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The Challenge of Platonism

The Indispensability Argument, the most important contemporary argument in support of Platonism, is summarized and explained. At the heart of that argument lies (i) a criterion of ontological commitment which takes first-order logical quantifiers and singular terms to be devices of ontological commitment and (ii) the claim that some sentences containing such devices are literally true. A taxonomy of anti-Platonist responses, both realist and anti-realist, to generic indispensability arguments is provided. Realist responses include absolute creationism and divine conceptualism. The first is a sort of modified Platonism which holds abstract objects to be created by God. The second substitutes for abstract objects divine thoughts of various sorts, which play the roles usually assigned to abstract entities. Anti-realist responses which challenge the criterion of ontological commitment underlying indispensability arguments include free logic, neo-Meinongianism, and neutralism. These views reject either one or both of the devices of ontological commitment typically assumed by Platonists. Anti-realist responses which treat abstract object talk as not literally true include fictionalism, figuralism, and pretence theory. On these views abstract talk is taken either to be false or to be non-literally true or merely imagined to be true. Arealism is ruled out as a theistic response, leaving us with various realist and anti-realist options.

Keywords: Indispensability Argument, Platonism, criterion of ontological commitment, arealism, realism, anti-realism

Introduction

The heart of the contemporary debate over Platonism is to be found in the philosophy of mathematics, and so this field shall be the focus of our investigation. The ancient problem of universals, though still discussed by contemporary metaphysicians, does not lie at the centre of today’s debate—and with good reason, I think. The principal motivation for belief in universals is that it provides a solution to the ancient problem of the One over the Many. That problem concerns how to give an explanatory account of the resemblance between two things. The basic idea is that when two objects resemble each other in some respect—for example, they are the same colour—then there must be something which they have in common. This is said to be a universal or property, say redness, which the particular objects share. For the Platonist, talk of universals is not just a façon de parler. There really exist these abstract objects which particular things somehow have in common.

On the face of it, this account of resemblance sounds bizarre. Certainly, brown dogs and big elephants exist, but are we really to think that brownness and bigness are existing things, abstract entities existing in addition to elephants and dogs? Few contemporary philosophers think that the problem of the One over the Many provides an adequate reason for postulating the existence of such entities. Explaining the resemblance between two things is not really the issue, for if the anti-realist can explain, for example, why the dog is brown, then he can account for the dog’s resemblance in colour to the chair simply by saying of the chair what he said about the dog.1 (p.45) For the anti-

realist, talk of properties may be regarded as just a convenient, and perhaps indispensable, façon de parler not to be taken ontologically seriously.

If asked to explain why the dog is brown, the anti-realist can offer a perfectly plausible and, I think, adequate scientific explanation in terms of the dog’s absorbing and reflecting various wavelengths of light, and so forth. By contrast, it does nothing at all to explain why the dog is brown to say that the dog has the property of brownness. Indeed, how does standing in a mysterious relation to a non-spatial, causally unconnected, abstract object make an otherwise colourless dog brown? It seems that Platonism has no explanatory advantage over anti-realism. It is therefore little wonder that, in the words of the philosopher of mathematics Mark Balaguer, ‘The One Over Many

argument is now widely considered to be a bad argument.’2

The Indispensability Argument

Rather, in the contemporary debate over Platonism, there is one argument that is predominant: the so-called Indispensability Argument. The argument’s claim is that we are committed to the reality of abstract objects by many

of the statements we take to be true, such as mathematical statements like ‘1 + 1 = 2’. Although the Indispensability Argument for Platonism originated with W. V. O. Quine, a giant of twentieth-century philosophy, subsequent discussion has largely overtaken Quine’s version of the argument, which was predicated upon a number of idiosyncratic and now widely rejected theses.3 Today there is a variety of versions of Indispensability Arguments on tap, free of Quine’s more controversial theses. Balaguer nicely epitomizes such arguments as follows:

Let us unpack these premisses for the sake of those unfamiliar with the terminology.

Premiss (I): A Criterion of Ontological Commitment

Premiss (I) states a criterion of ontological commitment. It does not tell us what exists, but it does claim to tell us what must exist if a sentence we assert is to be true. It is intended to reveal to us what our discourse commits us to ontologically. According to premiss (I), we make ontological commitments in two ways.

