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How real are real numbers? (2004)

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Re: How real are real numbers? (2004)

#21

Post by qsort »

andrewla wrote: post #17
qsort wrote: post #15

Earlier quoted context omitted.

I don't think your position is silly, but this is not a great argument for it. > But when we say things like "the rationals are discrete" In the usual topology they are not? > In the practical sense the rational numbers are quite continuous -- between any two rational numbers there are an infinite (unbounded) number of rational numbers -- there's no notion of a "leap" the way there is with the integers. This characte…

No, the rationals are not discrete in the usual topology. They end up being discrete when we consider continuous mappings from R->Q though. That is the "technical" sense that I refer to. The rationals, as you say, are dense in R but they are also dense in the computables. The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable. All the real construction techn…

> The big Cantorian leap that we make is when we use the diagonal argument to argue that the rationals are countable.

Again, I don't think your position is indefensible, but this doesn't strike me as particularly convincing. The usual definition of R is that there exists a unique ordered complete Archimedean field up to isomorphism. We get the kitchen sink from the least upper bound property. As a constructivist you're gonna say that I don't get to define R like that, but you can't pretend it's done for no reason or that it buys nothing.

> I certainly learned about areas in geometry

And how were they defined? In elementary geometry we just sweep the question under the rug, usually...

If you get to say that being able to articulate why the measure of Q is 0 is unimportant and uninteresting, then I get to claim that the supposed problems with the usual definitions are also unimportant!

Saying that the non-constructive world leads to worse problems is a respectable position. Pretending the usual way of doing things is completely arbitrary isn't very honest.

Re: How real are real numbers? (2004)

#22

Post by vee-kay »

dhosek wrote: post #2

I’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…

[dead]

Re: How real are real numbers? (2004)

#25

Post by testaccount28 »

thaumasiotes wrote: post #6
andrewla wrote: post #4

Earlier quoted context omitted.

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.

You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch. You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot. dhosek is saying that in a quantized space, there are reals that cannot be demon…

"suppose that the minimum distance is one inch. well, that's one of something. so now imagine half of that! there you go: one half. a physically unrealizable number."

Re: How real are real numbers? (2004)

#26

Post by jonahx »

Norman Wildberger is a required mention on this topic.

Here's a great discussion on Curt Jaimungal's podcast:

https://www.youtube.com/watch?v=l7LvgvunVCM

And a good debate on the topic with Daniel Rubin, who takes the more orthodox position:

https://www.youtube.com/watch?v=edh5bbgSKqo

Wildberger has tons more on this topic on his own channel. His arguments are thought-provoking, even if you don't agree with them.

Re: How real are real numbers? (2004)

#28

Post by testaccount28 »

xelxebar wrote: post #19
andrewla wrote: post #3

I 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…

> What do "real" numbers buy you? They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers. The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to c…

> equality is undecidable

equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.

Re: How real are real numbers? (2004)

#29

Post by ttctciyf »

I've long admired Chaitin for his original thinking and especially his ability to clearly convey his ideas about foundations, complexity and information in concise and digestible short proofs.

I'm a little surprised, however, to see him here proselytizing for a particular side in the constructivism debate. I associate him more with what he has described as a "quasi empirical" approach to mathematics[0] where the adoption of new axioms (such as for example the axiom of choice) is justified by their resulting in new, interesting mathematics.

But here, it seems his goal is to arrive at a somewhat Platonic conclusion, that either the reals are valid numbers or (seemingly he prefers) not.

My lay and naive take would be: if you adopt these rules (this Formal Axiomatic System) then you can have Big Fun in the playground of ever more esoteric and complex infinite cardinals, or if you adopt this other FAS you get to discover which results can and can't be obtained under a strict constructivist regime, and whichever FAS you choose it's just the same process of choosing axioms and applying valid deductive steps to arrive at a result you find interesting, with no more "existence" implied than the thoroughly non-Platonic existence of a solution to a problem, which can be demonstrated by solving it: if you take such-and-such steps then such-and-such result will follow.

I suppose the point being made is that avoiding axioms which imply the "existence" of the reals is more useful for doing physics, but that seems non-obvious in a field which for the last 200 years has seemingly sought to progressively make more and more phenomena intelligible by means of differential equations from infinitesimal calculus!

0: see, for example: https://arxiv.org/pdf/math/0303352, 1.9 Is Mathematics Quasi-Empirical

Re: How real are real numbers? (2004)

#30

Post by dhosek »

testaccount28 wrote: post #25
thaumasiotes wrote: post #6

Earlier quoted context omitted.

You're taking an anomalously narrow view of the parent comment. Say the minimum distance is one inch. You want to say that the concept of half an inch lacks physical representation, but that isn't true. You can easily demonstrate it as the ratio between one inch and two feet, compared to the reference ratio between one inch and one foot. dhosek is saying that in a quantized space, there are reals that cannot be demon…

"suppose that the minimum distance is one inch. well, that's one of something. so now imagine half of that! there you go: one half. a physically unrealizable number."

Except that you’re assuming that 1 must necessarily correspond to that minimum distance. Keeping that situation, we can just say that 1 corresponds to two inches and then realize 1/2 as that number.

Classical geometry (a la Euclid) allows for constructing a lot of numbers. We get natural numbers pretty cheaply, negative integers through adding in a concept of directionality (zero is a bit of an imaginative leap which is why it was absent from Western mathematics for so long). Constructing arbitrary ratios is possible through similar triangles and square roots through right triangles, but some basic algebraic numbers like cube roots cannot be constructed with a ruler and straight edge (which raises the question of whether, in a quantized universe, whether irrational cube roots actually exist). Of course there’s no guarantee that the quantization is going to be uniform and we also have the ɣ factor of special relativity (1/sqrt(1-v²/c²)) which gives us a non-Euclidean space to complicate things, but it’s not clear that if you can find a value for 1 that allows you to get a measurement for every irrational number.

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