
Are real numbers really real? - d3ckard
https://arxiv.org/abs/1803.06824
======
myWindoonn
There exists a notion of "computable analysis", based on the delightful fact
that Turing machines (or other Turing-complete enumerable computers) can be
viewed as computing real numbers. These so-called _computable_ reals are a
little limited, but still usable for most tasks [0].

[0]
[https://en.wikipedia.org/wiki/Computable_analysis](https://en.wikipedia.org/wiki/Computable_analysis)

~~~
chx
It's a real PITA to work with computable numbers because even their
equivalence is not computable.

Also, the field is really young. It was only in 2000 that Weihrauch has proven
that every computable real function is continuous.

~~~
voxl
As opposed to classical reals which have a computable equivalence? (Hint: they
don't)

~~~
chx
But when working with reals we normally don't bother with computability.
Whereas with computable numbers you don't even have anything else really.

------
ajross
I'm frustrated that everyone here seems to be hung up over an argument about
the definition of computable numbers and some nattering about Cantor, when the
real meat of the abstract seems to be this:

> This alternative classical mechanics is non-deterministic, despite the use
> of deterministic equations, in a way similar to quantum theory.

Wait, what? What does "a way similar to quantum theory" mean? Is he able to
derive quantum postulates from information theory here, or is this just
handwaving becuase "uncertainty" emerges like it did for Heisenberg?

Not expert enough to read the paper. Anyone?

~~~
thraway180306
There are no equations and no citations to relevant formalisms so your guess
is good as anyone's.

The author had a spat with arXiv who, while not being a peer-reviewed venue do
their best to keep out the crackpottery, rejected some of the stuff coming
from his vicinity. Gisin's complaint was loud, ended up published in Nature,
and now it seems anything goes.

Then went a flood of similar gems. Such like over-the-top argument for
physical time (deep problem formal in nontrivial ways), because thinking
requires time thus obviously
[https://arxiv.org/abs/1602.01497](https://arxiv.org/abs/1602.01497) (doubtful
any creative time went into that). I think figure 5 therein anwsers your
question best as the author can.

~~~
jessriedel
> [https://arxiv.org/abs/1602.01497](https://arxiv.org/abs/1602.01497)

Note that this is in the "history and philosophy of physics" section of the
arXiv, so it's not flooding the arXiv in general.

I've heard a small handful of practical physicists complain that Ginsin is
getting too much status by posting on the arXiv, but I haven't heard any
philosophers of science complain that his papers are particularly bad, much
less that they are so obviously worthless that they are a net negative due to
taking up space on the daily list of what's new in that section.

------
acjohnson55
I don't believe in the physical reality of real numbers, but that doesn't
detract from the their extreme usefulness as an abstraction. For that matter
the negative numbers don't really exist. Nor do fractions or zero. They're all
just convenient because they do a great job modeling situations we care about
in a consistent way. Or at least consistently enough for everyday needs.

It was a bit of a mind bender initially coming to this conclusion, but
_shrug_.

~~~
cup-of-tea
Roger Penrose gives a great explanation in _The Road to Reality_. We know some
subset of the natural numbers are real because we can count things. And we
therefore know some subset of the rational numbers are real because we can cut
things into pieces and count the pieces. But the reals? Well they are invented
purely so we can take the square root of two. And the complex numbers?
Invented so we can take the square root of minus one.

Mathematics is a tool we've developed to help us understand reality. But it
itself is not reality.

~~~
acjohnson55
The rational numbers aren't physically real, because that would imply the
ability to divide objects into exactly fractions, which generally isn't
possible in a verifiable way, which also must be consistent with the
operations we want to perform. Fractions are a useful abstraction. The reality
is just relations between natural numbers.

The real numbers aren't invented to capture the square root of two. You can
deal with that using a simpler set. The real numbers are necessary for common
approaches to calculus and other forms of math that depend on a continuum of
quantity.

I would describe math as a tool for modeling reality, rather than
understanding it. Because we really don't _know_ much at all. All we've got
are rules that are pretty damn good at predicting things under everyday
conditions.