Singular Terms

According to the first part of premiss (I), we make ontological commitments by means of singular terms. What are they? Singular terms are words or phrases which are used to single out something. They include proper names like ‘John’, ‘HMS Bounty’, ‘Blue Velvet’, and so on; definite descriptions like ‘the man in the grey suit,’ ‘your sister-in- law’, ‘my worst nightmare’, and so on; and demonstrative terms like ‘this pancake’, ‘that boy’, and so on.

In premiss (I), Balaguer uses ‘a’ as a logical constant for which we may substitute a singular term to form a simple sentence. ‘F’ stands for any property which we might predicate of the individual picked out by the singular term, and ‘R’ any relation in which that individual may be said to stand. For example, the simple sentence ‘The Queen Mary II is a huge oceanliner’ has the form a is F, ‘a’ standing for the (p.47) Queen Mary II and ‘F’ standing for a huge oceanliner. The simple sentence ‘The Queen Mary II docks in Southampton’ has the form a is R-related

to b, ‘a’ standing for the Queen Mary II, ‘b’ for Southampton, and ‘R’ standing for the relation docks in. In the following, we may ignore without detriment simple sentences involving relations and focus on those which are predications of a property to an individual.

The claim of premiss (I) is that, if a simple sentence of the form a is F is literally true, then the object denoted by its singular term exists. In our example, the term ‘Queen Mary II’ denotes the Queen Mary II (or, in other words,

the Queen Mary II is the denotation of the singular term ‘Queen Mary II’).5 Accordingly, since the sentence

‘The Queen Mary II is a huge oceanliner’ is literally true, the Queen Mary II exists.

Notice Balaguer’s qualification that the sentence must be literally true. Sentences employing metaphors and other figures of speech may be true, but they are not literally true. It may be true, for example, that ‘It’s raining cats and dogs!’, but it would be obtuse to think that someone asserting such a sentence believes that there are animals falling from the sky. He simply means that it is raining hard. Singular terms employed in non-literal speech are not ontologically committing for their user. But premiss (I) states that, if a simple sentence with a singular term is literally true, there must exist an object corresponding to the singular term used in the sentence.

Why simple sentences? What Balaguer is trying to avoid is not complexity, but sentences involving what are called intensional contexts. Intensional contexts are non-extensional contexts. Extensional contexts are sentence phrases which have two characteristics. (i) Singular terms referring to the same entity can be switched without affecting the sentence’s truth value. For example, ‘The morning star is Venus’ is an extensional context because one could substitute the co-referring term ‘the evening star’ for ‘the morning star’ without affecting the sentence’s truth value. By contrast, ‘Ancient Babylonian astronomers believed that the morning star rises in the morning’ is (p.48) an

intensional context because substitution of ‘the evening star’ for the co-referring term ‘the morning star’ would yield a false sentence. (ii) One can quantify into such contexts from outside the context. For example, ‘Mars has two moons’ permits us to infer, ‘There is something which is a moon of Mars’. By contrast, ‘Le Verrier sought to discover Vulcan between Mercury and the Sun’ involves an intensional context because we cannot infer that ‘There is something that Le Verrier sought to discover’.

The criterion of ontological commitment proposed by (I) does not apply when it comes to intensional contexts. For example, use of the singular term ‘the bogeyman’ in the true statement ‘Johnny fears the bogeyman’ does not commit us to the reality of the bogeyman. The inapplicability of (I)’s criterion of ontological commitment to intensional contexts implies that it will be useless for vast stretches of human discourse, for intentional attitudes (like fearing, hoping, believing), modal operators (like ‘necessarily,…’ or ‘possibly,…’), and temporal operators

(like ‘it was the case that…’ or ‘it will be the case that…’) all establish intensional contexts.6 The criterion applies

only to sentences which are extensional.

