4 ms·

An axiom is simply something made up. At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empt

An axiom is simply something made up. At the end of the day, all of our Mathematics rests on foundations that are made up. It's difficult to say what the empty set is. Because it isn't really anything at all.

throwawaymath 8y ago
There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered. Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?
- 12elephant 8y ago
  The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
  - throwawaymath 8y ago
    In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.
xamuel 8y ago
Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity. Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.
- 12elephant 8y ago
  This is the same response I gave to throwawaymath: The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
  - xamuel 8y ago
    >In mathematics you don't understand things. You just get used to them. Yep, one of my favorite math quotes and highly under-rated. One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)
zazen 8y ago
Well, yeah. Axioms are not deduced, they are the starting points for the various theories that are deduced from them. (There is not much you can deduce ex nihilo.) Did you have some further point? If you were hoping that mathematics would have - or thinking that it _should_ have - some ontological foundation that is not in this sense "made up", I'm afraid it simply doesn't.

throwawaymath 8y ago

There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered. Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?

12elephant 8y ago
The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
- throwawaymath 8y ago
  In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.

xamuel 8y ago

Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity. Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.

12elephant 8y ago
This is the same response I gave to throwawaymath: The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism". What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down. If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement: > In mathematics you don't understand things. You just get used to them.
- xamuel 8y ago
  >In mathematics you don't understand things. You just get used to them. Yep, one of my favorite math quotes and highly under-rated. One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)

zazen 8y ago

Well, yeah. Axioms are not deduced, they are the starting points for the various theories that are deduced from them. (There is not much you can deduce ex nihilo.) Did you have some further point? If you were hoping that mathematics would have - or thinking that it _should_ have - some ontological foundation that is not in this sense "made up", I'm afraid it simply doesn't.