5 ms·

Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does

by mkl 5y ago

Well, no numbers exist in any real sense. Integers, real numbers, complex numbers, hyperreals, etc., they're all imagined, and built from axioms. Keisler does have a companion book that goes into more of the theory of hyperreals, and its bibliography (p209) points to deeper theoretical work too: https://people.math.wisc.edu/~keisler/foundations.html https://people.math.wisc.edu/~keisler/foundations.html

andi999 5y ago
They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers. Otherwise you do not easily know if your axioms are actually consistent.
- mkl 5y ago
  Yes, but what I meant is that there are axioms at the base of this structure, so everything is ultimately built on them. Some people put axiom foundations higher up too, like axiomatic real numbers.
  - andi999 5y ago
    Yes, there are of course axioms, but if you have a rich enough set of axioms like zfc, then one should stop piling more axioms (if not absolutely necessary). So of course the question is existence in zfc
Koshkin 5y ago
Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.
- xyzzy123 5y ago
  I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes: I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood. Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner. Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).
  - andi999 5y ago
    Anumeric cultures might not really have existed (not conclusive evidence, but some evidence: https://www.jstor.org/stable/10.1086/667452?seq=1 https://www.jstor.org/stable/10.1086/667452?seq=1 )
  - Koshkin 5y ago
    The integers represent perhaps the simplest case. Another example of their actual existence is the fact that properties of atoms and their nuclei critically depend on the number of the nucleons of particular types in them; note that this fact does not in any way depend on whether a culture is anumeric or not. With some effort you can extend this idea to other abstractions (this process is usually greatly hindered by one's leaning towards solipsism).
    - xyzzy123 5y ago
      The existence proof you describe is in terms of a yet more complex abstraction; we can imagine aliens might describe a more useful atomic theory than our own which does not involve nucleons or their discrete counts at all (ok, unlikely as I acknowledge that may be). I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument. I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up". I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.
Y_Y 5y ago
I agree with you (for some values of "exist") and furthermore think this is a good reason why "imaginary" is a terrible name for the number set they refer to. (I prefer "pure complex", but that's got its issues too, of course.)

by mkl 5y ago

andi999 5y ago

They are not all build from axioms but from definition (apart from some models of natural numbers, or you go back to ZFC). Like C is build from R^2, R from the powerset of Q, Q from integer^2, integer from natural numbers. Otherwise you do not easily know if your axioms are actually consistent.

mkl 5y ago
Yes, but what I meant is that there are axioms at the base of this structure, so everything is ultimately built on them. Some people put axiom foundations higher up too, like axiomatic real numbers.
- andi999 5y ago
  Yes, there are of course axioms, but if you have a rich enough set of axioms like zfc, then one should stop piling more axioms (if not absolutely necessary). So of course the question is existence in zfc

Koshkin 5y ago

Well, the real existence of abstractions is a pretty involved topic in philosophy of knowledge, and so you can’t simply say that, for example, integers do not ‘really’ exist. Because they indeed do, in a very important (and, in fact, obvious) sense: you need two cats to have a fight.

xyzzy123 5y ago
I flip between agreeing and disagreeing with you! I'm not well qualified to comment so I expect an informed rebuttal to this will be instructive for me. So here goes: I still find myself in the "number systems are a cultural artifact" camp. We choose which details to keep and which to throw away in our abstractions. There are anumeric cultures and it's not clear to me at all that the existence of integers is obvious (beyond what humans can subitize) unless you're immersed in a culture that has impressed them upon you since early childhood. Do you have "2 cats" or actually just "this cat and that cat"? This cat is a bit bigger but that one looks meaner. Deep down I think that an insistence on the "reality" of integers versus reals (say) is purely aesthetic. (Ignoring for the moment that our conventional constructions build one from the other in a certain way).
- andi999 5y ago
  Anumeric cultures might not really have existed (not conclusive evidence, but some evidence: https://www.jstor.org/stable/10.1086/667452?seq=1 https://www.jstor.org/stable/10.1086/667452?seq=1 )
- Koshkin 5y ago
  The integers represent perhaps the simplest case. Another example of their actual existence is the fact that properties of atoms and their nuclei critically depend on the number of the nucleons of particular types in them; note that this fact does not in any way depend on whether a culture is anumeric or not. With some effort you can extend this idea to other abstractions (this process is usually greatly hindered by one's leaning towards solipsism).
  - xyzzy123 5y ago
    The existence proof you describe is in terms of a yet more complex abstraction; we can imagine aliens might describe a more useful atomic theory than our own which does not involve nucleons or their discrete counts at all (ok, unlikely as I acknowledge that may be). I don't want to descend into solopsism and apologise if I veer too far in this direction. Also thanks for taking the time to rebut what is probably a sophomoric argument. I suppose my fundamental objection is that if integers are "real" then it seems to me that quaternions must be similarly "real" (since I can describe useful things with them) and so on, I can't see a boundary which would let me say "ok integers are the real deal but infinitesimals are just a thing we made up". I think we have a zone of possible agreement if we decide either that all these abstractions are "real" or none of them are.

Y_Y 5y ago

I agree with you (for some values of "exist") and furthermore think this is a good reason why "imaginary" is a terrible name for the number set they refer to. (I prefer "pure complex", but that's got its issues too, of course.)