Ask HN: Math books that made you significantly better at math?

So, I decided I wanted to study maths for the maths. I was in the fortunate position of being able to self fund myself through the Open University (uk based) Maths and Statistics BSc. One module at a time I’m now on my last module. There many things I’d studied before (calculus, sequences) and many new to me (group theory, graph theory)

My fear with machine translations is that subtle errors here and there might throw me off in something like math where things are precisely stated.

A year of part time study sounds doable though

Wouldn't that depend on what level of skill we're talking about? I believe that most people who need to learn just, say, high-school algebra/geometry/trigonometry/etc. can probably do so with "just books" if they are motivated.

Heck, I'd even venture that most anybody who is motivated enough can learn at least through Calc III on their own using online resources. I'm going through Calc III right now actually, using a combination of books and online videos, and there hasn't been anything that has struck me as particularly challenging so far. This after only making it to Calc I before I dropped out of school "back in the day".

Getting a degree in math is a good start, but even then it will be limited if you don't work with other people, go to office hours, form relationships with professors---embed yourself in the culture, so to speak.

I think these things would be critical if one wants to become a mathematician. But for somebody who just wants to learn more math for the purpose of reading (non math) research papers (maybe in machine learning to pick an obvious, if possibly trite, example), solving problems in everyday life or at work, or possibly even for doing research in another (non math) field and then writing up their research, I would think of all of those things as being "nice, sure, if you have access, time, money etc. But not even close to absolutely essential".

Unfortunately the OP didn't really say what their goal is, so it's hard to say what advice makes the most sense for them.

Also unfortunate is that for probably most people who are working career professionals, there likely isn't enough free time available to go back and do an actual math degree in school. So learning on ones own from books, videos, etc. is probably the only viable choice.

My response was based on the reader saying something about "mathematical thinking" in their post. In my experience, programmers talking about "mathematical thinking" usually have math insecurity and are referring to more advanced topics, thinking they need to consult tomes of deep mathematical wisdom to correct this deficiency (wrong). Could be totally off base, of course. But note some of the other replies here, suggesting very sophisticated books. Are they appropriate for a random HN poster who is soliciting random book suggestions to improve their "mathematical thinking"...? Seems unlikely to me...

One caveat, though:

... Also unfortunate is that for probably most people who are working career professionals, there likely isn't enough free time available to go back and do an actual math degree in school. So learning on ones own from books, videos, etc. is probably the only viable choice.

I'm not so sure about this. If, like you say, someone is in a job where they need to read some papers with some math in them, they should be organizationally near someone who can help them out. If possible, I think a better strategy (better even than getting a degree) would be to find these people and develop a relationship with them to the point where you can ask them questions. They should then be able to explain any unclear notation, unfamiliar (simple) concepts, etc. Possibly some of that person's advice might come in the form of "watch a Khan Academy video on Topic X". But this will be a far more productive use of time than self-directed learning in this case.

On the other hand, if they aren't near anyone with those skills but they're reading these papers... something is probably amiss. For example, if they're reading a machine learning paper which requires a significant knowledge of "engineering math" (Calc 3, linear algebra, etc.), and there is no one with that knowledge nearby... having them read that paper is probably a waste of time from an organizational perspective.

There's also the question of why they're reading that paper when they don't have those basic mathematical skills. Without those skills, it is unlikely they will be able to do very much that is useful with it. If they want to implement the algorithm because they think it will be suitable for some task, I would argue that without those skills they are not in a good position to be able to accurately assess whether the algorithm will perform well. Part of what you learn in an advanced degree is how to read a paper---i.e., how to sniff out the bull shit, what to be wary of, etc.