
How real are real numbers? (2004) - caustic
https://arxiv.org/abs/math/0411418
======
tel
For those interested in constructive and intuitionistic approaches here
Dummett's [0] Elements of Intuitionism is an extremely good read.

Intuitionism is a form of a constructive foundation for mathematics which (a)
notes that any attempt to deny the uncountability of reals leads to
difficulties and (b) any attempt to internally define them violates
constructivity.

The resolution proposed is to posit the existence of "free choice sequences".
These are essentially "unpredictable" sequences of potentially infinite binary
choices. As there is no a priori reason to believe that they can be predicted
they are able to be much larger than what is computable and thus can be used
to give a characterization of the reals. Atop this you build a constructive
understanding of free choice reals which behaves very nicely (at least
foundationally... it points out all kinds of weirdnesses about what we assume
classically to be the structure of reals).

What's very nice about this solution is that it sidesteps the difficulty. Free
choice is a weaker thing to ask for than finite/constructive reals, but
finite/constructive reals could be transparently encoded into free choice
sequences and all the math would just work.

[0] [https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-
Gu...](https://www.amazon.com/Elements-Intuitionism-Oxford-Logic-
Guides/dp/0198505248)

~~~
pron
It sidesteps the difficulty by making the math itself (at least for the time
being) much more difficult, and that is the reason it was rejected by most
mathematicians in the Hilbert/Brouwer debates. Because here's the question:
suppose you can't philosophically justify the "existence" of the real numbers,
yet they coincide perfectly with observation and result in math that is much
simpler than constructive math. Should you reject them? Brouwer -- who was the
first to recognize just how much ordinary math relies on non-constructive
principles -- said yes, because non-constructive math is philosophically
wrong, period. Hilbert -- who was a finitist -- instead suggested that the
propositions of math come in two flavors: _real_ propositions, those that are
finitary and can be taken to say something about physical reality, and _ideal_
propositions, that are not. He said that as long as the ideal propositions are
consistent with the ideal ones, they should not be rejected on a priori
philosophical grounds even if no finitary meaning can be assigned to them.
I.e. they are philosophically justified after the fact by virtue of their
consistency with the real propositions. This philosophical classification of
mathematics into "real" and "ideal" is called formalism, because it does not
require that the ideal propositions be assigned a finitary meaning beyond
their _formal_ statement (as a finite string of characters).

Of course, most mathematicians are not finitist, so they require neither
intuitionism nor formalism -- both essentially finitist philosophies -- and
are Platonists, believing that even ideal objects that are beyond physical
reality and computation have a "real existence" in some Platonic sense.

BTW, I think that after Turing (who used Brouwer's choice sequences in his
construction of computable numbers) it is no longer necessary to rely on "free
choice" (or _lawless_ ) sequences because of the halting theorem, and both
lawlike and lawless sequences can be unified, and Brouwer's "creating subject"
identified with a Turing machine. But I'm not sure about that. Turing -- who
was a mathematical philosopher himself -- rejected any dogmatic a priory
philosophy of mathematics, except for common sense, as the one true
foundation, and suggested that the value of a formal system be derived not
from its a priori philolosphy but from its ad hoc utility.

~~~
threepipeproblm
Wonderful to know that Hilbert was a finitist, and that
[https://en.wikipedia.org/wiki/Finitism](https://en.wikipedia.org/wiki/Finitism)
is an official camp.

Without knowing too much about about the subject, I've vaguely wondered about
this idea for a long time, now, but I figured it likely an un-respectable
position. I think it's too bad that beginners are often shielded from
controversies in foundations.

