Related. Others?
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=24029791 - Aug 2020 (112 comments)
How real are real numbers? (2004) - https://news.ycombinator.com/item?id=14080024 - April 2017 (265 comments)
How real are real numbers? (2004)
51–60 of 108 posts
Re: How real are real numbers? (2004)
#52His second "proof" of uncountability is very poorly explained. Strictly speaking it is false, since all his reasoning applies equally to the rationals. What he shows is that a countable set would have zero measure. You have to also show that (say) the real interval [0,1] has measure 1 (or at least positive measure) to get a contradiction. That requires some more work. You have to be using some property of the reals in order to prove uncountability, as of course the Cantor diagonal argument does.
Re: How real are real numbers? (2004)
#53His second "proof" of uncountability is very poorly explained. Strictly speaking it is false, since all his reasoning applies equally to the rationals. What he shows is that a countable set would have zero measure. You have to also show that (say) the real interval [0,1] has measure 1 (or at least positive measure) to get a contradiction. That requires some more work. You have to be using some property of the reals i…
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Re: How real are real numbers? (2004)
#54I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist. But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Simil…
> Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense.
You may enjoy a recent update on that story [0] that maybe avoids a few paradoxes and looks at things other than navels.
Re: How real are real numbers? (2004)
#55https://ar5iv.labs.arxiv.org/html/math/0411418 for the in-browser html rendering
Re: How real are real numbers? (2004)
#56Earlier quoted context omitted.
>What do "real" numbers buy you? That is, what do you get for the trouble of building your formalism around numbers that are not computable? I guess the naive answer is completeness. Every Cauchy sequence converges to a member of the space. For example, quantum mechanics relies on the formalism of Hilbert space, defined as a complete inner product space. This gives us nice things like the spectral theorem for unbound…
Bad example. You can do all of this with constructivism. Any constructable Cauchy sequence converges to a constructable member of the space. What you get for the formalism around computable numbers is this. Every mathematical object in the theory is something that can be, at least in principle, actually written down. When we say that it exists, this existence is of the most tangible form that any mathematical thing c…
Having constructible Cauchy sequences doesn't guarantee that we can construct unbounded operators. I'm no expert, but the little searching I've done suggests this is an open research question.
I don't see the benefit of being able to write something down "in principle." A number can only ever be computed to a finite number of digits in practice. If we're talking about finite approximations, then the standard approach using numerical solutions to the Schrödinger equation handles this just fine, no alternative mathematics needed. If we're talking about theories, then we should choose whatever abstraction is most convenient for expressing the theory.
Personally, I don't believe numbers "exist." The physical universe exists, and numbers are abstractions that we invent to describe it. In that sense, uncomputable numbers are just as "real" as computable ones.
Re: How real are real numbers? (2004)
#57Earlier quoted context omitted.
in what way does drawing a diagram have anything to do with physically realizing a number?
I didn't say a diagram? I'm talking primarily about physical objects. And I'm just explaining how to use the same methods as the comment I replied to.
"Make a line of 10 apples and declare it to be 2 units long" <- what number does this physically realize?
Re: How real are real numbers? (2004)
#58Earlier quoted context omitted.
I didn't say a diagram? I'm talking primarily about physical objects. And I'm just explaining how to use the same methods as the comment I replied to.
"Make a line of 10 apples and declare it to be 2 units long" <- what number does this physically realize?
Both ten and two. And it lets you model fifths.
But don't ask me about that. That part wasn't my idea at all. Ask dhosek about that way to do fractions. My contribution was the square root and the subtraction.
Re: How real are real numbers? (2004)
#59I’ve had this paper downloaded for about a decade and haven’t gotten around to reading it, but thinking about it, especially if space and time are quantized (an undetermined question last I checked and almost certainly still so), there would exist numbers in ℝ that cannot be expressed as physical quantities, even with an infinite universe. It’s possible that even the algebraic numbers include numbers that are non-phy…
If reality is quantized and there is a smallest number that is physically relevant, you don't need the reals to break it. Take that smallest number and divide it by two, and now you have a physically meaningless number using only the rationals. This isn't fair for what quantization means in reality, but I'm just pointing out that you don't have to introduce the real numbers to get physically meaningless quantities.
Take the smallest number corresponding to a physically meaningful distance in meters, divide it by two, and that number is still a physical meaningful distance if you switch the unit to decameters or kilometers.
Any rational number has some meaning if you just add the right unit, even if the units become increasingly ridiculous. But for reals that trick does not work
Re: How real are real numbers? (2004)
#60I think people overindex on the continuity problem with the reals. I'm personally a bit of real-number denier myself as a constructivist / intuitionalist. But when we say things like "the rationals are discrete" or "the computable numbers are discrete" these are very specific claims in the domain of measure theory, a theory which yields almost nothing of value except endless paradoxes and naval-gazing nonsense. Simil…
> Similarly when people say "the rationals are countable" and "the computable numbers are countable" this is taking for granted the Cantor notion of measuring cardinality by bijective correspondence, once again, a theory that yields nothing of value except endless paradoxes and naval-gazing nonsense. You may enjoy a recent update on that story [0] that maybe avoids a few paradoxes and looks at things other than navel…
Thanks! That was a nice read.