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I don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinit

I don't understand what you mean by this. What would it mean for an infinity or any number to "exist", does the number 2 "exist" somewhere in a way that infinities do not? In any case our best physical models right now are full of infinities. Space looks like it is infinitely big to an absurd degree of precision. The spectrum of the hydrogen atom contains two different infinities! A countable infinity of bound states and a continuum of free electron states at higher energies.

rolisz 3y ago
Well, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity. Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"
- eigenket 3y ago
  I see, so the belief is that improved theories of physics will remove all the infinities we appear to see around us? I guess as religions go, that one is no worse than many, but there isn't really any evidence for it now. Space looks really infinite, the hydrogen atom spectra are very well described by infinite series.
  - nairboon 3y ago
    > will remove all the infinities we appear to see around us? Can you actually show us one of these apparent actual infinities? All one can reasonably postulate is a potential infinity, but this is completely different from Cantors actual infinities.
    - eigenket 3y ago
      The spectrum of the hydrogen atom and the volume of the universe both appear to be actually infinite based on our best physical models.
      - rolisz 3y ago
        According to the Big Bang model the universe started with a super small volume. When did it get to infinity? As far as I know, we can see/measure that the universe is about 93 billion light years across. We literally can't know what's outside, so we don't know how much bigger it is or if it's infinite.
        eigenket 3y ago
        This is a common misconception, in standard models of big-bang cosmology the universe was always infinite. It was hotter and denser earlier, and is cooler and less dense later but always infinite. A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively. If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.
        rolisz 3y ago
        How can you prove that? We literally can't see further than 98 billion light years. How's that not also a religion? :p
        eigenket 3y ago
        That's not how physics works. We can't prove anything. All we do is come up with worse or better models, and the best model anyone has come up with so far has the universe being infinite. Finite universe models have some problems, since the global curvature of the universe is (as best we can measure) zero the universe is either flat or pretending to be flat. Finite volume flat spaces can be characterised and they're a bit weird (1). The best we can say is that if the universe is not infinite it appears to be doing a very good job of pretending to be infinite. (1) for a good idea of what "weird" means here imagine a 2d 1km by 1km square with geometry like Pac man lives in (toroidal), so if you go off the top of the square you appear at the bottom, and if you head off east you appear on the west. Now start at the middle and leave a rock at your current position. Head due east until you cycle round and hit your rock again. You'll have walked 1km (500m to the east edge and 500m from the west edge back to the middle). Now do the same experiment but walk north-east. You'll hit your rock after sqrt(2) km of walking (1/sqrt(2) takes you to the north east corner and the same to get back to your rock). In other worlds the pac-man torus space is not isotropic some angles are special and more important than others. Essentially the same thing happens in other finite flat geometries you can invent.
    - mistermann 3y ago
      One example would be the infinite number of ways there are for humans to form unsound conclusions. But of course, this is not "real" reality that can be tidily corralled with math/science so it doesn't count, but it's still interesting.
- paganel 3y ago
  Are you sure you have two apples on your desk? What happens if you take a small bite? Is it still two apples? When will said apple "cease" to exist as an unit while you're eating it? If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them. [1] https://en.wikipedia.org/wiki/Parmenides https://en.wikipedia.org/wiki/Parmenides
  - nairboon 3y ago
    Isn't that a completely different question? OP was demonstrating that the existence of natural numbers is trivial to show, unlike actual infinity.
  - rolisz 3y ago
    I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. At some point some will start disagreeing, but with two big full apples, all of them will agree. Or is it more helpful if I say I have 10^27 atoms in my body? Quarks? Gluons? Strings from string theory? Still finite number, not "biteable", not infinite.
    - falserum 3y ago
      > I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. That’s useful, but very vague, and very movable criteria for “exists”. (Approx. Must be based in physical reality + enough people subjectively agree on it) Do following finite numbers exist: -1, 0, 0.5, PI, 2^300 (more than particles in observable universe), sqrt(-1)? Do individual digits exist? If PI exist, how many digits does it have? Do models and algorithms in general exist? Do model existance depend on limits of your/somebodies capacity to understand them? I would propose alternative, but useful way to look at this. “Two” and “infinity” are models. Both these models are useful, but “two” is just more common one. (Still, various infinities are useful for bunch of people)
  - ludston 3y ago
    So what you're trying to say is that he has infinite apples?
- soVeryTired 3y ago
  But this is mathematics, not physics. It's about axioms and logical contradictions, not apples.