Within the past few years I ran across an alternate approach to calculus,
which if I recall correctly, achieves the same basic results, but without the
same notion of infinitely small slices and so on... now I can't find it to
link.

~~~
pron
Yep, and as you can see in that Wikipedia article, both Brouwer and Hilbert
were finitists. It is not that it is an unrespectable opinion as much as
mathemticians these days -- from what little I know -- are not generally
expected to hold _any_ dogmatic opinion on philosophical foundations, but
maybe that attitude will shift again.

> I think it's too bad that beginners are often shielded from controversies in
> foundations.

Mathematicians generally shouldn't worry about foundations, as it is the
intention of foundations to stay hidden -- except in cases where the
foundation requires a rewrite of much of math, as in the case of Brouwer's
intuitionism, but constructive math is pretty advanced anyway. Foundations are
usually the concern of logicians, but from what little I've seen in online
discussions, it seems like many logicians aren't interested in philosophy,
either, and that's a real shame.

> I ran across an alternate approach to calculus, which if I recall correctly,
> achieves the same basic results, but without the same notion of infinitely
> small slices and so on

There's constructive analysis[1], which recreates analysis within the
framework of constructive math (by finite means etc.), and there's also non-
standard calculus[2] (which I know nothing about) that makes treat
infinitesimals as actual numbers, which may be the opposite of what you meant,
but maybe not -- I saw something about there being constructive versions of
non-standard analysis, too.

[1]:
[https://en.wikipedia.org/wiki/Constructive_analysis](https://en.wikipedia.org/wiki/Constructive_analysis)

[2]: [https://en.wikipedia.org/wiki/Non-
standard_calculus](https://en.wikipedia.org/wiki/Non-standard_calculus)

------
whatshisface
_" So, in Borel’s view, most reals, with probability one, are mathematical
fantasies, because there is no way to specify them uniquely." (Paraphrasing,
because there are only countably many possible math papers that might describe
a number.)_

I think Borel has confused names with things. The fact that we can only write
down only countably many _expressions_ for numbers doesn't mean that there are
numbers that we may never write expressions for - only that in a single
symbolic system we can't have expressions for all of them _at once_.

Besides, if you confuse extant with useful you might end up believing that
some random large integers aren't "there!"

~~~
enugu
I think you (and some other commenters) are responding to something different
- potential vs actual infinity - not being able to write down all natural
numbers, but we can potentially write any number(with some assumptions like an
infinite universe).

Whereas the point being made is different and needs some math background which
is the work of Cantor. Countable can include all natural numbers that you
count until infinity, and the reals are uncountable in that they exceed even
this.

The proof of this fact(Reals are uncountable), has the same idea involved in
Turing machines halting problem and also Godel's theorem. The general version
is the power set of a set(set of subsets of a set) cant be put in one to one
correspondence with original set. If you assume such a correspondence, then
(<guess the clever trick>) leads to a contradiction.

Reals are sequences of naturals which include sequences of 1's and 0's which
can be interpreted as subsets of naturals(a sequence represents the subset of
indexes which get assigned the value 1). So reals are bigger than the power
set of naturals.

Also, the infinite countable union of countable sets is countable. The number
of grid points(integer coordinates) on a plane is the same as the number of
grid points on a line. You can set up a zig-zag correspondence. A salesman can
visit all grid points on a plane in infinte time. This is used to prove
rationals are countable. So even countably infinitely different symbolic
systems doesnt help.

~~~
btilly
The proof of the reals being uncountable depends on the idea that one can
build a number that depends on being able to make an infinite number of
choices, each of which depends on the absolute truth or falseness of a
statement.

But what happens if we open it up to have statements be true, false, or
currently unknown? That is we develop a system of mathematics that could be in
principle done inside of a Turing machine?

Then Cantor's diagonalization argument falls apart because of all of those
"currently unknown" options. See
[https://news.ycombinator.com/item?id=13843725](https://news.ycombinator.com/item?id=13843725)
for a previous explanation that I gave of this.

~~~
mbid
Mathematician here. If you mean to say that you cannot constructively prove
"the real numbers are not countable", then you're wrong. As a rule of thumb,
you can usually prove negative statements constructively as you would prove
them classically.

A constructivist would probably state the result more positive (and stronger,
constructively): To every countable set M of real numbers, there is a real
number not contained in M.

~~~
hzhou321
I am not a mathematician (physicist). I think the concept of infinity is a con
that mathematicians have pulled on us (as there isn't an easy reality to map
on to). I can understand _arbitrarily big_ set; however, I never managed to
make the jump from arbitrarily finite to infinity. Mathematicians made that
jump and glossed over, then continue to show the difference between countable
infinity and infinity beyond. Since I never could make that jump, it all
sounds nonsense to me.

I have a similar (and I believe to be equivalent) problems with infinitesimal
as well. How does arbitrarily small but non-zero become infinitesimal? Since I
have problem with infinitesimal, I find the differentiation of real numbers
and rational numbers equally non-sensical.

So mathematician, take a pause, could you explain what is _countable_
infinity? Since you never can finish counting (all the natural numbers), how
does it make the set countable? What do we mean exactly by countable here?

Wikipedia refers to the idea of one-to-one correspondence. But since you can
never exhaust the correspondence, what do we mean by one-to-one? Give me any
unique real, I'll give you a unique natural number, and we can go on forever,
so how does that not count as one-to-one correspondence?

Unlike mathematicians, physicists are fine with _unresolved_ :)

PS: I guess countable can be defined as there is a definite way of ordering
the set, which is true for natural numbers but questionable for real numbers.
I still don't see how the ordering connects to the size of infinity and one to
one correspondence. Even for the set of real numbers, I can have an algorithm
continuously generate random numbers (discarding re-occurring ones so it will
be a unique sequence) and prove there is an order of the set (non-exhaustively
defined, same as the set of natural number). The ordering may not be
describable though. But non-describable ordering is still an ordering, right?
Just as a real number that cannot be exhaustively described is still a number.
I don't have to describe it, I can hand-wave it just as the way mathematicians
hand-waved the infinity.

