Abstract objects

The distinction between universals and particulars is parallel to the more general distinction between abstract objects and concrete objects. There is no universal consensus regarding the definition of abstract objects; however we can list some general properties that are usually attributed to them. Abstract objects are typically assumed to exist outside space and time, where the notion of “being outside” should not be interpreted in the spatiotemporal sense. This is usually explained in the form of the requirement that abstract objects cannot be subjects of true tensed predications which are also essential (excluding such predications as "It is true of number 6 that yesterday I thought about it", which is not essential for 6). One consequence of this assumption is that abstract objects cannot undergo genuine changes. But it may be claimed that there are objects which exist in space and time, and yet are not concrete things. An example can be the centre of mass of the solar system. However, the centre of mass lacks another important characteristic of concrete objects: it is namely not causally efficacious. Abstracta are assumed to be causally inert; they do not participate in causal interactions. This criterion of abstractness has to presuppose some philosophical conception of causation. According to the most popular approach causation is a relation between events. But this may suggest that things are not concrete, for they cannot literally cause one another. One solution is to extend the notion of causal interactions: a thing x participates in a causal interaction iff some event e which is constituted by x stays in the causal relation with some other events.

The third attribute of abstract objects is considered to be their ontological dependence on other objects. For instance, it can be claimed that the abstract object “direction” is ontologically derivative from and dependent on the existence of parallel lines. But this can be questioned by Platonists, who claim that if there is any ontological dependence at all, it goes in the opposite direction: it is concrete objects that depend on abstract objects. As we will see later, this approach can be further supported by the so-called bundle theory of particulars. Typical examples of abstract objects include properties, relations, meanings, propositions, values, and mathematical objects (numbers, sets). The status of tropes is somewhat controversial. Some insist that they are concrete, since they exist in space-time. But it is unclear whether they can interact causally.

Nominalists criticise the notion of abstract objects by applying the following two arguments. They point out that it is unclear how we can acquire knowledge about abstract objects (the epistemological problem) and how we can refer to them (the semantic problem). Underlying these two problems are the assumptions of the causal theories of knowledge and of reference. According to the first one, in order to know something about an object x we have to interact causally (directly or indirectly) with x. According to the causal theory of reference, for an expression t to refer to some objects, a causal link has to be established between one sample object belonging to the extension of t and the later utterances of the expression t. Because they are causally inert, abstract objects are excluded from causal theories of knowledge and reference. There is no consensus regarding what alternative theories can be adopted in the case of abstracta.

The most important category of abstract objects is the category of mathematical objects. A simple argument based on mathematical practice can be given in favour of the existence of mathematical objects:

(1) Mathematical statements are true,
(2) Mathematical statements imply that mathematical objects exist.
Therefore
(3) Mathematical objects exist

The nominalist can meet this challenge by denying either (1) or (2). Let us start with the strategy that tries to question (2). This is essentially to claim that mathematical theorems can be reformulated in such a way as to eliminate their ontological commitments to abstract objects. One possible way is to try to interpret mathematical statements as being about concrete things. This may work in the case of simple arithmetical truths, such as 2 + 3 = 5. This equation can be restated as expressing the fact that if there are two objects of the kind A and three objects of the kind B, and no object is both A and B, then there are five objects of the kind A or B. Crucial to the success of this strategy is the fact that statements of the sort “There are exactly (at most, at least) n objects of the kind A” can be expressed in first-order language without any reference to number n. For instance, the sentence “There are exactly two objects with property P” can be interpreted as “There is an x and a y such that x is distinct from y, x has P and y has P, and for all z, if z has P, then z is identical with either x or y”. This reformulation is more awkward, but does not contain any reference to the number 2. But this strategy cannot be directly applied to more abstract theorems, such as the statement that there is no greatest prime number.

A more general nominalistic method of paraphrase is possible. Let S be any mathematical theorem. Then the implication “If there are mathematical objects, then S” does not carry any commitments to mathematical objects. However, the problem is that a material implication is true if its antecedent is false, hence the nominalistic interpretations of even false mathematical statements will always be trivially true. This leads to the following modification: instead of material implication we should use strict implication “It is necessary that if there are mathematical objects, then S”. This is known as modal interpretation of mathematics. There are two main problems with this interpretation. Firstly, it is unclear whether a satisfactory semantic analysis of the modal operator of necessity can be given in purely nominalistic terms (without any reference to abstract objects). Secondly, in order to maintain that some strict implications of the above form are false we have to assume that the antecedent “There are mathematical objects” is not necessarily false. But what sense can the nominalist make of the hypothesis that abstract objects might exist? Under what conditions would this be true?

Fictionalism is the approach which denies premise (1). Mathematical statements are literally false, but they are useful. The main challenge to fictionalism is given in the form of the indispensability argument whose premise is that mathematical theories and notions are applied in empirical sciences (physics, chemistry, biology, etc.). If we confirm empirically a given scientific theory, this confirmation should also reach to its mathematical part. Thus we should conclude that the best explanation for the empirical successes of a scientific theory is that the mathematical theorems used in it (such as the theorems of mathematical analysis or group theory, etc.) are true. Hartry Field in his 1980 book Science without numbers set out to defend nominalism against the indispensability argument. His strategy, in rough outline, is to find, for a given physical theory T, two theories Tp and Tm such that Tp contains only physical, nominalistically acceptable notions, while Tm is a mathematical theory used in T. T has to be logically equivalent to the conjunction of Tp and Tm. If finding such Tp and Tm were possible, then in the next step we could appeal to the logical fact that all mathematical theories are conservative with respect to non-mathematical vocabulary. This means that whatever logical consequence A of Tp + Tm can be expressed in the non-mathematical vocabulary, A should follow logically from Tp itself. Thus Tm does not have to be considered true, and its role is reduced to a mere simplification of logical deductions. The main problem with Field’s strategy is to find the nominalistic version Tp of a given physical theory T. Field showed how to do this in the case of classical mechanics, but it is unlikely that his method could be applied to more sophisticated theories, such as quantum mechanics, quantum field theory or general theory of relativity.

Further reading:

E.J. Lowe, "The abstract and the concrete", pp. 366-385, A Survey of Metaphysics.