Mathematics

What Hacker News says

1. xiande04Mar 2025
   If you like this approach, I highly recommend Mathematics: It's Content, Methods, and Meaning by Kolmogorov. He uses this same approach, but applies it to many more concepts in math (about 1,000 pages!). In fact, I think I actually heard about that book on this site, so I guess I'm paying it forward.

   This approach was to align with the Soviet philosophy of dialectical materialism, which claims that all things arise from a material need. Not sure I'm fully onboard with the philosophy as a whole, but Kolmogorov's book was really eye opening.

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2. rsj_hnApr 2021
   The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Grad...

   but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.

   The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that

   Integral over the boundary = Integral over the interior of the divergence, or more poetically

   Int_(dA)A = Int_A dA

   Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.

   So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a).

   The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).

   I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.

   I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

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3. rramadassApr 2021
   >Some of these books look amazing for someone trying to revive his passion for math, decades after college.

   If you have never read any of the Soviet era Science books, you absolutely should get hold of some and go through them. They were succinct and to the point with the density of information being quite high. By comparison, most modern text books are so verbose and sometimes dumbed down that it becomes quite a chore to sift the wheat from the chaff.

   Dover publications has some good ones starting with Mathematics: Its Contents, Methods and Meaning by Kolmogorov et.al. There is also the I.M.Gelfand series; Functions and Graphs, The Method of Coordinates, Algebra, Trigonometry etc.

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4. harshrealityOct 2019
   Feynman was a unique teacher of course, and pure math isn't quite as easy to teach intuitively.

   Something that might be close would be the survey _Mathematics: Its Contents, Methods and Meaning_ by Aleksandrov, Kolmogorov et al. https://www.goodreads.com/book/show/405880.Mathematics

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5. whitepoplarJan 2019
   This book is fantastic and pretty much takes you through an entire undergrad mathematics course: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
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6. synthmeatMar 2018
   I’m going to go with a few assumptions here:

   a) You don’t do this full time.

   b) By “bottoms up” you just mean “with firm grasp on fundamentals”, not logic/set/category/type theory approach.

   c) You are skilled with programming/software in general.

   In a way, you’re ahead of math peers in that you don’t need to do a lot of problems by hand, and can develop intuition much faster through many software tools available. Even charting simple tables goes a long way.

   Another thing you have going for yourself is - you can basically skip high school math and jump right in for the good stuff.

   I’d recommend getting great and cheap russian recap of mathematics up to 60s [1] and a modern coverage of the field in relatively light essay form [2].

   Just skimming these will broaden your mathematical horizons to the point where you’re going to start recognizing more and more real-life math problems in your daily life which will, in return, incite you to dig further into aspects and resources of what is absolutely huge and beautiful landscape of mathematics.

   [1] https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...

   [2] https://www.amazon.com/Princeton-Companion-Mathematics-Timot...

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