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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?

#111

Post by nicklaf »

Todd wrote: post #50
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…

I came here to suggest the same thing. I started with real analysis but realized that I was lacking the mathematical maturity to understand things back to first principles. My suggestion is to start with learning how the foundations of mathematics were rebuilt starting from axiomatic principles in the last part of the 19th century and early 20th century. The book How to Prove It helped me tremendously. Get a good boo…

That is excellent advice. If the student is struggling to write proofs in her or his real analysis text, s/eh should definitely seek out "How to Prove It".

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

#112

Post by playing_colours »

Many people here advised Algebra Done Right by Sheldon Axler. There is 3rd edition of this book available now in electronic form, and paper books will be available in a week. They are just beautiful, colour: http://www.springer.com/mathematics/algebra/book/978-3-319-1... .

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

#114

Post by trendroid »

nicklaf wrote: post #104
vishnugupta wrote: post #93

Earlier quoted context omitted.

Thank you very much for taking time to post many elaborate, useful comments! I straight away went and purchased "Real Mathematical Analysis" thanks to your recommendation. > One thing to keep in mind: do not be mislead by similar notation between mathematics and physics: they are very, very different subjects. This has always bothered me especially while tackling analysis. Isn't analysis' origin rooted in Physics pro…

The difference is the emphasis: mathematicians focus on proofs, and therefore have to write proofs of theorems that ensure their validity in all cases. Many of these cases are what physicists would call "pathological", and so they basically ignore them when introducing calculus as a tool for students of physics. If you read the table of contents of virtually any book that professes to introduce the "mathematical meth…

Thank you for all the comments. I really hope you consider either writing series of blog posts or a small book on the topic of teaching maths. You might think there is already lot of stuff on this and there is nothing new you could add but trust me its not enough and people can use more ideas on 'how to go about learning maths'.

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

#115

Post by jorenb »

tel wrote: post #89

There is no royal road to math. There are instead, roughly, between 4 and 50 branches of mathematics which each start and "end" in different places with different goals and philosophies and styles. What makes this all "math" is that almost inexplicably these branches tread the same ground over and over. Which is to say: learning one branch can dramatically improve your ability to understand another branch. Learning s…

'Measures, Integrals and Martingales' by R. Schiller is a very good introduction to measure theory and some of its applications to probability theory.

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

#116

Post by pneumatics »

tel wrote: post #89

There is no royal road to math. There are instead, roughly, between 4 and 50 branches of mathematics which each start and "end" in different places with different goals and philosophies and styles. What makes this all "math" is that almost inexplicably these branches tread the same ground over and over. Which is to say: learning one branch can dramatically improve your ability to understand another branch. Learning s…

| I don't know a good introductory book, unfortunately.

I liked Eric Schechter's Classical and Non-Classical Logics for an eye-opening view into how logical systems are constructed from axioms. Might be a satisfying read for OP, as well.

http://press.princeton.edu/titles/8119.html

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

#117

Post by senderista »

It presupposes a certain degree of mathematical maturity, but Robert Geroch's _Mathematical Physics_ (don't let the title fool you!) has the most intuitive explanations (with diagrams) I've ever seen of definitions, concepts, and proofs. It's roughly at the first-year grad level, but is almost completely self-contained, and uses category theory to motivate the entire organization of the book (whatever object is being treated in the current chapter generally has a forgetful functor to an object in the previous chapter).

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

#118

Post by [deleted] »

marcosdumay wrote: post #94
pixelperfect wrote: post #57

Earlier quoted context omitted.

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…

For somebody that wants a career on electronics? "Sure" is not enough to express it... Your question is almost equivalent to "Is it worth it to spend time studying electronics"? The only thing I don't knwo is if the book goes deep enough, or if he'll need another couple of them. Now, if he wanted only to get good at the software realm, I'd recommend a more superficial understanding of calculus, saving time to study d…

[deleted]

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

#119

Post by richsherwood »

nicklaf wrote: post #63
pixelperfect wrote: post #57

Earlier quoted context omitted.

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…

Well, if you really take to heart what Spivak and Pugh have to say (and are thinking hard about the problems), there is a very good chance that you will be inspired enough to do further research that will lead you to tangentially related mathematics. So, in all likelihood, you will start to branch out even before you manage to finish your first serious math book. (Taking this to the extreme, Paul Halmos once pointed…

I would really like to thank you for posting this. Seeing how excited you get when you talk about mathematics is really inspiring and nice to see. If I may just chime and ask for my own selfish reasons, for someone who is interested in statistics, is there a starting point you would recommend? Thanks again.
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