2.1 What Does ‘Abstract Object’ Mean?

In the debate over the existence of abstract objects, the word ‘object’ is used in different ways by different philosophers. In the sense that is important for introducing the topic, ‘object’ simply means ‘thing’ or ‘entity’. In this sense of ‘object’, an object is not a special type of thing: everything is an object. If we discover that abstract objects exist, then we can debate what type of thing they are. But the more fundamental question is whether there are any abstract objects at all.

The debate over the existence of abstract objects therefore belongs to the part of metaphysics known as ‘ontology’, in which we try to find out what exists. There are well-known challenges to the legitimacy of ontological inquiry; I will ignore them here (see Reference Chalmers, Chalmers, Manley and WassermanChalmers 2009 for a survey). Similarly, some philosophers maintain that there is more to reality than what exists. They claim that there are some things which do not exist (see e.g. Reference ParsonsParsons 1980). Here I will presuppose that what there is and what exists are the same.

There is much more to be said about ‘abstract’. The term has no standard definition: philosophers sometimes claim to be repeating the standard definition, but the definitions they give are all different! Often philosophers use the term without introducing what they mean by it. Since ‘abstract’ is a technical term which different philosophers use in different ways, it doesn’t make sense to debate which meaning is the correct one. There are simply different ways of using the term (Reference RosenRosen 2009: Section ‘Introduction’). However, a closely related question is worth considering: which way of using the term is the most fruitful?

To answer that question, we should begin by asking which way of using the term is most popular. When we engage in ‘abstract object’-talk, we should try to mean the same as the other participants in the debate, or miscommunication will result. Also, we should bear in mind what sorts of objects are commonly thought of as clearly abstract and clearly not abstract: if our way of using the term does not respect those classifications, that is evidence that we have failed to latch on to the prevailing usage. When discussing the meaning of ‘abstract’ it is convenient to write as if all the candidates for being abstract or not abstract exist. That saves writing ‘if they exist’ all the time. So I will follow that policy for the remainder of this section.

Material objects, such as chairs, are commonly thought of as clearly not abstract. Numbers are commonly thought of as clearly abstract. By ‘number’ here I do not mean physical inscriptions, such as the numbers on the dial of my watch, but the objects to which these inscriptions refer.

Historically, a prominent way of introducing the notion of abstractness was in terms of a psychological operation whereby we ‘abstract’ away from some differences and focus on similarities (Reference RosenRosen 2009: Section ‘The Way of Abstraction’). But now abstractness is commonly introduced in terms of causality and spatio-temporal location. Call something ‘acausal’ if it does not take part in causal interactions. The notion of abstractness is often now introduced along these lines:

Popular variations replace ‘acausal’ with ‘causally impotent’ or ‘lacks causal power’: on these definitions, abstract objects do not take part in causal interactions because they cannot. That difference is probably not very significant, because there is little plausibility to the thought that good candidates for being abstract objects are capable of causal interaction but somehow never do so. For instance, if we said that numbers can start fires, then we would be under pressure to explain why they never do so. It is neater to say that they can’t.

Within the philosophy of mathematics, it is common to hold that numbers are acausal and tables are not. But many philosophers working on causation would say that only events are causes, so numbers and tables all cause nothing. If that is right, then we need to explain how some non-events can count as causally active even though they are not causes. (Here I echo Reference RosenRosen 2009, Section ‘The Causal Inefficacy Criterion’.)

The phrase ‘lacks spatio-temporal location’ raises deep complexities. Philosophers seldom explain what they mean when they use it. Does something that lacks a location in space automatically count as lacking spatio-temporal location? If so, then the ‘temporal’ part is redundant. So presumably this is not what is meant.

Abstract objects are sometimes characterised as ‘beyond’ or ‘outside’ time. But this talk is puzzling: it is not clear how to cash out these metaphors. Believers in abstract objects think that they exist now. It is natural to think that numbers have always existed and will always exist. On this view, numbers are especially old and especially long-lasting objects. But that need not mean that they are ‘outside’ time in any sense. Perhaps they occupy time in the same way as, for instance, chairs do – they just occupy it for an especially long period. The same goes for many other abstract objects. So the notion of being ‘outside’ time needs fleshing out; and the claim that abstract objects are ‘outside’ time in this sense will then need defending.

Philosophers of religion draw a distinction between sempiternality and eternality which might be useful here. To be ‘sempiternal’ is to exist at all moments of time. ‘Eternal’ is harder to define; there is a great deal of discussion of how best to do so and whether the notion really makes sense. (Reference Stump and KretzmannStump and Kretzmann 1981: 430 introduce ‘eternality’ as ‘the condition of having eternity as one’s mode of existence’, and go on to mount a classic defence of its coherence.) What is clear is that eternality is meant to be an alternative to sempiternality. Philosophers who are attracted to the idea that abstract objects are ‘outside’ time could draw on this debate in order to flesh out the idea and test it.

