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I know this is kind of a meme on boards like this, but is it actually possible to learn university-level mathematics on your own? Not a full mathematics program, just the equivalent of some select courses. How would one go about it?


What type of math were you looking to learn? Learning anything on your own is possible, you just have to be motivated enough.
You'll have to learn all the math that comes before the select courses you want to learn about.


One could read a textbook on the subject and do relevant problems.
It's how i learned most math in university anyways.


yes, but unless you van back that up with some actual tests or practical applications nobody is gonna believe you.


I'd like to start with geometry, because I remember being very bad at it in school. After that probably calculus, single and then multi-variable, plus differential equations, with the goal of studying physics in mind.

I have learned a lot about programming from books. But the difference there is that you have the compiler and you can run your code, both of which guarantee that you have understood things pretty well. I'm afraid that with maths I will get lazy and just reassure myself that yeah I understood it and can continue reading only to realize a few pages later that I have completely lost track and need to reread half of the book again and then I get discouraged, feel like a failure and give up.

I don't care about that, it's not for my CV or something.


I'd say algebra is the most important building block, the rest should come easy. Study order of operations really well. Once you have some single variable calculus under your belt you can start getting into physics.


Do you have anything to recommend to master it? Books, study methods, whatever that comes to your mind?


I don't have any books in mind but I found this for you.


Same person here. If you feel stuck feel free to ask me about it and I'll reply whenever I can.
The site says that if you get 33/36 problems correct you can move onto college level stuff. Once you do that I guess I can recommend something higher level. Good luck


If you can do relevant problems, that would guarantee that you have understood things pretty well.
You should be able to find solutions for introductory problems.


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Cool, another autodidact mathanon!

I'm trying to do the exact same thing as you - university-level maths for pure enjoyment of the abstract knowledge. I stopped studying maths formally at age 16.

I don't know what kind of maths you like the look of, but bear in mind that there is a lot out there - whole, gigantic, overarching fields - which they don't even tell you exist during high school. Topology. Group theory. Linear algebra. Mathematical logic. If you haven't already done this, I'd recommend scouring popular maths content for inspiration about what to learn. Some example sources:
>YouTube (Numberphile, especially)
>books (e.g. Simon Schama's Fermat's Last Theorem is excellent)

If you have a feeling for the fields that are out there, you might be able to skip some high-school stuff and move straight onto the more abstract fruits of mathematics. I don't know shit about trigonometry and, so far, that hasn't harmed me in any way. At university level, my impression is that a 2D triangle is a laughably specific object - the things being studied by university students nowadays are much more general in nature. However, as another anon said, algebra is everywhere so definitely get an intuitive feeling for manipulating equations.

Logic and set theory. You might have heard of these subjects in the context of the 'foundations of mathematics'. It is pretty much accurate to say that all other fields pre-suppose them, in the sense that almost all mathematical constructions (real numbers, vector spaces, etc.) are, ultimately, set-theoretical constructions. And, of course, mathematical proofs must use logic.



If you can spare some time, I would familiarise yourself with the conventions of first-order logic and Zermelo-Fraenkel set theory. Paul Halmos' book Naïve Set Theory is short and covers a lot of useful stuff. (Even just the first 15 sections would be sufficient for a good grounding in the subject.)

My friend, who did maths at undergraduate level, also said that linear algebra is important for understanding other areas, so it's what I picked to study this year (along with some topology if I have time).

Apart from doing exercises, you can also really test your understanding by programming mathematical algorithms - e.g. an algorithm to produce the prime factorisation of any natural number.

Some links:
https://mathoverflow.net/questions/28158/a-learning-roadmap-request-from-high-school-to-mid-undergraduate-studies - introductory textbook recommendations from MathOverflow users

https://libgen.fun - Library Genesis gives you free access to basically any textbook you want. This portal downloads the book you request via IPFS.



I should also have mentioned lectures on YouTube. Several universities post full undergraduate lectures under a Creative Commons licence. MIT does this with its OpenCourseWare channel. Here is a lecture series about topology and geometry, given by one of the academics who appears occasionally on Numberphile: https://www.youtube.com/watch?v=SXHHvoaSctc


What about proving things, how can I make sure that I am not making mistakes or even deluding myself in a proof?


If you read a lot of proofs in the textbook, you will gradually get a feeling for the level of precision required for a proof.

If you are paranoid about your answer to a particular exercise, pretend that you are a sceptical mathematician examining the purported proof. Ask yourself things like:
>does statement X really imply statement Y?
>can I find a counter-example to the statement Z which I'm not sure about?
>am I allowed to assume this?
For the last one, basically: if it's part of the statement of the theorem you are asked to prove, then yes. If not, then you have to prove the assumption as well.


Thanks anon. The course looks interesting, I'll give it a try. Do you usually take notes while watching these? I guess for courses it is a must, but what about the "edutainment"?


I know hobby programmers can contribute to open-source and post their programs on github, but what do hobby mathematicians do? Is there some forum where they share the proofs they come up with or something?


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>Is there some forum where they share the proofs they come up with or something?
Yes! There is a small effort to make "massively multiplayer mathematics" a reality, where randos on the internet shoot out ideas about unsolved problems:

It's a bit dead, but still.


I was thinking of something more analogous to rewriting fizzbuzz for the thousandth time, but this is really cool.


I started reading a geometry book for high schoolers, I thought it would be an easy warmup before more serious maths. But it turned out to be really hard for me, it is basically an endless collection of proof exercises and I progress very slowly. It makes me feel like I am too stupid for maths.


lol, I find proofs hard too. But I'm kind of resigned to the fact that learning maths is going to take me a very long time, since I'm not doing it for school or university.

What kind of geometry is it, out of interest? 2D Euclidean geometry? I'd like to be able to understand non-Euclidean geometry one day.


It's Birkhoff's "Basic Geometry", it was in one of those "learn maths from scratch" guides with the obligatory anime girl. It's Euclidean but based on Birkhoff's own axioms. I was always bad with geometry and proofs, that's why I chose to study this, to correct some deficiencies so to speak, but it gets pretty frustrating.


why do you wanna learn maths?

Is it for fun?


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Any recommendations on maths for physics? Single/multi-variable calculus, differential equations, things like these?


I recommend Precalculus Mathematics in a Nutshell by George F. Simmons.


I took a quick look, it looks nice and short, I might read parts of it…


Why does maths take so long to learn? Compared to programming books, books on mathematics seem to be at least twice as long and take ten times as much time to work through. Is maths just this much harder? Or am I just untrained for it, and it will get better with time? And there's so much I would love to learn, it's really overwhelming.


One of my profesors could look at pretty much any equation and solve it without doing any calculations on paper in a few seconds. He told us that in order to do this, you just need to practice.


That sounds amazing but does it transfer to new, unfamiliar parts of mathematics?


I don't think they take longer to learn, but it probably depends on 1.) your skill of thinking abstract 2.) Your integrated RAM
For Maths you need to think way abstracter than for programming. Sure programming is abstract as well, but you always get something out of it, that is translated into an action or some kind of feedback on the computer. While in maths all you get out of it is maths. It's like programming but never running the code.
And by integrated RAM I just mean how long you can go on with one thought without forgetting the start of it


It's just strange that people spend 12 years in school learning maths only for university-level mathematics to be completely different.

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