true randomness in mathematics

[Thomas Klimpel]

On Wed, Mar 16, 2022 at 7:51 PM José Manuel Rodríguez Caballero wrote:
> I am not aware of a source of true randomness other than quantum mechanics:
> all other sources of true randomness may rely on quantum phenomena at the fundamental level.

For chaos in classical mechanics, the initial conditions (or rather
the current state) effectively acts as a source of randomness. In the
real world too, the current state contains unimaginably enormous
amounts of information from which randomness can be extracted.

You might argue that it was QM which created the randomness that is
now contained in the current state, so at the fundamental level we
still rely on quantum phenomena. This is fine for me. But let me also
point out the role played by the huge number of atoms here, which are
responsible for preserving that information long enough to "be
useful". After all, the "classical approximation" is useful, therefore
the aspects of the real world which are responsible for its usefulness
should not be ignored.

> The fact that quantum mechanics is a source of true randomness
> seems to be a consequence of the Copernican principle.
> Indeed, considering the branching of the universe after quantum measurements,
> most positions will be truly random just because of combinatorics.
> In virtue of the Copernican principle, we should be in a branch that is not special,
> i.e., a random branch. Therefore the randomness of quantum mechanics is just a
> consequence of the fact that we are at a random position in the branching of the universe.

I see a two step process here. First, the Copernican principle is used
to argue that we should be in a generic position. And then this
generic position is used to argue that the branching "inherent in the
Wolfram model and also in MWI" gives rise to true randomness. The
second step is non-trivial, because you don't just need to explain the
emergence of randomness, but also why it is distributed according to
the Born rule. I am aware of ways to argue for this in the case of
MWI, so this is not a fundamental problem. But it hints at a
fundamental difference between a "generic position" and a "random
position".

The "correct distribution" for a "random position" is often highly
symmetric, to the point of achieving some sort of uniqueness. If you
try to understand how this arises in the real world, you often see
laws of large numbers at work. For a "generic position", the trick is
rather that you first specify an extremely small number of explicit
conditions to be avoided, and then argue for the plausibility that
they won't be violated based on their extremely small number. But
catastrophe theory shows that even this limited goal often fails, as
soon as you stop looking at that generic position in isolation, but
also look at it as a function over time, or as other parameters are
varied. Also a notion of "most generic position" would even risk to be
self-contradictory in many cases. Things like the anthropic principle,
or the observation that interesting phenomena (like life) often happen
at interfaces between different phases (we live at the surface of
earth, for example) weaken the arguments for truly "generic positions"
in the real world.

---

Oh well, now I have written so many objections, and all mostly from my
own point of view. Maybe I can also manage to be a bit more
positive...

> There are some results about extracting true randomness from big data
>
> https://www.nature.com/articles/srep33740

Interesting, thanks for the link.

>  My definition of true randomness is grounded on the theory of computability and algorithmic information theory:
>
> Given a black box generating a sequence of bits, i.e., 0s or 1s, x_1, x_2, x_3, ..., we say that this black box
> is pseudo-random if there is a (deterministic) algorithm generating the same sequence of numbers.
> If this is not the case, we say that this black box is truly random.

I initially misunderstood your "deterministic" and "pseudo-random" in
the sense that you consider classical chaos to be only pseudo-random.
In the sense of algorithmic information theory, you are completely
right that this is how pseudo-randomness is typically understood.

> A characterization of true randomness using the Kolmogorov complexity and the notion of time is as follows:
>
> (i) a sequence of bits x_1, x_2, x_3, ... is pseudo-random if and only if the Kolmogorov complexity ...

While thinking about your paper, I was a bit confused why you used
"descriptive complexity"/"Kolmogorov complexity" instead of
"algorithmic complexity"/"prefix-free Kolmogorov complexity". After
all, you mention Gregory Chaitin multiple times, the case r=0 would
have been less of an exception (it would have been simply "the
descriptive complexity of x_n is equal to a constant plus r n plus the
number of binary digits of n"), and I also think it plays nicer in the
context of randomness.
I guess your reason was that it would have made your paper more
complicated and more subtle, without any real benefit other than
demonstrating that you mastered those subtleties. I think you made the
right decision, your paper is short and understandable, and focuses on
one specific aspect of one specific class of models.
Basically you show that the typical observer experience in 2^r-regular
Wolfram models are distinguishable for different r. Then you
"implicitly" notice that for r >= 1 this distinction fails in
practice, because you would need to forbid linear rescaling of time.
Therefore you go on to show that it would even fail for r = 0, if
logarithmic rescaling of time would be allowed. And then you notice
that the "old" models from "New Kind of Science" and maybe even "'t
Hooft's models" can be interpreted as 2^0-regular Wolfram models.
So I think it is a nice paper, and the way it uses algorithmic
randomness is completely appropriate for its purposes. And the one
time where it mentions "true randomness" is also completely
appropriate: "Unlike the Copenhagen doctrine, where true randomness is
imposed in the universe as a postulate, ..."

Kind regards,
Thomas