Perhaps the idea that some abstract objects are ‘outside’ or ‘beyond’ time can be clarified and defended. But it is unlikely that this could be true of all abstract objects, because some of them seem to be created by human activity. Consider Pride and Prejudice – not individual physical copies, but the novel itself of which these are copies. That is plausibly an abstract object; and before we start thinking systematically about the metaphysics of fiction, we are strongly inclined to say that it came into existence only when Jane Austen created it. Numbers may well have existed since the dawn of time, but nineteenth-century novels probably not. Abstract objects created by human activity are not even sempiternal, so they will have little chance of being ‘beyond’ time.

This example shows that, even if we want to say that some abstract objects are ‘beyond’ time, it is unwise to build the idea of being ‘beyond’ time into the definition of ‘abstract’. It also shows that the notion of sempiternity is of limited use here (Reference HaleHale 1987: 48–9 makes a similar point). Perhaps some abstract objects – numbers, for example – are sempiternal. But it is unwise to make it a matter of definition that all abstract objects exist sempiternally.

Most definitions do neither of those things. But many of them incorporate the idea of lacking a spatio-temporal location, which remains to be clarified. It is notable that in the contemporary debate over the existence of abstract objects, the notion of time plays only a minor role (Reference BakerBaker 2003 is a notable exception). Many of the arguments would look just the same if ‘abstract’ were defined as follows:

I conclude that building a temporal concept into the definition of ‘abstract’ is more trouble than it’s worth.

Things that lack a spatial location are sometimes said to be ‘outside space’. But this is a dangerous metaphor: it encourages the thought that abstract objects are infinitely far away from us, in some ‘abstract realm’. Because abstract objects lack a spatial location, they are not the sort of things that stand at a distance to other things. They are neither close nor far away: such concepts do not apply. As Reference RosenRosen (1993: 152) points out, abstract objects are not ‘elsewhere’ but ‘nowhere’.

Indeed, we can go further and point out that what is important here is not the idea of lacking a spatial location, but that of lacking a particular spatial location. Perhaps numbers are spread out through the whole of space: we cannot link them to particular locations, because each one of them is everywhere. That option is seldom discussed. I suspect it would leave the debate much as it is.

As I said, numbers are often thought of as clear cases of abstract objects, and material objects, such as chairs, as clear cases of objects that are not abstract. We should check that (D2) classifies these cases correctly. Let us start with numbers.

No-one ever makes their food more delicious by sprinkling into the saucepan some numbers, and no-one is ever rushed to hospital after a nasty collision with the square root of minus one. It’s not just that there is no empirical evidence that such things ever happen; rather, we are confident that no such things ever happen, even without checking the empirical evidence. That’s because we think of numbers as the sort of things that cannot be sprinkled, tasted, or collided with. We think of numbers as acausal.

Mathematics teachers do not take their pupils on school trips to get close to numbers. They do not even try to. We never describe numbers as being in a particular place; and it would be ludicrous to claim that π is moving slowly south-west. In other words, we do not think of numbers as being the sorts of things that are spatially located.

So our ordinary thinking about numbers portrays them as acausal and lacking spatial location. Until a sufficiently powerful challenge to our ordinary thinking arises, (D2) tells us that numbers are abstract. Since numbers are paradigms of the abstract, that is the right result.

Tables, on the other hand, are both spatially located and causally active. There is one in this room and it is reflecting light into my eyes right now. So (D2) tells us that tables are not abstract – again, the right result.

From here on, when I call something ‘abstract’, I will mean that in the sense given by (D2). For brevity and variety, philosophers sometimes borrow from Latin and call abstract objects ‘abstracta’ (singular: ‘abstractum’).

Many different things are probably abstracta. The following are plausibly thought of as acausal and lacking spatial location, hence abstract: games (e.g. chess); languages (e.g. Swedish); concepts (e.g. the concept of a triangle); theories (e.g. Newtonian mechanics); fictions (e.g. Hamlet); musical works (e.g. Beethoven’s First Symphony); biological species (e.g. the King Penguin); recipes (e.g. for Peach Melba); and possible worlds. Of course, for any object, there is room for debate over whether it really is abstract.

Often we do not know enough about an object to be sure whether it counts as abstract. Not all mathematical entities are clearly abstract. Consider the singleton set of Edinburgh (written ‘{Edinburgh}’) – that is, the set whose only member is Edinburgh. It is a matter of debate whether this object is located in Scotland, or not spatially located at all. If we decide {Edinburgh} is spatially located, we cannot say it is abstract. We should not assume that all sets lack a location, or that all sets have a location. Perhaps some sets, such as {Edinburgh}, are spatially located, whereas other sets, such as the empty set, are not. There is also a good question about whether {Edinburgh} is acausal. Is {Edinburgh} causally active? Or is it something causally inactive that contains as a member something causally active? What we say about this will have implications for whether to count {Edinburgh} as abstract.

