A Course of Pure Mathematics

What Hacker News says

1. maydDec 2024
   Hardy's "A course of Pure Mathematics" has been highly regarded since it was first published in 1908 because it was an innovative text: rigorous, modern, well-written. Its intended readership was always first year "honours" mathematics students. This book inspired innovation in subsequent generations of textbook writers.

   However, in the 21st century, this book really can no longer be recommended for its original teaching purpose. As a textbook it is outdated (a term I hate, but it is true). It is now an historical curiosity - although one which I am pleased to own, and the exercises in the book are still worth a look.

   Calculus teaching has progressed considerably since 1908. The construction of the real number system in Hardy's book, using the Dedekind Cut method is overly complicated - the use of the of Least Upper Bound is much simpler and clearer. Hardy defines the concept of integral solely as the anti-derivative; there is no discussion of Riemann sums, or Darboux sums, etc. I am sure I would not want to take Hardy's approach today.

   I think we are better off recommending books are more modern.

   I will start by recommending "Calculus" by Michael Spivak.

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2. gen220Apr 2024
   I "knew" enough calculus to get good grades in math courses through high school and college as an engineer.

   But I didn't really "know" calculus until I read a book not too dissimilar from this one, "A Course of Pure Mathematics" from 1908 (!), which constructs calculus up from number theory (I think the fundamental theorem of calculus comes halfway through the book?). From that point it's impossible to forget.

   As for why it's not taught this way today, I'd blame our testing regime and large classrooms, which incentivizes temporary memorization of key formulae and knowing where to mindlessly apply them, over deep/lasting/semantic understanding. I'd also blame the fact that we have a different maths teacher every year, so students come in with heterogenous understandings of the pre-requisites for the next year's material, so the first part of a section is spent reviewing + consolidating.

   It takes maybe 10-20% more time to get a rich understanding of the subject that lasts a lifetime. But we value compression and instantly-measurable results more than actual learning. :/

   Encountering material like this makes you really happy, but it's kind of bittersweet because it makes you realize that the modal state of modern pedagogy is pretty abysmal.

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3. ekm2Jan 2024
   What math textbooks used in the late 19th or early 20th century are still great right now?Something like G.H Hardy's A Course of Pure Mathematics
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4. jjgreenNov 2022
   G.H. Hardy, A Course of Pure Mathematics. Old but beautiful, redoes results which you may well know but with rigour.
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5. harshrealityFeb 2022
   Maybe Calculus Made Easy by S. Thompson (more recently revised by Martin Gardner, but that new edition is under copyright). Maybe Hardy's Course of Pure Mathematics. However...

   Modern books use, in some areas and cases, more modern notation and approaches. That's helpful because math is not just knowledge but a language for communicating that knowledge.

   If money is an issue[1], either of those Gutenberg texts (re-done with LaTeX), or the OpenStax calculus book are perfectly adequate. There's also no reason you couldn't use an AP Calculus review guide, or Schaum's Outlines, which aren't free but are in the <$25 range. Combined with resources like Youtube math channels, Wikipedia, Mathworld, and Sage and Mathematica (version 12 is freely available for the Raspberry Pi, though it'll be limited by hardware performance), anyone could learn calculus from the most dull and stilted and antiquated-notation calculus books ever written.

   Math presented in a slightly obtuse way that doesn't lend itself to instant understanding might actually be better, as it forces the reader to think about the concept more, increasing retention.

   Nobody needs Spivak's Calculus. I think it's a bit like the subtle meaning of Frost's Road Less Traveled. Everyone self-rationalizes about how their favorite math book is obviously the best, but in reality most of them probably learned the core of that subject from a different, less elegant book. It's not that it doesn't matter at all; better presentations might avoid an occasional unnecessary struggle over a concept, or lead to faster intuition of a visualization here and there, but it ultimately doesn't matter very much.

   However, if you can buy a book (or pirate it) and want a modern digital alternative less like OpenStax or Stewart's, and more like Spivak (which sadly is only available in scans), Velleman's Calculus isn't bad, and you can supplement that with a modern analysis book like Tao, Pugh, or Abbott.

   [1] Unless you are flat broke—in which case there's no reason even the most IP-respecting person should feel guilty about raiding libgen for learning material—the cost of a textbook approaches irrelevance when you'll spend 50, 100, or even more hours working through it. The opportunity cost of that time at minimum wage is many times the price of the textbook. What are you trying to accomplish by finding something free? Nobody needs to buy expensive textbooks anymore, but if you're opposed to paying money for textbooks on principle, you might ask yourself why.

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6. gen220Jul 2020
   If they like calculus and can stand proofs, I’d recommend a Course of Pure Mathematics by Hardy. It totally blew my mind when I was that age, to see how everything was “connected” by proofs, starting with real numbers. Despite being proof heavy, I found the writing style singularly legible and comprehensible.
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