~~~
tree_of_item
The rational numbers are physically real because you can compute them. They
just map on to the naturals.

~~~
acjohnson55
Depends on what you mean by physically real. I'm talking about in the sense of
tangibility, not computability.

------
noobermin
The author makes statements with qualification of "almost all" even though
"almost all" physically allowed systems (that is, solutions to the PDEs under
question) are not physically relevant or beneficial. I can only see this
applying to chaotic systems like the weather, which is a valid point. The key
argument that finite volume contains finite information seems potentially true
but suspect. Case in point, take a unit square, it will have a diagonal who's
digits have a long "infinite" string of digits but sqrt(2) isn't "infinite
information", it's very finite (ie., one number!)

This desperately needs to be peer reviewed by people who study the foundations
of physics.

~~~
edna314
> The key argument that finite volume contains finite information seems
> potentially true but suspect. Case in point, take a unit square, it will
> have a diagonal who's digits have a long "infinite" string of digits but
> sqrt(2) isn't "infinite information", it's very finite (ie., one number!)

I share the suspicion you have, but I think your counter example is not valid
because there is no unit square in (physical) reality. If you would try to
measure the diagonal of the unit square you would at some (finite) point run
into trouble with the uncertainty principle.

~~~
posix_me_less
All measurements have limited precision (number of significant decimal digits
gained), so the amount of data that can be gained indeed is limited. But there
is no general law preventing increasing the precision with better instruments
and better methods, so as measurement gets more advanced, more data (such as
more decimal digits) can be obtained. The uncertainty principle postulates
that uncertainties of coordinate and its conjugated momentum cannot be
simultaneously arbitrarily low. It is not clear to me how that postulate would
prevent arbitrary increase in precision of position measurements. In theory
one could have uncertainty 10E-100m in position of an electron, if uncertainty
in its momentum is of the order 10E66 or higher.

~~~
spiralx
> But there is no general law preventing increasing the precision with better
> instruments and better methods, so as measurement gets more advanced, more
> data (such as more decimal digits) can be obtained.

That's not true, as at the Planck length both quantum and general relativistic
effects are equally important, preventing arbitrary measurements over smaller
scales - that's the case even for a continuously-divisible universe. In most
quantum gravitational theories today the Planck length truly is a minimum
possible distance in the same way the Planck time is the minimum possible
duration, representing a quantized space-time with hard limits to the
precision of measurements.

~~~
posix_me_less
Quantum gravity "theories" are speculative extrapolations of current knowledge
for extreme situations, not accepted general laws of physics. They have
problems with consistency (discretization of lengths and other quantities
breaks relativity postulates) and even bigger problem with lack of experience
with phenomena that they describe (minimum possible lengths/times/energy/etc).

~~~
spiralx
Yes I know, that's why I made the general point first, then talked about how
quantum gravity theories impose even harder limits if true.

------
geebee
I'm curious about something, figured I'd ask since there are some math people
on this thread (I was an undergrad math major but wouldn't include myself in
that group, for the purposes of this question).

The abstract contains the statement: "moreover, a better terminology for the
so-called real numbers is "random numbers", as their series of bits are truly
random."

Has this been proved? I suppose another way to ask it is, if the sequence
never repeats, does that mean it is random?

For instance, if the sequence was

.1121231234123451234561234567... isn't random, but does it resolve to a
rational (can this sequence bet represented as the ratio of two integers)? It
would be a never repeating sequence that has an upper and lower bound that
approaches zero as the sequence continues, right?

If my example turns out to be a rational, is it possible to construct a non-
random sequence that would still never resolve to a rational?

I might try to look this up and read about it a little, it's been a while.

~~~
ouid
If the sequence doesn't repeat, and goes on forever, it cannot be a rational
number. [https://math.stackexchange.com/questions/61937/how-can-i-
pro...](https://math.stackexchange.com/questions/61937/how-can-i-prove-that-
all-rational-numbers-are-either-terminating-decimal-or-
repe?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa)

This does not "prove" the statement given in the abstract, however, as the
statement isn't a mathematical statement. It cannot be proven.

~~~
geebee
Here's the question I have:

Are the bits of a real, non-rational number random.

I suppose another way to think of it would be

Are the bits of a real, non-rational number expressed as a decimal random
integers between 0 and 9?

I'm out of my depth here, but it seems that is a statement that could be
proven.

~~~
andrewla
I think generally speaking, the answer is no.

A normal number [1] is pretty close to what you're describing. Although there
are a lot of normal numbers, theoretically, they are very difficult to
construct, and equally difficult to prove normality. Unfortunately, proving
non-normality is also difficult, although the wikipedia article has some
examples of base-10 normal numbers and a couple of examples of irrational non-
normal numbers.

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

~~~
geebee
Wow, yes, that's exactly it. Normal numbers.