~~~
jonreem
A set being "countably infinite" only means that you can write a function that
maps each distinct entry in the set to exactly one natural number (0, 1, 2,
etc.) without duplicates. That's it.

So for example, the set of natural numbers is countably infinite and we know
this because we can write a function that maps each natural number to exactly
one natural number: the id function.

We can extend this and say that the set of even natural numbers is countably
infinite because it has a mapping function of x => x / 2.

The same is true for all integers (natural numbers + negative numbers): x =>
if (x < 0) { x * -1 * 2 } else if (x == 0) { 0 } else { x * 2 + 1 }, i.e. if
it's negative map it to an even number and if it is positive map it to an odd
number, if it's zero map it to itself.

You can even write a function that maps all rational numbers to the natural
numbers, since each rational number can be written as a fraction of two
integers. (Figuring out the function is a fun exercise but it is also easy to
google)

However, you can't write a function that maps any real number to a natural
number. The easiest to understand proof of this is Cantor's Diagonal
Argument[0], which is a proof by contradiction that shows that any attempted
function must exclude some real numbers. Therefore, the real numbers are not
countably infinite, and we call them uncountably infinite.

EDIT: In response to your edit, Cantor's Diagonal Argument basically shows
that for any given function (and you have to define the function completely
ahead of time - that's key) I can give you a real number that is not included
in the domain of your function.

[0]:
[https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

~~~
bjourne
Here is how you do it. Have a function p(r) which evaluates to the previous
real number. Then your mapping function is:

    
    
        f(r) = if (r == 0) { 0 } { else f(p(r)) + 1 }
    

If your objection is "You can't determine what the previous real number is."
Then my counter-objection is "Please prove that you can't." Which I don't
think is possible without first assuming reals are uncountable.

~~~
jamelk
Your definition of f is circular: to calculate f(r) we need to know p(r),
which in turn depends on f(r).

Cantor's diagonal argument shows that any mapping from the natural numbers to
the real numbers must necessarily miss some real numbers out. It takes some
time to get your head around if you aren't used to mathematical proofs, but
it's definitely worth looking it up and trying to work through it if you're
interested in this subject.

~~~
bjourne
p(r) is given a real number and retrieves its predecessor. Why is it circular?

~~~
abcdefgxdf
Suppose there exists an r' such that for some real number r, p(r)=r', where
p(r) gives the real number that precedes r. What does that mean? Does it mean
there are no numbers between r and r'? Because that's what I think predecessor
means, even though it's trivial to prove that there are numbers between r' and
r. For example, (r'+r)/2, the average of r and r', is between the two numbers.
And there are also numbers between r and (r'+r)/2, and between r' and
(r'+r)/2\. So there can't possibly be a real that is the predecessor to
another real, because there are always more real numbers between any two
reals.

------
CogitoCogito
The continuity of real numbers provides a clean theoretical basis for
continuity of functions. I personally view it as more of a theoretical tool
that seems to do pretty well and instead steer clear of the philosophical
questions.

One thing that I think is important to note though is that the jump from real
numbers to complex numbers is nothing compared to the jump from
integers/rational numbers/etc. to real numbers. The complex numbers come about
by simply adding one dimension whereas the real numbers come about from an
abstract "completion" of (say) the rational numbers in a very specific
mathematical sense.

My point is that deriding "imaginary numbers" (as many do) is total nonsense
if you accept real numbers.

~~~
j2kun
> The complex numbers come about by simply adding one dimension

Complex numbers are also best thought of (in my opinion) as an abstract
completion of the reals under the operation of taking roots of polynomials.

Because otherwise, what would be the difference between the real plane and the
complex numbers? They are topologically identical, after all. There has to be
something more substantial than simply adding a dimension.

~~~
nilkn
If by real plane you just mean the standard vector space R^2, the answer is
algebraic rather than topological. R^2 as a vector space has an addition
operation, but no multiplication operation. The complex plane is obtained by
simply picking the appropriate multiplication operator.

