
What are the 'real numbers', really? - dhammack
http://www.math.vanderbilt.edu/~schectex/courses/thereals/
======
Bahamut
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.

~~~
igravious
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 beguiling little creatures if you take the
time to get to know them. I'd be happy to elaborate or leave a few links here
if anybody is interested.

~~~
protonfish
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](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.

~~~
ColinWright
_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.

~~~
protonfish
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.

~~~
ColinWright
Actually, you are agreeing with me. You are saying that doing calculus was,
for you, much easier using the infinitesimal approach. I'm not disagreeing
with you. In fact, you'll find that advanced mathematicians think in that way,
although they can drop back to epsilon-delta work if they need to (which they
often do).

So we are in agreement. My point is that if you teach calculus that way you
have immediately ham-strung anyone who might go on and do anything other than
engineering or physics. In fact, there are deep theoretical arguments in
physics where you need to use the standard approach, and the non-standard
approaches are much more difficult.

My point is that if all you want is calculus then it's very likely that the
non-standard approach is fine. I'm also arguing that this is limited thinking.
Clearly you were never going to go further in these sorts of subjects - does
that mean that everyone else should also be taught in a similarly limited way?

I also observe that limiting arguments are essential in anything other than
the most direct and practical versions of engineering, so again, the point
isn't in the calculus, the point is learning about limits.

Many people don't need any math at all beyond arithmetic, and I know a lot of
people who proudly announce that they can't even do that. And to some extent
it's true - most people don't need any math at all. Why were you bothering to
take calculus? I'm sure you've never needed it.

But let me add that if all you want to do is arithmetic, why bother? Just use
a calculator. If all you want to be able to do is differentiate, why bother?
Feed it to Wolfram Alpha. If all you want to do is program, why bother? Hire
someone to do it.

But yes, if all you want to do is high-school calculus, there are easier ways
to learn the processes to jump through the hoops, pass the exam, and get the
piece of paper. For most people that's all they care about. We probably agree
on that.

------
gweinberg
The problem with "points on a number line" as a definition for real numbers is
that it's not clear how you can tell if you have all of them. You can populate
a number line as densely as you care to using just rational numbers, but
that's not all of them, you're missing out on numbers like the square root of
two. You can toss in the non-intergral powers of rational numbers, but you
still won't have all of them, you're missing out on col numbers like pi (or
tau, if you prefer). Even after you toss in every solution to every
differential equation you can name, and every number you can generate using
well defined finite or infinite serieses, there's probably some horrible
diagonaliztion proof that says you still don't have all of them.

~~~
ColinWright
If you assume that there is no number bigger than zero but smaller than every
positive number (basically the Archimedean property) then you can prove that
"you've got them all." You use Dedekind cuts.

Suppose there's a location on the line that's somehow missing - call it x. Let
A be all the numbers less than x, let B be all the numbers greater than x, and
that gives you your Dedekind cut. That Dedekind cut is, in a very real sense,
x, and that means x is a real. QED.

That needs tidying up and formalising, but it does work.

~~~
lmm
If you're using the Dedekind cut definition why use the line at all? Just say
a real is any set of rationals bounded above, with arithmetic defined the
obvious way; defining equality is slightly fiddly but it's fiddly with a
number line too. What does the line visualization gain you?

~~~
ColinWright
Because it was asked how we knew we "got them all", referring to points on the
line. The reals are a way of modelling the line, the line is a way of
visualising the reals. Each is complementary to the other.

And besides, the rationals are totally ordered, and their completion is
totally ordered, so it makes sense to think of them as arranged in a line. The
problem is that the reals are very, very strange in some ways, and people do
get seduced into thinking they understand them, whereas usually it's just a
case that they've got used to them.

------
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.

~~~
thaumasiotes
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](http://en.wikipedia.org/wiki/Thomae%27s_function)

~~~
bjornsing
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.

~~~
thaumasiotes
I don't understand the point you're trying to make?

The Cantor set shares the property that "for every two [points in the set]
there are infinitely many [points in the set] in between", but no one would
describe it as looking like a line. It's rather sparse.

