Khan Academy. Not a book per-se, but have worked through the courses over the past few years, I'm confident that my college and university results would be about 2 or 3 grades higher we I to retake them.
Ask HN: Math books that made you significantly better at math?
301–310 of 323 posts
Re: Ask HN: Math books that made you significantly better at math?
#302Earlier quoted context omitted.
I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing. I think this error in thinking comes from the fact that Sigma notation can often be trivially implemented as a for loop. Programming languages are designed to describe a specific computat…
> I find one of the biggest mistakes programmers have about mathematical notation is that it's somehow just a terse, badly implemented programming language. But this is a very poor understanding of what mathematical notation is doing. No, we think that because proofs and programs are isomorphic[1]. It's not a mistake: traditional mathematical notation provably is a terse badly implemented programming language. Actual…
I get the feeling that you're stuck in Terry Tao's 'rigourous phase' of mathematical understanding, where everything in the end is a computation and has to be carried out according to a set of rigorous steps and definitions.
I get that, but it does miss a bit the cultural context of how mathematically fluent people use mathematics to communicate with each other. When you're discussing maths with colleagues in front of a blackboard, you're often not really trying to prove anything, but discussing the relationship between mathematical objects. In this context the ambiguity and implication in the notation is almost a requirement, otherwise the communication speed tanks.
Having a mathematical discussion between a group of people all fluent in the context and terminology is a wonderfully fluid thing.
Re: Ask HN: Math books that made you significantly better at math?
#303Earlier quoted context omitted.
As a programmer I really wish math notation was more rigorous: less ambiguity, more explicit typing, no implicit variables, etc. So much of it would never pass code review. We programmers figured out that code should be optimized for readability, not writtability ; I wish mathematicians did too.
Couldn't agree more. Mathematician are masters at whipping up random notations and then not adhering to it rigorously. Higher level math would be an order of magnitude easier with machine checked syntax.
> Higher level math would be an order of magnitude easier with machine checked syntax.
This just isn't true, at least in terms of developing new mathematical ideas. There are already tools (e.g Coq) for providing mathematical syntax checking etc, but nobody uses these in developing new ideas because it would cripple the process of doing mathematics.
Particularly in the early stages of work, mathematics is very intuition and heuristic driven, you tend to sketch out an idea by actively using the ambiguity in the notation. Once you've got something that looks broadly correct you progressively try to shore it up by using more detailed and rigorous argument.
Maybe as an analogy, think of how architects design houses. They don't start by placing each brick according to correct civil engineering practice. They design something that broadly makes sense and then the civil engineers 'shore it up'.
Re: Ask HN: Math books that made you significantly better at math?
#304Earlier quoted context omitted.
Couldn't agree more. Mathematician are masters at whipping up random notations and then not adhering to it rigorously. Higher level math would be an order of magnitude easier with machine checked syntax.
> Higher level math would be an order of magnitude easier with machine checked syntax. This just isn't true, at least in terms of developing new mathematical ideas. There are already tools (e.g Coq) for providing mathematical syntax checking etc, but nobody uses these in developing new ideas because it would cripple the process of doing mathematics. Particularly in the early stages of work, mathematics is very intuit…
You are right that it wouldn't be true in regards to new mathematical ideas. That's not what I meant, I should have been clearer. What I meant was 10x easier to learn.
Coq isn't about mathematical syntax checking. Coq is about encoding the whole proof in such a way Coq can machine verify it. That's 100 steps further then what I'm talking about. Just a simple syntactic check. Similar to what gofmt does. Is what you are writing considered valid syntax. Not whether what you are writing is correct.
Re: Ask HN: Math books that made you significantly better at math?
#305I got a lot out of "Unknown Quantity" by John Derbyshire. Subtitle a real and imaginary history of algebra. I particularly enjoyed the lead up to the Chapter "Assault on the Quintic".
Also, I hold "The Dictionary of Curious and Interesting Numbers" close to my heart for the endless fun it brought me.
Re: Ask HN: Math books that made you significantly better at math?
#306I would not call myself great at math – I struggled with it in school, in fact – but in recent years I’ve begun “correcting” my lack of mathematical knowledge. The single best decision I’ve made is to first start with the philosophy of mathematics. Maybe it’s because my background is in philosophy, but I also think that for certain people like myself, understanding what math is makes me far more interested in underst…
I have a very similar background, did my undergrad in Philosophy and feel that I need to learn some basics. Do you have any pointers on where to move after this?
I just started with that SEP article and then googled around for some other books and videos. There are some excellent lectures on YouTube, this one for example:
Also, you might find that symbolic logic is a good introduction to thinking mathematically. I used Klenk’s Understanding Symbolic Logic for a course a decade ago and really enjoyed it.
For actual mathematics lessons, Khan Academy is pretty solid.
Re: Ask HN: Math books that made you significantly better at math?
#307V.I. Arnold "Problems for children from 5 to 15" [0].
The book was discussed on HN in 2021 [1] (325 comments)
If you have kinds and teach them math this book has mind-opening problems that even curious adults would enjoy.
[0] https://www.imaginary.org/sites/default/files/taskbook_arnol...
Re: Ask HN: Math books that made you significantly better at math?
#308Earlier quoted context omitted.
Yeah, I've been doing the same thing, but with the rule that I have to do "a math problem". Right now I'm going through a stochastic processes book a bit at a time that way and really enjoying it.
In my experience, nothing beats solving problems. Even if you get help via solution manuals. I went from being a B/A student to top of my classes (engineering) by solving many problems. I would do my homework and then go back before exams and redo the homework twice for a total of solving the problems 3 times. Never failed me. I actually managed to get a perfect score from a professor who wrote notoriously difficult…
Yep, a book with proofs and worked exercises is great for this because you can try to do the proof and then look at the solution to see if you got it reasonably correct. In my case the book covers a lot of stuff that I passively "know", but working problems has helped get be back to an active understanding.
Re: Ask HN: Math books that made you significantly better at math?
#309Nonlinear Dynamics and Chaos by Strogatz is a work of art
Re: Ask HN: Math books that made you significantly better at math?
#310Nonlinear Dynamics and Chaos by Strogatz - does a great job at explaining very complex mathematical topics with great examples. Not for beginners. Every applied math student should read this cover to cover.