ANÁLISIS DEL PROCESO

For all that has been said, I think there is another more devastating ob- jection to Priest’s argument. In part 2.5.1 we saw that the idea that there would only be one formalised counterpart of informal mathematics would not hold any water. However, it was only on this reading that it seemed acceptable to treat informal mathematics as if it were a formal theory, at least superficially, stemming from the fact that there was one ‘body’ of infor- mal mathematics and one formalisation thereof. Nonetheless, having been discussing the difficulties involved in formalising theories, it should now be clearer that there was something fishy going on in this step of the argument. The objection is the following: by moving from informal proof to a for- malised version thereof, Priest’s argument is guilty of changing the subject. The argument intended to show that informal proof was inconsistent, and not just coincidentally but inherently so. Yet, almost immediately in the reasoning, to get the application of the incompleteness results off the ground, Priest needs the subject of his argument to be a formal theory. The answer, therefore, is that mathematics is not a formal theory and that transforming it to be one will do an injustice to its source material. The argument speaks as if the multiple representations that informal mathematics can have as formal systems are identical to the informal mathematics itself, but this is just a confusion of distinct things.

While Priest was looking to demonstrate that informal mathematics was inherently inconsistent, an option that is now on the table is that mathemat- ical reasoning is inherently informal, a view common in the mathematical

practice literature (e.g. Larvor 2012), or that it may be inherently incom- plete, or indeed both. The thought would then, in these cases, be that no formal system would suffice to adequately capture mathematics in its entirety. Indeed, this is the traditional lesson that people take from the in- completeness results, but this standard result relies on the question-begging move from consistency to incompleteness. Now, though, we have seen inde- pendent motivations for thinking so and rejecting the argument, motivations stemming from mathematical practice and paying attention to formalisation as a process.

Priest’s challenge was looking to adjust the balance between consistency and completeness in favour of the latter over the former. But now, by con- sidering the third axis of formality and informality, we have obtained a way to defend incompleteness over inconsistency in the formal setting without begging the question, and to see incomplete and inconsistent systems as both serving purposes which may be justified by pragmatic principles. For the argument relies on a number of assumptions about the nature of for- malisation which allow one to easily and without injustice take informal mathematics into formal mathematics. I have, to the contrary, argued that this distinction runs deep and cannot be bypassed lightly, meaning that ar- guments that work for formal theories cannot be straightforwardly applied to informal mathematics, and ultimately that Priest’s argument does not go through.

Chapter 3

Mathematical Concepts:

Open-Texture, Dialectics and

Engineering

Proof suggests new mathematics. The novice who studies proofs gets closer to the creation of new mathematics. Proof is mathe- matical power, the electric voltage of the subject which vitalizes the static assertions of the theorems.

— Philip J. Davis & Reuben Hersh (Davis & Hersh 1981, p. 151)

3.1

Introduction

In this chapter I will explore something deeply connected to the over-arching theme of proofs and their formality: the nature of mathematical concepts. For proofs feature and operate on mathematical concepts and as such the degree of formality or informality of a proof is closely related to the exactness of those concepts it deploys. If one believes, like the Formalist-Reductionist, that informal proofs correspond to formalised counterparts, then one should also see mathematical concepts as having exact, formal definitions which can be deployed in those formal proofs. Conversely, rejecting this idea leads to the opportunity to be more historically sensitive in seeing that mathe- matical concepts develop over time. Furthermore, we can also be more open

to the possibility of mathematical concepts not being fully fixed in their applications.

The philosophy of mathematical practice is heavily influenced by Imre Lakatos on precisely these issues. Lakatos thought that mathematical con- cepts were not fully fixed, but instead are changed and developed through the proofs they appear in. One way to fill out what this means could be through Waismann’s notion ofopen texture, where a concept is open-textured when it is not fully delimited for all potential applications. The first half of this chapter will bring out the connection between Waismann’s open-texture and Lakatos’s dialectical approach to the philosophy of mathematics. With the door hereby open to ongoing conceptual development in mathematics, in the second half of the chapter I look to connect these ideas to recent work on

conceptual engineering. I will suggest that deploying particular distinctions and strategies pertaining to conceptual change found in the conceptual en- gineering literature is a promising route for integrating the formal/informal axis discussed in the previous chapter with change of concepts in math- ematics. This will ultimately leave us with three major questions. One that underlies the conceptual engineering literature concerns whether the concepts are to be revised or replaced. A second question is whether all mathematical concepts need to be changed or just some. Finally, we can wonder if conceptual change is needed for all mathematical contexts or just some restricted range of them. Of course, the answers relate to one another and jointly contribute to a view on mathematical conceptual change. I shall return to these questions at the end of the chapter.

The precise plan for this chapter is as follows. In section 3.2 I will start by explaining what Waismann holds open texture to be, and comparing it to Shapiro’s more recent usage of the term, showing that the two are very close but are different in the particular phenomena they intend to pick out. Next, in section 3.3, we shall explore Lakatos’sProofs and Refutations, focusing on the roles of concepts and proofs. Following this, in section 3.4 I will briefly cover G. T. Kneebone, a figure who has received little to no attention but pre-empts Lakatos in several crucial respects in proposing a dialectical philosophy of mathematics. In section 3.5, I will bring the three figures together and discuss how Waismann’s notion of open texture is a useful way of describing aspects of the Lakatosian and Kneebonian accounts. However, we will also see that open texture is just one tool in

the toolbox for the contemporary undertaking of conceptual engineering, introduced in section 3.6. Indeed, it has been argued in (Corfield 1997) that Lakatos fails to appreciate the amount of formal and axiomatic work that is involved in modern mathematics. In response to this I suggest that we can draw on this new work in conceptual engineering to supplement the Lakatosian picture of concepts. In sections 3.7 and 3.8, I consider two examples of this: Haslanger’s distinction between manifest and operative

concepts and Scharp’sreplacement strategy, showing how these might apply to the mathematical concepts found in mathematical practice.