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What are the 'real numbers', really?

math.vanderbilt.edu

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Re: What are the 'real numbers', really?

#71

Post by thaumasiotes »

bjornsing wrote: post #64

I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing…

Well, consider the ruler function[1], which is continuous on the irrationals and discontinuous on the rationals. The real numbers really are denser than the rationals; that's why something like the ruler function is possible (notably, a conceptual reverse, continuous on the rationals and discontinuous on the irrationals, cannot exist -- the rationals are too far apart). I'm pretty sure this is precisely the phenomenon the quote you extract is referring to: if you were standing, infinitely magnified, at a point on the ruler function, then the function would be continuous ("look like a line") if your point was irrational, but if your point was rational, there would be a measurable gulf separating you from the rest of the function.

[1] http://en.wikipedia.org/wiki/Thomae%27s_function

Re: What are the 'real numbers', really?

#72

Post by bjornsing »

gizmo686 wrote: post #68
bjornsing wrote: post #64

I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing…

I don't think that works. The rational numbers are a dense subset of the real numbers. Informally this means every real number is either a rational number, or is arbitrarily close to a rational number. This means that at any magnification, if there was a hole that is filled by a real number, then their would also be a rational number that is arbitrarily close to that real number.

Well, I'm not a big fan of this "infinite magnification" idea in the first place, but "arbitrarily close" is typically one of those things that infinity can beat.

(Compare e.g. with the fourier transform of a function. It consists of a sum series which comes "arbitrarily close" to the function, but "at the limit" when the number of terms approaches infinity the function and its fourier transform is one and the same.)

Re: What are the 'real numbers', really?

#73

Post by bjornsing »

thaumasiotes wrote: post #71
bjornsing wrote: post #64

I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing…

Well, consider the ruler function[1], which is continuous on the irrationals and discontinuous on the rationals. The real numbers really are denser than the rationals; that's why something like the ruler function is possible (notably, a conceptual reverse, continuous on the rationals and discontinuous on the irrationals, cannot exist -- the rationals are too far apart). I'm pretty sure this is precisely the phenomeno…

Infinity is a pretty strange concept. :) I'm not sure arguing over it in this format is meaningful, but for the fun of it:

Consider that the integral of the ruler function from 0 to 1 is 0 (as is stated in your reference 1). In layman's terms you could express this as "there are infinitely more irrational than rational numbers between 0 and 1". At the same time, "for every two rational numbers there are infinitely many rational numbers in-between them". What sort of "picture" is this compatible with?

I still think that the only picture that really makes any sense is a solid line at any finite magnification, yet empty space at infinite magnification.

Re: What are the 'real numbers', really?

#74

Post by totemizer »

GFK_of_xmaspast wrote: post #58
totemizer wrote: post #32

" It seems that any proper theory of real numbers presupposes some kind of prior theory of algorithms; what they are, how to specify them, how to tell when two of them are the same. Unfortunately there is no such theory." http://njwildberger.wordpress.com/2012/12/02/difficulties-wi...

Guys like that in general have never seemed all that convincing to me.

I admit, some of his ideas are a bit.. well, I don't like when people talk about God seriously, and he sometimes mentions it, very rarely. But aside that, everything I can understand from what he says is true. It's a philosophical debate and if you are on the "real numbers" bandwagon (where most people are), you would lose integrity and your reputation might suffer even if you would speak to Wildberger about real numbers. It's a shame really how people don't see why it's bad to use abstractions which are so general that they can be fit for any kind circumstances. Even if you real this article about the real numbers, there is wishful thinking (where he says that the real numbers would look like a line even at infinity but the rationals wouldn't. well, I don't see why the rationals would stop especially given what he says later...), cherry picking / the whole axiom selection stuff for proving it... and yeah, the axiom idea is generally bad anyway. etc.

Re: What are the 'real numbers', really?

#75

Post by jfarmer »

protonfish wrote: post #12
igravious wrote: post #5

Earlier quoted context omitted.

