
How real are real numbers? (2004) - MindGods
https://arxiv.org/abs/math/0411418
======
Smaug123
I know it's completely unrelated to the article, but I would like to take a
moment to plug an absolutely beautiful proof of the uncountability of [0,1],
which I first saw in
[http://people.math.gatech.edu/~mbaker/pdf/realgame.pdf](http://people.math.gatech.edu/~mbaker/pdf/realgame.pdf)
.

Briefly: Alice and Bob play a game. Alice starts at 0, Bob starts at 1, and
they alternate taking turns (starting with Alice), each picking a number
between Alice and Bob's current numbers. (So start with A:0, B:1, then A:0.5,
B:0.75, A:0.6, … is an example of the start of a valid sequence of plays.) We
fix a subset S of [0,1] in advance, and Alice will win if at the end of all
time the sequence of numbers she has picked converges to a number in S; Bob
wins otherwise. (Alice's sequence does converge: it's increasing and bounded
above by 1.)

It's obvious that if S = [0,1] then Alice wins no matter what strategy either
of them uses: a convergent sequence drawn from [0,1] must converge to
something in [0,1].

Also, if S = (s1, s2, …) is countable then Bob has a winning strategy: at move
n, pick s_n if possible, and otherwise make any legal move. (Think for a
couple of minutes to see why this is true: if Bob couldn't pick s_n at time n,
then either Alice has already picked a number bigger, in which case she can't
ever get back down near s_n again, or Bob has already picked a number b which
is smaller, in which case she is blocked off from reaching s_n because she
can't get past b.)

So if [0,1] is countable then Alice must win no matter what either of them
does, but Bob has a winning strategy; contradiction.

~~~
avani
In a similar vein, Cantor's original diagonalization proof of uncountability
is one of the most satisfying results in real analysis. From
[https://www.cs.virginia.edu/luther/blog/posts/124.html](https://www.cs.virginia.edu/luther/blog/posts/124.html):

"Any real number can be rep­re­sented as an inte­ger fol­lowed by a dec­imal
point and an infi­nite sequence of dig­its. Let’s ignore the inte­ger part for
now and only con­sider real numbers between 0 and 1. Now we need to show that
all pair­ings of infi­nite sequences of dig­its to inte­gers of neces­sity
leaves out some infi­nite sequences of dig­its.

Let’s say our can­di­date pair­ing maps pos­i­tive inte­ger i to real number
r_i. Let’s also denote the digit in posi­tion i of a real number x as x_i.
Thus, if one of our pair­ings was (17, 0.651249324…) then r_17^4 would be 2.
Now, con­sider the spe­cial number z, where z_i is the bot­tom digit of r_i^i
+ 1.

The number z above is a real number between 0 and 1 and is not paired with any
pos­i­tive inte­ger. Since we can con­struct such a z for any pair­ing, we
know that every pair­ing has at least one number not in it. Thus, the lists
aren’t the same size, mean­ing that the list of real numbers must be big­ger
than the list of inte­gers."

~~~
jwilk
The proof on this page has a small gap: it fails to take into account that
some numbers don't have unique decimal expansion.

For example, if all digits on the diagonal were 8, then z = 0.(9) = 1, which
is not between 0 and 1.

~~~
eru
Yes. Though that problem is repairable.

------
karmakaze
> The first person to notice this difficulty was Jules Richard in 1905, and
> the manner in which he formulated the problem is now called Richard’s
> paradox. Here is how it goes.

> Since all possible texts in French (Richard was French) can be listed or
> enumerated, a first text, a second one, etc., 2 you can diagonalize over all
> the reals that can be defined or named in French and produce a real number
> that cannot be defined and is therefore unnameable. However, we’ve just
> indicated how to define it or name it! In other words, Richard’s paradoxical
> real differs from every real that is definable in French, but nevertheless
> can itself be defined in French by specifying in detail how to apply
> Cantor’s diagonal method to the list of all possible mathematical
> definitions for individual real numbers in French!

I followed this up to "and produce a real number that cannot be defined".
Would this be from some computation on some/all of the diagonalized reals? I
can't see how to guarantee that this generated number wasn't already in the
set.

~~~
missblit
> I can't see how to guarantee that this generated number wasn't already in
> the set.

The trick here is the diagonal argument [1].

The set (S) contains all real numbers that can be described by a valid French
sentence (enumerating French sentences in some fixed order). For example we
could come up with an enumeration such that the first few elements are:

S_0 = .7139847654

S_1 = .111111111111

S_2 = .93939

S_3 = .313331333133 repeating

...

We can construct a diagonal number r such that the ith digit of r differs from
the ith digit of S_i for all S_i in S:

r = 0.8204... (8 = 1 + S_0[0], 2 = 1 + S_1[1], 0 = S_2[2] + 1, etc)

For any element S_i in S; r and S_i differ in at least one digit (because we
constructed it that way); which is why r is guaranteed to not be in S.

(But this argument can be translated to French, hence the paradox!)

