Formalized Mathematics

It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical. 1 History and Philosophy Mathematics is generally regarded as the exact subject par excellence. But the language commonly used by mathematicians (in papers, monographs, and even textbooks) can be remarkably vague; perhaps not when compared with everyday speech but certainly when compared with the language used by practitioners of other intellectual disciplines such as the natural sciences or philosophy. Trybulec andŚwiȩczkowska (1992) point out that in a way this is natural: since the underlying semantics of mathematics is generally clearer than that of other disciplines, the mind is naturally trammelled into a precise mode of thinking, and terminological exactness is less important. But a knowledgeable reader is sometimes needed to separate rhetorical flourish from real content, and to appreciate the context in which results are asserted. Some everyday phrases are imbued with a particular significance; observe for example the crucial distinction between a ‘mapping into X’ and a ‘mapping onto X’, or the precise (albeit context-dependent) meaning of woolly-sounding expressions like ‘for almost all x’. And on the other hand many issues that the author feels are obvious or unimportant may be glossed over. Bourbaki (1968) stresses the importance of ‘abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability’, but many of the conventions go beyond mere abuse of language. Consider the following examples taken from the early pages of Matsumura (1986): If f A B is a ring homomorphism and J is an ideal of B, then f J is an ideal of A and we denote this by A J ; if A is a subring of B and f is the inclusion map then this is the same as the usual set-theoretic notion of intersection. In general this is not true, but confusion does not arise. When we say that R has characteristic p, or write char R p, we always mean that p is a prime number. In definitions and theorems about rings, it may sometimes happen that the condition A is omitted even when it is actually necessary. Trybulec and Świȩczkowska (1992) remark that the language of mathematical texts isn’t incontrovertibly a natural language at all; but it invariably contains a substantial admixture of natural language, and that has all the usual potential for ambiguity and imprecision. This might not be a problem if mathematics were a small, unified subject easily graspable by a single practitioner. But on the contrary mathematics is heading increasingly in the direction of specialization, and it’s not realistic to expect mathematical physicists, say, to appreciate deeply all the theoretical results in topology, differential geometry, numerical analysis and what not that they use. There is also the question of the correctness of mathematical reasoning. Mathematical proofs are subjected to peer review before publication, but there are plenty of well-documented cases where published results turned out to be faulty. A notable example is the purported proof of the 4-colour theorem by Kempe (1879); the error in this proof was eventually pointed out in print by Heawood (1890), and it is only with the work of Appel and Haken (1976) that the theorem has finally come to be ac-