4 ms·

It is easy to prove for finite sets (just count) but much harder for infinite ones. For example, which has more subsets, the integers or the positive integers?

It is easy to prove for finite sets (just count) but much harder for infinite ones. For example, which has more subsets, the integers or the positive integers? How about the integers and the reals? If you answered "the integers" to the first question, you aren't thinking about this right, as the integers and the positive integers have the same number of elements to start with. source: https://en.wikipedia.org/wiki/Aleph_number https://en.wikipedia.org/wiki/Aleph_number

The_rationalist 7y ago
Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be an optional extension to the mathematical framework instead of being the default, in order to not pollute discussions.
- new2628 7y ago
  This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise. See e.g. https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism Doron Zeilberger would like to pat you on the back.
  - baddox 7y ago
    How do finitists describe the decimal representation of the fraction 1/3?
    - improv32 7y ago
      I don't think finitists have a problem with convergent sequences, they just wouldn't talk about infinities, rather than extending the sequence gets you as arbitrarily to 1/3 as you wish to go
- RossBencina 7y ago
  Being "optimally useful" is not the goal of mathematics. See for example, https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
- eximius 7y ago
  Tell all of the engineers using calculus that they don't need infinity. And you can't easily only partially include infinity.
  - The_rationalist 7y ago
    Could you give a use case?
    - Filligree 7y ago
      All of calculus.
      - roywiggins 7y ago
        It's possible to do quite a lot without infinities in calculus and trigonometry, whether it makes sense to bother is another question. https://en.m.wikipedia.org/wiki/Rational_trigonometry https://en.m.wikipedia.org/wiki/Rational_trigonometry
        Cogito 7y ago
        > It's possible to do quite a lot without infinities in calculus and trigonometry This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points." Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them. You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating. [0] https://math.stackexchange.com/questions/387234/how-far-can-one-get-in-analysis-without-leaving-mathbbq https://math.stackexchange.com/questions/387234/how-far-can-... [1] https://www.dpmms.cam.ac.uk/~wtg10/reals.html https://www.dpmms.cam.ac.uk/~wtg10/reals.html
  - tgflynn 7y ago
    Computers handle calculus quite nicely using numerical algorithms, which don't involve any infinite sets. > And you can't easily only partially include infinity. Sure you can. You just need to use a dx that's small enough for the particular functions you're working with and the degree of precision you need.
- voxl 7y ago
  Since you feel so strongly about it, let me argue that you're simply wrong: Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers. In analysis, it's much easier to reason about infinitely differentiable, or smooth, surfaces then very rigid and complicated services. The fact that infinities cause some complications in the foundations itself is a drop in the bucket to the practical application and simplifications of day to day mathematics.
  - tgflynn 7y ago
    > Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers. I would argue that a few extra lines in a proof is a small price to pay to avoid the Godelian catastrophe.
- roywiggins 7y ago
  The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything without real numbers, which he doesn't believe make physical or logical sense and therefore shouldn't be used as a basis for anything. He's very nearly a minority of one, mind you.
- tgflynn 7y ago
  Yes, my current inclination would be to define truth (in a practical sense) as that which is computable. Everything else belongs to the realm of fiction or fantasy. Note that I don't mean to imply that this is the entirety of my philosophy. Just that one should distinguish between what is practically relevant and what is not and I think the current state of mathematics and CS does a poor job of making this distinction.
- xelxebar 7y ago
  A formalist might say that mathematics has no infinities. Even infinite sets are just definitions of finite length, even if expanded all the way to ZFC axioms. The problem is your intuitive understanding. A realist might say that mathematics exists within this universe, therefore it is equally subject to its constraints just like anything else. Problems arise via misapplication or misinterpretation of the math. A model theorist might point out that even in models of math with only a countable collection of objects, it's still true that the reals are uncountable. The problem isn't with infinitie, it's that the finitary objects have subtle interactions. A logician might object that ZFC already does start of with an intuitive notion of infinity, i.e. the counting numbers is a natural collection. The problem is that this inevitably has unintended consequences. A historian might gently point out that this debate already occurred vigorously about 150 years ago, and the verdict was that we just gotta live with the weirdness of infinities. The problem is that we lose too much useful math by trying to throw them out. Etc.

