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Ask HN: How or where to begin learning mathematics from first principles?

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Re: Ask HN: How or where to begin learning mathematics from first principles?

#51

Post by WallWextra »

cma wrote: post #46
WallWextra wrote: post #14

I got started on "real" math with Spivak's Calculus. Some people start with Topology by Munkres, which is not a difficult book but is very abstract and rigorous so makes a good introduction. If you feel like you have ok calculus chops, maybe Real Mathematical Analysis by Charles Pugh. Other good books are Linear Algebra Done Right by Axler, or the linear algebra book by Friedberg, Insel, and Spence. Maybe even learn…

I'd start with Discrete Mathematics before Spivak. From what I remember, Spivak jumps in to constructing the reals out of set theory pretty quickly. A discrete math text will drill you over a lot more basic proofs involving set theory that would help in understanding his construction.

Construction of the reals is in an appendix at the back. He starts with ordered field axioms.

Frankly there's something about the presentation in "discrete math" books that I find much more confusing and difficult than Spivak, which is just talking logically about numbers and their properties.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#52

Post by WallWextra »

nicklaf wrote: post #41

I absolutely, positively second the recommendation of "Real Mathematical Analysis" by Charles Pugh (don't miss the advice he relates from his colleague, on pages 9&10, with the heading "Metaphor and Analogy", which could easily form the basis for a dissertation on the psychology of mathematical intuition and inspiration). Pugh does an exquisite, uncommonly good job of avoiding a pitfall that >99.9% of mathematics aut…

Pugh taught my ODEs course in college and basically turned it into a dynamical systems party. Great class, but I still don't know anything about ODEs. Really cool guy.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#53

Post by jumasheff »

ArkyBeagle wrote: post #5

There is some distance from this to antenna design, electromagnetism, etc, but I think you have to be fluent in proofs to actually follow along on those. Math is a big subject; ymmv. A Transition to Advanced Mathematics by Douglas Smith (Author), Maurice Eggen (Author), Richard St. Andre (Author) ISBN-13: 978-0495562023 ISBN-10: 0495562025 Edition: 7th It shows up on Abebooks which could help with the price. It's a s…

@ArkyBeagle Thank you for the book recommended!

@smtucker I'm also a math noob, and I'm a half way through Udacity's "College Algebra" (free) course:

https://www.udacity.com/course/viewer#!/c-ma008

There is also a good introductory class on Statistics:

https://www.udacity.com/course/viewer#!/c-st101

After these classes, hopefully, reading "Transition to Advanced Mathematics" will not be a pain.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#54

Post by sumgy »

azmenthe wrote: post #27

I wanted to give some recommendations before I hijacked your thread with my own question but I was going to suggest What is Mathematics, How to Prove it and Naive Set Theory which have already have been mentioned. I'm actually in a related situation in which I'm competent in analysis (bachelors in physics) but I struggle with all the category theory inspired design patterns in functional programming. Every book/artic…

What language specifically are you looking at monads in? If it's anything other than haskell, I can give advice, having recently had the concept click, and gone through similar frustrations.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#55

Post by begriffs »

If you really want to go back to first principles, try "Foundations of Analysis" by Edmund Landau. It builds the integers, fractions, Dedekind cuts, and the real and complex numbers from scratch.

It's totally rigorous and starts from, "the ability to read English and to think logically -- no high-school mathematics, and certainly no advanced mathematics."

Re: Ask HN: How or where to begin learning mathematics from first principles?

#56

Post by sekon »

Thanks for putting this question. I too hope to do this someday .. I am stuck with knowing what are the fundamental topics so that i can apply what i learn to as many domain as i see fit, with minimal learning of basic concepts. I have unfortunately not made that much headway.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#57

Post by pixelperfect »

nicklaf wrote: post #41

I absolutely, positively second the recommendation of "Real Mathematical Analysis" by Charles Pugh (don't miss the advice he relates from his colleague, on pages 9&10, with the heading "Metaphor and Analogy", which could easily form the basis for a dissertation on the psychology of mathematical intuition and inspiration). Pugh does an exquisite, uncommonly good job of avoiding a pitfall that >99.9% of mathematics aut…

Thanks so much for your insightful post.

I was working through the first couple of chapters in Spivak's Calculus recently, and was struck by 1) what a great book it was, and 2) what a time commitment it would take to complete it properly! If I could choose a book to take to a tropical island for a year, Spivak might be it. But is it worth spending hundreds of hours working through Spivak and Pugh from the standpoint of developing a professional skill? For someone like OP already out of college and wanting to learn to think mathematically to apply it to programming/electronics, is working through these books as a basis to pursue further mathematical studies overkill? Or worth it?

Re: Ask HN: How or where to begin learning mathematics from first principles?

#58

Post by galeos »

I can recommend 'Who is Fourier?' By the Transnational College of Lex. It assumes very little prior knowledge and introduces the reader to, among others, the concepts of trigonometry, calculus, imaginary numbers, logarithms and Fourier analysis.

I wish I'd known about this book when I was studying maths at school.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#59

Post by jpfr »

Wanting to learn mathematics from "first principles" brought a lot of comments from graduate-level mathematicians. While their advice applies very much for mathematics students, I can't recommend going down that road for engineering types.

In mathematics, everything is connected. One can build up a specific topic from first principles only. But with a too narrow focus one looses these lovely connections between different fields that allow to change the perspective on how we think about problems.

I was in a similar situation some 2 years ago. Try "Vector Calculus, Linear Algebra, and Differential Forms" by Hubbard and Hubbard. You will not be disappointed. Yes, you get (enough) rigor and a lot of first principles mathematics. Nonetheless, the authors have found a lovely way to integrate a wealth of important results from many fields into a coherent text that has one goals: letting you understand the connections and letting you solve the problems.

Re: Ask HN: How or where to begin learning mathematics from first principles?

#60

Post by creatio »

azmenthe wrote: post #27

I wanted to give some recommendations before I hijacked your thread with my own question but I was going to suggest What is Mathematics, How to Prove it and Naive Set Theory which have already have been mentioned. I'm actually in a related situation in which I'm competent in analysis (bachelors in physics) but I struggle with all the category theory inspired design patterns in functional programming. Every book/artic…

How To Prove It is definitely the best (it's the only I read till now though, apart from online tutorials and course notes) book on how to learn on your own. Especially since it details the thought process on how to prove something, has around 20 questions per subsection with answers (especially this is great)!

Another book that I think is great is 'Getallen: van natuurlijk naar imaginair', but its in Dutch.

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