
Khinchin's Constant - goldenkey
http://mathworld.wolfram.com/KhinchinsConstant.html
======
kilovoltaire
My favorite thing about this constant is the counter-intuitive fact that if
you uniformly randomly choose a real number, there's a 100% chance it will
have the Khinchin behavior, and yet there are almost no known numbers which
actually do.

Or as Wikipedia puts it:

> Although almost all numbers satisfy this property, it has not been proven
> for any real number not specifically constructed for the purpose.

~~~
pash
This reflects basic properties of the real numbers, e.g., that almost all of
them are irrational, almost all of them are transcendental, almost all of them
are uncomputable, etc.

The familiar real numbers, those that people (including mathematicians)
actually use, and that our machines use, are atypical of the set of real
numbers as a whole in many important respects. The real numbers are a formal
abstraction to an extent that few people realize.

~~~
chriswarbo
> The real numbers are a formal abstraction to an extent that few people
> realize.

This is what makes me dislike the name "real" so much; I long ago grew tired
of mathematicians making non-intuitive claims about the world which turn out,
in fact, to be claims about the real numbers.

As Kronecker said, "God made the natural numbers; all else is the work of
man".

~~~
wyager
Tons of stuff in physics is real-valued (or complex-real-valued). Phase
difference between eigenstates is real. The energy, momentum, etc. of an
unbound particle has an uncountably infinite number of eigenstates, so you
need to index them with a real number.

~~~
chriswarbo
Correction, tons of physicists _use_ real values (or complex real values) _in
their calculations_. That doesn't imply that the corresponding values in the
world are irrational, transcendental, infinitely-precise, etc.

There are a few ways we can avoid the reals in physics.

Abstractly, there are results such as Skolem's "paradox" (
[https://en.wikipedia.org/wiki/Skolem%27s_paradox](https://en.wikipedia.org/wiki/Skolem%27s_paradox)
) which tell us that theories involving uncountable sets (such as the reals)
can be satisfied by countable models.

More concretely, physicists never manipulate real numbers directly; only
symbols which may represent real numbers. Such symbolic manipulation could, in
principle, be performed by a turing machine, and hence any value obtained from
such theories (e.g. experimental predictions) must be in the set of computable
numbers. Hence the reals are just an 'implementation detail' of any theory,
and their purported presence cannot have any influence on the theory's
predictions.

As I said previously, I have nothing against the reals as a useful
mathematical tool; I just disagree with the name, and claims that they are
physically real. Others go further though, arguing that discrete mathematics
is more general and useful than continuous, e.g.
[http://www.math.rutgers.edu/~zeilberg/OPINIONS.html](http://www.math.rutgers.edu/~zeilberg/OPINIONS.html)

~~~
wyager
Nothing you said in your post has actually demonstrated that it's unlikely
that reals underly physical reality. It sounds like you just have some sort of
aesthetic agenda.

Your entire premise (that we might replace reals in current physical theory
with some other structure and get the same result) is trivially debunked. Pi
shows up all over the place, and e.g. rationals are not closed under
exponentiation by e.

The schroedinger equation applied to physical hilbert spaces is inextricably
real-valued.

~~~
chriswarbo
> Nothing you said in your post has actually demonstrated that it's unlikely
> that reals underly physical reality.

Sorry, I was taking it for granted that the "reals" are problematic and
unphysical (based on the sentiment of the grandparent). For clarification, the
real numbers are a superset of the "computable" numbers (those numbers which
can be approximated, to any given precision, by a universal turing machine).

The computable numbers are countable (since there are countably-many turing
machine tapes), but the reals are uncountable. That means "almost all" of the
reals are uncomputable.

Since all the laws of physics are computable, all physical processes must be
based on computable numbers; even if some initial condition of the universe
resulted in an uncomputable number, e.g. if the digits of the muon's magnetic
moment are the same as the digits of the 100th busy beaver number, we can
never know since we'll never compute what that busy beaver number is (we could
test such predictions empirically, but it could always just be a coincidence).

