Climbing up the math tree

Earlier today a person on Quora asked me the following question:

“For a long time I have wanted to learn everything about math topics, but I don’t know where to start. What is the topic to start with and what are the following? MY GOAL is to be very good at math, so I appreciate don’t skip topics.”

I thought this was a really interesting and ambitious question, so I decided to answer it in some detail, and leave my response here for my Medium followers to see. In this response I assumed the person asking had yet to receive a college degree [in mathematics].

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For better or for worse, the truth of the matter is that there is simply too much math out there to learn all about all of it, but this is the right attitude if you want to say, pursue the career path of an academic mathematician, or some other research-oriented career (such as working in a lab).

So, what topics should you start with? I’m going to assume you’ve gotten through the high school math curriculum up through two semesters of calculus and are entering college. If not, at least get through two semesters of calculus (through Taylor series and integration techniques) before proceeding on to what I am going to suggest next.

In my opinion (and some may disagree with me on the point), the quintessential course to learn about proofs is an introductory course in real analysis (on the real line). While you can teach the basics to proof writing in the context of a mathematical logic…