5 ms·

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
  Except for all that numerical analysis around floating point arithmetic...
- eximius 7y ago
  Calculus would likely not be invented without infinity. Many theorems, identities, and techniques may be approximated but ultimately rely on proofs using infinities - I doubt we would have discovered them quickly or at all without infinity. After all, the very concept of a limit evokes the concept of an infinite sequence.
  - tgflynn 7y ago
    That is true historically but I doubt that it is true necessarily. I mean I suspect that starting from what we now know it should be possible to reconstruct calculus (at least for all practical purposes) without reference to infinities or infinitesimals. One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals. The later, at least, has been my experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.
    - Enginerrrd 7y ago
      >One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals. Don't derivatives count? That's a pretty important and trivial calculation. Sure, you can approximate it when it's nicely behaved, but they aren't always. There's also lots of verrrrrrry slowly converging series that can't be easily computed numerically.

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
Except for all that numerical analysis around floating point arithmetic...
eximius 7y ago
Calculus would likely not be invented without infinity. Many theorems, identities, and techniques may be approximated but ultimately rely on proofs using infinities - I doubt we would have discovered them quickly or at all without infinity. After all, the very concept of a limit evokes the concept of an infinite sequence.
- tgflynn 7y ago
  That is true historically but I doubt that it is true necessarily. I mean I suspect that starting from what we now know it should be possible to reconstruct calculus (at least for all practical purposes) without reference to infinities or infinitesimals. One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals. The later, at least, has been my experience. I think I have a fairly decent practical intuition for calculus based on imagining dx becoming smaller and smaller until it's small enough, but I don't think my brain has any actual representation of "true infinitesimals" and my intuition breaks down completely if I try to imagine things like the relationship between the rational and irrational numbers. Maybe that's due to my intellectual limitations, but I wonder if it isn't because these concepts might be over-elaborate abstractions that don't really exist in our world.
  - Enginerrrd 7y ago
    >One argument in favor of this belief is that neither practical computations nor analytic intuition require actual infinitesimals. Don't derivatives count? That's a pretty important and trivial calculation. Sure, you can approximate it when it's nicely behaved, but they aren't always. There's also lots of verrrrrrry slowly converging series that can't be easily computed numerically.