### Pluralism in Mathematics: A New Position in Philosophy of Mathematics

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The problem of understanding what mathematics brings to scientific explanation has recently received new impetus from debates about explanatory indispensability arguments for mathematical platonism.

## Pluralist Theories of Truth

These new indispensability arguments build on the traditional indispensability arguments found in W. Quine and Hilary Putnam. However, advocates of an explanatory indispensability argument insist that explanatory benefits from pure mathematics are significant enough by themselves to warrant our belief in mathematical entities. Critics of these new arguments either complain that the account of explanation at work here remains unclear, or else they offer their own proposals for how to make sense of mathematical explanation in science.

Acknowledgments: The authors would like to thank Hein van der Berg and an anonymous referee for their valuable suggestions. There is as yet no major monograph on mathematical explanation that brings together the historical tradition of discussing explanation in mathematics and science, contemporary debates about explanation in pure mathematics, and the contemporary examination of the significance of mathematics in scientific explanation. Mancosu was the first thorough general overview to touch on each of these areas. He links an interest in mathematical explanation to the broader turn toward mathematical practice in the philosophy of mathematics.

### Work in Progress & Under Review

This movement emphasizes issues that arise outside of foundational areas like logic and set theory and tries to use these issues to illuminate traditional and novel philosophical questions about mathematics. Paolo Mancosu summarizes the rich history of debates about explanation in pure mathematics as well as several examples in which explanatory considerations seemed decisive for mathematical practice.

These authors emphasize explanatory factors in theory acceptance besides traditional proofs from accepted axioms. Mancosu and Mancosu provide comprehensive summaries of the discussions of mathematical explanation. The former article emphasizes the importance of mathematics in scientific explanation. In particular, Mancosu discusses the explanatory indispensability argument for mathematical Platonism and the corresponding question for pure mathematics of whether explanatory considerations might justify the ontological commitment to entities mentioned in the explanans.

## Pluralism in Mathematics: A New Position in Philosophy of Mathematics | Michèle Friend | Springer

Mancosu contains more information on the historical debates on explanation and on how debates on mathematical modeling and idealization are relevant to mathematical explanations in science. Mancosu, Paolo. DOI: Edited by Paolo Mancosu, — Oxford: Oxford University Press, The pluralistic and instrumentalistic themes that I will sound in the coming sections will be discernible for the most part to anyone familiar with the 20th century developments in logic.

So, to the extent possible, I will flesh them out with a minimum of specific attribution. While the principal focus is present-day pluralism, at various places I will cut to MacColl, or segue to instrumentalism, when doing so will assist in testing the RR-claims.

I want to sweep away some of the forms of pluralism which, while not devoid of interest, are not of much moment for present purposes. He is an original and protean thinker, multi-tasking outside the box and beyond the reach of his technical powers. His pioneering work ranges far and wide, and his writings anticipate some of the most important developments in the work that was yet to come. In his writings we have the modal systems of strict implication, we have many-valued logic, relevant logic, and connexive logic, and we have probability logic. MacColl as we might say, is a many-one pluralist.

Less clear than that is the reality of the connection to instrumentalism. There is a well-known dispute in logical theory as to when a formalism achieves the status of logic. For most of his philosophical career, Quine was of the view that nothing is a system of logic beyond first-order quantification theory with identity [Quine ]. If we hold with Quine that only the usual logical particles 8 under their classical interpretations are genuinely logical terms, then we are logical-term unitarians.

Can you be a many-one pluralist without being an instrumentalist? One wonders.

Take Aristotle again. If this is what instrumentalism is, then even realists are instrumentalists, and we have an answer to our second question:. Of course, it is trivial that for instrumentalism in this sense the Neglect Thesis also fails. More briefly, the goodness of ends is imbibed by their means. This is hardly instrumentalism in any sense seriously deserving of the name.

Means-end instrumentalism stands to philosophically robust instrumentalism as white chocolate stands to chocolate. Accordingly, although MacColl is a many-one pluralist and a means-end instrumentalist, neither of these positions originates with him, and neither is a plausible basis for his neglect. This leaves us some further work to do, not least concerning the sense in which MacColl really was an instrumentalist—that is, a philosophically robust one. I want, as I say, to postpone the more important parts of the discussion of the instrumentalism question until later.

For the present, let me call a brief time-out for a few words about formalisms.

## (H/B) PLURALISM IN MATHEMATICS

Regarding the first of these claims, we could say that different logical symbols can express different concepts. Regarding the second claim, we could say that a given logical symbol can be given different interpretations which express different concepts. Many-one pluralism is as interesting as the unitarianism it rejects and the logics to which it extends its hospitality. But it bears repeating that in the present context it is hardly interesting at all. There are, as the saying has it, more logics than you can shake a stick at.

It is an unruly plenitude—crowded, chaotic and conflicted. It is a turbulence that over-runs its own quality controls. For every cluster of pairwise incompatible logics, at least half are false. The problem with blindness is want of subject matter, the absence of something to be about.

There are two ways of annulling these difficulties, each a form of denial. Seen the first way, the properties are bad but not present. Seen the second way, they are present but not bad.

source site Of these two forms, existential denial has a much larger provenance in the literature. Since time presses, I shall confine myself to it, reserving even-so denial for another occasion. Systems below line B are incompatible with RL. To see how, let me begin with a brief word about blindness. There are several logics in this cluster. This alone is offensive to unitarian leanings.

It is a Kantian complaint: Theories without subject matters are blind. We should try to tighten up. Perhaps concepts are the way to go.

It is widely agreed that the central target for logic is the concept of consequence. One way of specifying the subject matter of a logic is to equate it with its target concepts. But it does so in a way that appears to land us in the incompatibility problem. Consider, for example, S 5 and S2. It does so in the manner of existential denial. When G is a generic concept of which G 1 , When this happens, we may say that specific senses are tied to generic senses by a relation of meaning-subsumption.

This, too, appears to solve the incompatibility problem.

- MacColl’s elusive pluralism!
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Whereupon, we have a further specimen of pluralism, also of the existential denial sort:. An argument is valid when there is no case in which the premises are true and in which the conclusion is not true.