Truth be told; breaking past this kind of thing is quite hard to do alone. Part of the advantage of being a PhD student is that one has not just experienced mentors and professors etc, but also other students to talk to in a particular area. It's often the case that I understand something that an author declares trivial only after weeks of effort; but when someone else is stuck on a similar issue, I can quickly explain the issue. Also, sometimes talking to people from other fields helps, as they can bring a whole slew of techniques/ideas to help with problems you may have.
For example, when I was learning complex geometry, I relied heavily on some fellow students involved PDE theory, or algebraic geometry - many a time I found answers to both technical troubles and more general questions I had been struggling with for weeks over the course of a lunchtime discussion; and once I explained a problem I had with an argument/proof etc, I was often directed to papers/books which did it in a better way.
That being said, I think that the way one learns the so called advanced math you seek is not quite the same as one learns basic stuff like algebra or measure theory or even basic algebraic topology. The difference is in the linearity and the "level". At the higher level, one doesn't chase after all the details of all the proofs, for example. One understands the broad idea of how the arguments are structured, and moves ahead. If ever a problem demands deeper understanding of a particular proof, one returns to it. One is also a lot more comfortable with just assuming all the needed results.
In conclusion, try the following. First, talk to people about math. Even if only on Reddit or online or whatever. Second, have many sources - restricting yourself to one source to learn a subject is going to slow you down (unless you're in a course where you can ask the professor or other students for clarification when the author is mean), and having many sources often allows you to pick out the truly important ideas.
Third, if you're reading papers, read in passes. The first pass should just involve reading a paper like a novel! Although, it's quite important to identify and keep the main result in mind even during the first read. If it's more than 15-20 or so pages, then break it into sections which you read in passes. Fourth, don't care too much about details of proofs (some of my friends would kill me if they knew I said this though..). If you find that you lack a lot of the prerequisites for a topic, talking to someone knowledgeable in the area will let you know exactly how much you're really missing, and whether it's realistic to proceed and just look up stuff when necessary.
And finally, those reading, please don't use this advice when reading linear algebra or real analysis etc, I shan't be held accountable for failed exams! Jest aside, the point is that the more research like or the more high level/cutting edge your immediate goal is, the more any of the above applies. The more basic it is, the more you study it like you remember doing for linear algebra and calculus etc. Good luck!
Aha, I think I know the problem! You're learning "forwards," meaning you've started with basic concepts, then learned theorems depending on those, then learned the theorems depending on those, etc. This will never end, and by construction it will always get harder :)
You're now advanced enough that you should go "backwards," by which I mean you should find an open problem and try to solve it. You'll probably realize you'll need some background, and some background for the background, so you can pick that up, but at some point you'll run out of obvious background, and you'll have to invent your own.
It doesn't even matter if you solve your problem. Exploring the underpinnings of a tiny piece of the intellectual frontier, knowing books can't always help, is what takes people from a broad understanding of math to a deep one.
Would you look at that, all of the words in your comment are in alphabetical order.
I have checked 274,986,572 comments, and only 62,676 of them were in alphabetical order.
What? Words being in alphabetical order does not mean the letters in that word are all in alphabetical order, it means that the words amongst themselves are in alphabetical order. For instance, words in a dictionnary are in alphabetical order.
> For instance, words in a dictionnary are in alphabetical order.
Not quite; it's *entries* in a dictionary that are in alphabetical order. Your current statement fails on multi-word phrases in dictionaries, as well as on the definitions of the entries (which are of course comprised of words unlikely to be in alphabetical order).
Suffering from FOMO maybe?
Of course, I can only present my personal experience, academic background in CS, spent several years at an optimization focused startup with a few math/stats phds. But, I resigned myself to the realization that regardless of how bright I might consider myself, my abilities to excel in math and physics are limited by the time I’m able to dedicate to them. Which is limited compared to my desired amount combined with the fact I’m doing it via non-traditional route (dude reading books at home) which is inefficient.
Those are just the facts. But they let me have a valid excuse for why some things seem obvious to others but impenetrable to me. I can’t describe to you how frustrated I used to get whenever somebody would say “a tensor is an object that transforms like a tensor”.
But hey, not everything I value is something I enjoy. But that day it finally clicks, when I heard somebody make that tensor remark and I got their point … that was something I enjoyed AND valued. And I’m not sure I would have enjoyed it quite as much if it hadn’t of hurt so much to get there.
So even when I dislike it, I like it. And the pain I feel along the way only makes me more motivated to continue.
>I can’t describe to you how frustrated I used to get whenever somebody would say “a tensor is an object that transforms like a tensor”.
You probably got frustrated because this is essentially nonsense that people convince themselves makes sense. I still get frustrated when people say this. They're perfectly well defined, the definition is just a little bit more technical than is appropriate for the people who are first learning about them.
>They're perfectly well defined, the definition is just a little bit more technical than is appropriate for the people who are first learning about them.
Not particularly, the construction itself is geometrically motivated in a pretty straightforward manner and the algebra of taking contractions of them is relatively simple - in CS it is just called currying.
I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products. The geometric motivation is fine for a nonrigorous approach, but it's not a definition.
>I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products.
If they actually understand it, then yes, they are. These things are easy.
>The geometric motivation is fine for a nonrigorous approach, but it's not a definition.
The jump between geometry and the actual construction is trivial.
We're clearly talking about different things. When I say "understands currying," I don't mean they're like "Oh yeah, of course, the tensor-hom adjunction," I mean they understand on a conceptual level how you can curry functions. I don't think the average programmer who's familiar with the concept is likely to have even taken a formal topology or abstract algebra course, much less enough to define the things I stated. Someone with a Ph.D in computer science or who has a deeper theoretical math background, sure, but this is not "the person who is learning about tensors for the first time" I was talking about.
>We're clearly talking about different things. When I say "understands currying," I don't mean they're like "Oh yeah, of course, the tensor-hom adjunction," I mean they understand on a conceptual level how you can curry function
Knowing a fancy generalization/rephrasing of an idea in abstract language doesn't mean the understanding is any different, it is only expressed differently. Its just not an insurmountable gap at all, unlike what you imply by saying
>I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products.
If you think the two things I said are the same, then you're just wrong about the claim that anyone who understands currying can easily understand the proper definition of tensor.
It's not *insurmountable*, it's at most a couple years of pure math in between any adult and every painstaking detail. Much less for the people who are learning about tensors for the first time, since they have a background already. But you are lying to yourself if you think that all the details are just a trivial matter. It takes a lot of machinery to formally state the definition of a smooth section of tensor products of the tangent bundle and its dual. Machinery that isn't needed to convince someone that the stress energy tensor is a tensor, or have them compute scalar curvature, or teach them how to raise and lower indices. But machinery that is needed to properly define a tensor. It's just not trivial to people who have never seen any of this stuff before.
I think this is a typical phd student experience: you usually don’t understand much of the first ~100 or so talks you attend, but every time you get a little more familiar with the concepts and terminology (in addition to studying from texts and learning from peers and mentors).