It is common to take properties as clear examples of abstracta. But in fact it is controversial whether these entities meet the conditions for counting as abstract. There is a long-standing debate about the location of properties: are they where their instances are, or nowhere at all? Perhaps redness is where the red things are – or perhaps it’s nowhere at all (see Reference OliverOliver 1996: 25–33). Only those who deny properties spatial locations can claim properties are abstract. While the debate over the location of properties is unresolved, we should draw back from claiming that properties are abstract objects.

Propositions, too, are frequently classed as clear examples of abstract objects. But whether we say they are abstract depends partly on what we say about their nature. Consider the influential Russellian account of propositions. According to that account, propositions are tuples, some of which have chairs among their members. Is that enough to locate those propositions? If it is, then they are not abstract. Unless we have a good reason to deny that such tuples lack a spatial location, or to reject the Russellian account of propositions, we should keep an open mind about whether propositions are abstract. (On defining ‘abstract’, Reference LewisLewis 1986: Section 1.7 and Reference RosenRosen 2009 are essential reading.)

2.2 The Abstract/Concrete Distinction

So far I have just talked about whether objects are abstract or not. It is now time to bring in the notion of concreteness. As with ‘abstract’, many authors leave it unclear precisely what they mean by ‘concrete’, especially when they use the term in passing.

Abstract objects are contrasted with concrete ones: philosophers agree that nothing could be both, and speak of the ‘abstract/concrete’ distinction. In addition, physical objects such as chairs are regarded as clear examples of concrete things.

As we have seen, chairs are clearly not abstract. In order to count as abstract, they would have to be acausal, and they would also have to lack spatial location. Chairs fail both conditions, so they are not abstract. Since chairs are paradigms of the concrete, we should class anything that fails both conditions as concrete. But what about things that fail only one condition? Should we class them as concrete or not? In terms of Table 1, should we define ‘concrete’ as ‘(c)’ or should we define it as ‘(a), (b), or (c)’?

This question would have little significance if categories (a) and (b) were certainly empty – but they are not. The Equator seems to be spatially located but acausal, so it seems to belong to (a). Gods and Cartesian souls are naturally seen as causally active but not spatially located, so it is tempting to classify them as (b).

Both ways of defining ‘concrete’ seem to have limitations. The phrase ‘abstract/concrete distinction’ suggests that everything is either abstract or concrete. (That is, it suggests that the distinction is exhaustive.) If we define ‘concrete’ as ‘(c)’, then we will have to remember that there may be objects that are neither abstract nor concrete – everything in (a) and (b). That will complicate our reasoning: we will not be able to infer ‘x is concrete’ from ‘x is not abstract’, nor ‘x is abstract’ from ‘x is not concrete’. On this definition, it is misleading to speak of the ‘abstract/concrete distinction’: it would be clearer to speak of the ‘abstract/concrete/none of the above’ distinction.

Suppose instead that we define ‘concrete’ as ‘(a), (b), or (c)’, which is equivalent to defining it as ‘not abstract’. That definition implies that if something is not abstract, it is concrete – so, as a matter of definition, everything is either abstract or concrete. The objects in (a) and (b) are classed as concrete. The ‘abstract/concrete’ distinction is then an exhaustive one, so the phrase ‘abstract/concrete’ distinction will not mislead; and we are safe to infer ‘x is concrete’ from ‘x is not abstract’, and ‘x is abstract’ from ‘x is not concrete’. So this way of defining ‘abstract’ has some advantages.

The disadvantage of this definition is that it threatens to make concreteness uninteresting. Why should this disjunctive category have any theoretical significance? It is a rag-bag into which we stuff anything that fails, for whatever reason, to be concrete, rather than a unified category which makes for genuine resemblance.

This disadvantage is not a serious problem. When we are defining a distinction, we often want it to be exhaustive. A neat way to do that is to define the second term as the negation of the first, so that the distinction is between the Fs and the non-Fs. Typically the non-Fs will have little in common other than failing to count as an F. (There are many different ways of failing to count as an event, or of failing to count as a mental state.) This is the price we pay for the benefit of having an exhaustive distinction.

Another objection to defining ‘concrete’ as ‘not abstract’ is that it privileges the abstract over the concrete. We could just as well have defined ‘concrete’ as ‘spatially located and causally active’ – that is, as ‘(c)’ – and then defined abstract as ‘not concrete’. Why start with the abstract rather than the concrete?

