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Madeleine Birchfield (Aug 02 2024 at 17:22):split off from #community: discussion > ACT pedagogy and hype (cross post)
Madeleine Birchfield (Aug 02 2024 at 17:28):A question I have for everybody. What exactly are the real numbers?
There are so many different models of real numbers, such as Cauchy real numbers, Escardo-Simpson real numbers, Dedekind real numbers, MacNeille real numbers, the locale of real numbers, hyperreal numbers and other non-standard real numbers, Archimedean ordered Kock fields from synthetic differential geometry, Cauchy sequences of rational numbers from some parts of constructive mathematics, rational numbers, dyadic rationals, finite decimals, or floating-point numbers, most of which are provably not equivalent to each other.
Graham Manuell (Aug 02 2024 at 19:17):I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.
Graham Manuell (Aug 02 2024 at 19:21):Kevin Carlson said:
Graham Manuell said:
I guess wouldn't call what computers do to model real numbers the 'real world'. Computers might use fixed point or floating point numbers, but the real world generally does not. Sure, I agree that it is good for people to prove things about how well these models correspond to the real world, but it feels backwards to take these models as the fundamental notion.
What exactly is this "real world" for you that computers are not part of? It seems very intuitively clear to me that any kind of numbers, insofar as they're actually going to be used to model specific concrete phenomena (whether physical, social, whatever) are going to factor through some finite approximation of the real numbers.
Obviously computers are part of the real world, but this point seems somewhat obtuse, since by this standard anything that mathematicians do is also part of the real world. I have the opposite intuition to you regarding finite approximations. I don't know why you say that when you actually model phenomena you use finite approximations, because essentially all of physics does not do this.
Graham Manuell (Aug 02 2024 at 19:24):I guess the point behind both of my responses is that we already have an entire field of mathematics that allows us to connect idealised 'infinite' objects such as the reals to finite approximations thereof -- namely, topology.
Kevin Carlson (Aug 02 2024 at 19:34):Yes, I think I said “model” when I should have said something more like “measure”. I mean that any specific instance of an actual measurement or output of an actual calculation has to be in some finite space.
Kevin Carlson (Aug 02 2024 at 19:35):So, yes, of course physics models the universe as a real manifold, but I certainly wouldn’t confused with “the real world”.
Graham Manuell (Aug 02 2024 at 19:50):Kevin Carlson said:
Yes, I think I said “model” when I should have said something more like “measure”. I mean that any specific instance of an actual measurement or output of an actual calculation has to be in some finite space.
This is true, but using real numbers already allows us to handle this. Using would not be appropriate.
Graham Manuell (Aug 02 2024 at 19:54):It's not obvious to me whether the universal 'actually is' a manifold or not, but this model is certainly closer to reality than floating point numbers. This is what I was pushing back against. (I wasn't the one to introduce the real world terminology and don't really care to defend it, but I do think you know we mean by it.)
Notification Bot (Aug 02 2024 at 20:09):A message was moved here from #community: discussion > ACT pedagogy and hype (cross post) by John Baez.
Madeleine Birchfield (Aug 02 2024 at 20:32):Graham Manuell said:
I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.
What do you say to the predicative constructivists and type theorists who don't believe in the Dedekind reals or the locale of real numbers in their foundations for whatever reason?
Kevin Carlson (Aug 02 2024 at 20:34):OK, so it's probably clear that floating point numbers are worse than real numbers because reality seems to have some property that you can zoom in arbitrarily far or something. But a structure where you can take measurements valued in arbitrarily fine finite meshes looks like an inverse limit--say an inverse limit of rulers with twice as fine a resolution at every stage, with the transition maps rounding down. This inverse limit will be basically a Cantor space, viewed as the space of actual decimal expansions-- doesn't get identified with , there's just nothing in between them. This distinction comes because the inverse limit constructs the space of possible sequences of measurements, each refining the previous one. So you could actually get the sequence of measurements on your family of rulers, or you could get the sequence , and since the sequences aren't the same they end up different in the limits.
Kevin Carlson (Aug 02 2024 at 20:34):While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?
Madeleine Birchfield (Aug 02 2024 at 20:44):Kevin Carlson said:
While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?
In constructive mathematics, taking the quotient doesn't preserve Cauchy completeness of the pseudometric space, so one will still have to construct the Cauchy completion of the resulting metric space to get a more well-behaved real line with well-defined analytic functions. Or one can just not take the quotient set and keep Cauchy completeness, but have to deal with the fact that the space is only a setoid.
Graham Manuell (Aug 03 2024 at 10:40):Madeleine Birchfield said:
Graham Manuell said:
I think there is really only one answer, and that is the locale of real numbers (and so if you really want to consider the points, the Dedekind reals). They are the only ones that behave in a reasonable way imo. I don't understand how rational numbers of anything similar could possibly work, since they have the entirely wrong topology.
What do you say to the predicative constructivists and type theorists who don't believe in the Dedekind reals or the locale of real numbers in their foundations for whatever reason?
They should just work with the locale of reals via a presentation. This is essentially what formal topology is doing.
Graham Manuell (Aug 03 2024 at 10:44):Kevin Carlson said:
While there are lots of mathematical and pragmatic reasons to prefer to take the quotient of this Cantor space and get the usual real line, I'm very curious whether there's anything else--is there any good reason to think that models using the reals rather than the Cantor space is somehow actually "more correct" about how the world is?
As I said above for the rationals, Cantor space is clearly wrong because it has the entirely wrong topology. If you tried to use Cantor space to model reals you would be able to decide whether a given number is < 1 or not. This is not something that should be possible with measurements of finite precision.