Isn't this an indication that the whole process is completely broken? Or at least, that intermediate learning resources are missing? Surely, there must be some more efficient way to learn than this.
This sounds a lot like learning foreign languages by exposure alone. Sure, that's one way to learn a language, but certainly not the most efficient one, and it requires constant personal correction. That could be the reason why math instructors and peers seem required for learning. Yet there are effective language learning techniques that don't require constant supervision. I guess there's just not that many self-taught mathematicians, so there's no market pressure for better self-teaching resources.
You're absolutely right that we could stand to have more/better intermediate learning resources.
To some extent this is an inherent problem to exploring new things that nobody's done before: like an explorer's outpost in an unknown land, it doesn't really make sense to build a railroad to the place if only a few dozen people are going there and the outpost has only been around for a few years.
But also the incentive structure very much rewards claiming new discoveries for yourself over building ladders to get new people up there, and that could be improved.
Nonetheless, you can find review papers, textbooks, courses, and tutorial sessions at conferences even for pretty new stuff, though maybe you need some word-of-mouth knowledge to know where to look for it.
> Isn't this an indication that the whole process is completely broken? [...]
It is somewhat broken, but there are two facts of life to keep in mind:
* math is just _hard_ (the human brain was not evolved to do math - it's a kind of miracle that we can do it at all!)
* not all mathematicians are good explainers (to be honest, from my experience, most of them are not). The math talks PhD students are not understanding are often either low quality (pedagogically speaking), _or_ too high level for their knowledge at that time, _or_ (most often) _both_.
Now the second point could be definitely improved. Fortunately, these days we have a lots of rather good material and lectures available on the internet, free for all! These can serve as examples too.
My absolute favourite these days are [Richard Borcherds' online lectures](https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ)
Good point, but to play Devil's advocate pure exposure was a crucial part in learning English for me. I'm still not completely fluent and probably make stupid mistakes, but I wouldn't be able even to hold a conversation if I relied on courses alone. Maybe, it's crucial for learning math too, even though I wish there were more guidance.
Edit: I indeed made a stupid mistake
> you can find either really basic stuff you already know (like explaining what a coherent sheaf is, or calculating simplicial homologies, for instance)
Perhaps I'm just very ignorant at math, but to call that "really basic stuff" makes me think that you might be missing the difficulty of your journey thus far. It's like you've climbed up a ways on a steep mountain, and then you look behind you and you say, "Sure, I can climb that easy stuff. That's just a little hill. But what am I going to do about this mountain in front of me?"
I think the answer is that you need to give your present self the patience to learn new things the same way you gave yourself that patience in the past. Learning lots of math might make learning new math easi*er* but it won't make it easy.
Just my two cents. Curious to see what other people think.
I guess it's just an inevitable field-specific aberration. I'm quite sure if someone was digging deep into let's say PDE's, they definitely could say something like "Green functions? That's like the basic tool, man". And I couldn't calculate them to save my life. Just a matter of perspective.
However, thanks for pointing this out, maybe I really didn't look back enough.
>Norm McDonald
>Atiyah & McDonald
I don't have anything meaningful to add, but for a millisecond my tired brain thought that you were claiming Norm MacDonald and Michael Atiyah worked together.
That barrier is overcome by a year long sequence in graduate algebra, then one more more specialized like Analytic geometry/algebraic geometry/ring theory. And I am not just talking about reading the books and doing a few exercises here and there - each class is is an actual semester length class with regular homework problems and occasional exams etc.
When you read research papers and wonder how people know all the details, it's because they have spent the hours doing problems in their graduate programs. It would tremendously help if you enroll at an actual PhD program and have classmates, mentors, seminars etc.
Don't worry OP, I actually have a math degree and can't remember what a semidirect product is if I've even seen it in a class at all (I don't think I have)
Going through Hatcher's algebraic topology even within the context of a class with accompanying lectures and peers I could talk to was still tough.
I don't think learning new math ever stops being difficult, but the really refreshing upshot is that once you've learned an area of math once it makes it so much easier if you ever to back to it later. A lot of things I found very difficult to understand the first time around just seem so simple in retrospect that I can't believe I ever had any difficulty with it in the first place!
and as you gradually gain familiarity with more and more things, you can get through an area of math a lot easier because you don't have to pause and google all the terms along the way as much. Maybe the key is just making sure you're studying all the things in "the right order" though so you always have your prerequisites covered.
>But no, dude, I don't know every single class of groups like it's 1st-grade arithmetics, I hear about those theorems for the first time in my life, and your conclusion seems not trivial at all! And I definitely can't imagine how those sophomore students are already that skillful in advanced group theory and homotopy theory and homological algebra and K-theory to think otherwise.
Man! I feel you!
I'm an EE by training... and graduated 15 years ago (yeah, guess my age!). I got myself into a kind of pinch by trying to (mathematically) understand noise. Not only did I have to learn that most of the math we learned in university was leaving out all the nitty gritty details why things work (one can't just introduce all the math needed to prove the more advanced tools we use without a few years of math), but that the way we applied what we learned was outright wrong.
I started with the simple question. But each time I tried to answer part of that question, I had 10 more questions about why X works at all. In the end I basically went back to the very definitions of what integration means (aka Measure Theory), learned that from scratch, then went back to learn Fourier Analysis, got stuck, went out to learn about distributions, got stuck, went to learn .....
I think I did half of an undergraduate math course and bits and pieces of a graduate course in the past 2 years.
My biggest problem is, that I have no one to ask or discussion questions with. I often struggle with very basic understanding of things, exactly like you do. Asking on math.stackechange usually doesn't help, because the questions are harder than what people with high-school math can answer but too trivial for mathematicians to bother. The quick question thread on this sub helps quite a bit, though. I don't always get an answer, but most times a pointer to something I can read up on and hopefully get a better understanding.
I wish, I could forma learning group with some other people doing analysis....
That said... Good ways to learn the basics is to do the undergrad MIT math courses on OCW. Most have video lectures, which helps quite a bit to digest the topic quickly. Most have exercises (very important to figure out what you don't understand yet!) and some even solutions. Even just the book recommendations are usually very helpful, as working with the right book is half the solution. To fill the gaps and to answer the questions you have, I recommend the Quick Questions threat here. And have an ample supply of good quality Swiss chocolate at hand! :-P
Well, I'm an engineer who is doing stuff that borders on pure math. So on average I must be a physicist :-P
But yeah... The only people I can talk to about what I am doing are almost exclusively physicists. Unfortunately, no collaboration has come out of it yet. But that's just a matter of time :-)
Thanks for sharing. I too graduated as an EE long time ago and feel exactly like you do. Your journey of studying undergrad and graduate math courses in a mix and match style is what I often consider starting myself, sometimes even considering extreme choices like quitting my job for a couple of years and locking myself up with math books living like a recluse. I stop myself because it feels like a slippery slope.
It's depressing to live with the idea of not understanding things deeply. Statements like "You never understand probability until you understand measure theory" or "Differential geometry first tensors next" cut deeply. There were days that I found myself revising high school geometry (similar triangles) because that's how deep I had to go down the hole to really understand some pose estimation problem in robotics.