It's been 20 years since I majored in math, so maybe I did encounter these and
just don't remember. Looks like there is a proof that almost all real numbers
are normal, but where it comes to specific numbers pi, e, sqrt(2), they are
thought to be normal but no proof.

Well that was informative, thanks for the link!

------
recursive
"Real" numbers are mathematical abstractions. Perhaps it's unfortunate that
"real" and "imaginary" were used instead of more abstract sounding terms that
came with less attached connotation.

~~~
danharaj
The argument of the author is not that the real numbers are poorly named, it's
that the mathematical abstraction supposes physically impossible properties of
the model. Therefore physics ought to be based on an abstraction that has more
physically plausible properties instead of the real numbers.

Not that I necessarily agree with such an argument.

------
madmax96

        >I do not make any metaphysical claims about space, time nor numbers,
        >but notice that the mathematics used in practice is always finite
        >...I argue that the so-called real numbers are not really real.
        >More precisely, I argue that the mathematical real numbers
        >are not physically real, by which I mean that they do not represent anything physical
    

Well, isn't the statement 'only physical objects are real' a metaphysical
claim?

~~~
ckaygusu
I read it as "I do not make any _new_ metaphysical claims". The statement you
have put indeed is a metaphysical assertion, but you have to have a set of
metaphysical "axioms" before you can proceed with physics, hence it's
impossible to talk about physics without making implicit metaphysical
assertions.

~~~
madmax96
That's certainly true. But one doesn't require the axiom "only physical
objects exist" to do physics, and that principal is not universally accepted
among physicists.

This paper -- although I'm an outsider to physics, and I'm way more
comfortable evaluating mathematics and computer science research -- seems like
it has an interesting and novel approach to a theoretical problem in physics.
I don't understand why the author felt the need to couple this with the given
title and stress this point so much. It _almost_ felt like click-bait, until I
understood generally what was being done.

~~~
ckaygusu
You are right, just saying "he does not make new assumptions" does not explain
the author's act quite well enough.

On further deliberation, I have come to the following conclusion: there is a
difference between assuming "only physical objects exist, and they are finite"
and making an argument of that. The author is trying to say he is not making
an argument for the validity of the axiom but implicitly says he just assumes
it, to guide the focus of the reader to the main argument he is making.

I acknowledge that you definitely do not need to make that assumption, but as
I said, you have to start somewhere. I'd say the metaphysical assumption
captures a substantial subset of all physics, hence his ideas are worthy of
consideration.

------
jimhefferon
"Since a finite volume of space can't contain more than a finite amount of
information"

Is that perfectly clear? Not so, to me.

~~~
pjz
This stuck out to me too. Fractals teach us that infinite length can be fitted
into finite area, so I'm dubious about this claim. Though I guess if we're in
the realm of physics, the Planck constant determines a kind of minimum length,
so the original statement may be true?

~~~
JoeAltmaier
Pendant here: the Planck constant is simply a convenient length for all things
subatomic, like a hectare or a fortnight. No minimum whatsoever implied there.
Many, many things are smaller than the Planck constant.

~~~
jerf
"Many, many things are smaller than the Planck constant."

Do we have a proof of that? Honest question. I know strings are often
postulated to be smaller (by many orders of magnitude in some cases) but we
are not yet particularly confident they exist. I'm not aware of us being
reasonably confident that anything is smaller than Planck's size. ("Smaller
than Planck's size" isn't even all that easy to define.)

~~~
JoeAltmaier
Event horizon of a black hole is many times smaller for instance.

~~~
chii
The mass of a blackhole determines the size of the event horizon.

To have an event horizon smaller than the planck constant, the mass have to be
also small. While theoretically possible, i don't believe this has ever been
considered possible in reality - see
[https://en.wikipedia.org/wiki/Kugelblitz_(astrophysics)](https://en.wikipedia.org/wiki/Kugelblitz_\(astrophysics\))

------
NelsonMinar
See also
[https://en.wikipedia.org/wiki/Constructivism_(mathematics)](https://en.wikipedia.org/wiki/Constructivism_\(mathematics\))

------
carapace
Which real numbers? All of them? That's a kind of mental infinitude that I
don't think exists, although Kurt Gödel did.

Anyway _no_ numbers exist unless you're some kind of mystic or dualist. This
numeral "2" isn't the number two, nor are these two dots: . . Nor is the word
"two".

There are many twonesses, but they're not in the world, they are in our heads.

Like "tuesday" doesn't exist. The words exists, and if you ask somebody what
day is today they'll use that word, but there is no tuesdayness _in the real
physical world_. You can prove this my going to the a pole (North or South)
and wandering around. Now "tuesday" is stuck in gimbal lock.

If numbers are "real" it is in some "other" Universe, some realm of archetypes
and dream-stuff.

