Andrea Sereni

Equivalent explanations and mathematical realism. Reply to “Evidence, Explanation, and Enhanced Indispensability”

The author of “Evidence, Explanation, Enhanced Indispensability” advances a criticism to the Enhanced Indispensability Argument and the use of Inference to the Best Explanation in order to draw ontological conclusions from mathematical explanations in science. His argument relies on the availability of equivalent though competing explanations, and a pluralist stance on explanation. I discuss whether pluralism emerges as a stable position, and focus here on two main points: whether cases of equivalent explanations have been actually offered, and which ontological consequences should follow from these.

Indispensability and explanation: an overview and introduction

On November 19–20 2012, the Institut d’Histoire et de Philosophie des Sciences et des Techniques in Paris hosted a two-day workshop on Indispensability and Explanation organized by Marco Panza and Fabrice Pataut. Most of the papers presented on that occasion or, rather, their distant descendants, have been included in this special issue; some additional contributions came in at a later time. The workshop was structured so that a rejoinder followed the delivery of each paper and we have remained faithful to this structure for their publication even though some of the rejoinders have almost become stand-alone papers thanks to suggestions by our panel of reviewers, to whom we would like to express our gratitude for their invaluable work.

A Dilemma for Benacerraf’s Dilemma?

A major part of the debate in the philosophy of mathematics of the last forty years has been dominated by attempts at escaping the dilemma Paul Benacerraf suggested in “Mathematical Truth” (Benacerraf 1973).

On the Indispensable Premises of the Indispensability Argument

We identify four different minimal versions of the indispensability argument, falling under four different varieties: an epistemic argument for semantic realism, an epistemic argument for platonism and a non-epistemic version of both. We argue that most current formulations of the argument can be reconstructed by building upon the suggested minimal versions. Part of our discussion relies on a clarification of the notion of (in)dispensability as relational in character. We then present some substantive consequences of our inquiry for the philosophical significance of the indispensability argument, the most relevant of which being that both naturalism and confirmational holism can be dispensed with, contrary to what is held by many.

How to water a thousand flowers. On the logic of logical pluralism

Abstract How many logics do logical pluralists adopt, or are allowed to adopt, or ought to adopt, in arguing for their view? These metatheoretical questions lurk behind much of the discussion on logical pluralism, and have a direct bearing on normative issues concerning the choice of a correct logic and the characterization of valid reasoning. Still, they commonly receive just swift answers – if any. Our aim is to tackle these questions head on, by clarifying the range of possibilities that logical pluralists have at their disposal when it comes to the metatheory of their position, and by spelling out which routes are advisable. We explore ramifications of all relevant responses to our question: no logic, a single logic, more than one logic. In the end, we express skepticism that any proposed answer is viable. This threatens the coherence of current and future versions of logical pluralism.

The varieties of indispensability arguments

The indispensability argument (IA) comes in many different versions that all reduce to a general valid schema. Providing a sound IA amounts to providing a full interpretation of the schema according to which all its premises are true. Hence, arguing whether IA is sound results in wondering whether the schema admits such an interpretation. We discuss in full details all the parameters on which the specification of the general schema may depend. In doing this, we consider how different versions of IA can be obtained, also through different specifications of the notion of indispensability. We then distinguish between schematic and genuine IA, and argue that no genuine (non-vacuously and non-circularly) sound IA is available or easily forthcoming. We then submit that this holds also in the particularly relevant case in which indispensability is conceived as explanatory indispensability.

Benacerraf’s Arguments

One of the main strands in the contemporary debate on Plato’s problem has originated from two essays by Paul Benacerraf (1965; 1973). Before entering this debate, which will occupy us in Chapters 3–5, let us sum up some relevant aspects of Plato’s problem that have emerged so far. One way of formulating Plato’s problem is to ask an ontological question: are there mathematical objects? In order to properly understand this question, and to answer it, at least three points should be addressed: (i) what does it mean to assert (or to deny) the existence of abstract objects? (ii) under what conditions can be asserted of something that it is an abstract object? and (iii) what makes it possible to conceive of an object as mathematical?

The Indispensability Argument: Structure and Basic Notions

Most of the platonist strategies we have discussed so far are based on a clarification of the notion of mathematical objects. The strategy to be discussed in this and the following chapters is different, and is based on the so-called Indispensability Argument (henceforth, IA). IA is neutral on the nature of mathematical objects. It stems form the simple undeniable datum that mathematical theories are applicable, and actually applied to (empirical) science.1 Its premises are prima facie acceptable by nominalists and platonists alike, and it would offer, if sound, plausible grounds for the claims that mathematical statements are true and/or mathematical objects exist, or at least that we are justified in so believing. Not relying on its sleeves on a priori reasoning, IA is particularly suitable for supporting platonism even in an empiricist framework. IA is commonly credited to Quine and Putnam, though some precursors have been recognized in Frege, von Neumann, Carnap and Godel.2 The basic structure of the argument is suggested in many of Quine’s remarks, for it seems to flow naturally from some of Quine’s tenets.3 However, the first explicit formulation is in Putnam (1971). As mentioned earlier, IA can be considered as a non-conservative response to Benacerraf’s dilemma, at least insofar as endorsing it is meant to imply that that the truths of mathematics are not a priori, what contradicts the second part of condition (ii) of conservative responses.

Conservative Responses to Benacerraf’s Dilemma

In this chapter we continue our survey of responses to Benacerraf’s dilemma and discuss those that in § 3.3, following Hale and Wright, we have classified as conservative. Again, we will present four of them: Hale’s and Wright’s neo-logicism; Linsky’s and Zalta’s “Object Theory”; Shapiro’s ante rem non-eliminative structuralism; and Parsons’ version of non-eliminative structuralism.

Non-conservative Responses to Benacerraf’s Dilemma

In the present chapter we discuss some of the major responses to Benacerraf’s dilemma. In particular, we will discuss some of those that in § 3.3 have been classified, following Hale and Wright (2000), as non-conservative. We will consider four of these: Field’s nominalism; fic-tionalism, both in Field’s and in Yablo’s version; Hellman’s eliminative modal structuralism; and the version of platonism advanced by Maddy on cognitive grounds.