In general, completing the rationals into the reals is more complex than
constructing the complex plane from the real numbers. For the latter, you just
need to adjoin a single element (sqrt(-1)), enforce existing arithmetic rules,
and the rest falls into place. For the former, you can't just adjoin a single
new element like sqrt(2). Doing so will get you the ring (actually field)
Q[sqrt(2)], but not R.

If you take R and adjoin _two_ special new elements (sqrt(-1) and the point at
infinity), you do obtain a topologically different result: the Riemann sphere.
This sphere is in many ways the more natural domain for complex analysis than
the complex plane.

------
Sharlin
The author of this paper is Gregory Chaitin of Chaitin's constant fame, among
other things (I didn't know that Kolmogorov complexity is also known as
Chaitin-Kolmogorov complexity!)

~~~
surement
Also the author of _Meta Math! The Quest for Omega_ [0], where this topic is
discussed at length.

[0] [https://arxiv.org/abs/math/0404335](https://arxiv.org/abs/math/0404335)

------
scythe
>In addition to this mathematical soul-searching regarding real numbers, some
physicists are beginning to suspect that the physical universe is actually
discrete [Smolin, 2000] and perhaps even a giant computer [Fredkin, 2004,
Wolfram, 2002]. It will be interesting to see how far this so-called “digital
philosophy,” “digital physics” viewpoint can be taken.

Here is how far: Everything written _in words_ about the physical universe is,
by necessity, discrete. Thus all information that can be encoded in human
languages is discrete. Any non-discrete behavior of the physical universe
which causes a change in the discrete information available to us, must, by
assumption, have a component which is orthogonal to all of the prior discrete
information (otherwise it is fully discrete). Since this component is
independent of all previously available information, it looks like randomness.

In other words: from the viewpoint of a discrete (linguistic) observer, the
behavior of a continuous universe looks identical to that of a discrete
universe that contains random fluctuations.

What is interesting, then, is that _observationally_ , our discrete observable
universe is full of random fluctuations. Speculation as to their true
continuous underpinnings is, however, unfalsifiable, unless the randomness
itself can be made to disappear. I usually turn the question around: is it
inconceivable that there would be a continuous universe with inhabitants that
used a discrete language?

So:

>According to these ideas the amount of information in any physical system is
bounded

"the amount of _observable_ information in any physical system" \-- any
unobservable continuous information shows up as unpredictable changes in the
observable information.

~~~
mikhailfranco
One of the few comments here that shows great insight. I think you should push
the argument down a few layers, to talk about the discreteness of quantum
observables, the underlying unobservable continuous wavefunction, and the
randomness of the Born rule that maps between the two (Copenhagen collapse or
forking of Many Worlds).

Tegmark takes a similar, but rather extreme, MW approach in his book 'Our
Mathematical Universe'.

Personally, I struggle with Tegmark's use of Measure Theory, and proponents of
various Anthropic Principles, because they seem to have a completely broken
frequentist view of probability and inference.

------
mikhailfranco
For the parallel historical development of 'the continuum' in physics, I
recommend this readable survey:

Paper:

[https://arxiv.org/abs/1609.01421](https://arxiv.org/abs/1609.01421)

[https://math.ucr.edu/home/baez/continuum.pdf](https://math.ucr.edu/home/baez/continuum.pdf)

Blog summary with discussion:

[https://johncarlosbaez.wordpress.com/2016/09/08/struggles-
wi...](https://johncarlosbaez.wordpress.com/2016/09/08/struggles-with-the-
continuum-part-1/)

[https://johncarlosbaez.wordpress.com/2016/09/09/struggles-
wi...](https://johncarlosbaez.wordpress.com/2016/09/09/struggles-with-the-
continuum-part-2/)

------
gertef
Lawrence Spector (professor at CUNY, Manhattan) on this topic:
[http://www.themathpage.com/acalc/anumber.htm](http://www.themathpage.com/acalc/anumber.htm)

------
chairmanwow
> "the halting probability Ω, which is irreducibly complex (algorithmically
> random), maximally unknowable, and dramatically illustrates the limits of
> reason"

I really enjoyed the beauty of this statement.

~~~
prmph
My philosophical take: the halting problem illustratres how free will can
exist in a deterministic universe

~~~
Retra
Free will is just what it feels like to have a mind that can construct models
of realities that are not fact. And you can model nondeterminism in a
deterministic system just fine.

So I would argue that it illustrates nothing of significance under either of
those issues.

~~~
prmph
How do you model true non-determinism in a deterministic system? And if that
can indeed be done, would that not support my original point?

------
FabHK
I like the idea of encoding answers to all questions, or for that matter all
books written so far (or both, while we're at it), in one real number between
0 and 1. My favourite number, really.

