Two notions of sets

Sets are considered fundamental mathematical objects, because in principle all other mathematical objects can be defined in terms of sets (can be reduced to sets). However, the notion of a set can be given at least two unequivalent interpretations. In one interpretation, the set of physical objects X is a physical complex whose spatiotemporal parts are all objects X. Sets of that sort are called mereological (or collective). Mereological sets have the following characteristic properties. First, the mereological set consisting of one object is identical with this object. Second, there is no mereological empty set (for instance, the collection of all centaurs does not exist). Third, two mereological sets built out of numerically different objects can nevertheless be identical. For instance, the mereological set of two hydrogen atoms is identical with the mereological set of two protons and two electrons constituting those atoms. This example also illustrates the fact that the relation of belonging to a mereological set is transitive. Clearly, this follows from the fact that the relation of being a member of a mereological set is identical with the part-whole relation, and the latter is transitive (if x is a part of y and y is a part of z, then x is a part of z).

The second interpretation of sets is called distributive (or set-theoretical). Distributive sets are analogous to linguistic concepts. The distributive set of all people has people as its only elements. No proper part of a person (such as a hand or a leg) belongs to this set, since proper body parts are not humans. As concepts differ from the objects subsumed under them, the set consisting of only one object is numerically different from this object. There is an empty set (a set with no elements), since there are empty concepts (such as the concept of a unicorn). Actually, as we will soon see, it can be proven that there is exactly one empty set. If you have two sets which have different numbers of elements, you can be sure that these sets are different. Hence the set of two hydrogen atoms is distinct from the set of their two protons and two electrons. Distributive sets satisfy the principle of extensionality: two sets are identical if and only if they have exactly the same elements. (Actually, mereological sets satisfy this principle too: two mereological sets are identical iff they have the same parts. But it is still possible to describe one and the same mereological set as consisting of numerically different objects, as in the example with two hydrogen atoms). From the condition of extensionality it follows that there is exactly one empty set. The relation of membership is not transitive in the case of distributive sets: if x is a member of y and y is a member of z, x does not have to be a member of z (although it may). Example: 1 є {1} and {1} є {{1}, {2}}, but it’s not the case that 1 є {{1}, {2}}.

From the ontological point of view it is important to ask what kind of objects sets are and whether they can be accepted by a nominalist. Mereological sets don’t create much of a problem, since they are just spatiotemporal objects, provided that their elements are spatiotemporal. The only contentious issue is whether we should admit the existence of arbitrary collections of objects. Is there an individual object that consists of my left pinkie, the planet Venus, and the left hind leg of some particular dinosaur? But the status of distributive sets is more controversial. Arguably, distributive sets cannot be identified with spatiotemporal objects. The singleton consisting of one physical object x is not identical with x, and because it cannot be identical with any other physical object y (since its existence would be contingent upon the existence of y, and the only acceptable ontological dependence of {x} is on x), hence {x} cannot be a physical object. Thus the most common interpretation of distributive sets is that they are abstract objects, and as such are not acceptable to the nominalist. However, nominalists can make use of the notion of distributive set in certain contexts, for which it is possible to give a nominalistic paraphrase. For instance, the statement “Socrates belongs to the set of all philosophers” can be interpreted nominalistically as “Socrates is a philosopher”. The sentence “The set of all philosophers is a subset of the set of all people” is interpreted as “All philosophers are humans”, and an analogous interpretation of the sentence “The set of all people is disjoint from the set of all elephants” can be given as “No humans are elephants”. Thus it can be claimed that the nominalist can accept first-order sets of physical objects (so-called classes). But higher-order sets, and especially those founded on the empty set, are not so easy to eliminate from the discourse.

At the beginning of the lecture we mentioned the fact that mathematical objects can be reduced to (distributive) sets. But, as Paul Benacerraf has famously noticed, such reductions are not unique. For instance, natural numbers can be interpreted as sets in at least two ways. One interpretation is given by following identifications: 0 = Ø, 1 = {Ø}, 2 = {Ø, {Ø}}, 3 = {Ø, {Ø}, {Ø, {Ø}}}, etc. But an alternative interpretation can look like this: 0 = Ø, 1 = {Ø}, 2 = {{Ø}}, 3 = {{{Ø}}}, etc. These two interpretations, taken literally, cannot be true, for this would imply mathematical falsehoods, such as {{Ø}} = {Ø, {Ø}} (you can prove that this identity is false, given the principle of extensionality and the assumption that {x} is different from x). But mathematical practice does not tell us which identification should be preferred. It looks like some questions regarding numerical identity between mathematical objects are fundamentally undecidable, which calls into question the ontological status of mathematical objects as independent entities. One solution to this problem is encompassed in the so-called structuralist interpretation of mathematics. According to this interpretation, the fundamental objects that mathematical theories speak about are whole structures, not individual objects. There is no number 1 as an entity that exists separately from the entire structure of natural numbers, hence it does not make sense to ask what this number is identical with. Mathematical objects are just positions in a given structure. There is more than one way to interpret one structure (e.g. the structure of natural numbers) within another structure (the structure of sets). But an interpretation is just a homomorphism, i.e. a mapping which preserves the structure. There is no identity involved. The essence of natural numbers is exhausted in the structure of a linear, discrete order.

