Academic Guidance How to become a mathematician

Feb 2020
4
0
North America
Hi! I just created an account to ask this question - and I don't see any evidence my question has been asked and answered before. I am NOT asking how to learn e.g., Calculus; nor algebra; nor any other particular technical application of math. I am asking how to deliberately, systematically become an authentic mathematician. By way of backstory, presuming that might be helpful, I got through one semester of engineering calculus, one semester of statistics, and one semester of linear algebra some decades ago, the hard way, and eventually got degrees in biochemistry and computer science. I actually went back to re-take algebra II at one point in order to strengthen my appreciation for the axioms -- and tried to subsequently refuse to do problems merely by formula, until I could mathematically derive the formula based on axioms. This worked pretty well except in stats, where the teacher never taught the derivation of the formulas. I never had a class in proofs - and I figure this is a critical skill I will have to master if I want, now, to become a mathematician.

By the way, to make my goal more tangible I am aiming to initially learn enough math to actually understand General Relativity in terms of math. I believe this is possible for me, but I don't believe it is possible without a pretty detailed plan and the discipline and mentoring. I am considering going back to school but I don't know where to start and, again, I don't have the time to just take everything. I might, possibly, somehow fast track my way into a masters program by being strategic in what subjects, and in what order, I tackle them - but right now I don't even know what I don't know...and need to learn. That is, what mastery is dependent on what prior mastery - trying to short-cut through the stuff which could otherwise consume more time than I likely have left in this life.

My guess is that I absolutely need to be strong in proofs. I am intending to begin working through some introductory books, probably starting with Polya's "How to Solve It". (Better resource suggestions appreciated). Beyond that I will probably attempt to work my way through "The Haskell Road to Logic, Maths an Programming" to give myself some practice. Assume I have already reviewed trig through 1st semester calculus. Do I need to take a more solid course in statistics - one which really dives into how the formulas were derived vs merely how to apply them?

Beyond that, there seem so many distinct areas of math I have no exposure to that I am guessing i will really need to work my way through in order to approach actually understanding General Relativity. But which ones are really essential to really feel I 'get' (as opposed to merely have a technical facility with) math - and it what order should I approach them? Set theory? Graph theory? Group theory? Topology? Category theory? Combinatorics? Something else? I obviously will need more calculus - but how many semesters? I've read I might need "diferential geometry" to begin to understand relativity?

I am hoping I might get some solid guidance from those among you who already have found your identity as mathematicians - either by luck (having good mentors) or the hard way (plowing through 'required' courses in order to discover for yourself what was actually most helpful). I'm pretty serious about this, but I'm also really, really
overwhelmed at the prospect of trying to find my way when I don't have forever, this isn't my career, I don't have a mentor, and I don't even know books to pick up...nor in what order.

Thank you kindly for your experienced advice.
 

SDK

Sep 2016
793
540
USA
Are you asking how to become a professional mathematician (i.e. work as a researcher in math) or an amateur mathematician (i.e. do math as a hobby)? Your question seems to be focused on what math topics you must learn which makes me think the latter but you also ask about studying at a university which makes me think the former.
 
Aug 2012
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780
By the way, to make my goal more tangible I am aiming to initially learn enough math to actually understand General Relativity in terms of math.
Oh that's a much more modest goal. You barely need anything more a good class in differential geometry and tensor calculus; which frankly will serve your purpose far better if you take them from the physics department and not from the math department.

You don't need to be a mathematician for this and you don't want to be.

To understand the math of general relativity you need a solid course in multivariable calculus; some general topology; a little group theory; and a big heaping of differential geometry.

But when you study physics from a physics curriculum, they'll teach you all you need to know about these things without sending you on the abstract flights of generality beloved by mathematicians.

You want to study physics, not math. And not even that much physics. GR is upper division or early grad level material these days in the physics curriculum.

Physicist Leonard Susskind has a series of lectures on general relativity on Youtube. You might start there. He starts from first principles, you barely need to know anything at all. And he reminds me of George Carlin so he's never boring.

Start here, this is the first of a series of lectures. You'll see that physics is about physics, not math. You use math but the arguments are physical.

 
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Feb 2020
4
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North America
Thank you for your responses - especially the direction on courses required to understand GR. Maybe physics is a better path for me to follow, as I seem to understand things better when I can map them to the "real world" (whatever that is...sometimes I wonder).

In redirecting me, however, you've made me even more curious: what does it mean to be a "mathematician"? I've taken math classes (although obviously not enough to satisfy me, much less understand) and I've heard before that most of what passes for math is actually only applications of math (technician instead of master)? And I've always felt there was a lack in my education in particular since I never learned how to do complicated proofs (which bothers me, considerably). I'm still curious as to what it means - to those of you who are mathematicians - to be doing real math and, I guess, to consider yourself mathematicians? If that's too hard a question to answer, maybe you could direct me to a reasonable answer on the web or in a book?

I'm actually puzzled, I guess...I'm thinking I'm fundamentally confused...and I am thinking a large portion of the population must be as well. I've long felt like there was something lacking - and now I really feel, in a way, even worse...there is that saying about "not knowing that you don't know". I guess I'd like to know - or at least better appreciate -- what I don't know.

Thank you, again, for your guidance.
 
Feb 2020
4
0
North America
It looks like my local college has an M.S. in math but only a B.S. in physics. To answer SDK's question, I am want to do this for myself, because I feel an incompleteness in my understanding of the universe and that I "should" understand math and the universe better than I do; that it will make me a more complete person.

I am too old to do this for a career. But I keep buying math books which are over my head. That might sound like a compulsion and I need a therapist :-}. Except I believe I actually could learn to understand a lot of mathematics if I had a good mentor/teacher and the rigid schedule of a school which would force me to put in the time practicing. Trying to study on my own I keep getting stuck and discouraged. I think I am fundamentally smart enough to get through the classes -- although I accept that I probably would never get top grades in competition with those who are younger and smarter than I am. I had a tough time with Physical Chemistry, though, so maybe I am mistaken.
 
Feb 2020
4
0
North America
Oh, one more question. I've got a book "Category Theory for the Sciences" by David Spivak. It is supposedly more accessible to non-mathematicians, but it is filled with exercises and I presume will give me a good grounding on which I might build. My question: Will learning some "Category Theory" now be a helpful in branching into other areas like the "Topology" you mentioned? Or am I taking things out of order -- and is there a better/ easier order in which to get introduced to the more abstract areas of math?
 

SDK

Sep 2016
793
540
USA
If you are not talking about becoming a professional mathematician, then you should simply study whatever interests you. I'm not sure it will "make you a more complete person". But it's fun!