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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?

#31

Post by jeffasinger »

tiler wrote: post #25

On where to start => I'd start by getting a very solid grasp of graph theory. It is the bedrock of many algorithms and along the way you'll learn all kinds of useful mathematical notation, but in a way that should be easier for you to pick up than plain old 'pure-math.' I've found the following series of videos presented by Donald Knuth, aka The Christmas Tree Lectures, to be incredibly informative and inspirational…

If he's looking for a more of an actual first principles approach to linear algebra, I could not recommend starting with Linear Algebra Done Right by Sheldon Axler enough.

It starts from a few basics which most people would be comfortable with post calculus, and builds up to the most important theorems in linear algebra. It's probably not the book for you if you are actually interested in linear algebra algorithms.

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

#33

Post by jckt »

You seem quite resourceful so you might enjoy MetaMath[1]. This is, in my opinion, literally learning mathematics from first principles. Well, it's not really learning mathematics (insofar that lot of the concepts won't be particularly novel), but it's really showing you the "first principles" of mathematics, and how you can build that up into other stuff.

If you feel like MetaMath is your kinda thing, do visit the FAQ; it's quite good (and virtually required reading if you're going to do this alone).

Admittedly, it's not going to give you the tools suited for solving problems of antenna designs, or enlighten you about electromagnetism, but if you're into just pure recreational mathematics, it's worth a look.

[1] http://us.metamath.org/index.html

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

#34

Post by kellros »

haxiomic wrote: post #4

Try http://www.khanacademy.org (free), their math series starts from basic arithmetic and walks all the way through to undergrad-level mathematics. I personally preferred khanacademy to my math teaching at school and it's been handy during my degree. For more advanced stuff i've found Stanford's online courses ( https://www.youtube.com/user/StanfordUniversity/playlists ) and MIT OpenCourseWare ( http://ocw.mit.edu/in…

I second this and would strongly advise this approach.

The courses available on Khan Academy help you visualize the math and gain a better understanding on the 'why' (reasoning) while also teaching you the 'how' (application).

There's sufficient math courses available to teach you everything from pre/primary school arithmetic to first year university/college level calculus/linear algebra.

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

#36

Post by maroonblazer »

I found myself in a similar situation about a year ago and while I've made a lot of progress I still have a long way to go. Here's what I've found useful:

Precalculus, Coursera, https://www.coursera.org/course/precalculus

"Precalculus Mathematics In A Nutshell", George F. Simmons, http://www.amazon.com/Precalculus-Mathematics-Nutshell-Geome...

"The Hitchhiker's Guide to Calculus", Michael Spivak, http://www.amazon.com/Hitchhikers-Guide-Calculus-Michael-Spi...

http://www.khanacademy.com

http://www.reddit.com/r/learnmath/

http://www.math.stackexchange.com

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

#37

Post by kikowi »

I am also interested in getting better at math, especially algorithmic type of math. I though about playing TopCoder arena, which focuses on algorithmic problems and often requires a lot of math. What do you guys think about this approach? Is it realistic to solve those problems with google as primary resource of knowledge and get better at algorithmic math?

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

#38

Post by Tinned_Tuna »

I would highly recommend reading:

  - A Course of Pure Mathematics (G. H. Hardy). I read this before I started my undergrad in CS/Maths. Free Online.
  - University Calculus (Hass, et. al.). This was reading for my first year, and continued to be useful throughout. Expensive.
  - A Book of Abstract Algebra (Charles C. Pinter). I read this after my degree, but boy, do I wish I'd had it _during_ my degree. Fairly cheap.
  - Linear Algebra Done Right (Sheldon Axler). Moderately priced.
I can't remember which of the texts I had on Number Theory were good at this point, but I do remember that it was quite hard to locate one which was tractable. There's a whole heap of fields which I chose to avoid (woo for joint degree!) but now I kinda regret it -- although I wouldn't have easily given up any of the comp. sci. modules I did...

I would always recommend working through Hardy first, regardless of what else you do.

I don't know of any good websites for this stuff. You may be able to find reading lists on public-facing university module/course web pages, which will help bolster this list.

It may be cheaper to buy access to your local uni's library than try to buy all of the above books. The uni I attended is around £70 a year for a non-student.

Your best bet is to get a notebook and the book that interests you most and work through that book, then the next, and so on. If you get stuck, as someone with a maths degree what the heck's going on :-p

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

#39

Post by mdup »

jckt wrote: post #33

You seem quite resourceful so you might enjoy MetaMath[1]. This is, in my opinion, literally learning mathematics from first principles. Well, it's not really learning mathematics (insofar that lot of the concepts won't be particularly novel), but it's really showing you the "first principles" of mathematics, and how you can build that up into other stuff. If you feel like MetaMath is your kinda thing, do visit the F…

No, I don't think MetaMath is the kind of thing the OP wants. Sure, it builds mathematics from "first principles", but not in the same meaning as the OP means. When you read his question, "first principles" really mean "as a beginner".

Although I like how MetaMath brags about how easy it is to verify proof by yourself if you're not a mathematician, in the end I don't think it's useful for any people but math-oriented in the first place. It's not going to explain the meaning behind the math (it actually intends to take rid of the meaning) and certainly not solve daily problems like antenna design or machine learning. I rather see it like a collection of theorems declared true by a computer following dumbly (I mean that in a positive sense) a substitution rule and base axioms. It's another kind of beauty.

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

#40

Post by [deleted] »

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…

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