Mathematics: Form and Function

What Hacker News says

1. telOct 2023
   Part of the issue is that there is no simple Why. To invent one is to either petition history or invent some perspective. It's not a bad idea to do these things, didactically, but they're unnecessary for the material. Or, said another way, there's something to be said for discovering your own "why" through familiarity with the many wonderous properties the complex numbers enjoy.

   That said, that's a frustrating answer. An excellent book which does just what I said above and tells a lightly fictionalized "just so" story of the "history and development" of mathematics as an excuse to introduce everything in a motivated fashion is MacLane's Mathematics: Form and Function which I just recommend endlessly.

   https://www.amazon.com/Mathematics-Form-Function-Saunders-Ma...

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2. bmitcSep 2022
   Some interesting entries I know about:

   * Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being by George Lakoff and Rafael Nunez

   * The Mathematician's Mind by Jacques Hadamard

   * Mathematics Form and Function by Saunders MacLane

   * On the Brink of Paradox: Highlights from the Intersection of Philosophy and Mathematics by Agustin Rayo

   The preface and chapter 1 of A Book of Abstract Algebra by Charles C. Pinter has an excellent discussion of a certain perspective of mathematics. Hermann Weyl wrote a lot about this stuff too.

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3. telMar 2018
   Probably not a perfect book for someone at the HS level, but just a bit past that I would HIGHLY recommend MacLane’s Mathematics Form and Function.

   It’s a revisionist history of mathematics aimed to demonstrate how ideas flow from one to another. He eliminates a lot of dead-ends and takes a perspective as to one “ought” to move from one subject or discovery to another. OTOH, the historical perspective is both readable and often a missing piece which makes other math tougher.

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4. telApr 2015
   I really enjoy Saunders MacLane's Mathematics Form and Function for sight at the "answers" this article seeks. MacLane's argument is that mathematics arises from a relatively small number of meaningful forces and "good" mathematics arises when those forces all align in particular places. You could of course go further by asking what exactly it is that makes those forces all align—why in these fields?—but the point of his book isn't exactly to answer the philosophical question but more to provide a little bit of a humanistic POV on the development of mathematics with just enough historical happenstance stripped away to make things intelligible but not so much as to lose the line of sight on why things (probably) developed the way that they did.
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5. telFeb 2015
   Given everyone recommending books for math autodidacts interested in learning the "breadth" of mathematics, I cannot recommend more highly Saunder's MacLane's "Mathematics, Form and Function".

   It reads as a high level false-historical account of how math might have been developed, in its entirety, if it unfolded in the most direct and logical way from the human experiences we all share (guided by understanding of where mathematics has ended up in the modern age).

   The false-historical expose is fantastic. The goal becomes less to express things as accurately and comprehensively as modernly possible—which you'll experience as you take on graduate level books or modern papers—but instead to demonstrate large-scope intuition for why whole fields evolved as they did and how they are all intermingled.

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6. telJan 2015
   If you can find a way to get a hold of it, Saunders MacLane's Mathematics: Form and Function is a tremendous book for understanding how math arises from the world and reflects it. What's written here is a lot more specific than that, of course, but if this illustration interests you then it might be a good book to try digging into a little bit.

   On the other hand, while I am always cognizant and amazed by the POV of math as a human construction, there's something a little otherworldly about it from time to time. In the same way that you sometimes hit a code design which feels so damn good, Math, especially older math, is just a huge collection of these. Somehow, despite this all coming, apparently, from our minds, we hit on these design decisions that are just so sweet that they last millennia. This is what inspired things like Voyager---maybe it's hubris, but it just has to be the case that aliens speak mathematics.

   So, I encourage anyone excited by this: math isn't "hard", it's just big and wonderful. You'll never finish learning it, but the journey will be incredible.

   I'll leave this linking two more great resources (edit: to be clear, really the first one is the great resource... the second is just me talking, not really great at all). First, Paul Erdös, a famous mathematician who perhaps specialized in combinatorics, loved this idea I espouse above. In his mind, God had a small number of "proofs" in his mind when he designed mathematics. These are so wonderful that their beauty is completely self-evident. After Erdös "stopped doing math" (passed away) people compiled some of his Proofs from The Book along with others they imagine he would have so regarded into a great text book called, unsurprisingly, Proofs from The Book [0].

   Finally, I'll self plug a little essay I wrote, actually in another HN comment, a while ago about learning mathematics.

   http://jspha.com/posts/there_is_no_royal_road_to_mathematics...

   [0] http://www.amazon.com/Proofs-THE-BOOK-Martin-Aigner/dp/36420...

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