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I am interested in starting this discussion in imitation of Zappers fine forum on becoming a physicist, although i have no such clean cut advice to offer on becoming a mathematician. All I can say is I am one.

My path here was that I love the topic, and never found another as compelling or fascinating. There are basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis.

There are several excellent books available in these areas: Courant, Apostol, Spivak, Kitchen, Rudin, and Dieudonne' for calculus/analysis; Shifrin, Hoffman/Kunze, Artin, Dummit/Foote, Jacobson, Zariski/Samuel for algebra/commutative algebra/linear algebra; and perhaps Kelley, Munkres, Wallace, Vick, Milnor, Bott/Tu, Guillemin/Pollack, Spanier on topology; Lang, Ahlfors, Hille, Cartan, Conway for complex analysis; and Joe Harris, Shafarevich, and Hirzebruch, for [algebraic] geometry and complex manifolds.

Also anything by V.I. Arnol'd.

But just reading these books will not make you a mathematician, [and I have not read them all].

The key thing to me is to want to understand and to do mathematics. When you have this goal, you should try to begin to solve as many problems as possible in all your books and courses, but also to find and make up new problems yourself. Then try to understand how proofs are made, what ideas are used over and over, and try to see how these ideas can be used further to solve new problems that you find yourself.

Math is about problems, problem finding and problem solving. Theory making is motivated by the desire to solve problems, and the two go hand in hand.

The best training is to read the greatest mathematicians you can read. Gauss is not hard to read, so far as I have gotten, and Euclid too is enlightening. Serre is very clear, Milnor too, and Bott is enjoyable. learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

If these and other top mathematicians are unreadable now, then go about reading standard books until you have learned enough to go back and try again to see what the originators were saying. At that point their insights will clarify what you have learned and simplify it to an amazing degree.

Your reactions? more later. By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.

Remark: Arnol'd, who is a MUCH better mathematician than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's.

I intend to revise my linear algebra notes now to point this out.

My path here was that I love the topic, and never found another as compelling or fascinating. There are basically 3 branches of math, or maybe 4, algebra, topology, and analysis, or also maybe geometry and complex analysis.

There are several excellent books available in these areas: Courant, Apostol, Spivak, Kitchen, Rudin, and Dieudonne' for calculus/analysis; Shifrin, Hoffman/Kunze, Artin, Dummit/Foote, Jacobson, Zariski/Samuel for algebra/commutative algebra/linear algebra; and perhaps Kelley, Munkres, Wallace, Vick, Milnor, Bott/Tu, Guillemin/Pollack, Spanier on topology; Lang, Ahlfors, Hille, Cartan, Conway for complex analysis; and Joe Harris, Shafarevich, and Hirzebruch, for [algebraic] geometry and complex manifolds.

Also anything by V.I. Arnol'd.

But just reading these books will not make you a mathematician, [and I have not read them all].

The key thing to me is to want to understand and to do mathematics. When you have this goal, you should try to begin to solve as many problems as possible in all your books and courses, but also to find and make up new problems yourself. Then try to understand how proofs are made, what ideas are used over and over, and try to see how these ideas can be used further to solve new problems that you find yourself.

Math is about problems, problem finding and problem solving. Theory making is motivated by the desire to solve problems, and the two go hand in hand.

The best training is to read the greatest mathematicians you can read. Gauss is not hard to read, so far as I have gotten, and Euclid too is enlightening. Serre is very clear, Milnor too, and Bott is enjoyable. learn to struggle along in French and German, maybe Russian, if those are foreign to you, as not all papers are translated, but if English is your language you are lucky since many things are in English (Gauss), but oddly not Galois and only recently Riemann.

If these and other top mathematicians are unreadable now, then go about reading standard books until you have learned enough to go back and try again to see what the originators were saying. At that point their insights will clarify what you have learned and simplify it to an amazing degree.

Your reactions? more later. By the way, to my knowledge, the only mathematicians posting regularly on this site are Matt Grime and me. Please correct me on this point, since nothing this general is ever true.

Remark: Arnol'd, who is a MUCH better mathematician than me, says math is "a branch of physics, that branch where experiments are cheap." At this late date in my career I am trying to learn from him, and have begun pursuing this hint. I have greatly enjoyed teaching differential equations this year in particular, and have found that the silly structure theorems I learned in linear algebra, have as their real use an application to solving linear systems of ode's.

I intend to revise my linear algebra notes now to point this out.

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