~~~
gertef
The encoding of all books written so far (and will ever be written in finite
time), is a rational number. Don't need Reals

~~~
kazagistar
Thats kinda the whole point of the article, in fact. All possible encodings of
all possible thoughts, books, formal systems, and whatever, fit into the
rationals, and the reals are categorically outside that.

~~~
FabHK
All books written and will be written (in finite time) - yes, rational.

All possible questions (infinitely many) - no, that would be a non-terminating
non-periodic binary, right. From the article:

> 2.4 Borel’s know-it-all number

> The idea of being able to list or enumerate all possible texts in a language
> is an extremely powerful one, and it was exploited by Borel in 1927 [Tasi
> ́c, 2001, Borel, 1950] in order to define a real number that can answer
> every possible yes/no question!

> You simply write this real in binary, and use the nth bit of its binary
> expansion to answer the nth question in French.

> Borel speaks about this real number ironically. He insinuates that it’s
> illegitimate, unnatural, artificial, and that it’s an “unreal” real number,
> one that there is no reason to believe in.

------
gerdesj
"According to Pythagoras everything is number, and God is a mathematician.
This point of view has worked pretty well throughout the development of modern
science. However now a neo-Pythagorian doctrine is emerging, according to
which everything is 0/1 bits, ... , God is a computer programmer, not a
mathematician, and the world is a ... a giant computer" [p13 of the pdf]

If you only have one finger then zero and one _are_ just as real (ahem) as
numbers that arise naturally when you have 10 fingers. The 10 toes are a
bonus. I doubt that we can really know what Pythagoras really thought but
given some of the results attributed to him I think Chaitin does him a
disservice.

Getting wound up over whether French is a sophisticated enough language to
describe numbers and some of the odder consequences of allowing construction
to equate existence will probably only lead to a headache.

As a civilian wandering on the outskirts of all this philosophical foot
stamping, I believe there are a fair few pretty rigorous arguments out there
that can't be denied by resorting to "it looks wrong, cos reasons" style
illustrations in a 13 page pdf.

------
gerdesj
Richard's Paradox seems a bit shaky to me (p4): "Since all possible texts in
French can be listed or enumerated"

Unless I have completely missed the point then he has simply stated a way to
generate another member of the set of French texts which of course is part of
that set and so on.

You can easily squint hard enough to generalize to all texts in all languages,
now, earlier and possible then allow that grammar, spelling and so can be
pretty slack. Now translate that lot into numbers in some way (a bunch of IT
bods should be able to manage that!) To be honest French on it's own is
probably more than enough.

"How very embarrassing! Here is a real number that is simultaneously nameable
yet at the same time it cannot be named using any text in French."

The very act of naming the number (in French) constructs the French text that
adds to the set of possible French texts.

I think that the set of possible French texts is exactly as large as the set
of reals. So is the set of all language texts and that the "paradox" is merely
trying to use the Cantor argument backwards.

~~~
joe_the_user
It depends how you conceptualize things.

If one takes natural language as a means to definite sets, you quickly get a
plethora of paradoxes, for example Russel's paradox("Takes the sets of all
sets that don't contain themselves. Does that contain itself?"). So if one
takes "natural language" as one's system of defining set, one has to assume
it's inconsistent and any statement is provably true and false. Thus
"Richard's Paradox" is in the same boat as all statements in our "system of
natural language".

That said, I think the proof in the text falls apart in another way - it
neglects the distinction made in Skolem's Paradox. The Löwenheim–Skolem [2]
theorem show that any system definable with a finite alphabet has a countable
model. This is only an _apparent_ paradox because this model is only countable
when "viewed" from outside the model, _within_ the model, it is possible to
have ostensibly uncountable sets. So ones could certainly haves a countable
model of the real numbers while the real numbers themselves remained
uncountable as "countable" and "uncountable" were defined in the countable
model.

[1]
[https://en.wikipedia.org/wiki/Skolem%27s_paradox](https://en.wikipedia.org/wiki/Skolem%27s_paradox)
[2] The
[https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_...](https://en.wikipedia.org/wiki/L%C3%B6wenheim%E2%80%93Skolem_theorem)

~~~
gerdesj
"If one takes natural language as a means to definite sets, you quickly get a
plethora of paradoxes"

Which is why I think if you resort to natural language to refute the
consequences of a more rigorous treatment of the concept of number then there
will be trouble. The whole paper attempts to refute things like Cantor through
a weird recourse to French.