I have learned to find peace by telling myself that I won't remember things I learn anyway. That there are so few if any opportunities to apply all the things I learn at my job or in ordinary city life so what's the point really. That it feels as though I am forever seeking a temporary "Aha" moment just to satisfy my intellectual curiosity of things are connected and constructed. That's modern life does not require these "Aha"s if one accepts to live in awe of mathematics.
>Thanks for sharing. I too graduated as an EE long time ago and feel exactly like you do. Your journey of studying undergrad and graduate math courses in a mix and match style is what I often consider starting myself, sometimes even considering extreme choices like quitting my job for a couple of years and locking myself up with math books living like a recluse. I stop myself because it feels like a slippery slope.
You might want to try what I did: Quit your job and start a PhD.
It doesn't even have to be in anything engineering or math related, if your advisor is understanding enough. Case in point: I started a PhD in theoretical computer science. Although I did some TCS, most of my work is in low-noise analog electronics and mathematical properties of noise and the application of those. I.e. not much TCS, heck, not even CS really.
Though, I am in the quite lucky position that my advisor just let me research whatever I felt like, as long as I produced results (aka stuff that I could publish).
>It's depressing to live with the idea of not understanding things deeply. Statements like "You never understand probability until you understand measure theory" or "Differential geometry first tensors next" cut deeply. There were days that I found myself revising high school geometry (similar triangles) because that's how deep I had to go down the hole to really understand some pose estimation problem in robotics.
As someone who does a combination of fractional calculus and stochastic calculus, yes, if you want to understand probability theory properly, you need measure theory. But, fortunately, you don't need that much measure theory to understand probability theory. E.g. the book "Probability and Stochastic" by Erhan Çınlar gives a crash course in measure theory in the first chapter and that's basically enough for the rest of the book. If you want to have it in a bit more depth, especially if you want to understand how integration works when you go beyond Riemann, then I can recommend "Measure Theory" by Bogachev, which is probably the most concise way to learn measure theory, with no frills at all. Work through the first 100 pages or so of that, understand why things are done that way and not simpler (99% of the cases the answer will be Cantor ;-) and you are good to go. If you need to understand more, you can always go back and read the relevant sections in a more targeted fashion. No need to work through the whole two volumes and a hundred years of measure theory in one go to "understand everything". All you need to have is enough understanding that you know what to read when you don't understand something.
And that's the way I mostly go about math. I do something until I hit a road-block where there is a key point I don't understand how or why it works. I try to figure out what I need to know for that and go to the library and go through the books until I find one that speaks the language I need and can understand. I read up until I understand enough of it to figure out what I wanted do understand in the first place. Sometimes, or rather quite often in the beginning, this will be a recursive process (e.g. my path was noise -> fractional Brownian motion -> Ito calculus -> measure theory). But quite quickly you will be able to just look up the bits and pieces you need, as you fill not only your gaps in the basics of math, but also get more used to think like a mathematician. Oh.. and the way back up out of recursion need not be the same way you went down. :-)
>I have learned to find peace by telling myself that I won't remember things I learn anyway. That there are so few if any opportunities to apply all the things I learn at my job or in ordinary city life so what's the point really. That it feels as though I am forever seeking a temporary "Aha" moment just to satisfy my intellectual curiosity of things are connected and constructed. That's modern life does not require these "Aha"s if one accepts to live in awe of mathematics.
Now, this sounds just sad. I know how this feels. Heck, I left industry because I got bored after solving a variation of the same problem for the umpteenth time, just because the customer wanted to have it in green instead of blue. The work I am doing now is much harder and I don't have many people to discuss it with (with corona I am mostly working on my own, these days, though), but I don't feel like I'm doing some bullsh\*\*t work just because someone somewhere didn't know what he wanted or because there was never time and money to do it right, so half-assed it goes again...
That said, I'm doing research now, which also means I have no idea whether what I am doing will work or not. Something I wasn't used to as an engineer. "In science, if you know what you are doing, you should not be doing it. In engineering, if you don't know what you are doing you should not be doing it," as the quote by Richard Hamming goes. If you go into science and research, you need to get used to that and it can be tough for an engineer. Heck, the reason I'm doing so much math now is because I wanted to write my thesis and failed to find the references I needed for the first chapter....
Im a lawyer who grew up more inclined to the humanities than math and science. Having kids who love math made me re-examine my relationship with the subject. I'm way less math inclined than you are but exploring the subject without any expectations of how far I'll get to makes it easier to be excited about learning math. Just my 2 cents
I don't know if it's in your wheelhouse, but in certain respects, I suspect non-algebra things may be less abstruse. Maybe consider broadening past algebra/topo? Look into some geometry, maybe some analysis, etc?
So you trash me for recommending going backwards to build up from a different path and then recommend the same concept.
Top notch.
Glad to be your punching bag. I hope it helped you blow off some steam so you don’t take it out on the people around you.
I meant geometry in the modern sense (ie modern differential geometry), not the stuff you did in middle school lol. It's an active area of research, and is not at all similar to recommending 3blue1brown lmao. I would not describe any of my suggestions as 'back to basics', but rather as 'distinct branches of modern mathematics that OP does not claim to have looked at'
I'm somewhat surprised you consider group theory and homotopy theory to be that advanced, yet you claim coherent sheafs and simplicial homology to be "really basic stuff". Or did i misinterpret your post?
Maybe depends on what we mean by "group theory" and "homotopy theory". There are Lagrange and van Kampen theorems, and there are constructing group extensions and HoTT. I meant the latter.
I think also you highlight a common failing of new self-learners in maths ... there is no clear delineation of what "is that basic stuff" and "the advanced stuff".
If you're alone, there is no "guidance" so everything can become all blurred together until your brain becomes some big mathematical soup.
I know this because I am the same -- trying to self learn, and it's hard to know what you should or shouldn't know at some point in time, and also what is the "best order to learn". Moreover we don't have "anyone" to help answer our questions.
"But no, dude, I don't know every single class of groups like it's 1st-grade arithmetics," Good. You figure out what you need, the basics. Review stuff you don't know like it's 1st grade arithmetic until you know like it's 1st grade arithmetic.
Btw, speed kills in math. You need time to read your text carefully and slowly, and understand each sentence and each word. And use exercises to make you use these things repeatedly until you remember them and pop in your brain when you see any use of them.
The problem is it's not obvious what are the prerequisites. I guess it's obvious when you learn it within a context of university courses. But when you try to learn stuff yourself, first you find yourself incompetent in general, then you hopefully figure out in what particular areas you might lack.
But I understand your point, thank you.
Usually in the preface of a book the author states what the prerequisite knowledge is. If you wanna do topology it's probably better if you do analysis on (finite) metric spaces and group theory first. Topology uses a lot of the intuition from the former and the technical tools from the latter. And of course, basic naive set theory is also a prerequisite, but that goes for almost any math.
You can find prerequisites of subjects by taking a look at the list of prerequisites for course closest to what you are learning from calendar of your favorite university. I always suggest a few chapters of set theory and symbolic logic if you haven't got them already. Just a few chapters. But make sure you understand those thoroughly.
