Those who feel obliged to justify this insist on classifying structures up to isomorphism before using them to solve real problems. Even Aristotle had his doubts about excluded middle, which is more properly attributed to the Stoics, according to Jan L ukasiewicz. On the other hand, all intuitionistically definable functions R R are continuous. With hindsight, it is unfortunate that Brouwer chose analysis to attack. As we shall demonstrate throughout this book, logic and algebra can be presented intuitionistically, with no noticeable inconvenience, whereas most of the usual properties of the real numbers rely on excluded middle.
Every reform provokes reaction.
Often, however, we shall find it convenient to assume excluded middle in order to give some simple introductory examples of concepts, particularly when the most familiar form occurs in the context of R . The reader for whom constructivity is essential will be able to recognise the cases where these conventions apply. The increasingly abstract form of late nineteenth century mathematics led to the use of infinite families of choices, often with no conscious understanding that a new logical principle was involved. Giuseppe Peano did formulate, and reject, such an axiom in 1890, and Charles Sanders Peirce gave the following definition in 1893, but it was Ernst Zermelo who first recognised how widely it had already been used. It was in order to formulate his response that he found his famous axioms of set theory, which we shall discuss in Remark 2.2.9 .