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thelastmemeera · 28 days ago

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Hello!! As someone with essentially no formal training who taught themselves weird math no one cares about and now has published papers in a journal no one reads, I may be able to offer assistance here! Obligatory disclaimer: this is how I personally learn, I can't promise it's the best style for anyone else. Some of this is directly contrary to what I understand to be the "traditional" pedagogical methodology, but in my opinion that's necessary to a certain degree if you're going to be working on your own.

Gather as many sources as possible. I mean papers, textbooks (there are actually a ton of free textbooks available online), online course notes, youtube videos, answers on math exchange/overflow, anything you can get your hands on (obviously take some of these with a grain of salt) - if you're like me and don't know anyone else into math, you're going to have to get creative sometimes. But one of the beautiful things about math is there's almost always multiple different approaches to every problem. If you're having trouble understanding something, try to find something else that explains it differently.

Don't be afraid to engage with difficult material - and equally don't be afraid to disengage. This could just be the ADHD, but I learned/learn math by repeatedly diving in way above my head until things start clicking. You should always put a solid effort into trying to understand everything you're reading, but realistically that's often not going to happen right away and that's okay. If there's a paper that looks interesting but you're not sure you'll get it, just read until it stops making sense and then switch to a different source. Revisit it again when you feel you've got a slightly better grasp on the subject matter, and you might find you can get a bit further. Keep doing this; be persistent.

It helps to have a specific problem in mind. Learning math is typically very difficult without motivation in the sense of understanding the context of whatever it is you're learning. Like for example, matrix multiplication seems really strange and arbitrary until you figure out it's just composition of linear maps between vector spaces. Find a problem you're genuinely interested in - it could be something very concrete like "how many possible knight's tours are there on a chessboard?" or something very abstract and general like "why is differential topology so much harder in 4 dimensions than ≥5 dimensions? Shouldn't it be the other way around?" - and embark down the rabbit hole of figuring out everything you need to know to get a satisfying answer. Some lesser known math youtube channels I've found super helpful: VisualMath Math Curator Zanachan Richard Behiel eigenchris Richard E Borcherds - for some reason tumblr is not letting me embed this link but here it is: https://www.youtube.com/@richarde.borcherds7998

this might be a dumb question but like. how do you learn math without a class/curriculum to follow. i have a pretty solid calculus understanding and I want to pursue more advanced math but like im not sure where to start. what even is like category theory it sounds so cool but so scary???. do you have any recommendations on specific fields to begin to look into/whether its best to learn via courses or textbooks or lectures/etc.? any advice would be super appreciated!! dope blog by the way

thanks for the compliment!

first of all it's not a dumb question. trust me i'm the algebraic-dumbass I know what I'm talking about. okay so uh. how does one learn math without a class? it's already hard to learn math WITH a class, so uhhh expect to need motivation. i would recommend making friends with people who know more math than you so you have like, a bit more motivation, and also because math gets much easier if you have people you can ask questions to. Also, learning math can be kind of isolating - most people have no clue what we do.

That said, how does one learn more advanced math?

Well i'm gonna give my opinion, but if anyone has more advice to give, feel free to reblog and share. I suppose the best way to learn math on your own would be through books. You can complement them with video lectures if you want, a lot of them are freely available on the internet. In all cases, it is very important you do exercises when learning: it helps, but it's also the fun part (math is not a spectator sport!). I will say that if you're like me, working on your own can be quite hard. But I will say this: it is a skill, and learning it as early as possible will help you tremendously (I'm still learning it and i'm struggling. if anyone has advice reblog and share it for me actually i need it please)

Unfortunately, for ""basic"" (I'm not saying this to say it's easy but because factually I'm going to talk about the first topics you learn in math after highschool) math topics, I can't really give that much informed book recommendations as I learned through classes. So if anyone has book recommandations, do reblog with them. Anyways. In my opinion the most important skill you need to go further right now is your ability to do proofs!