My impression is that starting with the abstract fits in slightly better with how philosophers have tended to use the terms. It feels somewhat revisionary to call the objects in (a) and (b), each of which is either spatially located or causally active, ‘abstract’. But I don’t think that much would be lost if we started with the concrete.

Perhaps we could neaten up our system of definitions by showing that categories (a) and (b) are empty (or necessarily empty). But that would be a difficult task. To pose an objection to Cartesian dualism, Reference Kim and CorcoranKim (2001) argues that being causally active requires a spatial location. This echoes Elizabeth of Bohemia’s challenge to Descartes to explain ‘how the soul of a human being (it being only a thinking substance) can determine the bodily spirits, in order to bring about voluntary actions’ (Reference ShapiroShapiro 2007: 62).

In the rest of this Element, when I write ‘concrete’ I will mean ‘not abstract’. Concrete objects are sometimes called ‘concreta’ (singular: ‘concretum’). As well as chairs and other physical objects, plausible examples of concreta include events (e.g. the First World War) and mental state tokens (e.g. my belief that metaphysics is fascinating).

The view that there is at least one abstract object is often called ‘Platonism’ or ‘platonism’; those who hold it, ‘Platonists’ or ‘platonists’. ‘Nominalism’ is often defined as the view that there are no abstract objects; sometimes as the view that everything is concrete. Adherents of nominalism are called ‘nominalists’. What those definitions mean of course depends on the meaning of ‘abstract’ and ‘concrete’. In a debate about a particular sort of abstract object, ‘platonist’ and ‘nominalist’ can also be used to refer to those who believe, or do not believe, in the abstract objects of the sort in question. For example, one might be a nominalist about properties without being a nominalist in the first sense. In such contexts, it is convenient to call those who believe in no abstract objects whatsoever ‘global nominalists’. (In the specific context of debates about properties, ‘nominalist’ can also mean ‘someone who rejects the existence of universals’. But in this Element, I will not use the word in that sense.)

The terms ‘platonist’ and ‘nominalist’ suggest connections to ancient and medieval philosophy, respectively. They are misleading. In the sense considered here, the abstract/concrete distinction played little role in philosophy before the twentieth century (Reference RosenRosen 2009: Section ‘Historical Remarks’). Even a figure such as Frege, whose influence on the contemporary abstract objects debate is palpable, worked with a different group of ontological distinctions.

How does the abstract/concrete distinction relate to the modal distinction between metaphysical necessity and metaphysical contingency? It is tempting to assume that the relationship is simple: abstract objects exist necessarily, concrete objects exist contingently. That assumption fits in neatly with the appealing thought that tables are contingent existents, numbers necessary ones. But the assumption should be resisted.

The main reason for resisting it is that it is not clear that abstract objects, if they exist, exist necessarily. Often we lack evidence that an abstract object is a necessary existent. For example, one of the leading arguments for the existence of numbers rests on the idea that mathematics is indispensable to empirical science. (I will say more about this argument in the next section.) Now whether mathematics is indispensable to empirical science seems to be a contingent matter. So this argument at best gives reason to believe that numbers actually exist: it does not support the conclusion that they necessarily exist. (In other words, even if the indispensability argument succeeds, it has no implications for the modal status of the existence of numbers. To use indispensability considerations to argue that numbers exist only contingently, we would require further assumptions. See Reference MillerMiller 2012 for discussion.)

Moreover, we have some evidence that no abstract object exists necessarily. The evidence is simple: for each abstract object, we can imagine that object not existing. That is not conclusive or indefeasible evidence, but is evidence nonetheless. (Reference Rosen, Gendler and HawthorneRosen 2002: 287–95 discusses this and other arguments for the same conclusion; see also Reference CowlingCowling 2017: 82–4.)

So it is not at all obvious that the abstract/concrete distinction lines up with the necessary/contingent distinction in the way suggested. If we could establish that it does, we would have an elegant theory. But until a compelling argument comes along, we should resist the assumption that there is a simple relationship between abstractness and necessity. (Reference Sider, Bennett and ZimmermanSider 2013: 287 issues a similar warning.)

2.3 Who Has the Burden of Proof?

In the debate over the existence of abstract objects, who has the burden of proof? This important question is surprisingly little discussed. To find out where the burden of proof lies, we have to examine what we think of the matter before we begin to reflect on it philosophically and start to theorise about it. In other words, we need to find out what our pre-theoretical belief is.

Since the concept of an abstract object is a philosophical one, I doubt that we pre-theoretically think of things as abstract (in the sense used here). If that is right, then we do not pre-theoretically believe that there are abstract objects – and neither do we pre-theoretically believe that there are no such things.

However, there is a strong case that if there are no abstract objects, then many of our pre-theoretical beliefs are untrue. In this sense, we can be said to be committed to the existence of abstract objects.

