Re: unification programs in math

On 02/22/2025 02:32 PM, sobriquet wrote:

Op 22/02/2025 om 22:07 schreef Ross Finlayson:

On 02/22/2025 12:33 PM, sobriquet wrote:

>
It seems that programs like the Erlangen program or the Langlands
program seek to unify math by bridging certain realms, like number
theory, algebra, geometry and topology.
>
https://ncatlab.org/nlab/show/Erlangen+program
https://ncatlab.org/nlab/show/Langlands+program
>
https://www.youtube.com/watch?v=xuLCPv6smwo
>
Will AI be helpful in condensing math concepts in a unifying
framework? On an abstract level natural language and mathematics seem
similar in the sense of a large network of related
notions/ideas/concepts, where we seek to differentiate and identify
things in an optimal fashion.
Naively you would think that if you want to master a particular topic,
like differential geometry, you should be able to feed hundreds of books
on the topic or closely related topics into an LLM so it can crunch it
down to a single comprehensive overview that can be explored
interactively, where the level of detail in the presentation can be
customized based on your level of understanding and your background
knowledge.

>
How about "strong mathematical platonism",
with regards to a "heno-theory", that makes
bridges as you mentioned, bridge results,
what results each of the "fundamental" theories,
is one theory.
>
Then, "theories of one relation", like set theory
and part theory and ordering theory and identity,
have various ways of looking at them as fundamental,
then though that the resolution of mathematical paradox
makes to arrive at the extra-ordinary, of course.
>
The "strong" and "weak" are not necessarily reflective
terms in mathematics, and "growth" is sometimes "in-growth".
>
The, "mathematical platonism" is the usual historical
account of "a mathematics, the mathematics".
>
>

>
 From AlphaTensor to AlphaSheaf?
>
https://www.youtube.com/watch?v=TUJ9tHl4_vw
>

 From "Nothing" to "all of Mathematics", ....
You mention a great facility in report-writing
and a rich and fluid interrogative, with various
actors and agents in ecosystems including large
language models, yet, it can be much simpler than
that since all of mathematics since antiquity is
simply human work.
The fundamental question of metaphysics, where
there is something rather than nothing, then
that via axiomless natural deduction that there
arises an axiomless geometry and then furthermore
the theory of words or grammar, Comenius language, say,
is quite a usual sort of exercise for thinking beings
to establish that mathematics is what it is.
Mechanical inference can be great, sure. It may
remind one of the quoted difference between C and C++:
"in C++ it's harder to shoot yourself in the foot,
yet when you do it takes your whole leg off".
Then with regards to mechanical inference and as
well with regards to novel computing architectures,
have them explain axiomless natural deduction and
a heno-theory, a strong mathematical platonism that
also enforces a strong logicist positivism, a science.
The science, ....