Existential Quantification

Singular terms are not the only devices of ontological commitment, according to (I). The second part of (I) claims that we also make ontological commitments by means of existential sentences. What are they? As Balaguer makes clear in the second part of premiss (II), he is talking about sentences which involve so-called existential (or particular) quantifiers. In contrast to universally quantified statements, which are true with respect to all the members of the domain of quantification7, existentially quantified statements are true with respect to some of the members of the domain of quantification. For example, if our domain of quantification is bears, then the statement ‘Some bears live near the North Pole’ is true just in case at least one member of the domain lives near the North Pole.

(p.49) In ordinary English, there is a variety of informal existential quantifiers, such as ‘some’, ‘at least one’, ‘there is/are’, and ‘there exists’. All of these informal expressions are captured in formal logic by the formal quantifier ‘∃’. A sentence like ‘Some bears live near the North Pole’ has the form (∃x) (x is a bear & x lives near the North

Pole), which is to be read ‘There is some x such that x is a bear and x lives near the North Pole’. In order for the sentence to be true, there must be at least one thing in the domain that can be the value of the variable ‘x’. Balaguer is evidently talking about first-order logic, where the variables x, y, z, take individual things as their values.

Now the claim of premiss (I) is that the literal truth of simple sentences involving existential quantification of the form ‘There is an F’ commits us to the reality of the object which is an F. Here ‘F’ stands for a general term like ‘a man’, ‘buffalo’, ‘facts which science has yet to discover’, and so on. Such sentences can be symbolized as (∃x) (x is an F). The existential quantifier ‘∃’ is claimed to express existence, not in a metaphysically lightweight sense, but rather in a metaphysically heavyweight sense. It is a device for making ontological commitments. Thus, the person who makes assertions involving informal quantifiers like ‘some’ and ‘there is/are’ thereby commits himself to the reality of the things in the domain of quantification which, as Balaguer puts it in premiss (II), his quantifiers ‘range over’, that is to say, the things which are the values of the variables in sentences he regards as true.

So, in sum, what premiss (I) expresses is not an ontological claim about what exists, but a meta-ontological claim about how a person commits himself ontologically. The claim is that singular terms and existential quantification are devices of ontological commitment. The idea is very simple, really, and I hope that I have not made it unduly complicated. My explication is necessary to prepare us for discussion of the premiss in what follows.

Premiss (II): Abstract Objects

Turn now to premiss (II). It claims that the denotations or referents of certain singular terms in literally true, simple sentences, for example, ‘2 + 2 = 4’, cannot plausibly be taken to be concrete objects of any kind. For example, the referents of such mathematical terms (p.50) as ‘2 + 2’ and ‘4’ are clearly abstract objects. Premiss (II) also excludes taking such mathematical discourse to be some sort of figurative language, not to be taken literally. It claims that at least some abstract discourse is literally true and therefore commits its user to the reality of abstract objects.

In the second part of premiss (II), it is likewise claimed that there are literally true existentially quantified statements involving quantification over abstract objects. For example, ‘There is a prime number between 2 and 4’ or ‘There are prime numbers greater than 100’. Again, such statements are not to be construed metaphorically, along the lines of ‘There’s a bee in her bonnet’. So existentially quantified abstract discourse also commits its user to the reality of abstract objects.

From (I) and (II) the conclusion (III) follows, that abstract objects exist. If such objects exist a se, as Platonists claim, then the classical theist must reject either (I) or (II).

Responses to the Indispensability Argument

Most Christian philosophers, not to speak of theologians, are largely unaware of the wide range of responses to the Indispensability Argument which are on offer today. If we take mathematical objects as a case in point, then we can portray some of the many options as in Figure 1 (p. xii).

The various options can be classed as realist (mathematical objects exist); anti-realist (mathematical objects do not exist); or arealist (there is no fact of the matter concerning the existence of mathematical objects). As

Figure 1 illustrates, there are two brands of realism about mathematical objects: views which take them to

be abstract objects and views which take them to be concrete objects. Of realist views which consider mathematical objects to be abstract, absolute creationism is a sort of modified Platonism, holding that mathematical objects have, like concrete objects, been created by God, thus safeguarding divine aseity. Concretist versions of realism can take mathematical objects to be either physical objects or mental objects, the latter either in human minds or

in God’s mind. The most (p.51) promising concretist view is some sort of divine conceptualism, the heir to the view of Philo and the Church Fathers.