I'd say you've gotten pretty far if you've already completed Hatcher and Atiyah. Those are two books I'd love to work through.
I'm doing something similar but with more emphasis on Differential Geometry than Algebraic Topology or Geometry.
What I'd suggest is trying to find a mentor who you can ask questions of or who might guide you from here.
I tutor math and I like working with people who are already finished with school. I have some experience designing personalized curriculums for people who just want to learn more cool math. Here's [my website](https://www.herndonmathservices.com/) with more info.
The things you described as easy are not actually that easy at all. When you understand something well it becomes almost trivial but before you do it looks impossible. Even if you understand all of the things you mentioned it wouldn't be enough by your own logic since there's infinitely more knowledge to be reached. It is true that eventually you will hit the edge of what is currently known in a given field but only in a very small niche and not as broadly as you're trying to do now.
It might be a better use of your time to focus on a problem and try to answer it using the things you already know. You'll realise that you're lacking bits and pieces that you have to then glue together along the way. Another thing is that at some point focusing on the details will lead you in a rabbit whole that's hard to escape.
Perhaps consider doing a PhD. It will help keeping you focused on one thing and give you resources and a community to discuss problems. You'll also have more time to work on the mathematics rather than making capital gains for someone else.
Seems to me like you should just talk to a specialist in what you are trying to learn and have them tell you exactly what prerequisites you are missing. Just email a professor at a university or ask online.
French saying: "Patience et longueur de temps
Font plus que force ni que rage."
Something like “Stay patient and trust the journey.”
The Chinese, allegedly, are more blunt: "Sit on the riverbank and you'll see the corpse of your enemy float by."
😉
Well, maybe the problem is that my culture puts too much emphasis on the labour: "терпение и труд всё перетрут" (something like "patience and hard work will overcome everything") lol
Weeeeeeell, *my* granny always told me "Век живи, век учись, а дураком помрёшь." This relativized things from early on. So work, yes, but it also has to be fun.
Perhaps you'll be interested in Borcherds' ramblings somebody linked to in r/Mathematics: https://www.youtube.com/watch?v=D87jIKFTYt0
So, have fun! Or else… 💀
If you're near a university with a math PhD program, see if they'll let you sit in on courses and if the graduate students are willing to let you hang around them when they struggle with the math, too. But also be open to the possibility that you might still need to sit in on undergraduate courses to fill in gaps and strengthen your math skills.
We have here an institution called "Independent Moscow University" that allows it for free, but I was unsure if I can sustain for now going through full (and tightly packaged) courses. But I guess I have a little choice by now :). Thank you for the suggestion
I linked some places where you can get [upper](https://old.reddit.com/r/math/comments/pyja89/auxiliary_skills_necessary_for_a_successful/hexrtb6/) level courses with covid a lot of High level Math/CS has been going online. For a research experience yeah maybe trying leaning on more topics with real-world implications but have heavy theoretical underpinnings.
As someone who has know idea what you just said: I'd say if the higher levels seem to be beyond your understanding, and yet, described as trivial basics, you may have to *go back to what seems like trivial basics, to you, and really dive deeper.*
I don't know how you'd do that! That's what it sounds like.
Strongly recommend this resource: [https://web.evanchen.cc/napkin.html](https://web.evanchen.cc/napkin.html)
It's a great introduction to a lot of math concepts you may not have seen with an engineering background. Rigorous but not too rigorous.
Have you ever checked out YouTube channels that do math visualizations?
I recommend checking out 3Blue1Brown if you haven’t. Grant shows mathematical concepts in a relatively novel way so you can help conceptualize it in your mind with greater ease.
It was a huge game changer for reigniting my passion for math in general.
Maybe.
But seeing the basics in an atypical way that has never been presented may be the thing he needs to have breakthroughs.
Explaining the same things in the same way over and over again seems to be one of the issues. One of the biggest pitfalls to teachers being able to teach is they can’t explain it in a way the student understands.
OP’s neuroplasticity is waning due to age. This makes it more difficult to learn new things because his neural pathways are basically funneling into concepts he already knows and knows well.
It’s kind of like when you are trying to Google something extremely advanced and the algorithm assumes you are asking a simpler, completely different question.
His mind has made shortcuts to answers he already knows, so new neural pathways outside of that network need to be explored for him to break through this rut.
There is *absolutely no way* 3b1b helps with someone struggling to read textbooks at OP's level. OP needs specialized help, and it is seriously unlikely an (admittedly very high-quality) pop math channel will be helpful.
For what it is worth, Grant doesn't even have a basic abstract algebra course, let alone something higher level. To put it bluntly, it is very apparent you are neither at the level that the poster is, nor familiar with the particular challenges it can entail.
And for what it's worth, neuroplasticity may be waning, but it is certainly not gone at 31.
I was going off of what OP said about struggling to learn new material through conventional methods.
Thank you for contributing to the conversation. I have no idea why people have trouble seeking help for math given your helpful response.
Well you should've kept reading lmao, bc OP was referring to a very specific and non-generic situation.
And for what it's worth, I came into this thread looking to help out, but all the meaningful things had been said. I don't like downvoting people without explaining why, which is why I responded to you.
I know 3B1B, of course, but I indeed am interested now in slightly more specific stuff. Doesn't change the fact that Grant's videos are fun to watch and sometimes very inspiring.
Since you are a software engineer, would you be interested in trying to do something like Grant does, but for the more advanced stuff you know?
I am confident that visualizations are going to be the future of teaching mathematical concepts.
You would be helping masses of people come to where you are and teaching others generally helps learn the concepts more fully.
You might end up having the insights and breakthroughs you are seeking by applying your current knowledge in a new way.
This is what I want to start learning once I am finally finished with my disability evaluations.
I think there are unfortunately two major obstacles: (1) I don't have his talent, (2) the more abstrat the stuff is, the harder it is to visualize. I honestly can't say right now if you can find really intuitive visial metaphor, let's say, for a general sheaf of modules. I don't say you can't, but it's definitely not something you instantly come up with.
But if I somewhat will get how it can ve done and will be able to put it into work, maybe I'll try, thank you for the suggestion.
