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We’ve done an enormous amount of work to make mainstream mathematics work the way it currently works. We have sets which are infinite, but individual elements a

We’ve done an enormous amount of work to make mainstream mathematics work the way it currently works. We have sets which are infinite, but individual elements are not. In other words, you have an infinite set of numbers, but no number is itself infinite. Only after centuries of effort did we make this work rigorously. I think it’s worth remembering this, when we think about alternative foundations for mathematics. What if we discard the axiom of infinity? What if we discard the axiom of choice? Axiom of regularity? Each of these decisions leads us to a path which may be as fruitful as current mainstream mathematics, but which relatively unexplored. Or you could stick with existing formulations, and rework how you do calculus. To pick on the calculus example—mainstream calculus defines continuity using limits, which are defined by formulas quantified over infinite sets. What happens if, instead of using infinite sets, you use sets that contain infinite numbers? Voilà, you have nonstandard analysis… the “dx” and “dy” can now be actual infinitesimal numbers. Is this easier for students to understand? Or is it confusing? Does it formalize existing intuition about infinitesimals, and let you do things like treat dy/dx as an actual division? Same is true in the opposite direction. What if you have no infinite sets, no infinite numbers? How do you do calculus? What theorems still work, which don’t? The theorems that don’t work—do we miss them when they are gone?

kmeisthax 4y ago
We actually know a fair bit about what happens if you discard the axiom of choice; mostly because Banach-Tarski made it so controversial. As for calculus, it doesn't require the existence of infinities or infinitesimals. Set theory needs to care about infinities and the sizes thereof because they are creatures of the axioms they adopt. But in calculus "infinity" just means "it gets as big as you want" and "infintesimal" just means "it gets as small as you want". Everything's argued on top of limits that are done in the real number system where there aren't any infinities or infinitesimals. The thing that calculus really cares about is continuity: i.e. the idea that if you nudge the input to a function a little bit, the output changes by a little bit. If you try to do calculus on functions that don't work this way, then calculus doesn't work, either.
- klodolph 4y ago
  > We actually know a fair bit about what happens if you discard the axiom of choice; mostly because Banach-Tarski made it so controversial. Do you have any more information about this? Is there an overview somewhere? The major work I know in this area is Reverse Mathematics, but I otherwise thought it was relatively unexplored. Emphasis on “relatively”. My impression here is that Banach-Tarski reveals “flaws” not just in ZFC but in ZF, and that while we may want to reevaluate the axiom of choice, we should be looking at the rest of ZF as well. > As for calculus, it doesn't require the existence of infinities or infinitesimals. The question was never “can you do calculus?” but “HOW do you do calculus?” Maybe I should clarify what I originally said and elaborate. Forgive me for imprecise wording here, and forgive me for being too verbose, I’m trying to get across a specific point and I’ll be stomping across some flower gardens to get there. The current, mainstream form of calculus is based on real analysis and measure theory, which in turn, makes heavy use of infinite sets, excluded middle, and the axiom of choice. When I say that calculus is “based on” real analysis, what I really mean is that calculus was invented, and then later, people invented real analysis to provide a rigorous framework for doing calculus with real numbers. I might ask, “What are the most important theorems in calculus?” The second theorem which comes to mind is the Bolzano-Weierstrass theorem, which involves infinite sequences and is the foundation for many other core results in calculus. Real numbers are themselves typically defined in terms of infinity—either as Dedekind cuts (infinite sets) or Cauchy sequences (infinite sequences). You don’t have to do calculus on real numbers, but real numbers are surely a very convenient tool for the kind of calculus we want to do, and some of the most important properties of real numbers have backed us into a corner w.r.t. how real numbers are constructed—namely, constructions satisfying “complete ordered Archimedean field”. If you decide that you still want to do calculus but you don’t want to do real analysis, the most popular alternative is nonstandard analysis, which has actual infinitesimal numbers. The main advantage of nonstandard analysis is that it provides a rigorous framework for manipulating infinitesimals, which may perhaps be more intuitive that working with differential forms. With nonstandard analysis, you can eliminate limits from a lot of your definitions, because you can use actual infinitesimal numbers instead. At that point, you’re working with non-Archimedean fields. So, to summarize, two likely candidates for the foundations of calculus are: - Real analysis, with heavy use of infinite sets, - Nonstandard analysis, with infinitesimal numbers. I ask, HOW do you do calculus without either? What does the field of calculus look like without either? When you are doing calculus in a strict finitist world, what definition do you come up with for “real number”?
GoblinSlayer 4y ago
Calculus works on finite sets, computational mathematics does exactly that.
- klodolph 4y ago
  Yes, technically correct, but we still want to use real numbers, which is what I was getting at.

