This volume is a collection of papers on philosophy of mathematics whichdeal with a series of questions quite different from those which occupied theminds of the proponents of the three classic schools: logicism, formalism,and intuitionism. The questions of the volume are not to do withjustification in the traditional sense, but with a variety of other topics. Someare concerned with discovery and the growth of mathematics. How does thesemantics of mathematics change as the subject develops? What heuristicsare involved in mathematical discovery, and do such heuristics constitute alogic of mathematical discovery? What new problems have been introducedby the development of mathematics since the 1930s? Other questions areconcerned with the applications of mathematics both to physics and to thenew field of computer science. Then there is the new question of whetherthe axiomatic method is really so essential to mathematics as is oftensupposed, and the question, which goes back to Wittgenstein, of the sense inwhich mathematical proofs are compelling. Taking these questions togetherthey give part of an emerging agenda which is likely to carry philosophy ofmathematics forward into the twenty first century.