I can do basic single variable calculus which is essentially all you do at A Level in the UK. I also just read "What is mathematics?" by Richard Courant which I found very good. I would like to know where to go after I finish my proof writing and basic discrete maths books I am currently reading. Should I study real analysis? What book? Linear algebra?
I would appreciate advice on where to go next and what book would be good. I am currently 14 and my end goal would to become a pure mathematician.
The reason I ask this question is because I was trying to create a structure and I thought about the Gerard t'Hooft physics one or even the Pure mathematician or statiscian plan but they don't seem particularly specific in what you are meant to do where i.e they call a topic calculus and then a later one vector calculus and then analysis but as a pure mathematician I want to go straight into analysis, of reals obviously. In conclusion, I am asking this because I am going to go by steps now and choose the next topic as it comes.
Thank you in advance.
EDIT:
Would this work:
In order…
Elementary Discrete Maths,
Real Analysis,
Linear Algebra,
ODE's,
Probability,
Fourier Analysis,
Complex Analysis,
PDE's,
Graduate stuff which I will get to when needed.
I'm not sure if probability is necessary but I think it'll be interesting.
Best Answer
I think you are correct to lean toward analysis. It is the usual first step to pure math as you describe your inclination. Linear algebra can be done on several levels, but I think is best done in a rigorous context after real analysis.
In real analysis you will become familiar with proofs, begin to accumulate a working, ingrained math vocabulary as well as learn about topology, metric spaces, concepts such as convergence, continuity, etc. There is also usually a component presenting the material of calculus on a rigorous basis. This will give you a feel for the difference between operational mechanics and the pure math behind it.
I would suggest you take a look at this free set of notes of lectures given by Vaughan Jones (Fields Medal winner, equiv. Nobel Prize in math). They are really beautiful, are self-contained, and build nicely from a level that does not require prior experience.
https://sites.google.com/site/math104sp2011/lecture-notes