kmeisthax 4y ago

We actually know a fair bit about what happens if you discard the axiom of choice; mostly because Banach-Tarski made it so controversial. As for calculus, it doesn't require the existence of infinities or infinitesimals. Set theory needs to care about infinities and the sizes thereof because they are creatures of the axioms they adopt. But in calculus "infinity" just means "it gets as big as you want" and "infintesimal" just means "it gets as small as you want". Everything's argued on top of limits that are done in the real number system where there aren't any infinities or infinitesimals. The thing that calculus really cares about is continuity: i.e. the idea that if you nudge the input to a function a little bit, the output changes by a little bit. If you try to do calculus on functions that don't work this way, then calculus doesn't work, either.

klodolph 4y ago
> We actually know a fair bit about what happens if you discard the axiom of choice; mostly because Banach-Tarski made it so controversial. Do you have any more information about this? Is there an overview somewhere? The major work I know in this area is Reverse Mathematics, but I otherwise thought it was relatively unexplored. Emphasis on “relatively”. My impression here is that Banach-Tarski reveals “flaws” not just in ZFC but in ZF, and that while we may want to reevaluate the axiom of choice, we should be looking at the rest of ZF as well. > As for calculus, it doesn't require the existence of infinities or infinitesimals. The question was never “can you do calculus?” but “HOW do you do calculus?” Maybe I should clarify what I originally said and elaborate. Forgive me for imprecise wording here, and forgive me for being too verbose, I’m trying to get across a specific point and I’ll be stomping across some flower gardens to get there. The current, mainstream form of calculus is based on real analysis and measure theory, which in turn, makes heavy use of infinite sets, excluded middle, and the axiom of choice. When I say that calculus is “based on” real analysis, what I really mean is that calculus was invented, and then later, people invented real analysis to provide a rigorous framework for doing calculus with real numbers. I might ask, “What are the most important theorems in calculus?” The second theorem which comes to mind is the Bolzano-Weierstrass theorem, which involves infinite sequences and is the foundation for many other core results in calculus. Real numbers are themselves typically defined in terms of infinity—either as Dedekind cuts (infinite sets) or Cauchy sequences (infinite sequences). You don’t have to do calculus on real numbers, but real numbers are surely a very convenient tool for the kind of calculus we want to do, and some of the most important properties of real numbers have backed us into a corner w.r.t. how real numbers are constructed—namely, constructions satisfying “complete ordered Archimedean field”. If you decide that you still want to do calculus but you don’t want to do real analysis, the most popular alternative is nonstandard analysis, which has actual infinitesimal numbers. The main advantage of nonstandard analysis is that it provides a rigorous framework for manipulating infinitesimals, which may perhaps be more intuitive that working with differential forms. With nonstandard analysis, you can eliminate limits from a lot of your definitions, because you can use actual infinitesimal numbers instead. At that point, you’re working with non-Archimedean fields. So, to summarize, two likely candidates for the foundations of calculus are: - Real analysis, with heavy use of infinite sets, - Nonstandard analysis, with infinitesimal numbers. I ask, HOW do you do calculus without either? What does the field of calculus look like without either? When you are doing calculus in a strict finitist world, what definition do you come up with for “real number”?

GoblinSlayer 4y ago

Calculus works on finite sets, computational mathematics does exactly that.

klodolph 4y ago
Yes, technically correct, but we still want to use real numbers, which is what I was getting at.