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It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. For one thing, they're speculative

It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. For one thing, they're speculative: the current theories that give the most precise and accurate predictions within their respective domains of applicability are general relativity and quantum field theory. These theories are based on continuous space and time, and no attempt to base them on discretized space and time has been successful (AFAIU both QFT and GR rely on Lorentz invariance, which means there's no absolute rest frame, hence no absolute unit of time and space, but a discretized spacetime would require absolute unit values for space and time, hence an absolute rest frame). Should we conclude then that the reals are real, because they're components of our best current physical theories? Maybe, maybe not: these are features of our current best models, but we don't know, and possibly will never know, the ultimate nature of physical space and time. For another thing, even if space and time are fundamentally discrete, there's still no doubt that the mathematical theories based on real numbers are effective in making predictions, and we would still like to use them. That's means they should have some logical foundation which can guarantee that reasoning using them is correct.

NoahZuniga 1y ago
> It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).
- griffzhowl 1y ago
  You're right. The question is more about mathematical ontology than its logical foundations. I should have said something like "It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
  - deleted 1y ago
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  - Nevermark 1y ago
    My comment is on whether the total class of reals are in nature. Given the theme and title of the article. Given one of the primary (but often not emphasized) properties of the class of reals, is that it contains and is actually dominated by the un-constructible reals, an argument that the total class of reals isn't represented in nature seems unremarkable to me. Un-constructible reals are a highly exotic abstract concept. You can never actually identify or operate on one. I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality. It is a mishap of terminology, that the larger class, which is unnecessary for middleschool math, or any math most people will ever encounter, has a pithy name "real". While the practical concept, that directly correlates with arithmetic, algebra, calculus, diff eq, etc., has the unwieldily title of "unconstructible reals". So in order to avoid having to talk about constructible vs. unconstructible numbers to kids who shouldn't need to care, we use the pithy term and actually throw unconstructible reals into the mix, where it wasn't necessary at all. We tell kinds how the reals have a higher-order uncountable infinite cardinality, relative to integers. Which is true, but gives the impression that cardinality is somehow a concept necessarily linked to algebraic and other practical numbers that we introduce at the same time, which it is not. The set of constructible reals has the same countably infinite cardinality as the integers.
    - NoahZuniga 1y ago
      While the observable universe is in some sense finite, as far as I know it is definitely still possible (based on our understanding of science) that the universe is infinite (and has infinite matter in every direction). I'll use this definition: a constructible real is a real that you can describe uniquely in a purely mathematical way (and prove that your description identifies a unique real). IE the positive solution to x^2=2 identifies a unique real. Also, the first positive solution to sin x. Now if you accept that the universe is infinite (and has infinite matter in every direction), you could get representations of un-constructible reals in your universe. One pretty contrived way to get an un-constructible real from this infinite universe is this: start at earth with some velocity, lets say 0.01c in some direction. Start with r = "0." Every second, find the closest particle. Take the amount of meters (rounded down) that this particle is away from you and append it to r. So if after one second the closest particle is 145m away, r becomes 0.145. If after another second the closest particle is 0.14m away, r becomes 0.1450 The value this process converges to could be un-constructible.

NoahZuniga 1y ago

> It seems wrong to me to base foundations of mathematics on speculative theories about the nature of physical space and time. But this claim is nowhere made in the comment? Like clearly the transfinite ordinals aren't real, but no one would say that implies they aren't a very useful mathematical idea (and also just interesting in and of themselves).

griffzhowl 1y ago
You're right. The question is more about mathematical ontology than its logical foundations. I should have said something like "It seems wrong to me to base mathematical ontology on speculative theories about the nature of physical space and time."
- deleted 1y ago
  [deleted]
- Nevermark 1y ago
  My comment is on whether the total class of reals are in nature. Given the theme and title of the article. Given one of the primary (but often not emphasized) properties of the class of reals, is that it contains and is actually dominated by the un-constructible reals, an argument that the total class of reals isn't represented in nature seems unremarkable to me. Un-constructible reals are a highly exotic abstract concept. You can never actually identify or operate on one. I do believe all the constructible reals (what most people think of when they think of real numbers), are likely to be found in nature/reality. It is a mishap of terminology, that the larger class, which is unnecessary for middleschool math, or any math most people will ever encounter, has a pithy name "real". While the practical concept, that directly correlates with arithmetic, algebra, calculus, diff eq, etc., has the unwieldily title of "unconstructible reals". So in order to avoid having to talk about constructible vs. unconstructible numbers to kids who shouldn't need to care, we use the pithy term and actually throw unconstructible reals into the mix, where it wasn't necessary at all. We tell kinds how the reals have a higher-order uncountable infinite cardinality, relative to integers. Which is true, but gives the impression that cardinality is somehow a concept necessarily linked to algebraic and other practical numbers that we introduce at the same time, which it is not. The set of constructible reals has the same countably infinite cardinality as the integers.
  - NoahZuniga 1y ago
    While the observable universe is in some sense finite, as far as I know it is definitely still possible (based on our understanding of science) that the universe is infinite (and has infinite matter in every direction). I'll use this definition: a constructible real is a real that you can describe uniquely in a purely mathematical way (and prove that your description identifies a unique real). IE the positive solution to x^2=2 identifies a unique real. Also, the first positive solution to sin x. Now if you accept that the universe is infinite (and has infinite matter in every direction), you could get representations of un-constructible reals in your universe. One pretty contrived way to get an un-constructible real from this infinite universe is this: start at earth with some velocity, lets say 0.01c in some direction. Start with r = "0." Every second, find the closest particle. Take the amount of meters (rounded down) that this particle is away from you and append it to r. So if after one second the closest particle is 145m away, r becomes 0.145. If after another second the closest particle is 0.14m away, r becomes 0.1450 The value this process converges to could be un-constructible.