These pages are of various kinds, but they are nearly all attempts to show how mathematical ideas arise naturally, in the hope that some people will find them a useful supplement to university mathematics courses. Often they contain ideas that I have come across in one way or another and wish I had been told as an undergraduate. (Probably I was told several of them, and just wasn't concentrating enough to take them in at the time.)
Why study finite-dimensional vector spaces in the abstract when they are all isomorphic to R^{n}?
The relationship between theory and computation in linear algebra.
Why is multiplication commutative?
How to invent some basic ideas of Galois theory.
Lose your fear of tensor products
A dialogue concerning the existence of the square root of two.
What is wrong with thinking of real numbers as infinite decimals?
A dialogue concerning the need for the real number system.
How to solve basic analysis exercises without thinking. (Gradually being revised and expanded.)
Proving that continuous functions on the closed interval [0,1] are bounded.
Finding the basic idea of a proof of the fundamental theorem of algebra.
What is the point of the mean value theorem?
A tiny remark about the Cauchy-Schwarz inequality.
What makes non-Euclidean geometry interesting?
What makes hyperbolic geometry particularly hard?
What can I do to make it seem less hard?
How can the arc of a circle be considered straight?
Why did people want to prove the parallel postulate?
What is the historical importance of non-Euclidean geometry?
Why isn't it just obvious that a regular dodecahedron exists?
What is naive about naive set theory?
Paradoxes concerning definability.
A beginner's guide to countable ordinals.
Eight metamathematical statements with proofs that can be understood by non-logicians.
How to discover a proof of the fundamental theorem of arithmetic.
How to discover the statement and two proofs of Fermat's little theorem.
How to think of the Riemann zeta function and discover the product formula.
New: Is Cambridge biased against state-school applicants?
Watch this space.
The definition of `definition'.
Is the phrase `well-defined' well-defined?
What is `solved' when one solves an equation?
The implication of implication.
Does mathematics need a philosophy? (This is a talk I gave to the new Cambridge University Society for the Philosophy of Mathematics and Mathematical Sciences.)