Jean Paul Van Bendegem

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity

What is philosophy of mathematics and what is it about? The most popular answer, I suppose, to this question would be that philosophers should provide a justification for our presently most cherished mathematical theories and for the most important tool to develop such theories, namely logico-mathematical proof. In fact, it does cover a large part of the activity of philosophers that think about mathematics. Discussions about the merits and faults of classical logic versus one or other ‘deviant’ logics as the logical basis for mathematical theories, ranging from intuitionist over modal logic to paraconsistent logic, typically belong to this area, as do debates about the natural-number concept, its ‘nature’, its properties, especially its uniqueness, and so on. No doubt sociologists of knowledge could explain why philosophers of mathematics came to select these particular problems and deal with them the way they do. What it does imply, however, is that the question is meaningful whether different kinds of philosophies of mathematics are possible, and—why not?—perhaps even desirable. To a certain extent, the answer is trivial: look at what, e.g., phenomenologists have to say about mathematics and you must notice it does not fit into the scheme sketched above.1 Rather, the question should be whether there are forms that, on the one hand, reinterpret the whole enterprise, and, on the other hand, somehow remain related to the work of ‘mainstream’ philosophers of mathematics today.The book under review here does precisely that. Let me be a bit more precise. The general proposal, present throughout all the contributions, is that even at a first, superficial, glance at what mathematicians do when they do mathematics, it is far more complex than ‘just thinking about and …

The heuristics and strategies presented in Lakatos' Proofs and Refutations are well-known. However they hardly present the whole story as many authors have shown. In this paper a recent, rather spectacular, event in the history of mathematics is examined to gather evidence for two new strategies. The first heuristic concerns the expectations mathematicians have that a statement will be proved using given methods. The second heuristic tries to make sense of the mathematicians' notion of the quality of a proof