Ian Stewart, David O. Tall.
The Foundations of Mathematics.
OUP. 1977
This is a book for readers in transition from ‘school mathematics’
to the fully-fledged type of thinking used by professional mathematicians.
It should prove useful for first-year students in universities, polytechnics, and colleges,
to sixth-formers contemplating further study,
and anyone else interested in the critical change in mathematical thinking from intuition to rigour.
It develops the more formal approach as a natural outgrowth of the pattern of underlying ideas,
building on a school mathematics background to develop the viewpoint of an advanced practising mathematician.
The topics covered include: the nature of mathematical thinking;
a review of the intuitive development of familiar number systems;
sets, relations, functions;
an introduction to logic as used by practising mathematicians,
methods of proof (including how a mathematical proof is written);
development of axiomatic number systems from natural numbers
and proof by induction to the construction of the real and complex numbers;
the real numbers as a complete ordered field;
cardinal numbers; foundations in retrospect.