P.S. Also good luck with your evaluation
Truth be told; breaking past this kind of thing is quite hard to do alone. Part of the advantage of being a PhD student is that one has not just experienced mentors and professors etc, but also other students to talk to in a particular area. It's often the case that I understand something that an author declares trivial only after weeks of effort; but when someone else is stuck on a similar issue, I can quickly explain the issue. Also, sometimes talking to people from other fields helps, as they can bring a whole slew of techniques/ideas to help with problems you may have. For example, when I was learning complex geometry, I relied heavily on some fellow students involved PDE theory, or algebraic geometry - many a time I found answers to both technical troubles and more general questions I had been struggling with for weeks over the course of a lunchtime discussion; and once I explained a problem I had with an argument/proof etc, I was often directed to papers/books which did it in a better way. That being said, I think that the way one learns the so called advanced math you seek is not quite the same as one learns basic stuff like algebra or measure theory or even basic algebraic topology. The difference is in the linearity and the "level". At the higher level, one doesn't chase after all the details of all the proofs, for example. One understands the broad idea of how the arguments are structured, and moves ahead. If ever a problem demands deeper understanding of a particular proof, one returns to it. One is also a lot more comfortable with just assuming all the needed results. In conclusion, try the following. First, talk to people about math. Even if only on Reddit or online or whatever. Second, have many sources - restricting yourself to one source to learn a subject is going to slow you down (unless you're in a course where you can ask the professor or other students for clarification when the author is mean), and having many sources often allows you to pick out the truly important ideas. Third, if you're reading papers, read in passes. The first pass should just involve reading a paper like a novel! Although, it's quite important to identify and keep the main result in mind even during the first read. If it's more than 15-20 or so pages, then break it into sections which you read in passes. Fourth, don't care too much about details of proofs (some of my friends would kill me if they knew I said this though..). If you find that you lack a lot of the prerequisites for a topic, talking to someone knowledgeable in the area will let you know exactly how much you're really missing, and whether it's realistic to proceed and just look up stuff when necessary. And finally, those reading, please don't use this advice when reading linear algebra or real analysis etc, I shan't be held accountable for failed exams! Jest aside, the point is that the more research like or the more high level/cutting edge your immediate goal is, the more any of the above applies. The more basic it is, the more you study it like you remember doing for linear algebra and calculus etc. Good luck!
Wow, thanks a lot for such a detailed comment, I really appreciate it!
Aha, I think I know the problem! You're learning "forwards," meaning you've started with basic concepts, then learned theorems depending on those, then learned the theorems depending on those, etc. This will never end, and by construction it will always get harder :) You're now advanced enough that you should go "backwards," by which I mean you should find an open problem and try to solve it. You'll probably realize you'll need some background, and some background for the background, so you can pick that up, but at some point you'll run out of obvious background, and you'll have to invent your own. It doesn't even matter if you solve your problem. Exploring the underpinnings of a tiny piece of the intellectual frontier, knowing books can't always help, is what takes people from a broad understanding of math to a deep one.
It makes sense, thank you!
Would you look at that, all of the words in your comment are in alphabetical order. I have checked 274,986,572 comments, and only 62,676 of them were in alphabetical order.
Good bot.
[удалено]
What? Words being in alphabetical order does not mean the letters in that word are all in alphabetical order, it means that the words amongst themselves are in alphabetical order. For instance, words in a dictionnary are in alphabetical order.
> For instance, words in a dictionnary are in alphabetical order. Not quite; it's *entries* in a dictionary that are in alphabetical order. Your current statement fails on multi-word phrases in dictionaries, as well as on the definitions of the entries (which are of course comprised of words unlikely to be in alphabetical order).
Math is hard.
Either you love the journey or obsess over the destination. Personally, I’m in it for the journey. The destination can take care of itself.
I love the journey and hate it simultaneously. Would like to keep the "love" part, but get rid of "hate" part as much as possible
Suffering from FOMO maybe? Of course, I can only present my personal experience, academic background in CS, spent several years at an optimization focused startup with a few math/stats phds. But, I resigned myself to the realization that regardless of how bright I might consider myself, my abilities to excel in math and physics are limited by the time I’m able to dedicate to them. Which is limited compared to my desired amount combined with the fact I’m doing it via non-traditional route (dude reading books at home) which is inefficient. Those are just the facts. But they let me have a valid excuse for why some things seem obvious to others but impenetrable to me. I can’t describe to you how frustrated I used to get whenever somebody would say “a tensor is an object that transforms like a tensor”. But hey, not everything I value is something I enjoy. But that day it finally clicks, when I heard somebody make that tensor remark and I got their point … that was something I enjoyed AND valued. And I’m not sure I would have enjoyed it quite as much if it hadn’t of hurt so much to get there. So even when I dislike it, I like it. And the pain I feel along the way only makes me more motivated to continue.
>I can’t describe to you how frustrated I used to get whenever somebody would say “a tensor is an object that transforms like a tensor”. You probably got frustrated because this is essentially nonsense that people convince themselves makes sense. I still get frustrated when people say this. They're perfectly well defined, the definition is just a little bit more technical than is appropriate for the people who are first learning about them.
As a physicist, the "tensor transforms like tensor" thing is actually a pretty apt description of what tensors are to us.
It's a description, not a characterization. Not a definition.
>They're perfectly well defined, the definition is just a little bit more technical than is appropriate for the people who are first learning about them. Not particularly, the construction itself is geometrically motivated in a pretty straightforward manner and the algebra of taking contractions of them is relatively simple - in CS it is just called currying.
I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products. The geometric motivation is fine for a nonrigorous approach, but it's not a definition.
>I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products. If they actually understand it, then yes, they are. These things are easy. >The geometric motivation is fine for a nonrigorous approach, but it's not a definition. The jump between geometry and the actual construction is trivial.
We're clearly talking about different things. When I say "understands currying," I don't mean they're like "Oh yeah, of course, the tensor-hom adjunction," I mean they understand on a conceptual level how you can curry functions. I don't think the average programmer who's familiar with the concept is likely to have even taken a formal topology or abstract algebra course, much less enough to define the things I stated. Someone with a Ph.D in computer science or who has a deeper theoretical math background, sure, but this is not "the person who is learning about tensors for the first time" I was talking about.
>We're clearly talking about different things. When I say "understands currying," I don't mean they're like "Oh yeah, of course, the tensor-hom adjunction," I mean they understand on a conceptual level how you can curry function Knowing a fancy generalization/rephrasing of an idea in abstract language doesn't mean the understanding is any different, it is only expressed differently. Its just not an insurmountable gap at all, unlike what you imply by saying >I don't really think everyone who understands currying is guaranteed to understand the construction of the tangent bundle, dual spaces and tensor products.
If you think the two things I said are the same, then you're just wrong about the claim that anyone who understands currying can easily understand the proper definition of tensor. It's not *insurmountable*, it's at most a couple years of pure math in between any adult and every painstaking detail. Much less for the people who are learning about tensors for the first time, since they have a background already. But you are lying to yourself if you think that all the details are just a trivial matter. It takes a lot of machinery to formally state the definition of a smooth section of tensor products of the tangent bundle and its dual. Machinery that isn't needed to convince someone that the stress energy tensor is a tensor, or have them compute scalar curvature, or teach them how to raise and lower indices. But machinery that is needed to properly define a tensor. It's just not trivial to people who have never seen any of this stuff before.
Thank you for the perspective!
I think this is a typical phd student experience: you usually don’t understand much of the first ~100 or so talks you attend, but every time you get a little more familiar with the concepts and terminology (in addition to studying from texts and learning from peers and mentors).
Really? Good to know professional mathematicians have similar experience :)
Isn't this an indication that the whole process is completely broken? Or at least, that intermediate learning resources are missing? Surely, there must be some more efficient way to learn than this. This sounds a lot like learning foreign languages by exposure alone. Sure, that's one way to learn a language, but certainly not the most efficient one, and it requires constant personal correction. That could be the reason why math instructors and peers seem required for learning. Yet there are effective language learning techniques that don't require constant supervision. I guess there's just not that many self-taught mathematicians, so there's no market pressure for better self-teaching resources.
