true randomness in mathematics

[José Manuel Rodríguez Caballero]

Thomas Klimpel wrote:

> Such a distinction between true randomness and pseudo-randomness
> appears to me to depend on interpretative assumptions. In my opinion,
> mathematical existence gains relevance by describing idealizations of
> real life situations. Describing which idealized properties true
> randomness should have in such real life situations then allows us to
> accept more sources of randomness than just those based on quantum
> randomness.


In your reference you provide two definitions of true randomness:


   - *Quantum mechanics*: The *randomness *itself is nonlocal, and it must
   be *really random*, because otherwise this non-locality could be used
   for instantaneous signal transmission.
   - *Gambling and games*: A *truly random phenomenon* must produce
   outcomes that are completely unpredictable. And not just unpredictable for
   you and me, but unpredictable for both my opponents and proponents.

https://gentzen.wordpress.com/2022/02/28/true-randomness-and-ontological-commitments/

For the "Quantum mechanics" definition above, I do not use anything
physical, except the hypothesis that the universes branches after quantum
measurements, i.e., just a tree from graph theory:

Everett, Hugh. "The theory of the universal wave function." The many-worlds
interpretation of quantum mechanics. Princeton University Press, 2015.
1-140.
https://www.pbs.org/wgbh/nova/manyworlds/pdf/dissertation.pdf

 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.

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 of x_n is a logarithmic function of n for almost
all values of n.

(ii) a sequence of bits x_1, x_2, x_3, ... is truly-random if and only if
the Kolmogorov complexity of x_n is a linear function of n for almost all
values of n.

Notice that "Gambling and games" definition above can be a case of
pseudo-randomness, where the unpredictability is given by
complexity-theoretic bounds. This is what is used in complexity-theoretic
cryptography.

There are some results about extracting true randomness from big data

https://www.nature.com/articles/srep33740

but we need to consider the fact that we live in a world where random
events triggered by quantum mechanics happen all the time. These events are
amplified by the butterfly effect:

https://www.wikiwand.com/en/Butterfly_effect

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.

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.

Whether true randomness plays a role in the development of mathematics can
be experimentally checked by developing two scholastic simulations of the
work of a mathematician. One simulation should use a pseudo-random bit
generator and the other one, a quantum random bit generator. Comparing the
performance of these simulations with the empirical data about how
mathematics was historically developed may show some statistical
differences.

As far as I know, there are no results comparing the efficiency of
automatic theorem provers using pseudo-random generators and truly random
generators, using the definition of Kolmogorov complexity. It would be
interesting to know whether the results are universal (for all theories) or
specific (for some theories pseudo-randomness and true randomness are
equally efficient, but for other theories, true randomness is more
efficient).

Kind regards,
Jose M.
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