~~~
andrewla
> Which real numbers? All of them? That's a kind of mental infinitude that I
> don't think exists, although Kurt Gödel did.

I read the article as saying that real numbers as a mathematical concept is
not a useful physical concept, and that considering classical physics with
that as a first principle yields non-determinism even in classical models.

Which is to say that the number "5.78" (exactly) has no physical meaning, even
though the number "5.78" (measured) is useful. If you put something 5.7800 cm
from another object, and run an experiment that relies on something on the
seventh decimal point of that distance, the result will be random. From a
classical point of view, the argument would be that measurement error accrues,
and the randomness is extrinsic. The author argues that if you consider the
randomness to be intrinsic instead, then you potentially gain some insight
into the bridge between classical and quantum mechanics.

I'm not as convinced as the author, because quantifying where exactly the
intrinsic error is introduced is very closely related to the foundations of
quantum mechanics. There's no reason classically that you can't make an
arbitrarily sensitive measuring device to resolve the "randomness".

------
gbacon
“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk.”
— Leopold Kronecker[0]

[0]:
[https://en.wikipedia.org/wiki/Leopold_Kronecker](https://en.wikipedia.org/wiki/Leopold_Kronecker)

------
tntn
Loads of questions here that seem to be based on only reading the abstract.
Most of these are addressed in the paper.

------
fizixer
Unless your new classical mechanics predicts the outcome of an experiment that
other existing theories cannot predict, and that prediction turns out to be
true if such an experiment is conducted, your new idea is just another
hypothesis.

Furthermore, all the real physical quantities come from the fact that we
assume space and time to be continuous (i.e., similar to real numbers). So
your hypothesis boils down to the claim that space and time are discrete. And
that is not new. Plenty of people have proposed that hypothesis, and the idea
of planck length and planck time are conjectured to be the smallest units of
space and time. But the two values are of the order of 10^-35 m, and 10^-44 s,
respectively. This is way below our ability to prob smaller length and time
scales. And to the extent that we have been able to prob length and time
scales (roughly 10^-15), space and time appears to be continuous.

Finally, when you're giving up continuity in any sense, you're essentially
'quantizing a theory'. So it's no longer a classical theory but a quantum
theory. We have already quantized energy (quantum mechanics), as well as
physical fields (quantum field theory), and associated phenomena. But quantum
(i.e., discrete) space and time are only a hypothesis at the moment.

------
ah27182
Norman Wilberger has some interesting/controversial views on this matter.

Here is a link to his YouTube channel, where he has posted many lectures on
subjects through his viewpoint:

[https://www.youtube.com/user/njwildberger](https://www.youtube.com/user/njwildberger)

I will say, that he is viewed as a pretty notorious crank. Especially due to
his views on finitsm. But I feel like his lectures on rational trigonometry
are really interesting nonetheless.

~~~
gowld
"crank" is the wrong word. Mathematicians don't like his attitude, because he
is smug/brash and he rejects unphysical mathematics (which many mathematicians
find offensive or irrelevant), but they don't dispute his math.

------
Strilanc
> _almost all real numbers contain an infinite amount of information. Since a
> finite volume of space can 't contain more than a finite amount of
> information [..]_

The Kolmogorov complexity can't be higher than the state of the initial
system, the dynamics, and a time parameter. Even if the system's current state
involves real numbers, you would have a finite description of it as long as
the time values, dynamics, and initial conditions are computable.

I also think it's subtle to apply the Bekenstein bound here. Even if the
numbers going in have infinite precision, that doesn't mean the relevant
information quantity is infinite. For example, a qubit's density matrix has
three real-number parameters and yet its von Neumann entropy is never more
than 1 bit.

~~~
danharaj
This corresponds to type II computability of the reals whereas the paper, as I
skimmed it, is about type I computability.

------
kavalec
"Interestingly, both alternative classical mechanics and quantum theories can
be supplemented by additional variables in such a way that the supplemented
theory is deterministic."

This sounds like "hidden variable" and is not true for quantum physics.

~~~
Retric
It is, but you need to give up more and allow information to travel faster
than light speed.

It's _local_ hidden variables that are a problem.

