### Stage 3

At this stage, you begin to learn modern Pure Mathematics. Yup, this is the beginning. Focus on the proof. Don't expect you can solve a problem by plugging numbers into formulae. It often happens in advanced text that the author skips some steps in a proof or calculation, while elementary text gives detail explanations. But as G.F. Simmons has said, '[t]he serious student will train himself to look for gaps in proofs, and should regard them as tacit invitations to do a little thinking on his own.' Moreover, '[i]t is a basic principle in the study of mathematics, and one too seldom emphasized, that a proof is not really understood until the stage is reached at which one can grasp it as a whole and see it as a single idea. In achieving this end, much more is necessary than merely following the individual steps in the reasoning. This is only the beginning. A proof should be chewed, swallowed, and digested, and this process of assimilation should not be abandoned until it yields a full comprehension of the overall pattern of though.'

Also get maturity and learn how to write mathematics, by reading good books (?). I don't think Gauss or Galois can get full mark in mathematics assignments these days, for 1) something obvious to them may not be obvious to everyone else, and 2) they don't bother to explain to others. The point is, every steps in a proof should be logically related and everyone can go easily from the previous step to the next. This is not the case for non-pure mathematics. If you insist, I can show you a copy of full mark statistics assignment consists work of mere calculation and computation. (Of course not done by me!)

I had tried to list more books so that you can compare and choose one or two that fit you. Despite that most of following courses are 'pure', they can be applied to other graduated level science subjects. In the analysis class, Ian Doust even showed us an article in the Econometrica (Hildenbrad W. and Metrens J.F. Upper Hemi-continuity of the Equilibrium-set Correspondence for Pure Exchange Economies, Vol. 40, No. 1) talks about liminf, limsup, measure, weak topology and stuffs like that. If one insists that the pure stuff is useless, one is just ignorant, I don't bother to argue with those people anymore.