Carlo Celluci is a mathematician and philosopher of mathematics at the University of Rome ('La Sapienza'), who has written a series of provocative articles, widely discussed, arguing that no received view in the philosophy of mathematics is tenable. The basic problem, he thinks, is the failure of professional philosophers of mathematics to follow out seriously the implications of Goedel's work: "[M]athematical logicians are most illogical people," Celluci writes, "even when they agree that the axiomatic method provides a distorted picture of mathematical practice, they go on as before as if they had forgotten their own criticisms or the latter did not concern them."

It's not that he thinks Plato or Aristotle's views are satisfactory either; but he does seem to hold that a familiarity with the views of the ancients can help one to appreciate the diverse character of mathematics, as it is actually practiced:

As Wittgenstein points out, even centuries ago a philosophy of mathematics was possible, a philosophy of what mathematics was then. Indeed, centuries ago there were philosophies of mathematics incomparably more articulated than those produced by the foundational schools. For example, Plato and Aristotle developed views of mathematical knowledge where the latter is seen as part of scientific knowledge, forming an integrated system with it, contrary to the foundational schools where theoretical mathematics is treated as a secluded subject, essentially unrelated to the rest of scientific knowledge...The last line I take as implying a rebuke, also, to the Oxford Handbook of the Philosophy of Logic and Mathematics, for its shortsighted view of 'history'.

It is not accidental that the most passionate supporters of the view that mathematics is a profoundly uniform subject based on the axiomatic method should be found among mathematical logicians rather than among mathematicians. While for the latter the axiomatic method is just a tool, for mathematical logicians it is the essence of their subject: without the axiomatic method, there would be no mathematical logic. Thus for mathematical logicians to defend the axiomatic method is to defend their very right to be on the map.

The axiomatic method provides an extremely simplified view of mathematical knowledge. From the viewpoint of mathematical logic, this has the advantage that the method can be easily approached in terms of straightforward notions, such as set-theoretic consequence, formal system, or mechanical process. On the other hand, such notions are patently inadequate to dealing with the variegated features of past and contemporary mathematics. On account of this it appears amusing that Kreisel should claim that philosophy deals only with immature notions and that preoccupation with them draws attention away from genuinely rewarding questions. As a matter of fact just the opposite is true. As already mentioned, both Plato and Aristotle developed a philosophy of the mathematics of their time incomparably more articulated than the limited view of contemporary mathematics provided by mathematical logic. In particular Aristotle--the first theorizer of the axiomatic method--was wise enough not to claim that the latter is the method of mathematics: he only claimed that it is a pedagogical method. Contrary to his modern followers, he did not confuse the presentation of mathematics with the process of mathematical discovery. It is somewhat ironical that the misconceptions of mathematical logicians about the nature of mathematics should be rectified by historians of mathematics like Crowe rather than by professional philosophers of mathematics.

Cellucci's complete article, "Mathematical Logic: What Has It Done for the Philosophy of Mathematics?", may be found here . (The reference to 'Crowe' is: M. J. Crowe, "Ten misconceptions about mathematics and its history", in W. Aspray and P. Kitcher, eds., History and Philosophy of Modern Mathematics, Minnesota Studies in the Philosophy of Science, vol. XI, Minneapolis: University of Minnesota Press, 1988, 260-77.)

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