Online research in philosophy

It is sometimes alleged that “the reliability challenge” to moral realism is equally compelling against mathematical realism. This allegation is of interest. The reliability challenge to moral realism is increasingly taken to be the most serious challenge to moral realism. However, the specific considerations that are said to motivate it – such as considerations of rational dubitability and evolutionary influence – are widely held not to motivate an analogous challenge to mathematical realism. If it turned out that, in fact, they do, then one might have to choose between moral realism and mathematical realism.

Nevertheless, the relevant allegation has never been clarified, let alone evaluated. In this paper, I clarify and evaluate it. I argue that the allegation is plausible, but depends on theses in the philosophy of mathematics that are widely doubted. One upshot of the discussion is that mathematical realism faces challenges that have not been widely appreciated. Another is that the reliability challenge to moral realism may not be the most interesting epistemological challenge to moral realism.
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In Mathematical Thought and Its Objects, Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a “nature” than that confers on them.

This article examines one aspect of Thomas Aquinas' understanding of abstraction. It shows in which way, according to Aquinas, universal material objects and individual material objects are the starting point for mathematical objects. It comes to the conclusion that for Aquinas there are not only universal mathematical objects (circle, line), but also individual mathematical objects (this circle, that line). Universal mathematical objects are properties of universal material objects and individual mathematical objects are properties of individual material objects. One type of abstractio formae leads from individual material objects to universal mathematical objects, a second type from universal material objects to universal mathematical objects, and a third type from individual material objects to individual mathematical objects. Therefore, the concept of abstractio formae is ambiguous.

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist.

Traditionally, in the philosophy of mathematics realists claim that mathematical objects exist independently of the human mind, whereas idealists regard them as mental constructions dependent upon human thought.It is tempting for realists to support their view by appeal to our widespread agreement on mathematical results. Roughly speaking, our agreement is explained by the fact that these results are about the same mathematical objects. It is alleged that the idealist’s appeal to mental constructions precludes any such explanation. I argue that realism and idealism, as above characterized, are equally effective (or problematic) in accounting for our widespread mathematical agreement.Both accounts are descriptivist for they take mathematical statements to be true if and only if they correctly describe mathematical objects. By contrast, non-descriptivist accounts take mathematical statements to be rule-like and mathematical symbols to be non-referential. I suggest that non-descriptivism provides a simpler and more natural explanation for our widespread agreement on mathematical results than any descriptivist account.

This paper examines whether structural realism entails an anti-realist thesis about natural kinds. Structural Realism is the view that the scientific realist can only support a realist claim about the structure of reality rather than its objects. Ladyman (1998) (2002) & French & Ladyman (2003) motivate the claim that ontic structural realism eliminates ‘objects’ as a distinct ontological category, thereby eliminating any possibility of a metaphysical account of individual objects. This is empirically motivated by fundamental physics. Those inclined towards realism about the rest of the sciences (chemistry, biology, the medical sciences, economics and so on) might think the appeal of structural realism as a general metaphysics for all of the sciences limited. Nevertheless, recent literature argues that mature special sciences e.g. economics, can be equally described by mathematical/syntactic models making the appeal of structural realism a more general one for the metaphysics of all of the sciences. {Ross (2006)}. Given a commitment to ontic structural realism, if natural kinds are kinds of “object”, then anti-realism about natural kinds should follow. However, I examine two realist theses about natural kinds and argue that a commitment to structural realism is not straightforwardly inconsistent with either.

I address Grosholz's critique of Resnik's mathematical structuralism and suggest that although Resnik's structuralism is not without its difficulties it survives Grosholz's attacks.

I point out that conceptions of particles as mathematical, or quasi mathematical, entities have a longer history than Resnik notices. I argue that Resnik's attack on the distinction between mathematical and physical entities is not deep enough. The crucial problem for this distinction finds its locus in the numerical indeterminancy of elementary particles. This problem, traced by Heisenberg, emerges from the discovery of antimatter.

Baker (2005) claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework.

Penelope Maddy and Elliott Sober recently attacked the confirmational indispensability argument for mathematical realism. We cannot count on science to provide evidence for the truth of mathematics, they say, because either scientific testing fails to confirm mathematics (Sober) or too much mathematics occurs in false scientific theories (Maddy). I present a pragmatic indispensability argument immune to these objections, and show that this argument supports mathematical realism independently of scientific realism. Mathematical realism, it turns out, may be even more firmly established than scientific realism.