Scientific Realism and the Indispensability Argument for Mathematical Realism: A Marriage Made in Hell

Abstract

An emphasis on explanatory contribution is central to a recent formulation of the indispensability argument (IA) for mathematical realism. Because scientific realism is argued for by means of inference to the best explanation (IBE), it has been further argued that being a scientific realist entails a commitment to IA and thus to mathematical realism. It has, however, gone largely unnoticed that the way that IBE is argued to be truth conducive involves citing successful applications of IBE and tracing this success over time. This in turn involves identifying those constituents of scientific theories that are responsible for their predictive success and showing that these constituents are retained across theory change in science. I argue that even if mathematics can be shown to feature in best explanations, the role of mathematics in scientific theories does not satisfy the condition that mathematics is always retained across theory change. According to a scientific realist, this condition needs to be met for making ontological claims on the basis of explanatory contribution. Thus scientific realists are not committed to mathematical realism on the basis of this recent formulation of IA.

Acknowledgements

I should like to thank James W. McAllister for his encouraging feedback. I should also like to thank Alan Baker for valuable comments on an earlier version of the manuscript. I am also in debt to two anonymous referees for this journal for valuable feedback.

Notes

Colyvan is concerned to show that scientific realists ought to be mathematical realists, and as part of that project he wants to show that whatever criteria scientific realists appeal to when deciding between scientific theories will favor mathematical realism. These particular criteria mentioned here, are based on Colyvan (2001, 78–79) who observes that scientific realists typically look for virtues such as simplicity, etc., in theories. Colyvan refers to Kitcher (1981) for having established that explanation is unification/explanatory power. Whether this is correct or not is highly relevant to the general debate as it may turn out that mathematics does not in fact contribute in the ways that Colyvan suggests, or that it contributes in ways that are not evidence for truth (it can for example be contested that explanatory power is evidence for truth). For my purposes we may, however, grant Colyvan his suggested criteria, because my argument concerns whether mathematics meet demands other than maximizing all the mentioned theoretical virtues and it is therefore also indifferent to the issue of whether all the mentioned criteria are relevant.

See Busch (forthcoming-a) for an elaboration of the relationship between IA and IBE. I argue that indispensability is best understood in terms of being part of a ‘best’ explanation. On any conception of ‘best’, if an explanation exists with the same amount of explanatory value as a competing explanation, and one commits to fewer entities than the other, then we should choose the explanation that commits to fewer entities. So if mathematics does in fact feature in our best explanations, that just means that any explanation that does not commit to the existence of mathematical entities is a worse explanation by our measure of overall explanatory value. Thus the notion of indispensability does no real work on this understanding of the argument.

There are very different views on what notion of explanation is relevant here. Cartwright (1983), for example, argues that the only trustworthy explanatory inferences we can make involving reference to unobservable entities are causal inferences. Psillos (2007) suggests that IBE works for different kinds of explanation and delivers reliable inferences in different contexts depending on what kind of explanation would be relevant in that context. Clearly a thorough investigation of this would do a lot towards making headway in the current discussion but it is outside the scope of this paper to do so here.

I wish to thank an anonymous referee of this journal for this observation.

There are different formulations of the pessimistic meta-induction argument in the literature. The following formulation from Putnam is sufficient for our purposes here: ‘What if all the theoretical entities postulated by one generation … invariably don't exist from the standpoint of later science? … One reason this is a serious worry is that eventually the following meta-induction becomes overwhelmingly compelling: Just as no term used in the science of more than fifty (or whatever) years ago referred, so it will turn out that no term used now (except maybe observational terms, if there are such) refers’ (Putnam 1978, 25).

Psillos (1999) draws on Kitcher (1993) in order to formulate his own version of the strategy which he calls the divide et impera strategy. There is a multitude of current realist positions that opt for the same strategy in answer to the pessimistic meta-induction argument. Worrall's (1989) epistemic structuralist position, Chakravartty's (1998, 2007) semi-realist position, and recently Saatsi's (2005) eclectic realist position are all instances of the compartmentalization strategy.

A different example one could consider in this context is infinitesimals. An infinitesimal is a number whose magnitude is greater than zero but is still so small that it cannot be distinguished from zero. Infinitesimals have been discussed since antiquity and through the middle ages, and appeared in the study of slopes and areas, which developed into differential and integral calculi. Newton and Leibniz, who discovered calculus (independently of one another), both used infinitesimals. With the rise of formalism in the nineteenth century, infinitesimals soon became the subject of criticism. With Dedekind's and Cantor's published work on the construction of real numbers, infinitesimals were banished from mathematics and mathematicians stopped working on infinitesimals for years to come. It was only as late as 1960 that infinitesimals were once again taken up as a serious subject of study by mathematicians.

Pragmatic factors of the kinds mentioned by Baker are not considered relevant to the evaluation of whether one theory is a better explanation than another, according to scientific realists.

As was highlighted earlier, the history of an entity matters to a realist for the purpose of arguing for the reliability of IBE. So, regardless of whether we are today in a position where quaternion theory has stood the test of time—observing that it has had a past of drifting in and out of science raises the concern that it might do it again. That history matters to a scientific realist, allows only realism in retrospect and this involves a number of challenges. Realists ought to provide more detail here about the exact role of history, the length of time required for being realist about an entity, etc., and one might find cases of entities that we believe we have particularly good reason to believe exist despite their having been discovered quite recently—which would pose a challenge to a realists criterion. One might for example try to make a case for quaternion theory actually posing a challenge to a realists criterion. But as was pointed out earlier, that we can only be realists in retrospect is a price that the realists pays in order to maximize epistemic caution.

The historical details are presented and discussed more elaborately in Maddy (2007). See Busch (forthcoming-b) for an argument that utilizes the example of Euclidean rescues to argue that proponents of EIA do not have any good suggestions for accounting for why mathematics is not falsified when it seems it ought to be. As suggested below, Colyvan (2001) opts for a Quinean solution but EIA is supposed to be a version of IA that does not rely on holism.

I wish to thank an anonymous referee of this journal for pointing this out.

Additional information

Notes on contributors

Jacob Busch

Jacob Busch is at the Department of Entrepreneurship and Relationship Management, University of Southern Denmark