Re: it's a conceptual zoo out there

On 06/23/2024 07:07 AM, Ross Finlayson wrote:

On 06/23/2024 05:32 AM, FromTheRafters wrote:

sobriquet pretended :

In particle physics, people used to refer to the particle zoo since
there was such a bewildering variety of elementary particles that were
being discovered in the previous century.
Eventually things got reduced to a relatively small set of fundamental
fermions and bosons and all other particles (like hadrons or mesons)
were composed from these constituents (the standard model of particle
physics).
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Can we expect something similar to happen eventually in math, given
that there is a bewildering variety of concepts in math (like number,
function, relation, field, ring, set, geometry, topology, algebra,
group, graph, category, tensor, sheaf, bundle, scheme, variety, etc..).
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https://www.youtube.com/watch?v=KiI8OnlBTKs
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Can we kind of distinguish between mathematical reality and
mathematical fantasy or is this distinction only applicable to an
empirical science like physics or biology (like evolution vs
intelligent design)?

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I don't think so because regarding physics there is one goal, to model
reality, and I believe only one reality to deal with. With mathematics
there are endless abstractions such as the idea of endlessness itself in
its many forms.

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As models of relation, of course mathematics is associated
with any matters of definition and reason, as logic is,
or also il-logic, about the realm of the abstract.
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Then one may wonder there's a "Plato's ideal realm, of
mathematical objects", that it's a wider universe than
Earthly matters, and also whether it's the same.
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Besides the usual notion of analysis as "mathematics
of the infinite", i.e., of analytical character that's
only so in the infinite limit, then also the non-standard
analysis, has as its goal again: real analytical character.
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A nice thing about Goedel's incompleteness is "at least
it's open".
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Then, whatever matters of real analytical character that
mathematics provides, can also automatically equip physics.
(The theory, "mathematical physics".)
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About model theory and various theories of mathematical
objects regarding relations, numbers and counting, and
points and spaces, the equi-interpretable parts are
equi-interpretable.
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Then that various number-theoretic conjectures about
number theory with standard integers are un-decide-able,
i.e. independent, including that there are standard integers,
is great.
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Like all things Great it's Sublime.
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Seems he starts talking about "operator calculus".
https://en.wikipedia.org/wiki/Operational_calculus
Yeah, Groethendieck, or "Groot-en-deek", universes,
kind of aren't, and various conjectures in them
are independent, number theory, and furthermore,
number theory, is independent them, the algebraist's
"algebraic geometry", where there's a great divide
in "algebraic geometry" between geometers and algebraists,
then besides, there are non-standard models, about
that sometimes these "stronger" theories are sometimes
"weaker", you see.
The operator calculus is a great idea it's basically
applying continuous transforms over infinite ranges,
and reducing it to single terms, resulting neat analysis.
You might recall us here discussing topologies
and "relations that hold in all topologies", vis-a-vis,
those that don't.
About 32:48 he's talking about "completeness" which
is about making closed forms from infinite expressions,
which of course is pretty much the goal of analysis,
to arrive at algebraic forms that are manipulable their
quantities, safely.
"... the theory is very subtle ...", euh, ....
"Falting purity" is this idea that "modern algebraic
geometry is lacking Falting purity, i.e. it's dirty".
If going to Groethendieck then back to Hausdorff it's
like "Hausdorff has a strong geometrical attachment".
So, I could imagine you could call most those sorts
of approaches "conservative extensions, under equi-
interpretability", then some have that those are
particularly _sub_-fields of mathematics.