[ExI] Gödel and physical reality

[Jason Resch]

Hi Giulio,

Thank you for raising so many profound questions!

On Wed, Nov 3, 2021 at 5:10 AM Giulio Prisco via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> Gödel and physical reality. What are the implications of Gödel's
> theorem for fundamental science and metaphysics?
>
>
I think there are deep connections between Gödel's theorem and the question
of why anything exists. I write about some of the connections and
implications in this article:
https://alwaysasking.com/why-does-anything-exist/#20th_Century_Mathematics

In summary, Gödel's result implies an independence of mathematical truth
from any human theory of that truth, and I would argue, even makes
mathematical truth independent of the physical universe.

Then, there are a number of constructive arguments that show how the
existence of mathematical truth leads to the existence of entities who
believe and consider themselves to be parts of physically existing worlds.
In effect, all computational histories (of all programs) are a small part
of what exists in the set of all mathematical truth. This was discovered in
1970 as part of the MRDP-Theorem which solved Hilbert's 10th problem. There
exists a single Diophantine equation, whose solutions embody all possible
programs (all computable sets). See:
https://alwaysasking.com/why-does-anything-exist/#Universal_Equations


> What does Gödel’s theorem say about physical reality?



If you define physical reality as the observable universe, whose
information content is large but finite (say 10^120 bits), then our
universe lacks the computational and memory resources to even list all the
properties of the number "3". In this sense, the number 3 is a larger, and
more complex object than our physical universe, for there are an infinite
number of true statements concerning the number 3, not all of which can be
enumerated in our physical universe. The finiteness of our universe also
means there are axiomatic systems which contain more axioms than there are
atoms in our universe, so we could never conceive of them.


> Does Gödel’s
> theorem imply that no finite mathematical model can capture physical
> reality?


Since our physical reality allows the construction of computers, and since
the behavior of computers cannot be determined in general (you will always
need ever-more expensive mathematical theories to prove things about the
operations of certain computers), then in a very real sense, Gödel’s
theorem (and particularly in conjunction with the unsolvability of
Hilbert's 10th problem (undecidability)), it means certain physical
problems are not decidable either.

For example, consider if I have a computer searching for counterexamples to
the Goldbach conjecture
<https://en.wikipedia.org/wiki/Goldbach%27s_conjecture>, after which it
will turn on a light switch, because the computer is a physical system, I
could ask the physical question of: "*how long, if ever, will it take for
the light to turn on?*" And yet, it very well could be that we cannot
determine the answer to this problem, without developing a new more
powerful mathematical system that enables us to prove that there are no
counterexamples to the Goldbach conjecture. So physics and our physical
theories, because they are general enough to allow us to build computers,
are subject to the limits of undecidability, and ultimately are limited by
the power of our mathematical systems.


> Does the nondeterminism found in quantum and chaos physics -
> it’s impossible to predict (prove) the future from the present and the
> laws of physics - have something to do with Gödel’s incompleteness?
> Many scientists e.g. Stanley Jaki, Freeman Dyson, and recently Stephen
> Hawking ("According to the positivist philosophy of science, a
> physical theory is a mathematical model. So if there are mathematical
> results that can not be proved, there are physical problems that can
> not be predicted…") have formulated this intuition. But I'm not aware
> of any proof (or very strong semi-rigorous argument) that causal
> openness in physical reality follows from Gödel's theorems (or the
> related results of Turing, Chaitin etc.). Can anyone give me
> links/ideas?
>

There are a number of recent results that suggest the uncertainty we
observe in physics could be a direct manifestation/consequence of our
existence within an infinite ensemble of all possible computational
histories. I quote and cite some of these theories and theorists here:
https://alwaysasking.com/why-does-anything-exist/#Why_Quantum_Mechanics

If you have any questions about any of these ideas, I would be happy to
follow up in more depth, but the links from there should provide a good
starting point.

Jason
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