected and deeply concerning discovery, the consequences of which we still don't understand; but we're quite sure by now that this new picture, a discrete description of reality, is closer to what's actually out there than the "naïve" description that assumes some "continuous smooth space").