The main issues have nothing to do with mathematics in particular, so I deliberately avoid more than passing reference to mathematical examples.

Colours, shapes, sizes, masses are the repeatables or ‘universals’ or ‘types’ that particulars or ‘tokens’ share. A certain shade of blue, for example, is something that can be found in many particulars – it is a ‘one over many’ in the classic phrase of the ancient Greek philosophers. On the other hand, a particular electron is a non-repeatable. It is an individual; another electron can resemble it (perhaps resemble it exactly except for position), but cannot literally be it.²

Science is about universals. There is perception of universals – indeed, it is universals that have causal power. We perceive an individual stone, but only as a certain shape, colour and weight, because it is those properties of it that confer on it the power to affect our senses. It is in virtue of being blue that a body reflects certain light and looks blue. Science gives us classification and understanding of the universals we perceive and finds the laws connecting them – physics deals with such properties as mass, length and electrical charge, biology deals with the properties special to living things, psychology with mental properties and their effects, mathematics with ... well, we’ll get to that.

Aristotelian realism about universals takes the straightforward view that the world contains both particulars and universals, and that the basic structure of the world is ‘states of affairs’ of a particular’s having a universal, such as this page’s being approximately rectangular.³

Science is also the arbiter of what universals there are. To know what universals there are, just as to know what particulars there are, one must investigate, and accept the verdict of the best science (including inference as well as observation). Photosynthesis turned out to exist, phlogiston not. Thus universals are not created by (or postulated to account for) the meanings of words, nor can one make up more of them by talking or thinking. On the other hand, language is part of nature, and it is not surprising if our common nouns, adjectives and prepositions name some approximation of the properties there are or seem to be (just as our proper names label individuals), or if the subject–predicate form of many basic sentences often mirrors the particular-property structure of reality.

The main problem for nominalism is its failure to give an account of why different individuals should be collected under the same name (or concept or class), if universals are not admitted. According to ‘predicate nominalism’ for example (that is, nominalism that takes universals to be mere words), ‘The word “white” correctly applies to Socrates’ is prior to ‘Socrates is white.’ That is counterintuitive, since it appears that things were white prior to language existing, and that we apply the term correctly because of the commonality between white things. And our recognition of that commonality, which is a condition of our learning to apply the word correctly, arises from the ability of all white things to affect us in the same way – ‘causality is the mark of being’. A further problem is that the predicates or concepts relied upon by nominalists to unite the particulars are themselves universals – the word ‘white’ means not a particular inscription on a certain page, but the word type ‘white’ in general; thus predicate or concept nominalism simply pushes the problem of the ‘one over many’ back one stage.⁴

A serious attempt to show that mathematics can be done nominalistically, that of Hartry Field, will be examined in Chapter 7. It will be concluded that, although not Platonist, the project implicitly includes a realist view of quantitative properties.

Platonism (in its extreme version, at least, which is the version usually found in the philosophy of mathematics⁵) holds that there are universals, but they are pure Forms in an abstract world, the objects of this world being related to them by a mysterious relation of ‘participation’ or ‘approximation’. Thus, what unites all blue things is solely their relation to the Form of blue, and what unites all pairs is their relation to the abstract number 2. Mathematicians’ unreflective use of names like ‘2’, ‘the continuum’ and ‘the Monster group’, as if they name particular entities with which mathematicians have dealings, is felt to support a Platonist view of such beings.

The problems for Platonism, both ontological and epistemological, arise from the relational view of its solution to the ‘one over many’ problem. First, there is the difficulty of explaining the nature of the relation: ‘participation’ and ‘approximation’ are metaphors that it is hard to clarify, while if we consider examples such as the relation of pairs to the number 2, we seem to have no insight into the relation.⁶ Second, the relational nature of how the Form works means that it bypasses the commonalities between things that do unite them: if we imagine the Form of blue not existing, which of the individuals are the ones that would be united by their relation to the Form of blue, if it did exist? Surely there is something about them that makes those the ones apt for participating in the Form of blue, and distinguishes them from red ones? That is what perception suggests. Blue things affect our retinas in a characteristic way because the blue in the things acts causally, without any apparent need to consult a Form elsewhere before doing so.

Epistemologically too, Platonism has difficulties because of its relational nature. Either there is a perception-like intuition into the realm of the Forms, or we have knowledge of them through some process of inference such as inference to the best explanation. The first is not possible, since the realm of the Forms is acausal, so no messages can come from that realm to our brains, as happens from coloured surfaces to our retinas. How can humans ‘reliably access truths about an abstract realm to which they cannot travel and from which they receive no signals’?⁷ (It has been maintained that in mathematical visualization we do have direct access to a realm of mathematical necessities;⁸ as will be argued in Chapter 11, that is true but the necessities are realized or realizable in diagrams, not in a separate abstract world.) The second option, access to the Forms via inference to the best explanation, faces the initial problem that young children appear to have a great deal of direct mathematical knowledge from counting and pattern recognition, without the need for any sophisticated reasoning to abstract entities. The nature of that basic knowledge will be treated in Chapter 10, while more elaborate attempts to argue to Platonism from the indispensability of mathematics in science will be considered in Chapter 7.

A complete answer to Platonism must include an account of what the number 2, the continuum and other mathematical entities are, if they are not abstract objects in a Platonic world. An alternative, Aristotelian, account will be given in Chapter 3 and 4.

This is not the place for more detailed criticism of Platonism and nominalism, which has been extensively pursued in general works on metaphysics.

At this point it can be seen how the Platonist–nominalist dichotomy that has been assumed in most of the philosophy of mathematics is a false one. If Platonism is taken to mean ‘there are abstract objects’ and nominalism to mean ‘There aren’t’, then it can appear that Platonism and nominalism are mutually exclusive and exhaustive positions. However, the words ‘abstract’ and ‘object’ both work to distract attention from the Aristotelian alternative: ‘abstract’ by suggesting a Platonist disconnection from the physical world and ‘object’ by suggesting the particularity and perhaps simplicity of a billiard ball. Indeed, the concept of ‘abstract object’ that has had such a high profile in the philosophy of mathematics is a comparatively recent notion and a very unclear one. It is an artifact of the determination of nominalists (especially Locke) and Platonists (especially Frege) to carve up the field between themselves. In particular, the notion is a creation of Frege’s conclusion that since the objects of mathematics are neither concrete nor mental, they must inhabit some ‘third realm’ of the purely abstract.⁹

Aristotelians do not accept the dichotomy of objects into abstract and concrete, in the sense used in talk of ‘abstract objects’. A property like blue is not a concrete particular, but neither does it possess the central classical features of an ‘abstract object’, causal inefficacy and separation from the physical world. On the contrary, a concrete object’s possession of the property blue is exactly what gives it causal efficacy (to be perceived as blue).

Thus an entity of interest to the philosophy of mathematics – say the ratio of your height to mine – could be either an inhabitant of an acausal, ‘abstract’ world of Numbers, or a real-world relation between lengths, or nothing. The three options – Platonist, Aristotelian and nominalist – need to be kept distinct and on the table, or discussion will be confused from the beginning.

Because of the special relation of mathematics to complexity, there are three issues in the theory of universals that are of comparatively minor importance in general but are crucial in applying Aristotelian realism to mathematics. They are the problem of uninstantiated universals, the reality of relations, and questions about structural and ‘unit-making’ universals. The first of these, perhaps the most important, will be left to the next chapter.