~~~
bjornsing
What I take issue with is an "image" of _two_ rational numbers as two separate
dots, with empty space in-between. That's a very deceiving image IMHO, since I
cannot think of a sane way to produce it.

The Cantor set is very different. It's even easy to give an example of two
points in the set that can (sanely) be depicted with empty space in-between:
1/3 and 2/3\. If I'm not mistaken that example also disproves your stated
conjecture... ;)

~~~
thaumasiotes
First of all, let me point out that 1/3 and 2/3 are both rational numbers, so
if you can imagine them with empty space between, you've imagined two rational
numbers with empty space between.

> It's even easy to give an example of two points in the [Cantor] set that can
> (sanely) be depicted with empty space in-between: 1/3 and 2/3\. If I'm not
> mistaken that example also disproves your stated conjecture... [that between
> any two points in the set, there is a third one] ;)

Fair enough. Consider, then, the intersection of the Cantor set with the
irrational numbers (you can think of this as the "open Cantor set"). It is,
obviously, a subset of the Cantor set, and really does have the property
described.

Since I'm feeling embarrassed about that last time, a proof follows:

\-----

The Cantor set consists of all real numbers in the interval [0,1] which have a
"decimal" expansion in trinary which does not contain the digit 1. That is to
say, they can be expressed in terms of powers of (1/3) such that the
coefficient of each power of 1/3 is either 0 or 2. (1/3 would usually be
represented in trinary as 0.1, but is in the Cantor set because of its
representation as 0.02222222...)

Let _a,b_ be two irrational numbers in the Cantor set, _a_ less than _b_.
There is some decimal place at which they diverge, and since _a_ is smaller,
it has a 0 at that point, while _b_ has a 2. Since _a_ is irrational, it also
has a 0 at some _later_ point in its expansion (if every digit after that were
2, then _a_ 's expansion would be repeating and _a_ would be rational). The
number constructed by substituting a 2 for a 0 at that index is greater than
_a_ , less than _b_ , and in the Cantor set.

Graphical representation of the proof:

    
    
        a = 0.......0......
        b = 0.......2......
    

then

    
    
        a = 0.......0....0.....
        c = 0.......0....2.....
        b = 0.......2..........

------
snake_plissken
"Since (a,0)+(c,0)=(a+c,0) and (a,0)×(c,0)=(ac,0), the points along the
horizontal axis have an arithmetic just like "ordinary" numbers"

Holy hell that is clear, concise and compelling. If only my professors would
have explained it like this more often in my freshman calc class which was so
much more abstract and proof based than anything I had encountered before. The
only thing I remember form that time is hellishly long study groups late into
the night with my classmates.

------
chowells
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.

~~~
flebron
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.

~~~
gizmo686
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.

~~~
ColinWright
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, taking a mapping _k_ from the natural numbers: { 1, 2, 3, 4, ... } to
the set of computable numbers in [0,1).

Now let's construct a new number. In the first decimal place we put 1 if the
first decimal place of k(1) is 0, and 0 otherwise. In the second place we put
1 if the second decimal place of k(2) is 0, and 0 otherwise. And so on.

This results in a number that's not on the list, and is between 0 and 1. So it
must, by our assumption, not be computable.

 _Things become tricky._

So there's a choice to be made, and most mainstream mathematicians have
decided to talk about, use, study, and otherwise accept the existence of the
real numbers because it's convenient.

Feel free to choose otherwise.

~~~
gizmo686
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 constructables is no larger than the integers. We can also show
that all integers are constructable, so the amount of constructables is no
smaller than the integers). Now, take the set of all constructables, and use
the diagonal arguement to construct a new number. We can see that this number
is not constructable, however, it would appear that I have just unambigously
described it, meaning that it must be constructable.

The only potential hole I see is that the ordering of the constructables when
I apply the diagonal arguement is ambigous, but we can unambiguously order
them by the lexical ordering of their 'canocial' description, and we can
unambiguous define the canonical description as the smallest one when
translated into a base n integer.

I suspect that doing the above will run into problems with computable numbers
(as it likely involves the halting problem), however it appears to be an
unambiguous description of a real number that is not constructable. Obviously
there is some flaw in this reasoning.

