Re: it's a conceptual zoo out there

Op 23/06/2024 om 22:42 schreef Ross Finlayson:

On 06/23/2024 09:05 AM, Mike Terry wrote:

On 23/06/2024 16:37, sobriquet wrote:

Op 23/06/2024 om 14:32 schreef FromTheRafters:

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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I think there is still a general trend towards unification in both
math and science.
In both cases things get discovered and explored and when things are
explored in more detail, often connections are discovered between
seemingly unrelated fields that allow one to come up with a unified
framework that underlies things that initially seemed unrelated.
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https://www.youtube.com/watch?v=DxCWRAT0WKc
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What does happen is that lecturers teach their material to students year
upon year upon year, and over time the ideas and methods are distilled
to become more efficient from a teaching perspective. Theorems that were
once long and complicated are approached in a more efficient way, and
the proofs may turn out to be quite short.  Often the shortness hides a
wealth of smaller results, but still there is a big improvement in
understandability, and the connections between areas become better
understood.
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I doubt all the above would be unified into /just/ one concept, because
they reflect different interests in what is being studied.  That doesn't
mean they won't be seen as aspects of some simpler ideas - for example
when I studied maths all the above were seen as sets.  However that
didn't mean there was just one course (on set theory) that covered all
the above - even though you might say "aha - everything is just a set,
so that's it."  (At that time category threory was a bit too new to base
the entire degree on, but I imagine these days category theory provides
a similar (better?) unififying view of the various areas, like set
theory in my study days.  But still there are numbers, topologies, sets,
manifolds, rings etc.).
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Mike.
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 The (descriptive) set theory and (descriptive) category theory
in their usual formalisms are mostly "equi-interpretable" and
"conservative" with respect to each other, though there's a
particular development in category theory about the Identity
functor [0,1] that results a sort of "clock arithmetic" about it,
that in the set-theory-wise is put off to Jordan measure and
line-elements of the line-element in usually time-ordering.
 The idea of "univalency" in "homotopy type theory" in "category
theory", is basically for "infinite union" the illative where
otherwise there's only "pair-wise" union. It's said that this
would result a category theory with "the strength of ZF together
with two large cardinal axioms", where, large cardinals are neither
cardinals nor sets, in set theory. It's sort of like the
"projective determinacy" in the set theory, though, kind
of coming down instead of up, as it were.
 So, you usually won't find anybody saying there's anything at
all different between "descriptive set theory" and "descriptive
category theory", both as applications of "model theory", as
with regards to that usually it's deemed they are "equi-interpretable"
in the sense of only modeling each other, because, anything that
doesn't, has the consequences to the validity and consistency
of the theory, with regards to non-standard and extra-ordinary things.
 Shortcuts like "Dedekind cuts" and so on, some find un-palatable.
 Most of the exploration of the logical has been in terms
of set theories, where it may be reflected that Mirimanoff's
"extra-ordinary" reflects the standard and non-standard,
while Skolem is usually ascribed to extension and collapse,
of models, in terms of the transfinite, and about whether
there's not even a "standard" model of integers, at all.
(Only extensions and fragments.)
 

Maybe sets and categories are two complementary universal ways of encoding conceptual structure (where sets are focused on the internal structure of concepts, while categories are focused on the way concepts interrelate to one another).
https://www.reddit.com/r/AncientGreek/comments/146e725/pythagoras_all_is_number/
It also seems to echo claims from math history that everything can be considered a number (kind of like the observation that all content people exchange online are just a bunch of large numbers, regardless of potential interpretations as books, images, movies, apps, songs, 3d models, etc..).
https://www.youtube.com/watch?v=p_r0bXrbddw