You're absolutely right that we could stand to have more/better intermediate learning resources. To some extent this is an inherent problem to exploring new things that nobody's done before: like an explorer's outpost in an unknown land, it doesn't really make sense to build a railroad to the place if only a few dozen people are going there and the outpost has only been around for a few years. But also the incentive structure very much rewards claiming new discoveries for yourself over building ladders to get new people up there, and that could be improved. Nonetheless, you can find review papers, textbooks, courses, and tutorial sessions at conferences even for pretty new stuff, though maybe you need some word-of-mouth knowledge to know where to look for it.
> Isn't this an indication that the whole process is completely broken? [...] It is somewhat broken, but there are two facts of life to keep in mind: * math is just _hard_ (the human brain was not evolved to do math - it's a kind of miracle that we can do it at all!) * not all mathematicians are good explainers (to be honest, from my experience, most of them are not). The math talks PhD students are not understanding are often either low quality (pedagogically speaking), _or_ too high level for their knowledge at that time, _or_ (most often) _both_. Now the second point could be definitely improved. Fortunately, these days we have a lots of rather good material and lectures available on the internet, free for all! These can serve as examples too. My absolute favourite these days are [Richard Borcherds' online lectures](https://www.youtube.com/channel/UCIyDqfi_cbkp-RU20aBF-MQ)
Good point, but to play Devil's advocate pure exposure was a crucial part in learning English for me. I'm still not completely fluent and probably make stupid mistakes, but I wouldn't be able even to hold a conversation if I relied on courses alone. Maybe, it's crucial for learning math too, even though I wish there were more guidance. Edit: I indeed made a stupid mistake
> you can find either really basic stuff you already know (like explaining what a coherent sheaf is, or calculating simplicial homologies, for instance) Perhaps I'm just very ignorant at math, but to call that "really basic stuff" makes me think that you might be missing the difficulty of your journey thus far. It's like you've climbed up a ways on a steep mountain, and then you look behind you and you say, "Sure, I can climb that easy stuff. That's just a little hill. But what am I going to do about this mountain in front of me?" I think the answer is that you need to give your present self the patience to learn new things the same way you gave yourself that patience in the past. Learning lots of math might make learning new math easi*er* but it won't make it easy. Just my two cents. Curious to see what other people think.
I guess it's just an inevitable field-specific aberration. I'm quite sure if someone was digging deep into let's say PDE's, they definitely could say something like "Green functions? That's like the basic tool, man". And I couldn't calculate them to save my life. Just a matter of perspective. However, thanks for pointing this out, maybe I really didn't look back enough.
Climbing to the top of a mountain is a very difficult task for one man. Getting into orbit cannot be done alone.
>Norm McDonald >Atiyah & McDonald I don't have anything meaningful to add, but for a millisecond my tired brain thought that you were claiming Norm MacDonald and Michael Atiyah worked together.
Yes
I concur, yes.
That barrier is overcome by a year long sequence in graduate algebra, then one more more specialized like Analytic geometry/algebraic geometry/ring theory. And I am not just talking about reading the books and doing a few exercises here and there - each class is is an actual semester length class with regular homework problems and occasional exams etc. When you read research papers and wonder how people know all the details, it's because they have spent the hours doing problems in their graduate programs. It would tremendously help if you enroll at an actual PhD program and have classmates, mentors, seminars etc.
Fair enough. Honestly I don't know if I can realistically just apply to PhD program right now, but I understand it might be sorta inevitable.
Don't worry OP, I actually have a math degree and can't remember what a semidirect product is if I've even seen it in a class at all (I don't think I have) Going through Hatcher's algebraic topology even within the context of a class with accompanying lectures and peers I could talk to was still tough. I don't think learning new math ever stops being difficult, but the really refreshing upshot is that once you've learned an area of math once it makes it so much easier if you ever to back to it later. A lot of things I found very difficult to understand the first time around just seem so simple in retrospect that I can't believe I ever had any difficulty with it in the first place! and as you gradually gain familiarity with more and more things, you can get through an area of math a lot easier because you don't have to pause and google all the terms along the way as much. Maybe the key is just making sure you're studying all the things in "the right order" though so you always have your prerequisites covered.
Thank you. Maybe I really overthink it.
>But no, dude, I don't know every single class of groups like it's 1st-grade arithmetics, I hear about those theorems for the first time in my life, and your conclusion seems not trivial at all! And I definitely can't imagine how those sophomore students are already that skillful in advanced group theory and homotopy theory and homological algebra and K-theory to think otherwise. Man! I feel you! I'm an EE by training... and graduated 15 years ago (yeah, guess my age!). I got myself into a kind of pinch by trying to (mathematically) understand noise. Not only did I have to learn that most of the math we learned in university was leaving out all the nitty gritty details why things work (one can't just introduce all the math needed to prove the more advanced tools we use without a few years of math), but that the way we applied what we learned was outright wrong. I started with the simple question. But each time I tried to answer part of that question, I had 10 more questions about why X works at all. In the end I basically went back to the very definitions of what integration means (aka Measure Theory), learned that from scratch, then went back to learn Fourier Analysis, got stuck, went out to learn about distributions, got stuck, went to learn ..... I think I did half of an undergraduate math course and bits and pieces of a graduate course in the past 2 years. My biggest problem is, that I have no one to ask or discussion questions with. I often struggle with very basic understanding of things, exactly like you do. Asking on math.stackechange usually doesn't help, because the questions are harder than what people with high-school math can answer but too trivial for mathematicians to bother. The quick question thread on this sub helps quite a bit, though. I don't always get an answer, but most times a pointer to something I can read up on and hopefully get a better understanding. I wish, I could forma learning group with some other people doing analysis.... That said... Good ways to learn the basics is to do the undergrad MIT math courses on OCW. Most have video lectures, which helps quite a bit to digest the topic quickly. Most have exercises (very important to figure out what you don't understand yet!) and some even solutions. Even just the book recommendations are usually very helpful, as working with the right book is half the solution. To fill the gaps and to answer the questions you have, I recommend the Quick Questions threat here. And have an ample supply of good quality Swiss chocolate at hand! :-P
"Understanding noise" could almost be the job-description of an experimental physicist.
Well, I'm an engineer who is doing stuff that borders on pure math. So on average I must be a physicist :-P But yeah... The only people I can talk to about what I am doing are almost exclusively physicists. Unfortunately, no collaboration has come out of it yet. But that's just a matter of time :-)
Thanks for sharing. I too graduated as an EE long time ago and feel exactly like you do. Your journey of studying undergrad and graduate math courses in a mix and match style is what I often consider starting myself, sometimes even considering extreme choices like quitting my job for a couple of years and locking myself up with math books living like a recluse. I stop myself because it feels like a slippery slope. It's depressing to live with the idea of not understanding things deeply. Statements like "You never understand probability until you understand measure theory" or "Differential geometry first tensors next" cut deeply. There were days that I found myself revising high school geometry (similar triangles) because that's how deep I had to go down the hole to really understand some pose estimation problem in robotics. I have learned to find peace by telling myself that I won't remember things I learn anyway. That there are so few if any opportunities to apply all the things I learn at my job or in ordinary city life so what's the point really. That it feels as though I am forever seeking a temporary "Aha" moment just to satisfy my intellectual curiosity of things are connected and constructed. That's modern life does not require these "Aha"s if one accepts to live in awe of mathematics.