For instance, until we start to reflect philosophically on the matter, we are happy to say that there is a prime number between four and six. Presumably, that is because we hold the belief that there is such a thing. For reasons given above, this object, if it exists, is abstract. So if there are no abstract objects then that belief is not true.

That there is a prime number between four and six is an existential belief. Not all beliefs that seem to commit us to abstract objects have that form. For example, we seem to believe that chess is a game for two players. That seems to entail that there is such a thing as chess. Very plausibly, if chess exists, it is abstract. So unless this abstract object exists, the belief is not true. In the same way, the belief that five is a prime number seems to commit us to the existence of an abstract object, the number five.

Types (if they exist) seem to be abstract objects. For instance, tokens of the word ‘bread’ have spatial location and causal power, but the word ‘bread’ itself – the type – seems to have neither. (Arguably, types are properties and their tokens are instances of them. But we do not have to take a stand on that here.) Reference WetzelWetzel (2009: chapter 1) argues that ordinary and scientific language abounds in apparent references to types: types of word, types of animal, types of gene, types of computer, types of atom, types of subatomic particle, and many more. Even when we are talking about a token – a particular atom or a particular move in a particular chess match – we naturally talk about a type to which it belongs (Reference WetzelWetzel 2009: 16–17, 21–2). This is further evidence that our pre-theoretical beliefs imply the existence of abstract objects.

Hence, it seems that our pre-theoretical beliefs depict a platonist world, not a nominalist one, and so the nominalists have the burden of proof.

However, things are more complicated. In the words of Reference MeliaMelia (1995: 223):

Melia’s point is that platonism is implausible: when we are introduced to platonism, we do not regard it as an obvious consequence of our other beliefs, but as a striking, non-obvious thesis. Melia points out that there is ‘a clash between our … intuitions’ (Reference MeliaMelia 1995: 223). We can state these as an inconsistent triad:

Pre-theoretically, we believe (1). As soon as we gain the concept of an abstract object, we find both (2) and (3) plausible. But (1), (2), and (3) cannot all be true. To portray us as pre-theoretical platonists or as pre-theoretical nominalists is therefore too simple. Although it is uncommon to put it this way, the truth is that the existence of abstract objects presents us with a paradox: we want to say that there are no such things, but our beliefs imply their existence.

If this is right, then neither side in the debate bears the burden of proof. Neither platonism nor nominalism has the advantage.

But things are even more complicated, since there is evidence that we are not committed to the literal truth of claims such as (1). Reference Yablo, Boghossian and PeacockeYablo (2000: 224–6; Reference Yablo2001: 86–7, 89–90) has pointed to a body of evidence which suggests that our attitude to claims such as (1) falls short of belief in what they literally express. Yablo argues that there are many striking similarities between mathematical language and non-literal language. I will mention just two of them.

First, both mathematical language and non-literal language turn out to enhance our expressive capacities: they help us to say things we could not otherwise say, or help us to say what we want to more quickly or effectively. To use an example of Reference WaltonWalton’s (1993: 40): describing the town of Crotone as ‘on the arch of the Italian boot’ specifies its location briefly and memorably. In the mathematical case, the idea is that mathematical language enables us to say things about the non-mathematical world we would otherwise find it harder or impossible to say.

Second, both types of language give rise to identity questions which are very hard to answer. For instance, saying whether the hatchet I buried yesterday is the same as the one I buried today is no easier than saying which sets the natural numbers are.

These pieces of evidence suggest that mathematical language is non-literal language. That may be a surprising conclusion, but in its defence Reference Yablo, Boghossian and PeacockeYablo (2000: 218–24; Reference Yablo2001: 95) argues that we sometimes speak non-literally without it being obvious to us that we’re doing so.

If mathematical language is non-literal, then there is no reason to think that we regard (1) as literally true. Those who believe that there are such things as prime numbers would then have the job of convincing us that such things exist. In other words, nominalism would be the default view, and the burden of proof would be borne by platonists.

The evidence Yablo points to deserves much more discussion. At the present stage of the debate, there is no agreement over its significance. Reference StanleyStanley (2001: 50) claims that ‘Yablo’s analogies are contentious, in that many of them only someone with nominalist leanings would find compelling’, though he does not argue for this claim. Alternative explanations of some of the phenomena Yablo points to are suggested by Reference Rosen, Burgess and ShapiroRosen and Burgess (2005: 526–34). Moreover, Yablo’s position raises a difficult question: if our attitude to the literal content of (1) is not belief, then what is it? I will return to this matter in Section 3.5.

The whole issue of burden of proof deserves much more discussion than it receives. We might easily suppose that many researchers in this area think that platonism is the default position and nominalists bear the burden of proof. I have just argued that the situation is actually more complicated.

