[ExI] [Extropolis] Gödel and physical reality

Jason Resch jasonresch at gmail.com
Wed Nov 3 17:52:31 UTC 2021

On Wed, Nov 3, 2021 at 11:09 AM Giulio Prisco via extropy-chat <
extropy-chat at lists.extropy.org> wrote:

> On Wed, Nov 3, 2021 at 4:59 PM John Clark <johnkclark at gmail.com> wrote:
> >
> > On Wed, Nov 3, 2021 at 10:38 AM Giulio Prisco <giulio at gmail.com> wrote:
> >
> >> > 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.
> >
> >
> > Mathematics is the language of physics but like any language it can be
> used to write both fiction and nonfiction. English is also a language and
> it can be used to write a book about Quantum Mechanics but it can also be
> used to write a book about Harry Potter, and both books could be equally
> grammatically correct. I sometimes have the feeling that some of the more
> abstruse areas of modern mathematics might be the equivalent of
> mathematical Harry Potter novels.

We have the freedom to choose and make up mathematical theories, just as we
have the freedom to choose and make up physical theories. But we don't
bother with the ones that give nonsensical or wrong answers. We want our
mathematical theories to be able to prove things that are objectively
verifiable, such as whether or not a given program will halt or not. If a
mathematical theory failed to give a correct prediction to such a question
we would discard it, just as we would a physical theory that gave bad
predictions. I think we ought to not confuse the theories (human invented)
from the objects we try to describe using those theories. It seems you lump
both the theories and the objects together when you say mathematics, but
you don't make this same mistake when you speak of physical theories vs.
physical objects. I think Gödel's incompleteness result effectively proves
mathematical objects are necessarily distinct from any theory attempting to
describe them, for no finite (human invented) theory can ever answer every
question or provide every property of these mathematical objects.

> >
> >> > But it can also be argued that it makes mathematical truth *strongly
> >> dependent* on the physical universe.
> >
> >
> > That's why I would argue physics is more fundamental than mathematics,
> and that's why a book on advanced computational theory sitting on a shelf
> cannot add 2+2, the ideas in the book have to be implemented in physics
> before it can actually do anything. If there is an even number that is not
> the sum of two prime numbers but the number is so huge that it cannot be
> calculated by physics even in theory then I think it would be meaningless
> to say they're really "is" such a number.

Consider the existence of some little "game of life" universe that is 1,000
x 1,000 cells, and there is a simple automaton in there with a rudimentary
nervous system. It can contemplate 2+2=4, but its brain and Game of Life
universe is too small for it to conceive of numbers larger than a million.
What effect, if any, would you say the multiverse theory has for the
meaninglessness or meaningfulness of different numbers, which might be
comprehensible in some universes but not in others? If the Goldbach
counterexample is too large to represent in this universe, but not others,
does it then make sense to say there is such a number? If not, what role
does physical causality/interconnectedness have to do with existence or
non-existence of numbers?

> >
> I agree, physics is more fundamental than mathematics.
> >> > 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,
> >
> >
> > That depends on exactly how you define "complexity", but you wouldn't
> need to list everything the number 3 can do to uniquely defined it, but if
> you define the complexity of something as the minimum number of characters
> needed to define it then the first billion digits of π are more complex
> than all of the digits of π since all the digits of π are uniquely defined
> by just 24 characters,  4*(1-1/3+1/5-1/7+1/9 -...), but you'd need 1
> billion characters to uniquely define just the first billion characters of
> π.

We can point to "3" quite easily, but to perfectly define 3 requires the
ability to specify all of 3's properties. Many of 3's properties are
unknowable to us with our finite mathematical theories. For example, is "3"
more or less than the number of Goldbach counterexamples? We don't
currently know. It may not even be knowable using our existing mathematics.
In fact, it could require a mathematical theory with more than 10^120
axioms to decide. There will always be such properties of 3 that are
unknowable/unanswerable, no matter how big and complex a theory of
mathematics we develop, so in a very real sense, "3" is bigger than our
universe, and much bigger than the information complexity of the first
billion characters of π. Actually "π" is a perfect example of how
mathematical objects can have greater information content than the
universe, you can't even write down the digits of π in our universe. Where
then, do the digits of π exist?

> >
> >>  > 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.
> >
> >
> > A logical system is only as strong as its weakest axiom, that's why
> axioms need to be simple and intuitively obvious.

Principles in physics don't need to be intuitively obvious. I think it may
be a mistake to require the same for mathematical axioms. It may be that
very counterintuitive axioms are required to build more powerful and
effective mathematics, which should be the only and true test of a
mathematical theory.

> We could just add the Goldbach Conjecture as an axiom but it's truth is
> not intuitively obvious and it would be very embarrassing if after doing
> this a computer crunching through huge numbers happen to find an even
> number that was not the sum of two prime numbers, it would mean all the
> work done in mathematics since the axiom had been added was meaningless.

Yes, mathematicians tend to be conservative when it comes to adding or
changing axioms, ann not altogether without good reason, but I think Gödel
was on to something when he said empiricism has a place in mathematics.

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