Chris Wilcox - Thesis - Introduction

up to Reality \Some problems with the indispensability argument for Platonism and its Quinean ontological underpinnings

When I look out my window at the trees, at the buildings, at the people do I really perceive a world separate to me? Or rather am I just somehow made aware of certain creations of my own mind? The answers one gives to these sorts of philosophical questions (or perhaps more accurately, the reasons one gives in support of them) have a bearing on a topic in philosophy with a long history: the realism debate. Due to its complexity, it is very hard to give a brief general account of the realism debate. As a first approximation, it is useful to frame it the debate between those who hold that some subject matter (the external world in the preceding questions) “exists” in some sense and thse who deny this claim. {[lime You move from realism to existence … and then immediately back again. The point is that something is real, in the language of this debate, if it exists AND IS MIND-INDEPENDENT or something like that (below you say “independent of us”). ]} However, in recent times the accuracy/usefulness of this depiction has been called into question.\citet{Dummett:1978 and :1992 for example.} {[lime Expand or omit. ]}

One area of the realism debate with a particularly lengthy history concerns the subject matter of mathematics, in particular, mathematical entities (consider the number two or a particular ring or group as a first example — many more interesting and useful examples will follow later). In this context positions in the debate are generally divided between two camps: mathematical realists, who hold that mathematics is in some sense external to and independent of us; and nominalists, who deny this claim. Within the mathematical realist camp there is one position in particular that will be of central importance in what follows: mathematical Platonism (hereafter referred to simply as “Platonism”). The Platonist holds that mathematical entities are real (what this means precisly will be discussed in detail in Chapter 1).

In recent years, a major argument for Platonism has garnered a lot of attention: the Quine-Putnam indispensability argument. It is used it to argue for the existence of mathematical entities on the basis of their involvement in our best scientific theories. While there are various versions of the argument in the literature, I will be concerned here primarily with the sustained treatment offered by Mark Colyvan in his book, {[lime omit comma! ]} The Indispensability of Mathematics~:2001.

Early in his book, Colyvan specifies that the primary target of the indispensability argument as he conceives it is meant to be the scientific realist: those who hold that the theoretical entities of science “exist” in some sense independently of us. In particular, he wishes to target those scientific realists who base their realist commitmentsoin {[lime typo ]} some sort of inference to the best explanation style argument. Given what he claims is the already close relation between inferences to the best explanation and indispensability arguments in general (to be explored in Chapter 1), he believes that scientific realists should be much more amenable to the reasoning engaged in in the indispensability argument. Accordingly, at various points in what follows it will be useful to link the discussion back to this proposed target audience and also to Colyvan’s purported supporter base: Platonists.

Unfortunately, there are many worthy criticisms of the indispensability argument I do not have the space to deal with here.a starting point for critical discussion of many of these I highly recommend Colyvan’s book. He is a most generous, eloquent and even-handed opponent. {[lime excellent but maybe a bit over the top ]} Of these, the most significiant include denials that mathematical entities are indeed indispensable to our best scientific theoriesparticular see \citet{Field:1980 :1989.}, criticisms of confirmational holism and further criticisms of Quinean naturalism. I highly recommend these works to the interested reader. {[lime It’s traditional to pretend that the examiner has read everything relevant. ]}

I will begin my discussion in Chapter 1 with a detailed account {[lime I’d omit “detailed” and let the examiner judge that ]} of just how Colyvan’s indispensability argument is supposed to work, focussing not only on the definition of the terms involved but also on the (largely Quinean) theoretical underpinnings of the premises. In Chapter 2, I will focus on a criticism very much in the “Quinean spirit” in which the indispensability argument is put forward. Accordingly, I will largely hold the premises of the argument to be true, focussing instead on how well the mathematical entities involved in it match up with the expectation (both of the Platonist and of the IBE-based scientific realist) of what a mathematical entity is supposed to be and the implications of any mismatch. {[lime Need to define “IBE” as “Inference to the Best Explanation” ]} In Chapter 3 however, I will engage in some more fundamental criticism of the theoretical underpinings of the argument. In particular, building on arguments due to Carnap, Price, and Yablo, I will call into question the conception of ontological commitment at the heart of Quinean naturalism. Finally, I will argue that while Colyvan probably fails to do enough to keep his supporters (Platonists) happy or to convince any significant proportion of his target audience (IBE-based scientific realists) to make the move to Platonism, there is still much of value to be found in the considerations involved in the assessment of the indispensability argument for Platonism in mathematics (and indeed in the debate in general). To this end I will point to a possible context in which the debate may be viewed that will do justice to the insights on both sides.