Online research in philosophy

I start by reconsidering two familiar arguments against modal realism. The argument from epistemology relates to the issue whether we can infer the existence of concrete objects by a priori means. The argument from pragmatics purports to refute the analogy between the indispensability of possible worlds and the indispensability of unobserved entities in physical science and of numbers in mathematics. Then I present two novel objections. One focusses on the obscurity of the notion of isolation required by modal realism. The other stresses the arbitrary nature of the rules governing the behaviour of Lewisean universes. All four objections attack the reductive analysis of modality that is supposed to be the chief merit of modal realism.

Think of “locations” very abstractly, as positions in a space, any space. Temporal locations are positions in time; spatial locations are positions in (physical) space; particulars are locations in quality space. Should we reify locations? Are locations entities? Spatiotemporal relation- alists say there are no such things as spatiotemporal locations; the fundamental spatial and temporal facts involve no locations as objects, only the instantiation of spatial and temporal relations. The denial of locations in quality space is the bundle theory, according to which particulars do not exist; facts apparently about particulars really concern relations between universals. A “space”, in our abstract sense, consists of a set of objects, together with properties and relations defined on those objects. The objects are the locations of the space, and the distribution of the properties and relations over the locations defines the space’s structure. All spaces are thus quality spaces; when the relations are thought of as spatiotemporal then the space is also a spatiotemporal space. By not reifying locations one denies that these abstract spaces isomorphically represent the real world. The real world does in some sense have a structure that can be non-isomorphically represented by a space (or, more likely, a class of spaces), but the locations in those spaces do not correspond to anything real. We will examine modal considerations on reifying locations. Denying the existence of spatiotemporal locations excludes certain possibilities for spatiotemporal reality. Denying the existence of qualitative locations excludes certain possibilities for qualitative space. In each case the excluded possibilities are pre-analytically possible. Some of the possibilities can be reinstated by modifying the locationless theories, but at the cost of an unattractive holism. Do these modal considerations mandate postulating locations? That depends on whether modal intuition can teach us about the actual world..

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle.

Unrestricted Composition (UC) is, roughly, the claim that given any objects at all, there is something which those objects compose. (UC) conflicts in an obvious way with common sense. It has as a consequence, for instance, that there is something which has as parts my nose and the moon. One of the more influential arguments for (UC) is Theodore Sider’s version of the Argument from Vagueness. (A version of the Argument from Vagueness was first presented by David Lewis (1986), pp. 212–213). That argument purports to show that some plausible claims concerning the nature of vagueness entail (UC). In this paper I will suggest a response to this argument. I will show that the proponent of Supersubstantivalism (SS)—the view that material objects are identical to regions of spacetime—can reject a premise of Sider’s argument without denying the plausible claims concerning vagueness. Doing so requires only rejecting a certain view concerning the relationship between the proper sub-region relation and the proper parthood relation. So, proponents of (SS) are in a better position than many of us to side with common sense regarding composition. In the first section of the paper, I will present Sider’s argument. In the second section, I will introduce (SS) and briefly discuss some reasons one might have to believe that it is true. In the third section, I will show how the proponent of (SS) can avoid commitment to (UC) and reject a premise of Sider’s argument. Last, I’ll briefly consider and respond to some objections.

It is alleged that the causal inertness of abstract objects and the causal conditions of certain naturalized epistemologies precludes the possibility of mathematical know- ledge. This paper rejects this alleged incompatibility, while also maintaining that the objects of mathematical beliefs are abstract objects, by incorporating a naturalistically acceptable account of ‘rational intuition.’ On this view, rational intuition consists in a non-inferential belief-forming process where the entertaining of propositions or certain contemplations results in true beliefs. This view is free of any conditions incompatible with abstract objects, for the reason that it is not necessary that S stand in some causal relation to the entities in virtue of which p is true. Mathematical intuition is simply one kind of reliable process type, whose inputs are not abstract numbers, but rather, contemplations of abstract numbers.

After reviewing some different indispensability arguments, I distinguish several different ways in which mathematics can make an important contribution to a scientific explanation. Once these contributions are highlighted it will be possible to see that indispensability arguments have little chance of convincing us of the existence of abstract objects, even though they may give us good reason to accept the truth of some mathematical claims. However, in the concluding part of this paper, I argue that even though there is a valid indispensability argument for realism about some mathematical claims, this argument is problematic as it begs the question at issue. This challenge to indispensability arguments is then used to suggest that if mathematics is making these sorts of contributions to science, then it may be the case that mathematical claims receive some non-empirical support prior to their application in scientific explanation.

of my axiomatic theory of abstract objects.¹ The theory asserts the ex- istence not only of ordinary properties, relations, and propositions, but also of abstract individuals and abstract properties and relations. The.

Quine and Putnam's Indispensability Argument claims that we must be ontologically committed to mathematical objects, because of the indispensability of mathematics in our best scientific theories. Indispensability means that physical theories refer to and quantify over mathematical entities such as sets, numbers and functions. In his famous book 'Science Without Numbers' Hartry Field argues that this is not the case. We can "nominalize" our physical theories, that is we can reformulate them in such a way that 1) the new version preserves the attractivity of the theory, and 2) the nominalized theory does not contain quantifications over mathematical entities. I'm going to reconsider Field's nominalization procedure for a toy physic theory formulated in a first order language, in order to make a clear distinction between the following three steps: -the physical theory in terms of empirical observations. -the standard physical theory, which contains quantification over mathematical entities, as usual. -the nominalized version of the theory without any reference to mathematical entities. Having Field's nominalization procedure reconstructed, it will be clear that there is no difference between the original and the nominalized versions of the theory, at least, there is no difference from a formalist point of view. It is because the only difference would come from the different "meanings" of the variables over which the quantifications are running. The formalist philosophy of mathematics, however, denies that the variables have meanings at all. So, the formal systems as abstract mathematical entities are still included to physical theories; and this fact is highly enough for the structural platonist or immanent realist to apply the Quine-Putnam argument. Finally, therefore, I will suggest a completely different way for the objection to the Quine-Putnam argument.

The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their dispensability cannot be demonstrated and, hence, there is no good reason for believing in the existence of mathematical objects which are genuinely platonic. Therefore, indispensability, whether true or false, does not support platonism. Mathematical platonists claim that at least some of the objects which are the subject matter of pure mathematics (e.g. numbers, sets, groups) actually exist. Furthermore, they claim that these objects differ radically from the concrete objects (trees, cats, stars, molecules) which inhabit the material world. We take the standard platonistic position to include the claim that platonic objects lack spatio-temporal location and causal powers. Many (perhaps most) mathematical platonists subscribe to this view.1 But some who call themselves (or might be called) mathematical platonists..

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.