[1]
[https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

~~~
karmakaze
Ah got it. This seems different than the idea of there being 'more' real
numbers and is rather a quirk of an unclear specification of set membership.
This seems to be covered in 'Richardian Numbers'[0]

[https://en.wikipedia.org/wiki/Richard%27s_paradox](https://en.wikipedia.org/wiki/Richard%27s_paradox)

------
karmakaze
> We discuss mathematical and physical arguments against continuity and in
> favor of discreteness

I can appreciate the physical arguments which makes me think of this as a
question in physics. What might the mathematical ones be? Math is about
defining a system and playing it out, what would make continuous numbers off-
limits?

~~~
ojnabieoot
The article goes into significant detail, but the problem is that the
_computable_ real numbers are a tiny slice of all real numbers, and in general
even in principle mathematicians can only deal with a countable subset of the
reals. So (metamathematically) if almost every single real can't actually be
used in mathematics, it's hard to say the concept has much mathematical
validity. [Which makes its mathematical _usefulness_ a philosophical pickle.]

> Math is about defining a system and playing it out

A lot of mathematicians would consider this a very limited view of the
subject.

~~~
asdf_snar
> A lot of mathematicians would consider this a very limited view of the
> subject.

Could you provide an example of such a mathematician?

~~~
shezi
"One of the main themes of the book is the beauty that mathematics possesses,
which Hardy compares to painting and poetry.[5] For Hardy, the most beautiful
mathematics was that which had no practical applications in the outside world
(pure mathematics) and, in particular, his own special field of number
theory." \--
[https://en.m.wikipedia.org/wiki/A_Mathematician's_Apology](https://en.m.wikipedia.org/wiki/A_Mathematician's_Apology)

G. H. Hardy, A mathematician's apology. I think that would be a good example
here, since the focus is on finding beauty and fulfilment, instead of simply
visiting every niche.

------
montecarl
If it is not possible to compute most reals, then is there a number system,
with only computable numbers that extends the rational numbers to include
computable reals such as Pi, e, sqrt(2)?

~~~
andromaton
Yes, computable numbers (0)
0:[https://en.m.wikipedia.org/wiki/Computable_number](https://en.m.wikipedia.org/wiki/Computable_number)

------
dmch-1
Real numbers are a convenient abstraction which help us think about certain
things. They don't have to be any more real than that. (Have not read the
article though, sorry if this comment is irrelevant).

~~~
Koshkin
But an abstraction is always an extension of reality.

~~~
mike00632
The concept of "smooth" is fairly rooted in reality, right? This is one of the
concepts that a continuous set of numbers like the Real Numbers can
accommodate:
[https://en.wikipedia.org/wiki/Smoothness](https://en.wikipedia.org/wiki/Smoothness)

If it turns out that reality is actually jagged at the quantum scale that
doesn't mean that our conception of "smooth" is void.

~~~
Koshkin
Exactly so. When reality becomes too complex, we abstract some things away to
get a simpler model. (This we "forget" about the finite and get infinity;
ignore the jaggedness and get smoothness).

------
patfla
I read this intriguing thing once to the effect that: if space and time were
discrete, say, at the Planck scale, that the infinities that bedevil modern
physics would, magically, disappear. No more renormalization - no more many
things.

~~~
mike00632
We would still be able to think about a half a Planck length.

------
rurban
So the answer is unexpected: real enough.

------
gthtjtkt
Sounds real enough for me.

------
mike00632
This is pretty far into crank territory.

One of the issues with this paper is simply a misunderstanding of terms. The
word "real" as in "real numbers" is jargon. It's not about whether or not you
believe in them or even whether they are representative of reality. You can
think of them entirely in logical terms (whether logically constructed with
Cauchy sequences or Dedekind cuts) and then the disagreement is about the
axioms of set theory.

Numbers are just a theoretical framework, that is how they are used and
"believed in". If you want to say that real numbers are absurd or not fitting
of our world then simply propose an alternative theory that we could elect to
use instead of real numbers. Nobody is adamant about the idea of all real
numbers being "real" in every sense of the word.

Furthermore, a theory of numbers with a restriction on the size of the set is
arbitrary. Cantor's diagonalization argument and construction methods still
exist. To exclude uncountably infinite sizes of sets would make as much
logical sense as restricting the total number of numbers to something finite,
e.g. "there can be no numbers greater than a hundred billion, try not to think
of a hundred billion and one because that is bullocks!".

An interesting example of irksome numbers being remedied with better theories
is the concept of infinitesimals in calculus/analysis. The theory of
Nonstandard Analysis provides a rigorous definition of infinitesimals in terms
of more familiar numbers.
[https://en.wikipedia.org/wiki/Nonstandard_analysis](https://en.wikipedia.org/wiki/Nonstandard_analysis)

~~~
wrycoder
Gregory Chaitin

[https://en.m.wikipedia.org/wiki/Gregory_Chaitin](https://en.m.wikipedia.org/wiki/Gregory_Chaitin)

~~~
mike00632
Yes, this is the author of the paper. He is neither a number theorist nor a
physicist. So any claims he makes about upending physics and number theory
should be taken with a grain of salt.