However I think it is possible to reduce real numbers as being a sort of
subset of French purely through the same construct that Mr Cantor describes
because that's the way descriptions work. If you define a real in some way in
some form of symbolic language - I recall from GED that SSS might embody
"three" and so does "trois". So I don't see why French can't encompass reals
SSS can be considered exactly equivalent to trois.

I suspect I need to know and understand the formal, rigorous definition of
"real" before I really give it some.

------
jfaucett
This was a light and interesting read.

Here is the direct link:
[https://arxiv.org/pdf/math/0411418.pdf](https://arxiv.org/pdf/math/0411418.pdf)

"Indeed, the most important thing in understanding a complex system is to
understand how it processes information. This viewpoint regards physical
systems as information processors, as performing computations. This approach
also sheds new light on microscopic quantum systems, as is demonstrated in the
highly developed field of quantum information and quantum computation. An
extreme version of this doctrine would attempt to build the world entirely out
of discrete digital information, out of 0 and 1 bits."

It would indeed be quite a blast to discover we are to a high degree of
probability in a simulation.

~~~
klodolph
I don't see the connection between figuring out that the universe is discrete
and learning that the universe is a simulation. Not even in our universe are
all simulations done using digital computers, some are done using analog
computers.

~~~
Smaug123
But in our universe, if it's discrete on a macroscopic level, then it's either
inherently a natural number (it can be counted), or it's intelligently
created.

~~~
maverick_iceman
Our best theories, General Relativity and the Standard Model, say that the
world is a continuum.

~~~
klodolph
I'm a little uncomfortable with the language that the theories "say that the
world is" X. General Relativity and the Standard Model both model the world
using real numbers, but they're both known to be wrong, and the fact that they
are continuous is not a great reason to claim that the universe is continuous.

On the other hand, observations about Lorentz symmetry holding at distances on
the order of the Planck scale put a wrench in a bunch of discrete spacetime
theories. I don't really understand the math, though.

All this is somewhat tangential to the issue of real numbers. Real numbers are
not necessary for continuity.

~~~
maverick_iceman
_> Real numbers are not necessary for continuity._

You know of any continuum that doesn't include the real numbers? That will
contradict the continuum hypothesis.

~~~
klodolph
Let's avoid equivocating here: "the continuum" is sometimes used to refer to
the real numbers, but "continuity" in this context is a property of functions
between metric spaces (or possibly topological spaces). "The continuum
hypothesis" and "continuous functions" are actually from completely different
branches of mathematics.

This happens fairly often in mathematics, where similar-sounding terms are
used to describe completely different concepts, or the same term sometimes
means different things in context, or sometimes an Adjective Noun is neither
described by Adjective nor by Noun.

------
D_Alex
Infinity is a weird thing, isn't it?

Now: one of the proofs in the paper relied on an assumption that all possible
computer programs are countable, which I think implies that they are finite in
length. But it is fairly trivial to generate computer programs that are
infinitely long, say by assigning characters or expressions in some language
to the digits of transcendental numbers such as pi. It is also possible to
generate infinitely many such programs, simply by using pi/2, pi/3... etc.

Now, the proof as presented fails, since these programs cannot be ordered by
size.

Can the proof be modified to take account of this? I don't know... comments
invited.

~~~
bo1024
I'm not sure it's so easy to define or generate an infinitely-long program.
Such a thing doesn't sound to me like it would be either possible in practice
or equivalent to a Turing Machine in theory.

For example, you suggest an assignment of expressions to digits of pi. Now how
would you run such a program? Presumably by generating the digits of pi,
interpreting them as expressions, and evaluating the expressions, etc.

But the program you used to do that was finite. So are you running the
infinite program? I think it's more fair to say you are running the finite
one.

~~~
D_Alex
Yes - I was just thinking the same thing. I have not come to a conclusion one
way or another on whether it is possible to de-couple the generating program
from the generated programs for the purpose of analysing the proof. It seems
that there should be a way to do it... unfortunately I cannot spend more time
on this now.

~~~
jawarner
Does there have to be a generating program? The set of all finite programs is
countable, so it could not possibly describe the uncountable set of real
numbers - this would be a surjection from a countable set to an uncountable
set. In particular only the subset of computable numbers [1] can be described
by the countable set of finite computer programs.

Also, any infinite program either passes through a finite number of
instructions or never terminates. So any program which does not have a finite
representation will never terminate.

[1]
[https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf](https://www.cs.virginia.edu/~robins/Turing_Paper_1936.pdf)

~~~
drdeca
What if the function takes input, and for each input, the program halts, but
for every instruction, there is an input such that the program when run on
that input will execute that instruction? Would that imply a contradiction? It
doesn't seem like it to me. Yeah, no, that should be fine.