rolisz 3y ago

Well, I have two apples on my desk. I can't have an infinity of anything. In that sense 2 exists more tangibly than infinity. Our best physical models are just that - models. All models are wrong, some are useful. Infinity in physics is usually a shorthand for "it's so big I don't have to care about the edges"

eigenket 3y ago
I see, so the belief is that improved theories of physics will remove all the infinities we appear to see around us? I guess as religions go, that one is no worse than many, but there isn't really any evidence for it now. Space looks really infinite, the hydrogen atom spectra are very well described by infinite series.
- nairboon 3y ago
  > will remove all the infinities we appear to see around us? Can you actually show us one of these apparent actual infinities? All one can reasonably postulate is a potential infinity, but this is completely different from Cantors actual infinities.
  - eigenket 3y ago
    The spectrum of the hydrogen atom and the volume of the universe both appear to be actually infinite based on our best physical models.
    - rolisz 3y ago
      According to the Big Bang model the universe started with a super small volume. When did it get to infinity? As far as I know, we can see/measure that the universe is about 93 billion light years across. We literally can't know what's outside, so we don't know how much bigger it is or if it's infinite.
      - eigenket 3y ago
        This is a common misconception, in standard models of big-bang cosmology the universe was always infinite. It was hotter and denser earlier, and is cooler and less dense later but always infinite. A decent mental model is to think about an infinitely long ruler or measuring tape, where I have positions marked every (e.g.) 1 meter interval. You can imagine stretching or shrinking the ruler which would move the marks further or closer to each other respectively. If you think about a "cosmology" for this ruler where I keep stretching it further and further forever then the points I have marked will ger further and further away, but the ruler itself is always infinite.
        rolisz 3y ago
        How can you prove that? We literally can't see further than 98 billion light years. How's that not also a religion? :p
        eigenket 3y ago
        That's not how physics works. We can't prove anything. All we do is come up with worse or better models, and the best model anyone has come up with so far has the universe being infinite. Finite universe models have some problems, since the global curvature of the universe is (as best we can measure) zero the universe is either flat or pretending to be flat. Finite volume flat spaces can be characterised and they're a bit weird (1). The best we can say is that if the universe is not infinite it appears to be doing a very good job of pretending to be infinite. (1) for a good idea of what "weird" means here imagine a 2d 1km by 1km square with geometry like Pac man lives in (toroidal), so if you go off the top of the square you appear at the bottom, and if you head off east you appear on the west. Now start at the middle and leave a rock at your current position. Head due east until you cycle round and hit your rock again. You'll have walked 1km (500m to the east edge and 500m from the west edge back to the middle). Now do the same experiment but walk north-east. You'll hit your rock after sqrt(2) km of walking (1/sqrt(2) takes you to the north east corner and the same to get back to your rock). In other worlds the pac-man torus space is not isotropic some angles are special and more important than others. Essentially the same thing happens in other finite flat geometries you can invent.
  - mistermann 3y ago
    One example would be the infinite number of ways there are for humans to form unsound conclusions. But of course, this is not "real" reality that can be tidily corralled with math/science so it doesn't count, but it's still interesting.
paganel 3y ago
Are you sure you have two apples on your desk? What happens if you take a small bite? Is it still two apples? When will said apple "cease" to exist as an unit while you're eating it? If the Old Greeks [1] haven't been able to solve this problem I'm not sure we'll be able to do better than them. [1] https://en.wikipedia.org/wiki/Parmenides https://en.wikipedia.org/wiki/Parmenides
- nairboon 3y ago
  Isn't that a completely different question? OP was demonstrating that the existence of natural numbers is trivial to show, unlike actual infinity.
- rolisz 3y ago
  I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. At some point some will start disagreeing, but with two big full apples, all of them will agree. Or is it more helpful if I say I have 10^27 atoms in my body? Quarks? Gluons? Strings from string theory? Still finite number, not "biteable", not infinite.
  - falserum 3y ago
    > I'm happy to accept that I have two apples of my desk if all my friends and the guy I bought them from agree with that definition. That’s useful, but very vague, and very movable criteria for “exists”. (Approx. Must be based in physical reality + enough people subjectively agree on it) Do following finite numbers exist: -1, 0, 0.5, PI, 2^300 (more than particles in observable universe), sqrt(-1)? Do individual digits exist? If PI exist, how many digits does it have? Do models and algorithms in general exist? Do model existance depend on limits of your/somebodies capacity to understand them? I would propose alternative, but useful way to look at this. “Two” and “infinity” are models. Both these models are useful, but “two” is just more common one. (Still, various infinities are useful for bunch of people)
- ludston 3y ago
  So what you're trying to say is that he has infinite apples?
soVeryTired 3y ago
But this is mathematics, not physics. It's about axioms and logical contradictions, not apples.