~~~
jfarmer
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.

------
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.

------
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.

~~~
pedrosorio
"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."

Yes. Where I graduated, all engineering majors learn the axiomatic definition
of the real numbers including the "supremum (least upper bound) axiom" at the
beginning of the first calculus class.

------
snowwindwaves
Along a similar vein you may also enjoy
[http://arxiv.org/pdf/1303.6576](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.

~~~
anatoly
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 that if aliens were able to get into space, we can take it for
granted that they knew negative numbers (in addition to more advanced
concepts). I disagreed. Negative numbers are very convenient, but all the math
that's needed for modern physics can, I think, be built up without them in a
way that's more bulky and awkward, but not an order of magnitude bulky. This
paper is weak evidence of my position.

~~~
snowwindwaves
The only other ones I know of are on his website
[http://cliftonlabs.net/TechnicalArticles.html](http://cliftonlabs.net/TechnicalArticles.html)

------
mherrmann
This is a great article but unfortunately has one thing horribly wrong:
Democracy far preceded the Age of Enlightenment. A form of democracy was
already in place in ancient Greece at around 500 BC. Newton and the Age of
Enlightenment were much later, at 1600+ AD. See Wikipedia:
[http://en.wikipedia.org/wiki/Democracy#History](http://en.wikipedia.org/wiki/Democracy#History),
[http://en.wikipedia.org/wiki/Age_of_enlightenment](http://en.wikipedia.org/wiki/Age_of_enlightenment),
[http://en.wikipedia.org/wiki/Isaac_Newton](http://en.wikipedia.org/wiki/Isaac_Newton).

Other than that, a great article!

~~~
ddebernardy
Hehe. I balked at that too...

It also states that you cannot order the field of complex numbers. Whereas I
seem to recollect that there are ways to do so. For instance, z1 < z2 if x1 <
x2 or x1 = x2 and y1 < y2.

~~~
claudius
By your definition of <, 0 < i and i^2 < 0, however, the OP requires that 0 <
p AND 0 < q => 0 < p * q, which is not fulfilled by your < for p = i = q.

------
bglazer
Wow, vector multiplication suddenly makes sense. I had never seen it described
with polar coordinates.

It's wonderful to have this little insight now. It's unfortunate that my math
knowledge is so filled with holes.

------
totemizer
" 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...](http://njwildberger.wordpress.com/2012/12/02/difficulties-with-real-
numbers/)

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

~~~
totemizer
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.

~~~
GFK_of_xmaspast
So I have a phd in math and I do tend to think less of other mathematicians
who argue against infinite sets or uncountable sets and such, the argument's
been over for a hundred years, you lost, deal with it. It's mathematical
geocentrism.

------
igravious
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 the question, what are lines made up of? Are they
made up of anything? Is a point really where two (or more) lines would
intersect if they could intersect. Is this what is meant by a Dedekind cut?

~~~
Someone
_" 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](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)

~~~
igravious
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.

~~~
Someone
_" 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
set of Aleph-0 points can completely cover a line.

~~~
mcguire
" _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.

~~~
Someone
OOPS. Thanks

~~~
mcguire
Infinities are weird.

------
SourPatch
The contents of the linked page were the first lecture I had in my
undergraduate calculus course. At the end of the lecture, we all looked around
at each other wondering what we had just signed up for.

------
Stal3r
Can someone explain the setup of the 0=1 exercise? It's poorly worded. Is it
saying find: (Y,1,+,1,×) or is it saying find what "1" has to be to make it a
valid field?

~~~
ddebernardy
Given the field (Y, 0, +, 1, *), you need to show that either 0 != 1, or 0 = 1
is the only element in the set.

I don't remember the precise proof, but if memory serves it derives from the
existance of opposites and inverses, and 0 and 1 being unique in the set, due
the commutative properties of abelean groups.