3.1 Numbers

As I have already mentioned, numbers present us with a paradox:

Examining the case for (3) is the task of Section 4. Let us assume for the moment that (3) has not been securely established, so that it is reasonable to reject (3). Then we have a very simple argument for platonism, made of (1) and (2):

This is really a telescoped version of a slightly more complex argument:

Call this the ‘basic argument’. In (2), (2a), and (2b), ‘If … then’ expresses the material conditional (the truth-functional conditional). The same goes for ‘If … then’s in later, similar arguments. For the arguments to be valid, nothing stronger than material implication is needed.

There is obviously nothing special about (1): we can replace it with any plausible mathematical claim that seems to imply the existence of numbers, provided we adjust (2a) appropriately. For example:

• Some square numbers are even.

• If some square numbers are even, then there is a number.

• If there is a number, then there is an abstract object.

• Therefore there is an abstract object.

I will now show that some of the most important arguments for the existence of numbers have this fundamental form. They are elaborations of the following pattern, where ‘X’ takes the place of a sentence which seems to be true and seems to imply the existence of numbers:

Call this the ‘basic structure’. (In Section 3.2, I will argue that there is a more fundamental structure underlying a wide range of arguments for the existence of abstract objects.)

Let us start with the indispensability argument for the existence of numbers (and other abstract mathematical objects), because this is the argument for platonism which is currently the most discussed. The basic idea is that mathematics is indispensable to science – that is, our scientific theories would be considerably worse if they were non-mathematical – and that this establishes platonism.

Formulated in the simplest terms, the argument runs as follows:

The argument is sometimes known as the ‘“Quine–Putnam” indispensability argument’. This title is somewhat misleading: while appealing to empirical science to establish platonism is a Quinean idea, Quine’s argument for platonism involved the notion of regimentation (which I will introduce later in this section), whereas standard presentations of the indispensability argument make no mention of regimentation at all; and a case can be made that Putnam did not seek to establish the existence of abstract objects (see Reference LigginsLiggins 2008). The leading contemporary defender of the indispensability argument is Mark Colyvan (see Reference ColyvanColyvan 2001, Reference Colyvan2010).

The indispensability argument is an elaboration of the basic structure. The indispensability argument adds to the basic structure a reason for believing the instances of (N1): we should believe them because of their role in empirically confirmed scientific theories.

There are other possible justifications for the instances of (N1). For example, we might appeal not to empirical science but to mathematics itself. According to Reference MaddyMaddy (1997: 184), ‘mathematics is not answerable to any extra-mathematical tribunal and not in need of any justification beyond proof and the axiomatic method’. This view implies that we should believe claims such as (1) because mathematics tells us they are true.

Appeals to empirical science, or to formal sciences such as mathematics and mathematical logic, are called ‘naturalism’. Naturalism comes in various different strengths. ‘Strong’ naturalism says that empirical science (or mathematics) gives philosophers an indefeasible reason to believe claims such as (1): any defeater must come from within empirical science (or mathematics), not philosophy. ‘Weak’ naturalism says that it gives us a reason to believe claims such as (1), but that these reasons might be overcome by philosophical argument. To make a case for platonism, we only need weak naturalism.

In setting out the indispensability argument, I deliberately left it unclear what sort of mathematical claims are involved. Does the argument concern pure mathematical claims, such as ‘Five is prime’, or applied ones, such as ‘The mass of the Earth is 5.9736 × 10²⁴ kg’?

In some ways, it is simpler to take the indispensability argument to concern applied mathematical claims, because these are clearly part of scientific theories. Finding the value of some quantity – the mass of the Earth, or the charge of the electron, say – can be a major scientific achievement in itself. However, this version of the argument raises the issue of how to interpret such claims. It is highly plausible that ‘Five is a prime number’ implies the existence of a number; it is not so obvious that ‘The mass of the Earth is 5.9736 × 10²⁴ kg’ does. Perhaps it means that there is such a thing as the number 5.9736 × 10²⁴ and that it is identical to the mass of the Earth, measured in kg, in which case it does imply the existence of a number. But perhaps it means instead that the Earth has a particular mass – one that is helpfully picked out by using the numeral ‘5.9736 × 10²⁴’ even if there is no such thing as 5.9736 × 10²⁴. Friends of the indispensability argument need to rule out this latter interpretation.

Perhaps they are on stronger ground if they focus not on ascriptions of particular quantities, but on laws connecting them. For example, Newton’s Second Law says that the force acting on a body is the product of its mass and acceleration:

But if m and a here can be multiplied, then presumably they are numbers. They must be the values of the body’s mass and acceleration on some scales of measurement – unless masses and accelerations are the sorts of things that can be multiplied.