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.

Versions of nominalism

Metalinguistic nominalism proposes a more uniform method of paraphrasing statements containing abstract terms. The general idea is to replace terms referring to putative universals (properties, relations, kinds) by terms describing linguistic expressions. Thus the statement “This ball is red” can be explicated as “This ball satisfies predicate ‘red’”. “Triangularity is a shape” becomes “’Triangular’ is a shape predicate”, and the troublesome sentence “Courage is a moral virtue” gets translated into “’Courageous’ is a virtue predicate” (note that the word “virtue” is clearly ambiguous: in the first sentence it serves as a noun, and hence carries an unwanted commitment to properties, whereas in the second sentence it becomes an adjective, modifying the noun “predicate”). Similarly we can treat the sentence “This tulip and that rose have the same colour”, rephrasing it as “This tulip and that rose satisfy the same colour predicate”. But it is unclear what the ontological status of linguistic expressions is, and whether a nominalist can accept them in his ontology. First we have to make a distinction between types and tokens. A token of an expression is an individual inscription or utterance. Hence each word has more than one token which belong to one and the same type. In the above examples of metalinguistic paraphrases the subject terms are singular, not general, hence it looks like they refer to types, not tokens. But types resemble universals in all relevant aspects: they are entities that are common to all individual tokens of a given expression, hence they can be interpreted as the common property of all inscriptions (utterances).

Another problem with metalinguistic nominalism is that it trades the objective, independent notion of property for a language-dependent notion of predicate. But what with properties that are not expressed in any language? There are examples of properties that we discovered and named only recently, such as spin or charm. It is quite natural to expect that there are more properties of that sort which have yet to be discovered. Consequently, a metalinguistic nominalist can’t offer a satisfactory translation for the sentence “Every object has a property that we will never know”.

The initial motivation for metaphysical realism was provided by the existence of objective similarities between particulars. Resemblance nominalism tries to develop and apply the notion of similarity without any recourse to universals. It may be said for instance that to be red is to be sufficiently similar to a paradigmatic red object. But this simple interpretation won’t do. Clearly there may be non-red objects that are similar to a selected red thing (with respect to every property other than colour). A more sophisticated attempt to explain away the attribution of properties may be as follows. The resemblance nominalist may try to define resemblance classes which will roughly correspond to the realist’s properties. For instance, a resemblance class can be defined as a maximal class such that any two objects in this class are more similar to one another than they are to any object outside of the class (more formally this condition can be spelled out as follows: for all x, y, and z, if x and y belong to class K but z does not belong to K, then x is more similar to y than to z). The condition of maximality is needed, because we don’t want to qualify the class of two red objects as a resemblance class. However, three fundamental objections can be made against such a solution.

(1) As we have already indicated, it can be argued that you can find a non-red object which is more similar to a particular red object than this object is to another red thing. Think for example of a green sphere, a red sphere of exactly the same dimensions, and a red cube twice as big as the sphere. It can be argued that the spheres resemble each other more that the red one resembles the red cube.

(2) Let’s consider two properties P and Q such that all objects that have P have Q but not vice versa. In such a case P will not define a resemblance class, for the condition of maximality fails.

(3) The universal class (the class of all particulars) satisfies the condition of being a resemblance class. But it is debatable whether there is a (non-trivial) property that is common to all particulars.

It should be clear that the problems (1) and (2) are a direct consequence of the fact that the nominalist cannot distinguish between various aspects of the similarity relation (for instance we would like to say that the class of red objects is defined by the relation of similarity with respect to colour). One solution that promises to evade this difficulty is known as trope theory. It postulates a new kind of objects – tropes – that may be acceptable to nominalists. Tropes are individual properties: the redness of that rose, the shape of that tree. Two numerically distinct individuals can never share any tropes. However, their tropes can be similar. The idea is that resemblance classes can be defined on tropes, and not on particulars, so that the resemblance class corresponding to redness will contain all tropes of redness. It is easy to notice that problems (1) and (2) disappear in this approach. No non-red trope can be more similar to a particular trope of red than a different trope of red, because tropes don’t have any ‘aspects’: they are themselves aspects. If two tropes are similar, they are always similar in precisely one respect. Problem (2) disappears, because the class of tropes that correspond to one property is always disjoint from the class of tropes corresponding to a numerically distinct property, even if all objects that posses one property possess the other one as well.

An interesting question arises whether the Principle of the Identity of Indiscernibles can be reinterpreted in trope theory. A simple replacement of properties with tropes results in a trivialisation of the PII. In virtue of the definition of tropes, if two individuals share at least one trope, they are numerically identical. A more promising strategy is to reformulate the PII in the form of the requirement that if each trope of object x is similar to a trope of object y, and vice versa, then x is numerically identical with y. Another point worth mentioning is that trope theory cannot accommodate unexemplified universals, hence it is more appropriate for a reinterpretation of the Aristotelian version of realism rather than the Platonist one.

Further readings:

M.J. Loux, The Problem of Universals II, pp. 62-79 (Metaphysics. A Contemporary Introduction)

E.J. Lowe, Realism Versus Nominalism, pp. 355-365 (A Survey of Metaphysics)