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What are the 'real numbers', really?

math.vanderbilt.edu

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

#61

Post by Someone »

mcguire wrote: post #54
Someone wrote: post #37

Earlier quoted context omitted.

"But if points have zero extension then even an infinity of them cannot sum to anything greater than zero." Infinities are weird; anybody who wants to learn math has to accept that. 0.99999… does equal 1, there are as many even numbers as integers, etc. these things are 'true' not because they make sense initially, but because they make the most sense of all the other things we have thought of so far. Similarly, a se…

" a set of Aleph-0 points can completely cover a line " Aleph_0 is the cardinality of the integers. I don't think that'll cover a line. For that, you need the cardinality of the reals, C, which may or may not be Aleph_1.

OOPS. Thanks

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

#62

Post by pash »

ColinWright wrote: post #19
protonfish wrote: post #12

Earlier quoted context omitted.

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.

Added in edit to emphasise a point: If all you want to do is differentiate and integrate, then non-standard analysis is probably, for most people, a faster way to be able to do just that. Now read on ... Non-standard analysis has been put on a firm, formal footing. Theorems have been proven showing that (largely) it's equivalent to the regular form of analysis. Some things are easier to prove in standard analysis, so…

> If you want to go beyond calculus, almost everything (in this and related areas) is about sequences, limits, limiting processes, functions, and transformations. There, non-standard analysis tends not to help ...

Why do you say this? I ask because I've found internal set theory, Edward Nelson's axiomatic version of nonstandard analysis, to be a lovely tool for doing typical sorts of things in analysis.

You have to learn to wield the "standard" predicate [0], which is too dark an art for some mathematicians, I suppose. But, in my opinion, nonstandard characterizations of notions like convergence and continuity are delightfully simple and direct.

It also turns out that when you have nonstandard numbers at hand, infinity is an over-powerful abstraction for some purposes. Nelson came up with a new formalism for probability theory [1], for example, that makes finite spaces powerful enough to capture what's interesting for most purposes. Similarly, finite but unlimited sequences often are "long enough" to incorporate all the interesting behavior of infinite sequences.

0. Alain Robert's Nonstandard Analysis is a good starting point.

1. See his short book Radically Elementary Probability Theory. I love this book, and didn't much like probability theory before reading it.

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

#63

Post by baddox »

gizmo686 wrote: post #52
baddox wrote: post #38

Earlier quoted context omitted.

What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the…

I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions ex…

Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interesting_number_paradox.

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

#64

Post by bjornsing »

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 two rational numbers then they are connected by a line of other little dots (just like the reals). Perhaps you could argue though that at "infinite magnification" there are no rational numbers to be seen, it's just empty space, whereas the reals of course still make a nice line.

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

#65

Post by Patient0 »

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.

For me it was completely the other way around: I was "taught" calculus using the infinitesimal approach but without any rigour. Statements like "As dx gets really really small x+dx/x becomes 1" drove me crazy! Why was it sometimes ok to replace dx with 0!? The idea of an "infinitely" small number to me was always vague and suspect. So while I could do the calculations I never trusted the results.

This meant that maths stopped having the same appeal to me as computer programming.

It was only years later when I revisited the epsilon delta arguments that it finally made sense. It was a revelation to me that you could explain all of calculus without ever talking about "infinite".

I wish it had been taught to me rigorously the first time around: I would have been much better off.

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

#66

Post by scotty79 »

chowells wrote: post #6

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

IMHO real numbers are anything but. I believe there isn't a single thing in the universe that is represented by real number. Any physical law that involves pi should be considered as statistical in nature. There are no perfect circles. Only the things that are really well approximated by them.

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

#67

Post by gizmo686 »

baddox wrote: post #63
gizmo686 wrote: post #52

Earlier quoted context omitted.

I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions ex…

Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interes…

We can construct the set of all real numbers with a finite amount of information. However, that set contains elements which we cannot precisely describe with a finite amount of information.

The phrase "all undescribable real numbers" does not introduce any problems, because we have still not described any specific undescribable number. We would run into a problem with a phrase such as "the smallest undescribable real number", as that would be a description of a specific undescribable real number. Fourtuantly, that particular phrase does not raise any problems because we can simply conclude that their is no smallest undescribable real number, in the same way that there is no smallest real number in general.

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

#68

Post by gizmo686 »

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.

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

#69

Post by lmkg »

chowells wrote: post #6

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

The Reals are venerated because they're actually usable. Other numbers systems tend to be a gigantic pain in the ass to get any work done with.

The Reals are constructed specifically to be the smallest set that has some nice algebraic properties, like Least Upper Bounds. Sets that model the real world, like the constructables, countables, computables, etc. tend to be subsets of the Reals, and therefore don't have those properties. That absence makes life difficult.

The Real Number system, like almost everything in mathematics, is an approximation of reality that makes a trade-off between faithfulness and tractibility. As it turns out, gaining more of the former loses you quite a bit of the latter. It's generally not worth it.

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

#70

Post by twobits »

scotty79 wrote: post #66
chowells wrote: post #6

What are "real numbers"? A horribly misnamed fiction. Nearly all of them cannot be represented with a finite amount of information. I strenuously object to naming an uncountable set "real" when only a countable subset (measure 0 of the full set) can be worked with in any way at all. We need to stop venerating the "real" numbers and start focusing on sets that are actually usable.

IMHO real numbers are anything but. I believe there isn't a single thing in the universe that is represented by real number. Any physical law that involves pi should be considered as statistical in nature. There are no perfect circles. Only the things that are really well approximated by them.

Amen to that. The real world is discrete. The real numbers in our equations are just approximations.
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