That's right, proofs! Reasoning and stuff. All the math after highschool is more-or-less based on explaining why something is true, and it's really awesome. For instance, you might know that you can't write the square root of 2 as a fraction of two integers (it's irrational). But do you know why? Would you be able to explain why? Yes you would, or at least, you will! For proof-writing, I have heard good things about The Book of Proof. I've also heard good things about "The Art of Problem Solving", though I think this one is maybe a bit more competition-math oriented. Once you have a grasp on proofs, you will be ready to tackle the first two big topics one learns in math: real analysis, and linear algebra.

Real analysis is about sequences of real numbers, functions on the real numbers and what you can do with them. You will learn about limits, continuity, derivatives, integrals, series, all sorts of stuff you have already seen in calculus, except this time it will be much more proof-oriented (if you want an example of an actual problem, here's one: let (p_n) and (q_n) be two sequences of nonzero integers such that p_n/q_n converges to an irrational number x. Show that |p_n| and |q_n| both diverge to infinity). For this I have heard good things about Terence Tao's Analysis I (pdf link).

Linear algebra is a part of abstract algebra. Abstract algebra is about looking at structures. For instance, you might notice similarities between different situations: if you have two real numbers, you can add them together and get a third real number. Same for functions. Same for vectors. Same for polynomials... and so on. Linear algebra is specifically the study of structures called vector spaces, and maps that preserve that structure (linear maps). Don't worry if you don't get what I mean right away - you'll get it once you learn all the words. Linear algebra shows up everywhere, it is very fundamental. Also, if you know how to multiply matrices, but you've never been told why the way we do it is a bit weird, the answer is in linear algebra. I have heard good things about Sheldon Axler's Linear Algebra Done RIght.

After these two, you can learn various topics. Group theory, point-set topology, measure theory, ring theory, more and more stuff opens up to you. As for category theory, it is (from my pov) a useful tool to unify a lot of things in math, and a convenient language to use in various contexts. That said, I think you need to know the "lots of things" and "various contexts" to appreciate it (in math at least - I can't speak for computer scientists, I just know they also do category theory, for other purposes). So I don't know if jumping into it straight away would be very fun. But once you know a bit more math, sure, go ahead. I have heard a lot of good things about Paolo Aluffi's Algebra: Chapter 0 (pdf link). It's an abstract algebra book (it does a lot: group theory, ring theory, field theory, and even homological algebra!), and it also introduces category theory extremely early, to ease the reader into using it. In fact the book has very little prerequisites - if I'm not mistaken, you could start reading it once you know how to do proofs. it even does linear algebra! But it does so with an extremely algebraic perspective, which might be a bit non-standard. Still, if you feel like it, you could read it.

To conclude I'd say I don't really belive there's a "correct" way to learn math. Sure, if you pursue pure math, at some point, you're going to need to be able to read books, and that point has come for me, but like I'm doing a master's, you can get through your bachelor's without really touching a book. I believe everyone works differently - some people love seminars, some don't. Some people love working with other people, some prefer to focus on math by themselves. Some like algebra, some like analysis. The only true opinion I have on doing math is that I fully believe the only reason you should do it is for fun.

Hope I was at least of some help <3

#math #mathblr

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themathblogofsashanova · 3 years ago

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August 26, 2022: Transpositions, Commutativity, Association, and Disjointedness; False and True Mirror Worlds

Imagine a space where you can see the form of time from some standpoint, and imagine if you can only see if it you can tell it apart from a similar form that does not result in a second mirror world eating you up when you weren't noticing the details. This is how my continued foray into the following paper feels.

We begin with a quick clip by one of favorite mathematicians, Cohl Furey. At the time when I found out about her, 2018, I didn't have the spoons, the space, the time or the safety to further pursue my interest in what she taught. It is true that apolitical positions are privileged positions, and that some people are successfully perched on a safety that allows them to pretend like evading the boiling lava of politicization is a choice. That, or they are just completely blind to their own political positions and consequences, or simply in denial. Only now I have I finally ended the relationship that took me from several opportunities in analyticity. I revisit her work.