~~~
kgwgk
Non-locality != faster-than-light communication

~~~
Retric
As I understand it 'Spooky action at a distance' does not allow communication,
but in for hidden variable theory to work a particle needs access to another
particles hidden variables. Unless you want to suggest another model?

~~~
kgwgk
Why is non-locality a problem for hidden variables theories and not for the
"standard" interpretation of quantum mechanics?

~~~
Retric
Quantom mechanics is based on non-locality, that's one of the things people
have problems with.

Quantum key distribution for example is seems to be based on this
[https://en.wikipedia.org/wiki/Quantum_key_distribution](https://en.wikipedia.org/wiki/Quantum_key_distribution).

As to the truth well ¯\\_(ツ)_/¯

~~~
kgwgk
I thought you said that the hidden-variables extension of quantum mechanics is
"bad" (compared to the "vanilla" QM) because you need to allow information to
travel faster than light speed.

But I agree if you meant that QM (both non-local hidden-variables and standard
interpretation) requires non-locality, unlike local hidden-variables theories
which are ruled out by the violation of Bell inequalities which has been
established experimentally.

~~~
Retric
Hidden variables where specificly proposed as a solution to non-locality. So,
I just mean bad in that they did not solve that problem.

~~~
kgwgk
True. However, unlike the "standard" QM interpretation, the non-local hidden-
variables theories are deterministic (which is the reason why they were
mentioned in this thread a few messages back).

------
OskarS
Ok, I'm not a mathematician (or a physicist), but this seems... dubious...
Like... what?:

> Moreover, a better terminology for the so-called real numbers is "random
> numbers", as their series of bits are truly random

0.5 is a real number. Its series of bits is not random at all. sqrt(2) is also
a real number, and its digits looks sort-of random, but they're not really:
they're exactly the sequence of digits that forms a number that, when squared,
equals 2. You can easily develop very simple formulas or programs to generate
them, so in a "Kolmogorov complexity" sense, it's not random at all. Point is:
the concept of "having random digits" is in NO way intrinsic to "being a real
number". I guess what he meant to say was "non-computable numbers"? So he's
saying "in a finite volume, there can't be non-computable numbers"... or
something?

Again: not a mathematician or a physicist, but this comes off as pretty
crank:y to me. I would love to be corrected.

~~~
plopilop
Well, computable numbers are countable, so almost all real numbers are non-
computable (also they're dense in R).

So while your objection is valid, in practice any truly randomly selected real
number is non-computable.

~~~
zuminator
How does one go about truly randomly selecting a real number?

~~~
Analemma_
This sort of depends on whether you're talking about "randomly selecting a
real number" in the real word or in the mathematically abstract world. In the
real world, you can't do it, because almost all the real numbers are non-
terminating decimals. Or to put it another way, any selection process is a
computation, and almost all the real numbers are uncomputable and so can't be
found by any computation.

To randomly generate a real number in the abstract, I suppose you could do a
random walk of infinite steps on the real number line, with each successive
step randomly going left or right and shrinking the distance walked. Off the
top of my head I'm not sure if that's rigorous, but it sounds like it would
work.

------
zhte415
Since a finite amount of space - my stomach and associated crevices - can't
contain a finite amount of coffee, nor could the keyboard; I argue that 200ml
on the cup isn't relevant, and I can order more without deterministic effect,
perhaps.

Clever. Nice. Very unexpected.

------
shmerl
Terminology is rather artificial. I.e. mathematicians just had to come up with
some names for various abstractions. So real, irrational, imaginary and etc.
aren't supposed to be taken literally.

------
DArcMattr
Keep in mind, this is using terminology that comes from courses appropriate
for advanced undergrad/beginning graduate level math courses.

------
kruhft
"God (sic) created the integers, the rest, was left to people."

Can't remember.

"If it cannot be done with integers, it cannot be done"

Von Neumann

~~~
shmerl
Letters in Sefer Yetzira.

~~~
kruhft
What book is that?

------
satyajeet23
Is anything real?

------
codewiz
Wow. The author of this paper should be awarded the Ig Nobel Prize for both
math _and_ physics :-)

------
frankohn
Not serious, just bla bla, no equations are given and the author does not
suggest any finite representation to replace the real numbers.

This kind of article is like the old philosophers way, just words, words and
arguments, nothing exact or observable that can be verified or falsified.
Normally this was over when physics began: we need hard fact, experiments and
equations.

