Essays in Constructive Mathematics

He [Kronecker] was, in fact, attempting to describe and to initiate a new branch of mathematics, which would contain both number theory and alge­ braic geometry as special cases.—Andre Weil [62] This book is about mathematics, not the history or philosophy of mathemat­ ics. Still, history and philosophy were prominent among my motives for writing it, and historical and philosophical issues will be major factors in determining whether it wins acceptance. Most mathematicians prefer constructive methods. Given two proofs of the same statement, one constructive and the other not, most will prefer the constructive proof. The real philosophical disagreement over the role of con­ structions in mathematics is between those—the majority—who believe that to exclude from mathematics all statements that cannot be proved construc­ tively would omit far too much, and those of us who believe, on the contrary, that the most interesting parts of mathematics can be dealt with construc­ tively, and that the greater rigor and precision of mathematics done in that way adds immensely to its value.