Re: it's a conceptual zoo out there

On 06/23/2024 08:37 AM, 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).
>
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..).
>
https://www.youtube.com/watch?v=KiI8OnlBTKs
>
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)?

>
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.

>
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.
>
https://www.youtube.com/watch?v=DxCWRAT0WKc
>

"Knot Theory is Impossible Without These 9 things" - Di Beo's
https://www.youtube.com/watch?v=DxCWRAT0WKc
Uh, sheet bend, square knot / granny knot, shoelace knot, surgeon's
knot, half-bend, ..., some say knots only exist in 3 and 7 dimensions,
about things like Camille Jordan, though one often finds that
knots are learned as shoelace-tying and fishline-tying and
for merit badges and later the profession of the, "rigger".
Betti numbers, knot-untying is a pretty usual thing, with
regards to mostly getting loose end going, then as with
regards to loops and through, there's something to be
said for crochet and yarn-work, for knot-nets vis-a-vis
bend-ends.
Ah, excuse me, bends are not hitches and hitches are not bends.
https://en.wikipedia.org/wiki/Bend_(knot)
https://en.wikipedia.org/wiki/Hitch_(knot)
There are more "knots" than "tangles". Most
"mathematical knot theory" is "tangles".
In 1881 a paper "On the analytical forms called trees",
Am. Jour. Math., reflects also calling what we'd call
"branchings" or vertices as "knots", like tree knots.
Half-Windsor, full-Windsor. Don't forget Moebius strip.
The "Gordian knot" has a usual sort of approach to
reducing a problem, yet, doesn't fix knots in knots.
I.e., it always removes one knot, yet, on average
doubles the number of knotted lines to un-knot.
"Descriptive Differential Dynamics:  dogma, doubling" - Ross Finlayson
https://www.youtube.com/watch?v=JhfoDJ0M7Tc&list=PLb7rLSBiE7F5_h5sSsWDQmbNGsmm97Fy5&index=20