When we turn to anti-realist responses to the Indispensability Argument, we find a cornucopia of different views. Neutralism rejects the criterion of ontological commitment expressed in premiss (I), taking the use of singular terms and existential quantification to be neutral with respect to ontological commitments. Neo-Meinongianism similarly denies that existential (or particular) quantification is ontologically committing, though it agrees that there are objects referred to by abstract singular terms—with the caveat that these objects are nonexistent. Free logic, on the other hand, takes existential quantification to be ontologically committing but denies that the use of singular terms is a device of ontological commitment. Fictionalism accepts entirely the Platonist’s criterion of ontological commitment but denies that mathematical statements are true. Figuralism, by contrast, holds that mathematical discourse is true but denies that it must be taken literally. Pretence theory considers mathematical discourse to be a species of make-believe, so that mathematical objects are akin to fictional characters. There are also paraphrastic strategies, which hold that we can offer paraphrases of mathematical statements which will preserve their truth value

but without ontological commitment to abstract objects.8 And so on!

In between realism and anti-realism about mathematical objects is arealism, the view that there just is no fact of the matter about the reality of mathematical objects. Is this an option that might be available to theists eager to preserve divine aseity? The classic version of arealism was the conventionalism of Rudolf Carnap.9 He drew a fundamental distinction between what he called ‘internal questions’ and ‘external questions’, that is to say, questions which are posed within an adopted linguistic framework and questions posed by someone outsidethat framework. Carnap gives

the illustration of what he calls the ‘thing’ framework or language. Once we have adopted the linguistic framework in which we speak of observable things in a spatiotemporal system, we can meaningfully pose internal questions like ‘Is the moon a thing?’ or ‘Is a school of fish a thing?’ (p.52)Someone who rejects the language of things may choose to speak instead of mere appearances such as sense data.

Similarly, once we have adopted a linguistic framework involving terminology for abstract objects like numbers, internal questions like ‘Is there a prime number greater than 100?’ are meaningful. No one who has adopted the framework would seriously raise the question ‘Are there numbers?’, for their existence is necessary once one has adopted the number framework. For someone who is outside the framework, Carnap insisted, the question is meaningless. As a logical positivist and verificationist, Carnap was convinced that such metaphysical questions have no cognitive content. Whether one adopts the linguistic framework in which it makes sense to speak of numbers is just a matter of convenience, or convention; hence, the name conventionalism.

No philosopher today would defend Carnap’s verificationism; but his conventionalism does find an echo today in what is sometimes called ontological pluralism.10 According to these thinkers, certain ontological questions, though meaningful, do not have objective answers. The philosophers of mathematics Mark Balaguer and Penelope Maddy, for example, would deny that the question ‘Do mathematical objects exist?’ has an answer that is objectively true or false.11 On arealism, there just is no fact of the matter whether or not mathematical objects exist.

Now at first blush arealism might seem a quick and easy solution to the challenge posed by Platonism to divine aseity. If there is no objective truth about whether or not mathematical objects exist, then the use of mathematical

terminology is devoid of ontological significance. Internal questions about the existence of certain sets or numbers or solutions to equations may be answered affirmatively; but there just is no answer to the external question of the existence of (p.53) such entities. In that case, one cannot truthfully assert that there are objects which God did not create.

Alas, however, there is no succour for the theist here. For given God’s metaphysical necessity and essential aseity, there just is no possible world in which uncreated mathematical objects exist. Hence, there most certainly is a fact of the matter whether uncreated, abstract objects exist: they do not and cannot exist. Therefore, arealism is necessarily false, as is conventionalism about existence statements concerning mathematical and other abstract objects.

Still, Carnap’s discussion is not without merit, and many contemporary philosophers find his distinction between internal and external questions to be intuitive and helpful. We shall have occasion to return to it.

Putting aside arealism, then, we are left with a number of both realist and anti-realist solutions to the challenge posed by Platonism to divine aseity. We shall want to explore which solutions offer the most promise for an answer to the Indispensability Argument and to the challenge of Platonism. We shall begin with the realist alternatives.

Notes:

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