What is good mathematics?

This is a revised version of the course notes handed to each participant at the limits of mathematics short course, Orono, Maine, June 1994.

The future of mathematics is described, by using the WZ algorithmic proof theory as a parable.

We give an overview of issues surrounding computer-verified theorem proving in the standard pure-mathematical context. This is based on my talk at the PQR conference (Brussels, June 2003).

In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms of progress in mathematics that are not captured by formal proofs of theorems, especially in his own work in the theory of foliations and geometrization of 3-manifolds and dynamical systems.

This survey article on Hilbert's first and second problems is adapted from a one-hour colloquium lecture given at the University of Auckland in May, 2000, just three months before the 100th anniversary of Hilbert's lecture. It includes an annotated bibliography.

The unique and beautiful character of certain mathematical results and proofs is often considered one of the most gratifying aspects of engaging with mathematics. We study whether this perception of mathematical arguments having an intrinsic 'character' is subjective or universal -- this was done by having test subjects with varying degrees of mathematical experience match mathematical arguments with paintings and music: 'does this proof feel more like Bach or Schubert?' The ...

This text (in Spanish) hopes to offer the reader a starting point to deliver good talks on mathematical topics.

This essay considers the special character of mathematical reasoning, and draws on observations from interactive theorem proving and the history of mathematics to clarify the nature of formal and informal mathematical language. It proposes that we view mathematics as a system of conventions and norms that is designed to help us make sense of the world and reason efficiently. Like any designed system, it can perform well or poorly, and the philosophy of mathematics has a role ...

We will outline our ideas for teaching in the core mathematics disciplines. They are based on our own experience in teaching at a number of universities in the USA, as well as in Europe. While some of the core ideas stay and have stayed relatively constant over a long period of time, they must be varied in accordance with the needs and the demands of students, and they must constantly updated keeping an eye to current research and to modern international trends in technology....

25th anniversary and new building dedication Centre de Recherches Math\'{e}matiques Montr\'{e}al, Canada, October 1994