I’m doing a Mathematics degree with the Open University and I am in my first two of the 4 required level 3 subjects.
I have not really had as much time as I would like to devote to Maths, so, for the most part, I have been reading the material and finishing the course work with as little paper and pen as possible. This system of doing things was working fine up until a few weeks ago when I started to run into trouble with M381 (Number Theory and Mathematical Logic).
The questions in this course have demanded a rigour that has not really been set out in previous courses. When asked “Prove the Following” I was completely baffled and had no idea where to start. This might sound odd for those people who are up to speed with their Math but for me it was an eye opener. For those maths teachers out there you may recognise that I have become a calculator and not a thinker (my own fault) which is fine for doing sums but pretty useless for real Mathematics.
Anyway. Not to be outdone I decided that the only way to get up to speed was to subject myself to more Maths, specifically, more thinking about Maths. I went looking for a text that would have the rigour I was looking for and having searched high and low one text that seemed to appeal was A Course of Pure Mathematics by G. H Hardy
I have been trying to get to grips with this book over the last few weeks and so far I have made the following observations.
The book states in the preface:

This book has been designed primarily for the use of first year students at the Universities whose abilities reach or approach something like what is usually described as “Scholarship Standard”

I am “meant” to be covering 3rd year level material in my degree and a lot of the material in this book appears new to me. This begs the question[s].

  • Have I forgot what I learned previously?
  • Have I not understood the material in previous courses?
  • Has the standard slipped since Hardy’s time?

There are more questions I could ask but these questions alone scared me into checking Google groups for more information and I found the following article on Littlewood which made me think that perhaps the required standard was higher in those days.
Having thought about this for a while I decided that “standards being higher” was a weak excuse at best, and that I should really not be casting historical aspersions when I’m hardly capable of judging a decent book from a bad one due to my lack of mathematical maturity. This left me in a bit of a dilemma.

  • I want to get better at Maths.
  • I have only so many hours in the day (I have bills to pay).
  • I need to teach myself.

The criteria above mean I need to pick an area of mathematics in which I will continually find myself in, regardless of what maths I am doing. From what I could gather there are a few areas that seem to crop up all over the place. Two of these subjects are Calculus and Analysis (I know they are similar subjects), they seem to find their way into everything to some degree or another.
To this end I have decide to spend some of my spare time doing problems from calculus and analysis books. I have also bought Polya’s “How to solve it and am about half way through it” and so far it has been an interesting read.
I have ordered Spivaks Calculus book. I would also like a few other books to read around the area a bit. Again the source I went to for information on this was sci.math. The books I intend to investigate are as follows.

  • Calculus: T. M. Apostol
  • Calculus: Spivak (on order)
  • Calculus Answer book: Spivak (Answer book) (on order)
  • Introduction to Calculus: Courant, John
  • Mathematical Analysis: T. M. Apostol
  • Real Anlysis: Royden
  • Principles of Mathematical Analysis: Rudin (Baby)

I don’t intend to purchase all of the above books although it would be nice to have them. The above list seems to be the most popular recommendations on sci.math (particularly Spivak, it has an answer book so I ordered it). I intend to have a look at each of them and decide which ones would be best suited for me.
I have also heard good things about:
What is Mathematics: Courant Robbins Stewart
which I have been meaning to buy for a long time. Current books I own are:

  • A Course of Pure Mathematics: Hardy
  • Differential and Integral Calculus: Courant Vol 1 (Old Blackie edition).
  • Calculus Made Easy: Silvanus P Thompson and Martin Gardner
  • An Introduction To the Theory of Numbers: Niven and Zuckerman
  • An Introduction To the Theory of Numbers: Hardy and Wright
  • Fundamentals of Number Theory: William J. LeVeque
  • Advanced Mathematics (A pure Course): Perkins and Perkins
  • Engineering Mathematics: K. A. Stroud 4th Ed
  • How to solve it: Polya
  • Fermats Lat Theorem: Simon Singh
  • A concise History of Mathematics: Dirk J. Struik
  • Introductory Logic and Sets for Computer Scientists: Nimal Nissanke
  • A History of Mthematics: Victor J. Katz