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What are the 'real numbers', really?

math.vanderbilt.edu

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

#21

Post by acjohnson55 »

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.

Unfortunately, the set of computable numbers, while countable, is more difficult to work with than the much more conceptually reasonable set of reals. Luckily, we have pretty great theoretical tools for dealing with uncountability, so I don't think the fact that the vast, vast majority of real numbers are unidentifiable is really that big of a problem. That, and the continuum is a really useful concept, even if it very well might not have any basis in physical reality.

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

#22

Post by moyix »

gizmo686 wrote: post #10
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.

Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.

One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]).

[1] http://en.wikipedia.org/wiki/Computable_number

[2] http://en.wikipedia.org/wiki/Computable_analysis

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

#23

Post by cokernel »

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…

In linear algebra, lines are sets of points, but in modern geometry, points and lines remain undefined terms, implicitly defined by their incidence relations (which points are "on" which lines).

You may find the Fano plane (a three-dimensional finite projective space) interesting:

  * http://en.wikipedia.org/wiki/Fano_plane (brief description)
  * http://math.ucr.edu/home/baez/octonions/node4.html (connections with higher math)

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

#24

Post by igravious »

Someone wrote: post #15
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…

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cannot sum to anything greater than zero. So I ask you again, does it make sense to think of lines (or line segments) as composed of points, I reckon it does not.

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

#25

Post by jonsen »

igravious wrote: post #24
Someone wrote: post #15

Earlier quoted context omitted.

"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…

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…

Would it make a difference if you substituted "infinitesimal extension" for "zero extension"?

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

#26

Post by gizmo686 »

flebron wrote: post #20
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…

The 'problem' with the reals is that there are numbers that cannot be constructed.

Every number that we can construct can be constructed in a finite amount of symbols. For example sqrt(2) is an unambiguous description of itself. Without use of the sqrt function, we can also call it the number x such that x*x=2. However, every description is a finite string constructed from a finite alphabet. We can easily show that the set of all such descriptions is countably infinite. However, we can also show that the set of all real numbers is uncountably infinite. Therefore, there is an uncountable infinity of real numbers that cannot be constructed.

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

#27

Post by zodiac »

igravious wrote: post #24
Someone wrote: post #15

Earlier quoted context omitted.

"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…

Excuse me. Of course. I was using line and line segment interchangeably there. Which I should have not been doing if I am aiming for clarity but I think my point (ahem) applies to line segments and lines that extend indefinitely in one or two directions. Presumably people will contend that even a line segment "contains" an infinite number of points. But if points have zero extension then even an infinity of them cann…

> But if points have zero extension then even an infinity of them cannot sum to anything greater than zero.

Can you make this rigorous? Because using the standard definitions, this statement is not true. It's true that a countable number of points must have total length zero (and you can even give a rigorous proof of this) but not necessarily true for a non-countable number of points. The study of "lengths of sets of points" is called measure theory.

I think it is unnecessary, however, to bring in the whole concept of length when defining lines. For example, we could simply define a line as a set of points obeying some special properties.

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

#28

Post by protonfish »

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…

I disagree. The vast majority of students take math classes for the practical applications - science and engineering - not to continue theoretical pure math study. Therefore the focus should be on effective teaching of applied math. I am sure that if a student wishes to explore their studies in pure mathematics they will be clever enough to learn whatever they need in specialized classes.

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

#29

Post by gizmo686 »

moyix wrote: post #22
gizmo686 wrote: post #10

Earlier quoted context omitted.

Do you have any idea of what set we should use to replace them with? The rational numbers can do a lot, but we have discovered that there are numbers worth talking about (and which can be described) that are not rational. Whatever replacement you propose must be usable where ever we would use real numbers, and must be at least as simple to use.

One possible replacement is the computable numbers [1]; this includes the algebraic numbers and some common transcendentals (e, pi), and you can even build up something akin to standard analysis (computable analysis [2]). [1] http://en.wikipedia.org/wiki/Computable_number [2] http://en.wikipedia.org/wiki/Computable_analysis

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

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

#30

Post by igravious »

acjohnson55 wrote: post #16
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'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.

I wrote a short paper on the topic once upon a time[1] which you may find interesting. It's part history of math, part philosophy of math.

It's not a great paper and most of the insights in it come from others but here is some of the arithmetic of nilpotent[1] infinitesimals as shown in the appendix.

Imagine an entity which is not equal to zero but that when raised to the power of 2 or higher is equal to zero! Sounds odd, doesn't it, but it works! (ϵ is an infinitesimal)

ϵ != 0 but ϵ^n = 0 | n>1

ok? so we get:

(ϵ + 1)^n = 1 + nϵ thus: (ϵ + 1)^−1 = 1 − ϵ

e^ϵ = 1+ϵ

(ϵ + 1)(ϵ−1) = −1, or alternately (1 + ϵ)(1 − ϵ) = −1

and finally (for calculus): ϵf′(x) = f(x + ϵ)−f(x)

1: http://leto.electropoiesis.org/propaganda/The_Analyst_Revisi...

2: https://en.wikipedia.org/wiki/Nilpotent

edit: clarity, line breaks!

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