It is more usual to present the indispensability argument as an argument for the truth of some pure mathematical claims. Pure mathematical claims are not parts of scientific theories, at least not in any everyday sense of ‘part’, so we cannot argue that our best scientific theories entail any pure mathematics. However, it’s impossible to test a quantitative scientific theory without making pure mathematical assumptions. (Imagine trying to confirm that F = ma while remaining agnostic about what results from multiplying any two numbers.) Friends of the indispensability argument claim that when a scientific theory receives empirical confirmation, the pure mathematical assumptions we need to make in order to test the theory are confirmed too.

That claim follows from the principle of ‘confirmational holism’: this says that when a scientific theory receives empirical confirmation, all the assumptions we need to make in order to test the theory are also confirmed. For example, suppose I have a theory about the structure of metals. This theory has implications for the density of various metals, so it can be tested by measuring the mass and volume of some metal samples, calculating their density, and comparing it with the theory’s predictions. To test the theory, I need to make several mundane assumptions: for example, I have to assume that the samples really are samples of the metals in question, and I have to assume that the methods I use to measure their mass and volume are reliable. To calculate the density of each sample, I need to divide its mass by its volume, so I need to make mathematical assumptions too. (For example, if the mass is 2 kg and the volume is 0.1 m³, then I have to assume that 2 divided by 0.1 is 20.) Confirmational holism says that if the theory is confirmed, all of these assumptions are also confirmed, including the mathematical ones.

In this way, friends of the indispensability argument call on confirmational holism and naturalism to argue for mathematical claims. They then argue that these claims imply the existence of abstract objects. Their arguments are instances of the basic structure, elaborated by appeals to holism and naturalism in support of (1).

Let us now turn to the Fregean argument for the existence of numbers. Bob Hale presents the argument as follows:

Hale endorses the Fregean argument. For instance, he would say that ‘Five is a prime number’ is true, and that ‘Five’ functions in that sentence as a singular term: that is to say, the word has the function of picking out a particular thing, the number five. So Hale supports (1) and (2a).

Reference HaleHale (1987: 11) observes that (H3) is not yet platonism: to get from (H3) to platonism, we need to add the premiss that numbers (if they exist) are abstract objects. So he also supports (2b). Thus, Hale supports all three premises of the basic argument, and we can see that he would support many other instances of the basic structure.

The distinctive feature of Hale’s approach is the way he supports premises such as (2a). Along with other Fregeans, Hale develops criteria of singular termhood: ways of showing that a particular expression functions as a singular term (see e.g. Reference Hale, Hale and WrightHale 2001a, Reference Hale, Hale and Wright2001b). These criteria enable Hale to defend (2a).

Fregeans focus on the semantic structure of natural language sentences such as (1). Another option at this point is to argue that the best regimentation of (1) into a formal language involves a singular term for the number five. For example, we might regiment (1) into predicate calculus as the predication ‘Pf’, where ‘P’ is a predicate expressing primeness and ‘f’ a singular term picking out five. We could then argue that this formula is true and requires for its truth the existence of an abstract object. Such an appeal to regimentation is part of Quine’s case for the existence of abstracta (e.g. Reference QuineQuine 1960: 244–5, Reference Quine and Quine1969: 96–100).

The Quinean appeal to regimentation bypasses some of the linguistic complexities that Fregeans have to face, because there is no claim that the regimentation has the same meaning as the sentence it regiments. But that leads us straight to the question ‘What makes for a good regimentation?’. (For discussion, see Reference Sennet, Fisher, Harman and LeporeSennet and Fisher 2014.) As used here, regimentation is meant to have epistemic force, in the sense that if we regiment a sentence we take to be true, then we have some reason to believe the regimented version. (We should expect an account of regimentation to imply that it has such epistemic force.) If regimentation is understood in this way, then the Quinean accepts instances of the basic structure, and bolsters the arguments with a rationale for premises such as (2a).

As well as defending (2a) using criteria of singular termhood, Fregeans offer a priori philosophical arguments for (1). Their starting point is ‘Hume’s principle’:

There is a one-to-one correspondence between the Fs and the Gs if and only if the number of Fs is identical to the number of Gs.

For example, let the Fs be some electric lamps and let the Gs be the bulbs in these lamps. Suppose that there is a single bulb in each lamp. That is enough for there to be a one-to-one correspondence between these things. And so Hume’s Principle allows us to conclude:

That is a sentence that seems to imply the existence of numbers, and so can play the role of (1).

We have seen that some of the most important arguments for the existence of numbers are all elaborations of the same basic structure. This reveals some new ways of arguing for the existence of numbers. For example, one could support (1) by appeal to the role of mathematics in empirical science, and also develop criteria of singular termhood to defend (2a). That would represent a hybrid of Quinean and Fregean approaches. Or one might agree with Maddy that ‘mathematics is not answerable to any extra-mathematical tribunal’, thereby supporting (1) – but then regiment (1) to support (2a).