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Her teaching is void of unnecessary distraction, organized, preplanned, clear, and distinctly moving forward with an insight for the student in mind. She shows high comprehensive skill therefore.

Still curious about division algebras, I found out that they are fields that are associative, aka, more or less coupled with their cancelling function with the exception of the Cayley numbers. Oddly, these are another name for the octonions, though apparently they were discovered independently.

In addition, Cayley numbers are non-associative. That means that changes in groupings have real effects, even if the contents are the same. It sounds like what I am learning about in complexity science and emergence. In addition to being non-associative, we also have to remember that as a division algebra the multiplication operation doesn't necessarily commute, meaning that direction of multiplication may have an effect as well. So, so far we know there are real effects on direction and group. When I hear that, I immediately think time--direction going to physics, group going to finance, and the link between the two going to complexity theory.

Moving on, we touch on involution--where I take to mean the shape of something coupled with its functional instantiation results in the identity. My hypotheses are that this identity can be either considered the (a) category or (b) some kind of pivotal location necessitated by the system constraints for this form to both exist and operate. It is dark, but I think of a gothic cathedral, and a gothic cathedral when it is on fire, operating, made of stuff. Genes, and genes when they are in the burning, mortal version of a human life.

Learning about involution and inversion, I couldn't help but to think how a shift in direction of an operation related to the inverting function. It seems to me inversion is just multiplicative commutation but on a different dimension. I often love to teach inversion to my kids using the fraction's bar as the mirror, and the fractions jump across the mirror world to find themselves in the identity function 1. Interestingly, the identity function is, as we see in the following video, also the first generator of a potential Cayley graph...this is me looping back to how Cayley category theory ties into graph theory.

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This piece by Visualmath is amazing not only because it is visual, and most mathematics is best taught with the end in mind (the final picture; I always view the computational approach as an artist who sketches a lot of lines while drawing each individual piece of a picture, which is more than a little energetically expensive, and I view a mathematician as one who draws in a preplanned, certain way) but because I love that he laughs and says intimidating elements of math are actually really simple and easy to understand. I wish more math was taught this way. In fact, I wonder why it isn't?

Some things to note: I will be noting one after another the additions I have to make to my original reading to show why autodidactic, thorough mathematical endeavor has a serious complexity issue that I don't think the university system solves.

Initial inquiry http://www.csun.edu/~asethura/papers/DegreeDetVars.pdf

Resulting supporting inquiries while trying to solve the initial inquiry to be read as supplement while reading 1, added August 26, 2022 (https://inspirehep.net/files/1e02ad81af7a278f19b217e99715a973)

https://www.researchgate.net/publication/308943414_Commutativity_Theorems_in_Rings_with_Involution

Supplementary reading: Applications of category theory, to be looped into each entry as an extra challenge.

Other items to note; subvariety is a smaller item in a larger field. The question of codimension is how many dimensions are not shared...I find this an interesting use of the word co. So essentially, "what is the distance between that I am, and that which contains me, dimensionally speaking?" What's even more interesting is that, if I understand correctly, which I probably don't, the smallest possible difference between a subvariety and a field is the field's dimension, and the larger the subvariety becomes, the "greater" the codimension terminologically speaking, while it is actually numerically smaller. All I can think to explain this is that the term is referencing how much space they mutually compare. To me this seems like an inverse function. Anyone else irked by this?

Finally, I was also interested in the transposition here and how it relates to commutativity, associativity, and the real effect of direction and group--perhaps the group formed by the traces themselves.

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Well, we're one small paragraph into a six page paper, and already in this deep. I've always been one for a steep learning curve though...I've been celebrating completing my financial analysis specialization! My favorite quote or something like it being finance is, in the end, the study of time. Forever the metaphysician.

#Cayley theory #group theory #graph theory #octonions #transposition #math #math journal #mathematics #category theory #symmetry

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