On Knowledge and Practices

A Manifesto

Although the idea of emphasizing them is relatively new, and there is still some disunity concerning how to focus our analysis, mathematical practices are in the agenda of every practicing philosopher of mathematics today. Mathematical knowledge, on the other hand, has always figured prominently among the mysteries of philosophy. Can we shed light on the latter by paying attention to the former? My answer is yes. I believe the time is ripe for an ambitious research project that targets mathematical knowledge in a novel way, operating from a practice-oriented standpoint.

Let us begin by placing this kind of enterprise within the context of the philosophy of mathematics. During the twentieth century, we have seen several different broad currents in this field, which, simplifying a great deal, can be reduced to three main types: foundational approaches (logicism, intuitionism, formalism, finitism, and predicativism), analytic approaches (focused on questions of ontology and epistemology), and the so-called "maverick" approaches (to use Kitcher's colorful terminology), which have typically been anti-foundational and focused on history, methodology, and patterns of change. Mixed approaches have, of course, been present throughout the century, although one can say that they remained relatively uninfluential; early examples are the work of Jean Cavaillès in France during the late 1930s and that of Paul Bernays in Germany and Switzerland from the 1930s on. But in the 1980s and 1990s the situation — at least judged from an Anglo-American perspective — seemed to be one of confrontation between the anti-foundational maverick camp and the system-oriented camp.

It seems to be the case that a new generation of philosophers of mathematics has arisen whose work is superseding those distinctions. They follow upon the footsteps of Cavaillès, Bernays, Manders, and others. Examples of this new phenomenon are provided in some recent anthologies, such as Mancosu (2008), Ferreirós and Gray (2006), and van Kerkhove, de Vuyst, and van Bendegem (2010). These philosophers engage in an analysis of mathematical practices that incorporates key concerns of the "mavericks," without adopting their anti-foundational, anti-logical orientation. They are no longer obsessed with all-encompassing formal systems (e.g., axiomatic set theory) and the associated metalogical results, directing their attention instead to different branches and forms of mathematics — geometry ancient and modern; different ways of practicing analysis, algebra, topology, and so on. But thereby they do not imply — not at all, in my case — that there's nothing to learn about mathematics and its methodology from the crisp results of foundational studies. They keep considering the traditional questions of ontology and epistemology, but within a broader palette of issues concerning the evaluation of mathematical results (see the different aspects treated in Mancosu 2008) and the place of mathematics within human knowledge — one of my central concerns here.

All of that is meant when I say that I aim at providing a novel analysis of mathematical knowledge from a practice-oriented standpoint.

Notice that the new orientation in the philosophy of mathematics is highly interdisciplinary. Some authors emphasize knowledge of mathematics itself and logic, coupled with careful scrutiny of epistemological issues; some put an emphasis on combining philosophical issues with historical insight; some others stress the role of cognitive science (Giaquinto 2007) or sociological approaches (van Bendegem and van Kerkhove 2007); and the list goes on, with mathematics education, anthropology, biology, etcetera. My own approach, as will become clear, has a strong interdisciplinary bent. But one has to be quite clear and careful about the ways in which the different disciplines could or should contribute to the enterprise. Instead of trying to provide a principled discussion at this point, we shall clarify the matter as we go along. However, let me give an example of what I mean by "careful": it is highly relevant to establish contact with cognitive science and with the biological underpinnings of human knowledge, but I believe the time is not ripe for simply taking some 'established' theories or models from cognitive science and "applying" them to mathematics. While paying attention to what goes on in cognitive science and neuroscience, and aiming at convergence with that kind of research, a philosopher of mathematics can and perhaps should remain independent from the concrete current theories in those fields (see Chapter 3).

To briefly describe the crucial traits of the approach I shall put forward, I can say that this is a cognitive, pragmatist, historical approach:

• agent-based and cognitive, for it emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture;

• pragmatist or practice-oriented, as it places emphasis on the practical roots of math, i.e., its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices;

• historical and hypothetical, because it emphasizes the need to analyze math's historical development, and to accept the presence of hypothetical elements in advanced math.

Our perspective on mathematics will thus stress the provenance of mathematical knowledge from particular kinds of interplays between cognitive resources and cultural practices, with agents at the center, making such interactions and the development of new practices possible.

There is an aspect in which such an approach goes against well-established habits of philosophers of mathematics. It was customary to focus one's attention on a single mathematical theory assumed to be sufficiently broad to embrace all of current mathematics; the chosen system was commonly a form of axiomatic set theory based on classical logic. This tradition emerged as a result of foundational developments in the early twentieth century, but its roots lie deeper, in the centuries-old vision of Math as an Ideal Theory, given in some Platonic realm (often conceived as God's mind), fixed and static, offering a unified foundation which one may bring to light by excavating the imperfect glimpses of the "true" math that our current theories provide. Of course, this old tendency has been heavily criticized, both by the "mavericks" and by more recent proponents of a philosophy of mathematical practice. One of the reasons for such a critique is that set-theoretic reductionism blinded philosophers of mathematics to important phenomena, such as the hybridization of branches of mathematics, progress through...