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What are the 'real numbers', really?

math.vanderbilt.edu

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

#11

Post by graycat »

"Points on the line" is fine for the first, second, ..., tenth cut at a definition. Sure, completeness is the biggie for the reals compared with the rationals, algebraics, etc.

Still, as in the OP, mentioning Dedekind cuts is okay since it is one way to establish completeness, but there is much more, e.g., as in

John C. Oxtoby, Measure and Category.

and even that doesn't fathom all that is special about the reals. E.g., for just a little more, there is the continuum hypothesis, that little thing!

The OP wants to say that by mentioning Dedekind and completeness he is getting at what the reals really are; no, instead he is just cutting one layer deeper of something that has likely some infinitely many layers available.

Yes, yes, yes, I know; I know; the reals are the only complete, Archimedean ordered field, okay, after we have defined completeness, Archimedean ordered, and field and explained why these are important.

So, back to "points on the line" -- it's actually pretty good for a first cut.

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

#12

Post by protonfish »

igravious wrote: post #5
Bahamut wrote: post #3

I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.

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.

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

#13

Post by dnautics »

igravious wrote: post #4

a real number is "a point on the number line" These posts are always stimulating. My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs t…

this is kind of a nonsense, circular definition: what is a line? A set of points that can be mapped onto the reals. The more meaty answer is down below, with dedekind cuts (although "a set that fulfills the arithmetic axioms and least upper bounds" is also sufficient).

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

#14

Post by dnautics »

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.

imaginary numbers are also a horribly misnamed fiction. For decades, my dad was mesmerised by how if you plugged in a bigger than c value into the lorentz transformations (he had long since forgotten the form of the transformation) you would become "imaginary". To set him straight, I asked him, if instead we named them "Green" numbers, would you become "green" if you went faster than the speed of light?

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

#15

Post by Someone »

igravious wrote: post #4

a real number is "a point on the number line" These posts are always stimulating. My understanding of a line is that it is delimited by two points, but does not contain any points. To elaborate, no point could be "on" a line because a point has no extension, whereas a line does. This is the crux of the matter. Therefore a line is not "made up of" points. (By analogy a plane could not be made up of lines.) This begs t…

"My understanding of a line is that it is delimited by two points"

That is not how Euclid defined it and how it is still seen in geometry today. What you describe is called a (line) segment (http://en.wikipedia.org/wiki/Line_segment)

"but does not contain any points"

Lines extend indefinitely in two directions (if you go past Euclidean geometry, that 'indefinitely' changes meaning a bit)

One talks of a point being _on_ a line in geometry. 'contains' is something from set theory: "the set of all points on line l contains point P" is a perfectly valid expression (but "P is on l" is way shorter)

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

#16

Post by acjohnson55 »

igravious wrote: post #5
Bahamut wrote: post #3

I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.

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'm interested! I think Dedekind cuts are reasonably understandable, but infinitesimals are on the surface of much of our calculus syntax, so I'd be glad to understand where they become so tricky formally.

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

#17

Post by cokernel »

Bahamut wrote: post #3

I do have an issue with this line "Ultimately, infinitesimals were discredited and discarded by mathematicians (though they continued to be mentioned in some physics books many decades later)" Infinitesimals have been made rigorous with modern mathematics.

To be fair to the author, he does discuss Robinson's approach to infinitesimals (but not Skolem's) in the very next paragraph.

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

#18

Post by DArcMattr »

I have a Master's in Applied Math.

The comments about how "few students take [Real Analysis]" doesn't square with my experience and survey of an undergraduate mathematics education. Such a course is often called "Advanced Calculus", and is a required course for a Bachelors-level education in Math. I also understand in the European-style approach to teaching Math, students start off with a foundational approach to Calculus through Real Analysis, and not the hand-wavy & computation-driven Calculus course.

The equivalence class approach attributed to Cantor is more generalizable in discussing sets. The theoretical foundation of Fourier Transforms lies in a similar completion of functions.

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

#19

Post by ColinWright »

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.

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, some things are easier to prove in non-standard analysis, etc, etc.

However, this is only really of use if all you want to do is calculus. 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, and unless you've done calculus the standard way, you have to learn all this stuff in an unfamiliar and difficult-to-visualize, abstract area.

One of the main reasons for continuing to learn calculus in the epsilon-delta limiting process manner is exactly because it's not only formally sound, it's also giving you tools for moving beyond the rather limited world of differential calculus.

Speculating wildly from limited experience, it might also be the case that starting people with the non-standard approach in calculus is actually just as confusing. You may find that you really only got the insights you did because you had already struggled with the standard approach, and then were given something that made it all fall into place. Perhaps some people they think the non-standard approach is easier, but in fact it's only because they've actually got the foundations from the other. Just a thought.

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

#20

Post by flebron »

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.

This is a similar argument to sqrt(2) being "not a number", back in the BC's, because it was not rational. And yet, you can construct it in a straightforward manner by making a right angled triangle with catheti of length 1, giving a hypotenuse of length sqrt(2). I suppose this would have made you equally uncomfortable back then.

One can definitely "work with" numbers that aren't easy to write. a + (-a) = 0, and this is valid for every real number a, not just "the ones which I can describe with a finite amount of information", or the ones I've written down at some point in my life.

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