Naked Science Forum
General Discussion & Feedback => Just Chat! => Topic started by: varsigma on 28/02/2024 09:30:53
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Despite my obvious disdain for what passes for the usual sense of philosophers, the meaning of the word does still have a certain, um, categorical significance.
I'll say that the Theory of Categories does seem to be a kind of mathematical heaven, for people with PhDs.
I get that impression watching some lecturers bang on about modules and algebras and what have you.
I think some of the female lecturers are very good, reassuring about all the abstract ideas that her students will be tested on, so it's about the things they need to figure out by themselves, given what they know about rings, fields, groups etc. It's a bit like psychiatry for math majors.
Anyways, does anyone here have any interest in this stuff, or some understanding of how far it all goes, this attempt to codify all mathematical objects and relations between them? or how doing this appears to simplify things considerably, in that mathematics is actually mind bogglingly simple? The general linear group is everywhere, as they say>
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Hi.
Anyways, does anyone here have any interest in this stuff
Some interest but not a lot of expertise, sorry.
or some understanding of how far it all goes, this attempt to codify all mathematical objects and relations between them?
Mathematicians will always be keen to identify what their most fundamental mathematical objects must be. Physicist's are keen to know what their most fundamental physical object or "atoms" would be, mathematicians are much the same.
or how doing this appears to simplify things considerably, in that mathematics is actually mind bogglingly simple?
It is simple and Mathematicians are probably only working with and manipulating a very small class or category of structures, all built from what are probably just a small set of atomic objects.
On the other hand, the English language is just 26 different characters with some rules on how to combine them - how much more simple can you get? Yet, you can still generate propaganda that will start wars between nations, poetry that can help to end them and works of philosophy that examines the most profound issues of existence. Computers require only two atomic elements, a 0 (or "false", "off", or whatever you wish to call it) and a 1. With that they can capture and display photographs or the audio from an orchestra, or seemingly understand and respond to questions as demonstrated by ChatGPT.
How many different categories or types of structures do you think we would need?
Best Wishes.
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Mathematicians will always be keen to identify what their most fundamental mathematical objects must be. Physicist's are keen to know what their most fundamental physical object or "atoms" would be, mathematicians are much the same.
Well ok. It does seem to be the case that Category Theory has been hailed a success. It isn't "there" yet, likely since nobody knows where "there" is. But I would say it's at least as successful as modern Information Theory, with it's languages and finite state machines to go with. That is, information entropy is also an algebraic entity, no surprise there really because information is not a continuous field. Ahem.
The continuous part of IT is about communicating the stuff, and the Categorists want IT to be understood as part of their lexicon. I don't see a problem. Someone tell me there isn't a problem (just kidding)
It is simple and Mathematicians are probably only working with and manipulating a very small class or category of structures, all built from what are probably just a small set of atomic objects.
Indeedly. I wrote down a really nice way to construct a tensor product of group algebras, the groups are the symmetric groups namely S2, and S1. This is the trivial case, so you start down the path of induction into larger symmetric groups. There is a vector space that uses an S3 "distinguished" basis, i.e. you have the blindingly obvious (i.e. simple) {1,2}x{1} = {1,2,3}, a "set product", this is also pretty nice. Some permutation diagrams since I want the basis {1,2,3} and Perm{1,2,3}, the set of permutations.
All this is succinctly represented in some diagrams, with vertical and horizontal composition corresponding to inner "vector" and tensor multiplication, resp. with restrictions on the sets. You also invoke the categories Ind and Res as cocategories, it's like a one-page construction of a bunch of highly simplified mathematical thingies.
So yeah, let the good times roll
p.s. I didn't mention that it's been like the way I've kept my keyboard and guitar skills, through self learning. I'm self learning "higher" algebra, and discovering it isn't so hard. It is though, a bit like telling yourself to ignore that voice that says you won't do it.
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I'd like to apologise for the monologue here.
But I find that attempting to be expository about what I think I'm doing can be useful, so I'll carry on.
So the recipe if you will, for constructing a module and landing in the monoidal category of objects, starts with a permutable set, X.
My choice is the set of edges and vertices of an icosahedron, along with a map of four colors, because I have a puzzle made out of plastic which is a physical realization of the set.
If you've seen it, one of the Rubik's Icosahedron has a removable piece, when it's in place all the pieces are fixed.
This removable piece is the key to the notion of a group action, on three colored edges of the hole.
Moreover, since the hole has no color, its orbit is 1. The orbit of the empty face is the unit in the module.
The set of edges for the remainder of the faces has a 2-color map, so there is a subset of faces with a color 'vector space' where the group acting on edges is S2.
Finally, each face except the hole has a number from {1,2,3,4,5}. There is a red, a green, a blue and a yellow set of these five numbers. Since there is no order for colors, the numbers are similarly not ordered over any fixed sequence on the sphere. That is, a set of numbers is ordered by inclusion (the relation ≤), but colors don't have this, red isn't less than yellow, etc.
The only fixed element is the hole. You require this group action, but can you generalize the algebra so you can have the group acting anywhere, not just on the orbit of '1' . . .? Maybe, but it would sure be hard to engineer it physically, I would hazard. Is this a restriction I see? Not in mathematics, because doing unphysical things isn't an engineering problem.
Ed. I've just spent some minutes looking for an image of the puzzle I'm using for Perm{X}. I could take a photo and post it if anyone wants.
More: the icosahedral puzzle I have, is not as interesting in terms of finding a solution--there isn't one. You can do things like try to permute all the faces, one at a time of course, so you have no two adjacent faces with the same color, or the same number, or both. See how hard that is. Yeah, sure you want to find that out. Or maybe just not.
Notice the puzzle is 'planar'--the action moves faces around on the surface, the interior is essentially a solid ball, whereas a Rubik's cube and similar rotatable puzzles are sliced deeply.
There is a significance to this and I'm pretty sure it's relevant in terms of packing things into a space. Yes, you do want restrictions with that.
Maybe that's just me.
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Just a correction about the empty or missing face in the puzzle.
Its orbit, under the group action, is every position on the surface; there are 20. The other faces have 'equivariant' orbits, so all the orbits "sum to" one, notated as a mathbold 1, so the unit is understood to be an equivalence relation between the different orbits. But, you only need to know one of them because of the equivalence.
Moreover, although there are physical restrictions for the puzzle and its construction, no such restrictions exist for mathematics. I can choose to exchange (i.e. transpose) a pair of adjacent faces, so I get a 20-point diagram and transpositions. We're away laughing. I can also choose to give each triangular face a direction by decorating each triangle with an arrow or similar direction indicator, so demonstrating that each point in the diagram is a vector.
Then I just say these 20 points are embedded in the complex plane which is de facto a vector space, and then do some more group theory tricks, there is no problem doing this. It might all sound a bit like kids playing with toys, and yeah, having fun. The kind of fun I'd like to have is something that informs my understanding of computational algebras.
I would like to enrol again but I guess it will be part time. I'd like to talk to Vaughan Jones but he left the Auckland faculty yonks ago. Ah well.
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I'd like to apologise for the monologue here.
But I find that attempting to be expository about what I think I'm doing can be useful, so I'll carry on.
Previously, when i'd ask questions, i requested for short & simple answers.
Knowing some Expert took 15mins to type a loonnngggg detailed answer n only 2mins for the Cl0wn to read it seemed to fill me up with a sense of Guilt.
Not Anymore!
A.I. will scrape off each & every word on the world wide web someday.
ps - V all maybe Gone...
But what V said..
Shall forever Stay!
( : v : )