I agree. It would be truer to say that infinitesimals are studiously ignored by modern mainstream mathematicians because they feel that Dedekind and co. have put the calculus on a firm footing way back when. Anybody with a small bit of curiosity or a dashing of non-conformity will be suspicious of this narrative. If anything, infinitesimals in their various guises carry a certain explanatory heft, and are quite begui…

I loathed limit-based calculus in High School and College. Later I read Elementary Calculus: An Infinitesimal Approach http://www.math.wisc.edu/~keisler/calc.html and it all came clear in a fraction of the pages. It's infuriating that most math curricula won't drop those old, bloated, overly formal calculus tomes to improve the clarity and effectiveness of the instruction method.

Conversely, I find infinitessimals vague and woo-woo, especially the way physicists and statisticians often use them. Once the epsilon-delta style "clicked" for me, it felt like second nature.

How can you tell whether it's standard analysis that's confusing per se or you just had poor math teachers?

Re: What are the 'real numbers', really?

#76

Post by jfarmer »

gizmo686 wrote: post #51
ColinWright wrote: post #42

Earlier quoted context omitted.

Indeed, and the constructable numbers are studied as a subset of the reals, as are the algebraics, and the computables. You can make a choice as to the domain of discourse. If you like, feel free to restrict it to the computables (or the constructables). Then apply the diagonal argument. Take the computable numbers between 0 and 1, including 0, not including 1. These are countable, so we can write them in a list, tak…

I can't find a flaw in your arguement, but it seems like it leads to a contradiction. Let a constructable number be one which can be unambiguously described in a finite string. Because we are working from a finite alphabet, we can trivially see that their is a bijection between the constructables and the integers (if we have n symbols, then each string can be read as an integer in base n, so the amount of constructab…

You're using too many imprecise terms. First, what does it mean for something to be able to be "unambiguously" described? As opposed to ambiguously described? You have to define it.

Second, what does it mean for something to be described (unambiguously or otherwise) with a "finite string?" What is a "string" here?

You're playing too loose with these ideas and it's biting you. You have to start by defining them precisely. For example, I don't see at all how the new number not on your list is "described unambiguously." It's presumably not enough to say "there is some number not on my list, we will call it x" since we know there is more than just one such number. How is that unambiguous?

In any case, that's why you have to define these things precisely.

Re: What are the 'real numbers', really?

#77

Post by jfarmer »

bjornsing wrote: post #64

I have an issue with this (albeit parenthesised) line: "It turns out that, in some sense, the real numbers would still look like a line under infinite magnification, but the rational numbers would be dots separated by spaces." In-between any two rational numbers there's an infinite number of other rational numbers. So, in any reasonable sense and at any level of "magnification", if you can "see" two dots representing…

Well, "in some sense" gives a lot of wiggle room! I agree the magnifying glass analogy is bad, but here's one "sense" in which the rationals "have more holes" than the irrationals.

There exists a function of the reals which is continuous at every irrational point but discontinuous at every rational point. However, there is no function of the reals which is discontinuous on the irrationals but continuous on the rationals. In this sense, the irrationals are "more continuous" than the rationals.

That's about the best I can do, though, which I admit is a stretch.

Re: What are the 'real numbers', really?

#78

Post by jfarmer »

moyix wrote: post #49
gizmo686 wrote: post #29

Earlier quoted context omitted.

Unfourtuantly, there exist numbers which are definable but not computable.

Sure. Chaitin's Omega is a good example. The question is whether such numbers occur in the real world.

The trick here is defining "occur" and "real world" precisely. Are you saying the act of me writing down those symbols and expressing the idea does not count as "occurring in the real world?" ;)

Re: What are the 'real numbers', really?

#80

Post by snowwindwaves »

anatoly wrote: post #45
snowwindwaves wrote: post #2

Along a similar vein you may also enjoy http://arxiv.org/pdf/1303.6576 The foundations of analysis by Larry Clifton. I always enjoy checking out the references in his papers as they are often hundreds of years old or more.

What other papers did he author? This is a curious paper. It's a rigorous derivation of (positive) real numbers without the use of 0 or negative numbers anywhere. It isn't very useful, although the fact that this can easily be done is by itself interesting. I have sometimes thought about the possibility of us encountering an advanced alien civilization and trying to match our math to theirs. Someone told me recently…

The only other ones I know of are on his website http://cliftonlabs.net/TechnicalArticles.html
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