The_rationalist 7y ago

Another example of why mathematics are a wrong abstraction to be optimally useful. Mathematics should have bounds in the same way as our universe has bounds. Any theorem that has a different behavior if something is infinite doesn't matter at all and is a waste of time for real engineers who solve things in the real world. The niche of mathematics that describe things beyond what our universe has to offer should be an optional extension to the mathematical framework instead of being the default, in order to not pollute discussions.

new2628 7y ago
This is a fine and defensible (although not mainstream) viewpoint with several adherents among mathematicians and various levels of success in making it formal and precise. See e.g. https://en.wikipedia.org/wiki/Finitism https://en.wikipedia.org/wiki/Finitism Doron Zeilberger would like to pat you on the back.
- baddox 7y ago
  How do finitists describe the decimal representation of the fraction 1/3?
  - improv32 7y ago
    I don't think finitists have a problem with convergent sequences, they just wouldn't talk about infinities, rather than extending the sequence gets you as arbitrarily to 1/3 as you wish to go
RossBencina 7y ago
Being "optimally useful" is not the goal of mathematics. See for example, https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
eximius 7y ago
Tell all of the engineers using calculus that they don't need infinity. And you can't easily only partially include infinity.
- The_rationalist 7y ago
  Could you give a use case?
  - Filligree 7y ago
    All of calculus.
    - roywiggins 7y ago
      It's possible to do quite a lot without infinities in calculus and trigonometry, whether it makes sense to bother is another question. https://en.m.wikipedia.org/wiki/Rational_trigonometry https://en.m.wikipedia.org/wiki/Rational_trigonometry
      - Cogito 7y ago
        > It's possible to do quite a lot without infinities in calculus and trigonometry This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points." Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them. You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating. [0] https://math.stackexchange.com/questions/387234/how-far-can-one-get-in-analysis-without-leaving-mathbbq https://math.stackexchange.com/questions/387234/how-far-can-... [1] https://www.dpmms.cam.ac.uk/~wtg10/reals.html https://www.dpmms.cam.ac.uk/~wtg10/reals.html
- tgflynn 7y ago
  Computers handle calculus quite nicely using numerical algorithms, which don't involve any infinite sets. > And you can't easily only partially include infinity. Sure you can. You just need to use a dx that's small enough for the particular functions you're working with and the degree of precision you need.
voxl 7y ago
Since you feel so strongly about it, let me argue that you're simply wrong: Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers. In analysis, it's much easier to reason about infinitely differentiable, or smooth, surfaces then very rigid and complicated services. The fact that infinities cause some complications in the foundations itself is a drop in the bucket to the practical application and simplifications of day to day mathematics.
- tgflynn 7y ago
  > Infinities have the opposite effect then you might think, they make things simpler. It's much easier to reason about an infinite list of numbers then to reason about 64 bit numbers. I would argue that a few extra lines in a proof is a small price to pay to avoid the Godelian catastrophe.
roywiggins 7y ago
The most prolific contemporary proponent of this view is probably Norman Wildberger, who takes a very unconventional approach and tries to build everything without real numbers, which he doesn't believe make physical or logical sense and therefore shouldn't be used as a basis for anything. He's very nearly a minority of one, mind you.
tgflynn 7y ago
Yes, my current inclination would be to define truth (in a practical sense) as that which is computable. Everything else belongs to the realm of fiction or fantasy. Note that I don't mean to imply that this is the entirety of my philosophy. Just that one should distinguish between what is practically relevant and what is not and I think the current state of mathematics and CS does a poor job of making this distinction.
xelxebar 7y ago
A formalist might say that mathematics has no infinities. Even infinite sets are just definitions of finite length, even if expanded all the way to ZFC axioms. The problem is your intuitive understanding. A realist might say that mathematics exists within this universe, therefore it is equally subject to its constraints just like anything else. Problems arise via misapplication or misinterpretation of the math. A model theorist might point out that even in models of math with only a countable collection of objects, it's still true that the reals are uncountable. The problem isn't with infinitie, it's that the finitary objects have subtle interactions. A logician might object that ZFC already does start of with an intuitive notion of infinity, i.e. the counting numbers is a natural collection. The problem is that this inevitably has unintended consequences. A historian might gently point out that this debate already occurred vigorously about 150 years ago, and the verdict was that we just gotta live with the weirdness of infinities. The problem is that we lose too much useful math by trying to throw them out. Etc.