> Pi shows up all over the place, and e.g. rationals are not closed under
> exponentiation by e.

Pi and e are computable numbers (quite trivially so, there are _loads_ of
known algorithms to do it). Exponentiation is a computable operation (again,
the algorithm is pretty trivial; it's just repeated multiplication,
multiplication is repeated addition and addition is repeated increment, so you
only need 3 loops)

[https://en.wikipedia.org/wiki/Computable_number#Can_computab...](https://en.wikipedia.org/wiki/Computable_number#Can_computable_numbers_be_used_instead_of_the_reals.3F)

> The schroedinger equation applied to physical hilbert spaces is inextricably
> real-valued.

It's been a while since my physics master's, so I wouldn't want to refute this
face-on; however, it's trivial to notice that (by definition) we can only use
computable numbers in our calculations, and hence even if there are any non-
computable reals in a theory, they cannot affect the predictions of that
theory. If some prediction depended on a value outside of the computable
numbers, e.g. the 100th digit of Chaitin's constant, or the 100th busy beaver
number modulo 7, then that prediction would not be calculable and could only
be given in symbolic form, e.g. 1.8 * Chaitin(100) ^ 3.45. This is _not_ the
case for the schroedinger equation, which gives results that are analytically
and/or numerically computable.

> Your entire premise (that we might replace reals in current physical theory
> with some other structure and get the same result) is trivially debunked.

It's not my premise to "replace reals in current physical theory" (although it
seems that Doron Zeilberger might be pleased if that happened).

I was pointing out that physicists, mathematicians and many others claim to
use the reals, when in fact they're only using the computable numbers. This
leads to confusion when unintuitive results about the reals are applied to the
world.

For example, when choosing a real number randomly from the unit interval, the
probability of each point is 0, but the sum of their probabilities is 1.
That's an example of the reals being unintuitive, and that's fine. However,
trying to apply such statements to the world (e.g. "I can prove that the
chance you've measured exactly 1 litre of water is zero") leads to confusion.
Even worse, the confused person (the one making the claim, confusing
uncomputable numbers with reality) may then "prove wrong" some perfectly
justified refutations.

In case you can't see the problem, the "proof" that an amount other than 1
litre was poured relies on checking more and more decimal places until a
difference is found between the measured amount and 1 litre. However, we have
no way of knowing that the "keep checking decimals" algorithm will halt, so
the claim might be an unsound counterfactual based on the "result" of an
infinite computation. Any claim that the algorithm _will_ halt is pre-
supposing that there is a difference, and hence is not a proof. At any finite
time, we can only ever compare the amounts to within some epsilon > 0, hence
we can only ever claim the probability is arbitrarily small.

Dubious non-constructive, uncomputable results like this appear all over the
place when dealing with reals. This makes the reals an interesting
mathematical curiosity worthy of study, like the surreals, the hyperreals,
transfinite cardinals, etc. and their properties can certainly be useful when
required.

However, their status as the "default" set of (non-whole) numbers should have
been removed after Turing's "On Computable Numbers" paper.

> It sounds like you just have some sort of aesthetic agenda.

When it comes to the reals I'm very serious. When it comes to renaming the
"negatives" to the "other imaginaries", replacing spurious use of "integer"
with "natural" and outright banning the awful phrase "unsigned integer", then
_that 's_ an aesthetic agenda :)

~~~
wyager
Assuming the universe is computable using a UTM, I agree. Why do you think the
universe is computable? It would be nice if it were, but I have seen no proof.

~~~
chriswarbo
Physical theories can't be "proven" in the same way as mathematical
conjectures, so we have to rely on agreement with experiment, self-
consistency, "simplicity" (e.g. Kolmogorov complexity or some approximation),
lack of counterexamples/disproof, etc.

