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Stupid question form a ignorant person (me): Are reals actually real? Do they even exist? Almost all reals (besides some countable subsets, which are irrelevan

Stupid question form a ignorant person (me): Are reals actually real? Do they even exist? Almost all reals (besides some countable subsets, which are irrelevant compared to the many many more reals that aren't part of those sets) have infinite many non repeating decimal places, and what's even more annoying, almost all of them aren't computable. The physical universe we can experience is finite. How can more than infinite many "unspeakable" things "exist" in that universe? How does this make sense? You can't express most reals in any meaningful way! Not even by constructing imaginary "infinite" computers. So, when almost all reals aren't anything we could ever experience, or even name in any meaningful way, why do people believe they "exist" in the first place? I have no issue to imagine infinite things as long as I can construct some algorithm to name them. But when there is no such algorithm, how can we know that the thing that can't be named actually exists? I'm not trolling. It's a genuine question I have since some time but didn't have the chance to talk to some mathematicians to get things explained. Maybe someone here could try to explain why we should believe in the existence of real numbers, even there is nothing in the whole universe that relates to most of them? (Especially as the possibility to subdivide any interval infinitely gives rise to phenomenons like the Banach–Tarski paradox. But most people would agree: You can't make something out of nothing. This is imho another strong pointer that reals are an illogical idea in the end. They give rise to contradictions to basic things we assume to know about reality).

nh23423fefe 4y ago
> why we should believe in the existence of real numbers, even there is nothing in the whole universe that relates to most of them? Seems like you are defining existence here. Supposed I asked you if 7-dimensional triangles (simplex) exists what would you say? What do the reals have to do with the physical universe?
- still_grokking 4y ago
  One can construct 7-dimensional triangles. So they surely exist (somehow). > What do the reals have to do with the physical universe? That's kind of a tangent, but the point is: There is absolutely nothing in the universe that relates to most reals! Most reals don't "exist" in any meaningful way. So why should we believe they "exist" at all? (Subsets exist for sure, but that's not the point). Triangles and dimensions exist. One can easily extend this kind of existence to more dimensions. So one can create 7-dimensional triangles. But in contrast one can't, by no means, construct most reals. So my question is: Why should I believe that something that can't be ever "observed" "exists". That's like someone trying to convince me to believe in God… The relation to the physical world is: If someone would say that "there is something" but it can't be even named (out of principle!), not to mention described meaningfully, most people would likely ignore such a purely made up "thing". Because it would have exactly zero influence on anything that may exist in fact. The whole idea that there is something like "smooth space" is absurd given how "the true reality" works. Quantum physics tells us that there is no smooth space, and smooth space is actually impossible. But mathematicians seem to still want to use a "leaky abstraction" that has absolutely nothing in common with reality. Like I said, that's like they would strongly believe in God. Or to phrase it differently: Something that can't be, just can't be. Nobody can construct most reals. That's by definition. So one would need very very strong arguments to still assume that they "exist" (somehow). But there is no such argument at all afaik. That's what I'm asking actually: Does anybody know any, even remotely, valid argument why we should believe in reals? I didn't find any while researching that. Mathematics started as a way to precisely describe "what can exist". But most reals can't be described in any way. So my conclusion would be that they're not part of any valid description of "what can exist". But as most mathematicians seem to believe in reals I wanted to know why it's like that. What do I miss? Where does this faith come form?
  - nh23423fefe 4y ago
    Lots of things are unobservable though. Amplitudes of a wave function are unobservable. What does your idea of existence and relevance say about that? Why is constructibility of elements relevant for belief in a whole. You will never observe an amplitude, is the wave function then "non-real"? The inside of black hole is unobservable from beyond the event horizon. Does that mean the space isn't real? Why is this property of observability or constructibility of individual elements relevant for the existence of a set. Concretely. I believe in the reals because of taking analysis. I guess I don't understand the leverage of calling the reals a "leaky abstraction", it just reads like a skeptical argument taken too far. I hand you a box and say, "these are the reals, they behave this way". You respond, "I've looked in the box and almost all the elements have no recipe, this box doesn't exist." It seems like a non-sequitur to me.
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").

nh23423fefe 4y ago

> why we should believe in the existence of real numbers, even there is nothing in the whole universe that relates to most of them? Seems like you are defining existence here. Supposed I asked you if 7-dimensional triangles (simplex) exists what would you say? What do the reals have to do with the physical universe?

still_grokking 4y ago
One can construct 7-dimensional triangles. So they surely exist (somehow). > What do the reals have to do with the physical universe? That's kind of a tangent, but the point is: There is absolutely nothing in the universe that relates to most reals! Most reals don't "exist" in any meaningful way. So why should we believe they "exist" at all? (Subsets exist for sure, but that's not the point). Triangles and dimensions exist. One can easily extend this kind of existence to more dimensions. So one can create 7-dimensional triangles. But in contrast one can't, by no means, construct most reals. So my question is: Why should I believe that something that can't be ever "observed" "exists". That's like someone trying to convince me to believe in God… The relation to the physical world is: If someone would say that "there is something" but it can't be even named (out of principle!), not to mention described meaningfully, most people would likely ignore such a purely made up "thing". Because it would have exactly zero influence on anything that may exist in fact. The whole idea that there is something like "smooth space" is absurd given how "the true reality" works. Quantum physics tells us that there is no smooth space, and smooth space is actually impossible. But mathematicians seem to still want to use a "leaky abstraction" that has absolutely nothing in common with reality. Like I said, that's like they would strongly believe in God. Or to phrase it differently: Something that can't be, just can't be. Nobody can construct most reals. That's by definition. So one would need very very strong arguments to still assume that they "exist" (somehow). But there is no such argument at all afaik. That's what I'm asking actually: Does anybody know any, even remotely, valid argument why we should believe in reals? I didn't find any while researching that. Mathematics started as a way to precisely describe "what can exist". But most reals can't be described in any way. So my conclusion would be that they're not part of any valid description of "what can exist". But as most mathematicians seem to believe in reals I wanted to know why it's like that. What do I miss? Where does this faith come form?
- nh23423fefe 4y ago
  Lots of things are unobservable though. Amplitudes of a wave function are unobservable. What does your idea of existence and relevance say about that? Why is constructibility of elements relevant for belief in a whole. You will never observe an amplitude, is the wave function then "non-real"? The inside of black hole is unobservable from beyond the event horizon. Does that mean the space isn't real? Why is this property of observability or constructibility of individual elements relevant for the existence of a set. Concretely. I believe in the reals because of taking analysis. I guess I don't understand the leverage of calling the reals a "leaky abstraction", it just reads like a skeptical argument taken too far. I hand you a box and say, "these are the reals, they behave this way". You respond, "I've looked in the box and almost all the elements have no recipe, this box doesn't exist." It seems like a non-sequitur to me.

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").