it's a conceptual zoo out there

Post by sobriquet
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..).
http://youtu.be/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)?

There still is a "Particle Zoo", there's a "Wave Zoo", too.
"Resonance theory" is above wave theory about "particle-wave"
duality. Energy's conserved, yet it's also in a form, and
the interactions and transitions are various.

I basically look at it as geometry, arithmetic, algebra,
number theory, topology, and function theory, operator calculus.

I.e. the discrete is embedded in the continuous.

Ross Finlayson

2024-06-23 04:09:39 UTC

It's mostly the milieu of model theory what relates among them.
Logically then that's mostly relations.
The "stronger" and "weaker" theories
are both "weaker" and "stronger",
proving more with less, less with more,
less with less, more with more.
Then "Foundations" has the usual goal of being "a 'the' theory".
"A Theory"
http://youtu.be/WT7yUJYtTz8
There still is a "Particle Zoo", there's a "Wave Zoo", too.
"Resonance theory" is above wave theory about "particle-wave"
duality. Energy's conserved, yet it's also in a form, and
the interactions and transitions are various.
I basically look at it as geometry, arithmetic, algebra,
number theory, topology, and function theory, operator calculus.
I.e. the discrete is embedded in the continuous.

Here's some reading including from d'Espagnat and Badiou
about that there really is a "the mathematics" to attain to,
not just poesie and metaphor, yet a real strong lyrical metonymy.

It's a concept.

https://en.wikipedia.org/wiki/Concept

Kind of like Gadamer does, Badiou in his "Second Manifesto"
also arrives at a sort of stronger "amicus Plato".
Well, sort of, ..., for a while.

Then for constant, consistent, complete, and concrete,
for constancy and consistency and diversity and variety,
then for physics in the tendencies and propensities of
the oscillating and restitutive and dissipating and attenuative,
and as for symmetry-flex, quasi-invariance, and continuity laws
for symmetry-breaking, invariance, and conservation laws, that
are fuller symmetries, invariances, and laws.

The Higgs Boson: these days has also for Big Higgs, Little
Higgs, and sort of anti-Higgs, supersymmetr(ies), plural.
So, "unification", where "all energy is the same",
is contrived. Yet, there's still to be said that
the four sorts quantities of physics, arrange each
other about and around, in a field theory,
sum-of-histories, sum-of-potentials.

FromTheRafters

2024-06-23 12:32:18 UTC

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..).
http://youtu.be/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.

Ross Finlayson

2024-06-23 14:07:34 UTC

Post by sobriquet
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..).
http://youtu.be/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)?

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.

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.

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.

A nice thing about Goedel's incompleteness is "at least
it's open".

Then, whatever matters of real analytical character that
mathematics provides, can also automatically equip physics.
(The theory, "mathematical physics".)

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.

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.

Like all things Great it's Sublime.

Ross Finlayson

2024-06-23 14:42:41 UTC

Post by sobriquet
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..).
http://youtu.be/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)?

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.
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.
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.
A nice thing about Goedel's incompleteness is "at least
it's open".
Then, whatever matters of real analytical character that
mathematics provides, can also automatically equip physics.
(The theory, "mathematical physics".)
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.
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.
Like all things Great it's Sublime.

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.

Ross Finlayson

2024-06-23 15:34:59 UTC

Post by sobriquet
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..).
http://youtu.be/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)?

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.
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.
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.
A nice thing about Goedel's incompleteness is "at least
it's open".
Then, whatever matters of real analytical character that
mathematics provides, can also automatically equip physics.
(The theory, "mathematical physics".)
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.
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.
Like all things Great it's Sublime.

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.

The "almost mathematics" about "Falting's purity" is
that in the "later Bourbaki" "algebraic geometry", is
arrived at expressions that use measure-theoretic results
to establish "almost everywhere: a.e.", with regards to
completion, so that the result of a completion of an
infinite expression, can be said to "almost hold".

So, that's of course not the same thing as saying "holds",
that "almost everywhere" is then made a "manner-of-speaking",
that there's a sense that the countable completion makes
some finite measure, instead of "measure zero".

These days then "almost everywhere" shows up a lot,
that then the applicability then of treating the resulting
expression as an algebraic quantity, is carried forward
then treating it as a "whole quantity", about algebra
and quantities, yet, like many other notions like
"small-angle approximation" or any sort of numerical
method, linearizations or here "linearisations" to
indicate "almost linearizations", it's only a manner
of speaking.

It's about the same sort of manner of speaking as
"the differential's an infinitesimal and its sum
is perfect", though different. It results after
the sort of fact that that's a given, making that
it's a given again, though, how it's arrived at
basically does not meet and does not hold, that
it's all soundly so.

So, "almost mathematics" is considered a bit "dirty".

Then, "almost mathematics" shows up in a lot of other
places than algebraic geometry as "almost everywhere: a.e.".

For something like Cantor space, Borel and Combinatorics
quite disagree about at least some features of it,
one with "almost-everywhere" the other "almost-nowhere".
Yet, no-one disputes their inductive arrival, it's
instead some deductive analysis to arrive at that
there's independence of number theory from standard infinity,
as that there is the rulial and regular again,
multiple law(s) of large numbers.

Then as with regards to measure theory and Vitali and
Hausdorff and so on, it is so that "geometer's algebraic
geometry" and "algebraist's algebraic geometry", are
two different sub-fields, of mathematics.

Vis-a-vis "the mathematics: the universe of mathematical
objects: 'Hilbert's Infinite Living Working Museum,
of Mathematics'".

Ross Finlayson

2024-12-26 19:32:26 UTC

Post by sobriquet
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..).
http://youtu.be/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)?

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.
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.
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.
A nice thing about Goedel's incompleteness is "at least
it's open".
Then, whatever matters of real analytical character that
mathematics provides, can also automatically equip physics.
(The theory, "mathematical physics".)
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.
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.
Like all things Great it's Sublime.

sobriquet

2024-06-23 15:37:21 UTC

Post by sobriquet
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..).
http://youtu.be/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 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.

Mike Terry

2024-06-23 16:05:42 UTC

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..).
http://youtu.be/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.
http://youtu.be/DxCWRAT0WKc

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.

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.).

Mike.

Ross Finlayson

2024-06-23 20:42:24 UTC

Post by Mike Terry

Post by sobriquet
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..).
http://youtu.be/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.
http://youtu.be/DxCWRAT0WKc

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.
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.).
Mike.

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.)

sobriquet

2024-06-29 23:50:04 UTC

Post by Mike Terry

Post by sobriquet
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..).
http://youtu.be/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.
http://youtu.be/DxCWRAT0WKc

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.
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.).
Mike.

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..).

FromTheRafters

2024-06-23 17:14:00 UTC

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..).
http://youtu.be/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 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.
http://youtu.be/DxCWRAT0WKc

Another example is/was Fermat's Little Theorem which was thought at the
time to be a 'gewgaw' or a 'bauble' but now is important in information
theory.

Ross Finlayson

2024-06-23 21:04:46 UTC

Post by sobriquet

Post by sobriquet
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..).
http://youtu.be/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)?

"Knot Theory is Impossible Without These 9 things" - Di Beo's
http://youtu.be/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

Stefan Ram

2024-07-19 14:06:23 UTC