Example: consider a language where 0 means "if the input register is at most
0, halt with the answer "no", otherwise, decrement the input register and
continue to the next instruction" and 1 means the same thing except that it
gives the answer "yes" instead.

Then the fractional part of any real, expressed in binary, would be a program
that takes an integer as input, and answers whether that digit of the number
is 1. This seems like it would count as a program to me.

So, this would allow there to be uncountably many "programs", only, almost all
of them would be impossible for us to refer to specifically.

Hey, this way you could define a uniform measure over programs. Hah.

Wait, is that how Solomonoff induction stuff works? (Of course, with a richer
language than what I described)

------
throwaway5752
Doesn't this basically rehash stuff covered 100 years ago by Hilbert,
Whitehead & Russell, and Godel? If it wasn't such an eminent author, I would
give it a pretty solid eye-roll. As some other poster noted in a link they
provided, "pi" and "e" \- among the uncountably infinite transendentals - are
probably reasonable responses to this article in it's entirety.

And again, even the point about describing the world in 1s and 0s at the end
seems to me to be repetitive of Whitehead & Russell's Principia Mathematica
which (and please correct me) used logic to construct the integers?

~~~
surement
> "pi" and "e" \- among the uncountably infinite transendentals - are probably
> reasonable responses to this article in it's entirety.

The point of the article is that most reals are useless in practice; pi and e
might be transcendental and the transcendental numbers are uncountable, but
they are both computable, and the set of computable numbers is countable.

~~~
throwaway5752
Is pi computable? I would have thought it would be a good halting problem
example. I am admittedly not strong in modern developments in computability.
Was aware of Chaitin and some of his work prior to this, but that's about the
limit.

If his point is that the universe is finite and finite methods are a more
correct basis for physical sciences, then I'm open to that even if I'm not
particulary interested (theoretical math objects are perfectly interesting in
their own right to me). But if there's more to it than that, I'd appreciate
the help.

~~~
hawkice
A computable number one where there is a finite length representation of the
number, namely, there is the program that, given a number of digits, can
output the number it describes accurate to that many digits. There are a
tremendous number of ways to calculate pi and e.

~~~
throwaway5752
I've taken a minor personal interest in some of Ramanujan's more quickly
converging sequences for pi. Giving myself a crash course in computable
analysis/computable reals. Wish there was more out there about them
(particularly limitations vis a vis traditionally defined reals).

------
option
Note that the probability of randomly picking a rational number from [0,1] is
_exactly_ 0\. That is, with P=1 you will end up with an irrational number, not
representable in any modern computer.

~~~
textfile
This is a mind-blowing. So rational numbers are an invention of humans. That
is to say, natural numbers simply don't exist unless and until they are
explicitly defined. Am I understanding correctly?

~~~
hn_throwaway_99
Someone else responded to me in a recent Hacker News discussion that really
clarified this in my head: A real number essentially has an infinite number of
digits after the decimal place - the difference between a rational and an
irrational number is that the digits end up in a repeating pattern in a
rational number (you can think of rationals that terminate as really having an
infinite number of zeros, e.g. 1.5 as 1.5000000...). Thus, to randomly pick a
real number, you randomly select digits an infinite number of times after the
decimal. Clearly there is no way a random process will produce an infinite
number of repeating digits, thus the probability of picking a rational would
be 0.

~~~
prmph
Wouldn't it be simpler to explain it this way: there are infinitely many reals
between any two different reals, and thus the theoretical probability of
picking any number between them at random is 1/infinity, which we think if as
0.

But in that case, why is is more probable to pick an irrational number?

~~~
hn_throwaway_99
I don't think it's simpler that way at all, mainly because the ways in which
some "infinities" are larger than others, which explains why it's more
probable to pick an irrational number. The rational numbers are countable,
while the real numbers are not.

------
lngnmn
Numbers? Real?

Neither molecular biology nor the sane part of physics has any of em.

Btw, it is heuristic - if there are numbers involved then it is human made.
Reality as it is has no such notion. Biology does not count.

Numbers require an observer, which is a by-product of the processes in vastly
complex brain structures of the cortex, and cannot be the basis of anything in
the underlying universe.

Any good (which means Eastern) philosophy arrived at these simple conclusions
millennia ago.

