Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

introduction 5A characterization in broad strokes of the main features of the ‘maverick’tradition could be given as follows:a. anti-foundationalism, i.e. there is no certain foundation for mathematics;mathematics is a fallible activity;b. anti-logicism, i.e. mathematical logic cannot provide the tools for anadequate analysis of mathematics and its development;c. attention to mathematical practice: only detailed analysis and reconstructionof large and significant parts of mathematical practice can provide aphilosophy of mathematics worth its name.Quine’s dissolution of the boundary between analytic and synthetic alsohelped in this direction, for setting mathematics and natural science on a parled first to the possibility of a theoretical analysis of mathematics in line withnatural science and this, in turn, led philosophers to apply tools of analysis tomathematics which had meanwhile become quite fashionable in the history andphilosophy of the natural sciences (through Kuhn, for instance). This promptedquestions by analogy with the natural sciences: Is mathematics revisable? Whatis the nature of mathematical growth? Is there progress in mathematics? Arethere revolutions in mathematics?There is no question that the ‘mavericks’ have managed to extend theboundaries of philosophy of mathematics. In addition to the works alreadymentioned I should refer the reader to Gillies (1992), Grosholz and Breger(2000), van Kerkhove and van Bengedem (2002, 2007), Cellucci (2002),Krieger (2003), Corfield (2003), Cellucci and Gillies (2005), and Ferreiros andGray (2006) as contributions in this direction, without of course implying thatthe contributors to these books and collections are in total agreement witheither Lakatos or Kitcher. One should moreover add the several monographspublished on Lakatos’ philosophy of mathematics, which are often sympatheticto his aims and push them further even when they criticize Lakatos onminor or major points (Larvor (1998), Koetsier (1991); see also Bressoud(1999)).However, the ‘maverick tradition’ has not managed to substantially redirectthe course of philosophy of mathematics. If anything, the predominanceof traditional ontological and epistemological approaches to the philosophyof mathematics in the last twenty years proves that the maverick camp didnot manage to bring about a major reorientation of the field. This is notper se a criticism. Bringing to light important new problems is a worthycontribution in itself. However, the iconoclastic attitude of the ‘mavericks’vis-à-vis what had been done in foundations of mathematics had asa consequence a reduction of their sphere of influence. Logically trained