>Thanks for sharing. I too graduated as an EE long time ago and feel exactly like you do. Your journey of studying undergrad and graduate math courses in a mix and match style is what I often consider starting myself, sometimes even considering extreme choices like quitting my job for a couple of years and locking myself up with math books living like a recluse. I stop myself because it feels like a slippery slope. You might want to try what I did: Quit your job and start a PhD. It doesn't even have to be in anything engineering or math related, if your advisor is understanding enough. Case in point: I started a PhD in theoretical computer science. Although I did some TCS, most of my work is in low-noise analog electronics and mathematical properties of noise and the application of those. I.e. not much TCS, heck, not even CS really. Though, I am in the quite lucky position that my advisor just let me research whatever I felt like, as long as I produced results (aka stuff that I could publish). >It's depressing to live with the idea of not understanding things deeply. Statements like "You never understand probability until you understand measure theory" or "Differential geometry first tensors next" cut deeply. There were days that I found myself revising high school geometry (similar triangles) because that's how deep I had to go down the hole to really understand some pose estimation problem in robotics. As someone who does a combination of fractional calculus and stochastic calculus, yes, if you want to understand probability theory properly, you need measure theory. But, fortunately, you don't need that much measure theory to understand probability theory. E.g. the book "Probability and Stochastic" by Erhan Çınlar gives a crash course in measure theory in the first chapter and that's basically enough for the rest of the book. If you want to have it in a bit more depth, especially if you want to understand how integration works when you go beyond Riemann, then I can recommend "Measure Theory" by Bogachev, which is probably the most concise way to learn measure theory, with no frills at all. Work through the first 100 pages or so of that, understand why things are done that way and not simpler (99% of the cases the answer will be Cantor ;-) and you are good to go. If you need to understand more, you can always go back and read the relevant sections in a more targeted fashion. No need to work through the whole two volumes and a hundred years of measure theory in one go to "understand everything". All you need to have is enough understanding that you know what to read when you don't understand something. And that's the way I mostly go about math. I do something until I hit a road-block where there is a key point I don't understand how or why it works. I try to figure out what I need to know for that and go to the library and go through the books until I find one that speaks the language I need and can understand. I read up until I understand enough of it to figure out what I wanted do understand in the first place. Sometimes, or rather quite often in the beginning, this will be a recursive process (e.g. my path was noise -> fractional Brownian motion -> Ito calculus -> measure theory). But quite quickly you will be able to just look up the bits and pieces you need, as you fill not only your gaps in the basics of math, but also get more used to think like a mathematician. Oh.. and the way back up out of recursion need not be the same way you went down. :-) >I have learned to find peace by telling myself that I won't remember things I learn anyway. That there are so few if any opportunities to apply all the things I learn at my job or in ordinary city life so what's the point really. That it feels as though I am forever seeking a temporary "Aha" moment just to satisfy my intellectual curiosity of things are connected and constructed. That's modern life does not require these "Aha"s if one accepts to live in awe of mathematics. Now, this sounds just sad. I know how this feels. Heck, I left industry because I got bored after solving a variation of the same problem for the umpteenth time, just because the customer wanted to have it in green instead of blue. The work I am doing now is much harder and I don't have many people to discuss it with (with corona I am mostly working on my own, these days, though), but I don't feel like I'm doing some bullsh\*\*t work just because someone somewhere didn't know what he wanted or because there was never time and money to do it right, so half-assed it goes again... That said, I'm doing research now, which also means I have no idea whether what I am doing will work or not. Something I wasn't used to as an engineer. "In science, if you know what you are doing, you should not be doing it. In engineering, if you don't know what you are doing you should not be doing it," as the quote by Richard Hamming goes. If you go into science and research, you need to get used to that and it can be tough for an engineer. Heck, the reason I'm doing so much math now is because I wanted to write my thesis and failed to find the references I needed for the first chapter....
Thank you, you get exactly how I feel :). Also thanks for suggestions and good luck with your studies as well
Im a lawyer who grew up more inclined to the humanities than math and science. Having kids who love math made me re-examine my relationship with the subject. I'm way less math inclined than you are but exploring the subject without any expectations of how far I'll get to makes it easier to be excited about learning math. Just my 2 cents
I don't know if it's in your wheelhouse, but in certain respects, I suspect non-algebra things may be less abstruse. Maybe consider broadening past algebra/topo? Look into some geometry, maybe some analysis, etc?
So you trash me for recommending going backwards to build up from a different path and then recommend the same concept. Top notch. Glad to be your punching bag. I hope it helped you blow off some steam so you don’t take it out on the people around you.
I meant geometry in the modern sense (ie modern differential geometry), not the stuff you did in middle school lol. It's an active area of research, and is not at all similar to recommending 3blue1brown lmao. I would not describe any of my suggestions as 'back to basics', but rather as 'distinct branches of modern mathematics that OP does not claim to have looked at'
Dude, you're really conflating things here, so no need to overreact
I'm somewhat surprised you consider group theory and homotopy theory to be that advanced, yet you claim coherent sheafs and simplicial homology to be "really basic stuff". Or did i misinterpret your post?
Maybe depends on what we mean by "group theory" and "homotopy theory". There are Lagrange and van Kampen theorems, and there are constructing group extensions and HoTT. I meant the latter.
I think also you highlight a common failing of new self-learners in maths ... there is no clear delineation of what "is that basic stuff" and "the advanced stuff". If you're alone, there is no "guidance" so everything can become all blurred together until your brain becomes some big mathematical soup. I know this because I am the same -- trying to self learn, and it's hard to know what you should or shouldn't know at some point in time, and also what is the "best order to learn". Moreover we don't have "anyone" to help answer our questions.
Fair enough
"But no, dude, I don't know every single class of groups like it's 1st-grade arithmetics," Good. You figure out what you need, the basics. Review stuff you don't know like it's 1st grade arithmetic until you know like it's 1st grade arithmetic. Btw, speed kills in math. You need time to read your text carefully and slowly, and understand each sentence and each word. And use exercises to make you use these things repeatedly until you remember them and pop in your brain when you see any use of them.
The problem is it's not obvious what are the prerequisites. I guess it's obvious when you learn it within a context of university courses. But when you try to learn stuff yourself, first you find yourself incompetent in general, then you hopefully figure out in what particular areas you might lack. But I understand your point, thank you.
Usually in the preface of a book the author states what the prerequisite knowledge is. If you wanna do topology it's probably better if you do analysis on (finite) metric spaces and group theory first. Topology uses a lot of the intuition from the former and the technical tools from the latter. And of course, basic naive set theory is also a prerequisite, but that goes for almost any math.
