Ask HN: Which books helped you gain mathematical maturity?

[curious16]

Mathematical maturity is a term thrown around to mean that someone with it is well versed with the lingo and workings of the field of mathematics. You can't learn the entire field of mathematics in your lifetime. But what mathematical maturity gives you is the power to understand mathematics that is at the moment beyond your scope by searching the right things or by pattern matching.

You are confident that you can learn more and tackle challenging problems with the use of mathematics.

How and when did you gain that so called "mathematical maturity"?

[tmp4c71]

"How to Think About Analysis" by Lara Alcock (2014) because the first part (half) of the book is specifically dedicated this aim, "[focusing] explicitly on skills and strategies for learning advanced pure mathematics." (from the author's intro). The remaining text demonstrates how to do this for a full first undergraduate course in real analysis.

Also, "A Book of Abstract Algebra" by Charles Pinter (available as a Dover paperback). The introductory sections to each chapter draw motivating lines from earlier definitions / theorems.

Other books that are good mentions, but even better after the above include the 2nd edition of Sheldon Axler's "Linear Algebra Done Right" (it motivates each development similar to Pinter) and "Topology" by K. Janich.

Of all, I have admired how they motivate without compromising rigor, using expository text alongside formal statements of definitions, theorems and proofs.

[max_]

I haven't quite reached the "mathematical maturity" but what you describe seems like something I have be after too.

I think you need to have a look at Kolmogorov's Mathematics: Its Content, Methods and Meaning [0]

You can also get it in print [1]

[0]: https://archive.org/details/MathematicsItsContentsMethodsAnd...

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

[uptownfunk]

I think my first leap came when I realized mathematics is not about the formulaic plug-and-chug exercise type of mathematics commonly learned in grade school (particularly in the USA). E.g. x^2 + 1 = 5, x = -2,2. Great, now repeat this 100 times.

So then what is it about? Well it is about going deeper, asking why does this work. Instead of focusing on specific statements like solving the above problem it is about conjecturing what is the technique to solve all such problems (e.g., quadratic formula). And then proving that this technique works for all given numbers of a certain type.

After that it is about thinking even more deeply about the problem and saying, well does this work only for real coefficients? what about complex? what if we restrict ourselves only to remainders mod 5 (e.g. the numbers 0-4 mod 5) and so on. Why only polynomials with powers of 1,2. What about 3,4, and above? What about polynomial equations in more than one variable?

More broadly it is about appreciating the challenges experienced before the theory to solve some problem was developed. Then understanding that some new innovation was required. Generally, this takes the form of defining some new type of object, showing that the problem we are solving is but one instance of that object, and then saying, well based on these definitions and what we already know, here is what we can say about these objects. This is the process behind building a theory.

I think when you can start appreciating some of the ideas that the mathematics is saying, and the limited understanding of the world before that theory was developed, and then following a similar process to develop a mathematics of your own, then you have matured as a mathematician.

This is but one humble attempt to answer the question briefly.

[hnthrowaway0328]

I think a serious, difficult real analysis I course (sometimes they have I and II) or an equally difficult elementary number theory course will expose you to enough mathematical proofs.

[hgsgm]

That's not what OP is asking. OP is asking for an overall intuition for "all" of mathematics, not just what a proof looks like.

It takes a while undergrad degree worth of pure math study.

[warrenm]

See also https://news.ycombinator.com/item?id=35874448