In this sense, I agree with Robert Harper (e.g. Practical Foundations for
Programming Languages, section 22.2) that unprovable claims regarding
computability may be merely "theses" to logicians, but to scientists they
should be considered "laws" just as much as Newton's law of universal
gravitation.

In fact, other than purported claims of "hypercomputation" (
[https://en.wikipedia.org/wiki/Hypercomputation#Criticism](https://en.wikipedia.org/wiki/Hypercomputation#Criticism)
) or the anthropocentric "microtubule"-type theories popularised by Penrose (
[https://en.wikipedia.org/wiki/Orchestrated_objective_reducti...](https://en.wikipedia.org/wiki/Orchestrated_objective_reduction#Criticism)
), it seems that the computability of the universe is commonly accepted
'background knowledge' by most researchers in the field.

Even those making extreme claims in this area, like those refuted in
[http://www.scottaaronson.com/papers/npcomplete.pdf](http://www.scottaaronson.com/papers/npcomplete.pdf)
are presumably assuming that the universe is computable; there doesn't seem to
be much point debating computational efficiency of soap bubbles if I thought
(for example) that halting oracles existed!

~~~
wyager
Unless I'm mistaken, the computable numbers are not complete in the Cauchy
Sequence sense. Therefore, there can be no hilbert space based on the
computable numbers, because a hilbert space has to be complete in the Cauchy
sense. It would only be a pre-hilbert space. I believe this means one can no
longer correctly use limits in QM; are there other consequences of this?

~~~
chriswarbo
Very interesting, and the kind of technical aspect I tried to avoid by not
refuting "face-on" since I don't feel qualified to answer this without looking
into it further :)

Still, if there are reasons to allow uncomputable numbers as elements of a
quantum theory, that still doesn't invalidate the larger point that any
predictions of those theories are computable; it's just a constraint on which
axiom scheme we follow when manipulating our equations.

Again, even if a theory involves uncountable sets, it may still have countable
models (following either Skolem's "paradox", or a similar argument; since QM
operates at much higher level than formal systems, I'm not sure if it's first-
order, second-order, higher-order, etc.).

------
fdej
> However, the numerical value of K is notoriously difficult to calculate to
> high precision, so computation of more digits get increasingly slower.

It can be calculated to n digits in O~(n^2) time, which isn't really that bad.
Something like Brun's constant or the area of the Mandelbrot set is probably a
better example of a real number that is "notoriously difficult" to compute.

------
russdill
So then my question would be, are physics constants likely to be Khinchin,
computable, or just non-computable?

~~~
analog31
They're all 1, in appropriate units. ;-)

~~~
drdeca
There are unitless constants in physics though.

Have I misunderstood something?

~~~
LolWolf
Yes: the joke.

------
yannis
Interesting, but where can I use it?

~~~
Retra
I know some people are down-voting you based on the idea that math "doesn't
have to be useful," (which I strongly disagree with), but I've down-voted you
for a different reason and I'd like to explain.

Mathematics is _abstract_. That means something. It means when I say "I have
$50", you can think of those $50 as peices of paper, portions of goats,
amounts of hot dogs, or investment potential. $50 is not a _thing_ , it's a
model for things that have arithmetical properties and value.

That's how numbers work: they have properties, and you look at the world and
map those properties to things that are relevant to you.

So you read something about Khinchin's constant, you understand what the terms
are, understand the structural implications of it's properties, and then you
go and try to find things that behave similarly in the world. Sometimes it
fits well, sometimes you have to force it, sometimes it's just overly complex.
The point is that the behavior of something like this is a _model_ , which
means it will never map perfectly to reality, and also that it may apply with
varying degrees of success to _any_ problem.

So where can you use it? Everywhere. You just have to think in terms of its
properties. Do that enough and maybe you'll find out why it's stupid to use
for some problems and smart for others.

But there's no point in asking someone else to do all the thinking for you.

------
323454
The behaviour of Khinchin(e) is seriously crazy! Not only does it not converge
to K, it looks O(log(n))...