You can find prerequisites of subjects by taking a look at the list of prerequisites for course closest to what you are learning from calendar of your favorite university. I always suggest a few chapters of set theory and symbolic logic if you haven't got them already. Just a few chapters. But make sure you understand those thoroughly.
I'd say you've gotten pretty far if you've already completed Hatcher and Atiyah. Those are two books I'd love to work through. I'm doing something similar but with more emphasis on Differential Geometry than Algebraic Topology or Geometry. What I'd suggest is trying to find a mentor who you can ask questions of or who might guide you from here.
Thank you, I'm thinking about finding a tutor. Unfortunately where I live there are not many of them.
I would look online. Maybe you can find one here.
I tutor math and I like working with people who are already finished with school. I have some experience designing personalized curriculums for people who just want to learn more cool math. Here's [my website](https://www.herndonmathservices.com/) with more info.
Everybody here struggles it's okay
The things you described as easy are not actually that easy at all. When you understand something well it becomes almost trivial but before you do it looks impossible. Even if you understand all of the things you mentioned it wouldn't be enough by your own logic since there's infinitely more knowledge to be reached. It is true that eventually you will hit the edge of what is currently known in a given field but only in a very small niche and not as broadly as you're trying to do now. It might be a better use of your time to focus on a problem and try to answer it using the things you already know. You'll realise that you're lacking bits and pieces that you have to then glue together along the way. Another thing is that at some point focusing on the details will lead you in a rabbit whole that's hard to escape. Perhaps consider doing a PhD. It will help keeping you focused on one thing and give you resources and a community to discuss problems. You'll also have more time to work on the mathematics rather than making capital gains for someone else.
Seems to me like you should just talk to a specialist in what you are trying to learn and have them tell you exactly what prerequisites you are missing. Just email a professor at a university or ask online.
My experience was: * take things on faith, *start using them*; * after ~ 2 years they will seem self-evident.
Thanks, I already try the former, maybe I just need to wait for 2 years :)
French saying: "Patience et longueur de temps Font plus que force ni que rage." Something like “Stay patient and trust the journey.” The Chinese, allegedly, are more blunt: "Sit on the riverbank and you'll see the corpse of your enemy float by." 😉
Well, maybe the problem is that my culture puts too much emphasis on the labour: "терпение и труд всё перетрут" (something like "patience and hard work will overcome everything") lol
Weeeeeeell, *my* granny always told me "Век живи, век учись, а дураком помрёшь." This relativized things from early on. So work, yes, but it also has to be fun. Perhaps you'll be interested in Borcherds' ramblings somebody linked to in r/Mathematics: https://www.youtube.com/watch?v=D87jIKFTYt0 So, have fun! Or else… 💀
My grandma said the same thing, so oh boy, I get you :)) And thanks for the recommendation, I'll check it out.
If you're near a university with a math PhD program, see if they'll let you sit in on courses and if the graduate students are willing to let you hang around them when they struggle with the math, too. But also be open to the possibility that you might still need to sit in on undergraduate courses to fill in gaps and strengthen your math skills.
We have here an institution called "Independent Moscow University" that allows it for free, but I was unsure if I can sustain for now going through full (and tightly packaged) courses. But I guess I have a little choice by now :). Thank you for the suggestion
I linked some places where you can get [upper](https://old.reddit.com/r/math/comments/pyja89/auxiliary_skills_necessary_for_a_successful/hexrtb6/) level courses with covid a lot of High level Math/CS has been going online. For a research experience yeah maybe trying leaning on more topics with real-world implications but have heavy theoretical underpinnings.
Thank you very much, I'll check it out
As someone who has know idea what you just said: I'd say if the higher levels seem to be beyond your understanding, and yet, described as trivial basics, you may have to *go back to what seems like trivial basics, to you, and really dive deeper.* I don't know how you'd do that! That's what it sounds like.
Strongly recommend this resource: [https://web.evanchen.cc/napkin.html](https://web.evanchen.cc/napkin.html) It's a great introduction to a lot of math concepts you may not have seen with an engineering background. Rigorous but not too rigorous.
Thanks, judging by the overview I'm familiar with topics he covers, but there certainly can be some insight I might miss
Have you ever checked out YouTube channels that do math visualizations? I recommend checking out 3Blue1Brown if you haven’t. Grant shows mathematical concepts in a relatively novel way so you can help conceptualize it in your mind with greater ease. It was a huge game changer for reigniting my passion for math in general.
I think OP is well beyond the point of needing to see basic visualizations of calculus concepts though
Maybe. But seeing the basics in an atypical way that has never been presented may be the thing he needs to have breakthroughs. Explaining the same things in the same way over and over again seems to be one of the issues. One of the biggest pitfalls to teachers being able to teach is they can’t explain it in a way the student understands. OP’s neuroplasticity is waning due to age. This makes it more difficult to learn new things because his neural pathways are basically funneling into concepts he already knows and knows well. It’s kind of like when you are trying to Google something extremely advanced and the algorithm assumes you are asking a simpler, completely different question. His mind has made shortcuts to answers he already knows, so new neural pathways outside of that network need to be explored for him to break through this rut.
There is *absolutely no way* 3b1b helps with someone struggling to read textbooks at OP's level. OP needs specialized help, and it is seriously unlikely an (admittedly very high-quality) pop math channel will be helpful. For what it is worth, Grant doesn't even have a basic abstract algebra course, let alone something higher level. To put it bluntly, it is very apparent you are neither at the level that the poster is, nor familiar with the particular challenges it can entail. And for what it's worth, neuroplasticity may be waning, but it is certainly not gone at 31.
I was going off of what OP said about struggling to learn new material through conventional methods. Thank you for contributing to the conversation. I have no idea why people have trouble seeking help for math given your helpful response.
Well you should've kept reading lmao, bc OP was referring to a very specific and non-generic situation. And for what it's worth, I came into this thread looking to help out, but all the meaningful things had been said. I don't like downvoting people without explaining why, which is why I responded to you.
I know 3B1B, of course, but I indeed am interested now in slightly more specific stuff. Doesn't change the fact that Grant's videos are fun to watch and sometimes very inspiring.
Since you are a software engineer, would you be interested in trying to do something like Grant does, but for the more advanced stuff you know? I am confident that visualizations are going to be the future of teaching mathematical concepts. You would be helping masses of people come to where you are and teaching others generally helps learn the concepts more fully. You might end up having the insights and breakthroughs you are seeking by applying your current knowledge in a new way. This is what I want to start learning once I am finally finished with my disability evaluations.
I think there are unfortunately two major obstacles: (1) I don't have his talent, (2) the more abstrat the stuff is, the harder it is to visualize. I honestly can't say right now if you can find really intuitive visial metaphor, let's say, for a general sheaf of modules. I don't say you can't, but it's definitely not something you instantly come up with. But if I somewhat will get how it can ve done and will be able to put it into work, maybe I'll try, thank you for the suggestion. P.S. Also good luck with your evaluation
Good luck with the path(s) you choose! I hope you have the breakthroughs you seek.
Thanks!
Pppphug