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

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

#251

Post by nextos »

jsenn wrote: post #156

* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world. * Calculus by Spivak. This was used in my intro calculus course in university. It…

These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is:

* Basic Mathematics by Lang. Covers basic algebra and geometry at high school level.

Then one of these two, depending on your interests, or both:

* Vector Calculus, Linear Algebra and Differential Forms by Hubbard and Hubbard. Takes you through linear algebra, single-variable calculus and multiple variable calculus. Analysis is discussed in an appendix. All proofs have a constructive bias, so it's very algorithmic and natural for a CS-minded student. Solutions are in a separate volume.

* Program = Proof by Mimram. Discusses logic and computation, and takes you from the basics to depedent type theory and beyond. Uses OCaml and Agda. Freely available at: https://www.lix.polytechnique.fr/Labo/Samuel.Mimram/teaching...

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

#253

Post by Py-o7 »

Steele's The Cauchy-Schwarz Masterclass is actually quite good and seemingly designed for self study. (A lingering result of this book is I heavily use inequalities even outside of analysis.)

Artin's Algebra probably has had the most impact on my math thinking. The development of groups and rings while tightly linking them to linear algebra was rather brilliant.

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

#254

Post by fn-mote »

jdkee wrote: post #241
chriswarbo wrote: post #204

Earlier quoted context omitted.

Norman Wildberger's YouTube channels are the most thorough I've seen ( https://www.youtube.com/@njwildberger and https://www.youtube.com/@WildEggmathematicscourses ). There are hundreds of videos, organised in playlists, from undergraduate lectures ( https://www.youtube.com/playlist?list=PL55C7C83781CF4316 ) and research seminars ( https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF ) all the way to basic fundam…

"The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits." So no transfinite ordinal analysis or large cardinals? Hard to take him seriously.

> So no transfinite ordinal analysis or large cardinals

You could insist on sticking with the Axiom of Countable Choice if you wanted to avoid some of that.

I'd say it's pretty hard to avoid thinking about 'infinity' though.

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

#255

Post by fn-mote »

chriswarbo wrote: post #248
jdkee wrote: post #241

Earlier quoted context omitted.

"The reason I find them fascinating is that Wildberger doesn't agree with some of the conventional approaches, in particular with the use of infinity and taking limits." So no transfinite ordinal analysis or large cardinals? Hard to take him seriously.

More than that: no Real numbers, no pi, no square root of 2, no sine/cosine, etc. It's similar to 'reverse mathematics' (trying to find the minimum set of assumptions required to prove a known result)

The square root of 2 does not require any limiting process. It's the hypotenuse of a right triangle with legs of length one.

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

#257

Post by sn9 »

salusinarduis wrote: post #74
enriquto wrote: post #49

The classical stuff is great: * Geometry and the imagination by Hilbert and Cohn-Vossen * Methods of mathematical physics by Courant and Hilbert * A comprehensive introduction to differential geometry by Spivak (and its little brothers Calculus and Calculus on manifolds ) * Fourier Analysis by Körner * Arnold's books on ODE, PDE and mathematical physics are breathtakingly beautiful. * The shape of space by Weeks * So…

I've had this idea of starting back at basics and relearning math from the beginning since I never "really" learned it besides memorizing and skirting my way through it in school. Do you know a good path or book that's suitable for that?

The Art of Problem Solving series of books cover all of pre-college math and have complete solutions manuals available.

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

#258

Post by fastneutron »

binarymax wrote: post #42

“Mathematical Notation: A Guide for Engineers and Scientists”[0] really changed my abilities with being able to read papers and decipher what was going on. I had university math experience but it was a long time ago. When I started reading papers for algorithms later in my career I couldn’t get past the notation. Once the symbols are explained, as a programmer, I was able to grok so much more. This should be on every…

Another document with the same flavor is “Larry’s Speakeasy: A Handbook for Spoken Mathematics.” Basically a notation-to-English translation guide.

It’s actually listed as a technical report from Lawrence Livermore National Lab, but the only online PDF I can find is here: https://www.academia.edu/28253460/Handbook_for_Spoken_Mathem...

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

#259

Post by Wistar »

Not nearly as rarefied as many of the books cited here, for me it was John Saxon's excellent Algebra 1/2, Algebra 1, and Algebra 2. I didn't get a good enough grasp on basic Algebra in high-school. When I was in my early–mid 20s, a friend gave me these three algebra textbooks. In a marathon session lasting about two weeks, I went through all three books from end-to-end and really learned algebra. For whatever reasons, the Saxon books worked really well for me — better than any other learning I have ever gotten from a textbook. Although I was very motivated, I attribute a lot of my success learning algebra well to those three books. I still own them.
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