------
SeanLuke
Richard's Paradox for some reason reminded me of the proof that there is an
infinite number of interesting whole numbers. The proof goes like this. Assume
instead that the number is in fact finite. Consider the first number higher
than any of the interesting numbers, and thus bounding them. Now that's an
interesting number! QED.

------
threepipeproblm
Just wanted to thank the OP, caustic, for this. I first thought it might be
past my available background/resources for a casual read. But I found it
accessible and rewarding.

------
plg
I love LaTeX in general, but man alive, the default choice of font
style/size/face for headings & sub-headings looks god-awful

------
prmph
So is there full agreement on what a number is, in the first place?

Some would argue that PI is not an actual number; but that it is a concept,
like infinity

~~~
marplebot
A number is also a concept, and there are number systems that include
infinities (e.g.
[https://en.wikipedia.org/wiki/Surreal_number](https://en.wikipedia.org/wiki/Surreal_number)).
It all depends on your definitions/axioms. So in a sense there is not full
agreement on what a number is, but several sets numbers are generally agreed
upon: integers, rationals, computables, etc.

------
kingkawn
Mephistopheles from Goethe’s “Faust”: “Theory, my friend, is gray, but green
is the eternal tree of life.”

------
graycat
"The natural numbers were invented by God. All the rest are man made."

------
psyc
Are they real? Well, when a human says something about a thing, they can never
be completely sure whether they're saying something about an actual thing with
an independent existence, or whether they're only saying something about what
they say.

------
AndrewOMartin
Shut up and calculate.

------
snarfy
I tend to think of real numbers as a composite made of whole numbers and an
operator.

~~~
Smaug123
Which operator?

~~~
snarfy
It depends on the real. 0.5 would be 1/2 using division operator, while an
irrational like square root of 2 is using the power and division operators
(raising to the 1/2 power).

------
mrcactu5
i think it's no coincidence Godel's proof of uncountable reals, comes around
the same time as Lebesgue integration. As mathematicians started exploring
what the serious use of Fourier series

~~~
ska
I think you are mixing up Godel (incompleteness of axiomatic systems) and
Cantor (uncountable reals).

Timing wise Cantors proof was done the year before Lebesque was born; Cantor
was a generation before Lebesgue, and Fourier a generation before that iirc.
Godel's is a generation younger than Lebesgue. Lesbesgue and Borel were
working at the same time - Godel was very young when he published his
incompleteness theorem in the 1930s.

------
DanBlake
Unrelated, but I read a article a while back which said something similar, but
it was based on the fact that our entire mathematical system is designed
around "base 10" and as such is only relevant in many constructs to our
specific human 'ten fingered', interpretation of math.

~~~
unit91
Ironically the publisher of that article used a variety of radices to make the
claim that 10 is the only radix!

2 - to enter and transmit data on the machine

10 - radix in question

16 - color description on the page

etc.

~~~
gertef
Of course 10 is the only radix

[http://cowbirdsinlove.com/43](http://cowbirdsinlove.com/43)

------
dbcurtis
1\. Given any two real numbers on the real number line, you can find another
real number between those two points.

2\. The Planck length is the smallest unit of distance with any meaning.

3\. The universe has finite diameter.

Discuss.

4\. For extra credit: Given 2 and 3, above, it follows that both the diameter
and circumference of the universe can be expressed in Planck lengths as
integers with a finite number of digits. Discuss the concept that Pi is a
ratio of two finite integers.

~~~
AnimalMuppet
1-3: You can define a (mathematical) real number that cannot be interpreted as
a position in the universe that can be physically realized, taking the Planck
Length into account.

4: At the precision of the Planck Length, you have two integers that are the
closest physically-meaningful values whose ratio approximates pi.

In both cases, you're confusing mathematical abstractions with what is
physically realizable in a discrete system. You can't (meaningfully) do that.

~~~
tossaway322
The article unfortunately does not address the question of whether current
mathematical analysis is an appropriate framework for a description of
space/time. We are, in the end, dealing with elements "smaller than" (if
that's the right phrase!) the Planck length. Of course, you can choose to
ignore that and use what's already been provided, but to do so is "whistling
past the graveyard". This has ramifications for all of string theory, quantum
gravity et al.

------
Jason-Andrade
This entire subject is very academic and theoretical, and will never impact
anything in the real world. Using Big Fractions instead of floating-point
numbers is a far more concrete argument with definite real-world impact!
[https://news.ycombinator.com/item?id=13855198](https://news.ycombinator.com/item?id=13855198)

~~~
Retra
Mathematical structure can -- and has -- impacted the world with far reaching
and unparalleled effectiveness without ever even having to have had introduce
the concept of a number. Math really isn't about numbers.

