## “What is Mathematics?”

It might be a little annoying that a description of mathematical activity that satisfies everyone has not yet been found. People have been doing, teaching and learning mathematics for centuries, in schools, colleges and universities throughout the world. Math is something there for all to see. So, what is being asked for? Moreover, what could be learnt from any such description, given that it is of something already known? There are several reasons for this situation.

Usually, when such a kind of question in relation to some activity (medicine, for example) is asked, there is behind it some concern with a potential or actual source of professional intrusion, that is, a concern with some distortion undermining honest professional practice. I think that this is, in part, the motivation of some of Frank Quinn’s writings (see Suggested Readings). Quinn worries that “a century of difficult adaptation to precision might be eroded by a new wave of heuristic work”. According to this author, heuristics may play their role in “mathematical science”, but they should not be used to justify results in “core mathematics”, because results so obtained cannot, and in fact must not, be considered reliable. In contrast, another author, Carlo Cellucci, defends that mathematics cannot be understood without the ‘analytic method’, which makes essential use of heuristics, or rules of discovery (see the Suggested Readings, below). So there is a demarcation problem here. “What is mathematics?” stands for “Which means are to be considered mathematical and which not? Should some open-endedness be allowed so that as new means are discovered they can be added to the collection of those already known? How would one then decide whether candidate means deserve the right to join the previously accepted ones?”, and so on.

But it is also common that what is sought with this kind of question is to put the discipline in a broader context than the most immediate to its practice, to look at it from a different perspective, so to speak, and there are a host of possible choices of perspective (historical, sociological, philosophical, educational, … and any combinations and ramifications of these). Actually, there could be so many different expectations behind a single formulation of the question that it would be impossible that one answer could satisfy all of them at once. Gian-Carlo Rota described several senses of the question “What is mathematics?” under various circumstances in a chapter of the *Lezioni Napoletane*, ed. by F. Palombi, La Città del Sole, Naples 1999 (an English version of this chapter, entitled *What “Is” Mathematics?*, can be found here. I thank Cellucci for the reference). Rota concludes that the feeling of wonder expressed by this question is the start of a philosophical journey that will eventually disclose the ‘conditions of possibility’ of mathematics. Maybe Quinn would make the point that this ought to be a scientific journey, rather than a philosophical one. It might seem strange, however, that to understand or to explain what is mathematics one needs to offer an explanation in terms of ‘conditions of possibility’ (just imagine one had to give an account of carpentry or tailoring in such terms). This approach to mathematics springs from Kant’s transcendental idealism in relation to science and the general problem of knowledge as a response to Descartes’ methodological skepticism. But there is also a modern motivation for it. In connection with Gödel’s realism (I borrow here terms from Penelope Maddy) ‘common features of [mathematics] are taken to derive their justification’ from ‘Robust Realism’ for which ‘the justification — or lack of justification — for mathematical methods is based on a metaphysical account of its subject matter’ (Maddy [2010], p. 87). By calling for metaphysical or epistemologycal accounts, one attempts to leverage the exploration of the frontiers of incompleteness, trying to answer exactly the same questions as above but now with an entirely different purpose in mind.

It is, hence, natural that different answers exist to the question “What is Mathematics?” each of them in relation with a different purpose or throwing light on a different aspect of mathematical activity.

### Suggested Readings

Cellucci, Carlo

[2011a] ‘Top-down and Bottom-Up Philosophy of Mathematics’, forthcoming in Foundations of Science.

[2011b] ‘Philosophy of Mathematics: Making a Fresh Start’, paper presented at the 14th Congress of Logic and Philosophy of Science. Nancy. July 19-26, 2011.

[2011c] ‘Indiscrete Variations on Gian-Carlo Rota’s Themes’, in M. Pitici (ed.), The Best Writings on Mathematics 2010 (pp. 311-329). Princeton: Princeton University Press.

[2008a] ‘The nature of mathematical explanation’, Studies in History and Philosophy of Science, vol. 39, pp. 202-210.

[2008b] ‘Why proof? What is a proof?’, in R. Lupacchini and G. Corsi (eds.), Deduction, computation, experiment. Exploring the effectiveness of proof (pp. 1-27). Berlin: Springer.

Maddy, Penelope

[2011] Defending the Axioms. Oxford: Oxford University Press.

Frank Quinn

*A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today*, Notices of the AMS **59** (January 2011), 31-37.