3.2 Propositions and Other Abstract Objects

Let me now introduce the standard argument for the existence of propositions. Briefly, the idea is that we need to admit the existence of propositions in order to explain the validity of some clearly valid arguments, such as these:Bo believes that snow is white.Mo asserts that snow is white.Therefore, there is something that Bo believes and Mo asserts.Mo believes that snow is white.Jo asserts everything Mo believes.Therefore, Jo asserts that snow is white.

According to the standard argument, we should interpret ‘Bo believes that snow is white’ as saying that Bo stands in the believing relation to the proposition that snow is white; and more generally, we should interpret ‘X Vs that p’ as saying that X stands in the V-ing relation to the proposition that p. Thus these sentences are dyadic predications. This allows us to explain the validity of the arguments by formalising them in predicate calculus as valid deductions:

Precisely how does this provide an argument for platonism? The idea is this: to explain the validity of these arguments, we should take statements of the form ‘X Vs that p’ to imply the existence of propositions; since there are true statements of that form, there are propositions; and since propositions, if they exist, are abstract objects, we reach platonism. In other words, the argument has this basic form:

Premiss (P2a) is supported by the observation that it helps us to explain the validity of arguments such as the examples above.

This is an example of the more general pattern of argument:

In Section 3.1, we saw that leading arguments for the existence of numbers are based on instances of this pattern. It has also been used to argue for the existence of abstract objects other than numbers or propositions. For example, we can use it to argue for the existence of properties, understood as abstract objects:

(This is essentially what Reference EdwardsEdwards 2014: 4–6 calls ‘the reference argument’.) It is easy to see how arguments for the existence of other abstract objects – games, languages, concepts, theories, recipes, and so on – can be seen as instances of this basic pattern. Just replace ‘X’ with a sentence that (i) seems to imply the existence of at least one of the abstract objects in question, and (ii) seems to be true.

As before, the premises of such arguments can be supported in various ways. We can recruit science to support the first premiss, thereby creating an indispensability argument. For instance, we can base an argument for the existence of properties on evolutionary biology:

(Compare Reference SoberSober 1981: Section IV.) And we can argue that psychological claims such as (P1) are part of our best scientific explanations of human behaviour (Reference FodorFodor 1987).

Echoing the Fregean argument for the existence of numbers, we can argue that the word ‘patience’ is a singular term, thereby supporting (Q2a); or we can argue that the phrase ‘that snow is white’ is a singular term, thereby supporting (P2a).

A priori philosophical arguments for (1) have been offered in the case of propositions. The idea is that unless some claims such as (P1) are true, ‘the concepts of rational acceptability, of assertion, of cognitive error, even of truth and falsity are called into question’, so unless we uphold some ascriptions of beliefs (and other mental states), we will commit ‘cognitive suicide’ (Reference BakerBaker 1987: 134, 148).

Presented in the usual ways, arguments for platonism look very different from each other. For example, the indispensability argument for the existence of numbers, the Fregean argument for the existence of numbers, and the standard argument for the existence of propositions bear little resemblance. These differences conceal underlying similarities. We have seen that some of the most important arguments for platonism have the same basic structure, given by this pattern of argument:

They differ in the choices of X and F, and in how the premises are supported.

Although I have identified a pattern of deductive argument at the heart of the abstract objects debate, I am not saying that the debate proceeds only by deduction. Reasons offered for and against premises of these arguments are typically abductive. That is what we should expect, if contemporary metaphysics (and perhaps philosophy more generally) is fundamentally abductive, not deductive (Reference Nolan, Cappelen, Gendler and HawthorneNolan 2016; Reference WilliamsonWilliamson 2016).

Seeing these arguments as elaborations of the basic structure just given is illuminating, not just because it sheds light on how these arguments work, but because it helps us think about how to reply to them. We can classify responses to these arguments by identifying which part of the basic structure they contest. That enables us to think about these responses in a more general way – or, if you’ll excuse the pun, in a more abstract way. That is more illuminating than looking at responses to each argument one by one, because it connects together debates about different types of abstract objects. It also allows us to think about arguments not yet represented in the literature.

3.3 Responses to the Arguments: Concretism

Say that one ‘resists’ a premiss if one either denies it or refuses to endorse it. When presented with an argument based on the basic structure, one might resist the first premiss. Call this the ‘error-theoretic’ response. One might resist (2a): the ‘paraphrase’ response. Or one might resist (2b): the ‘concretist’ response. As usual, if you resist a premiss that your opponent argues for, you had better have something to say about their argument.

A nominalist is not obliged to reply to every argument for platonism in the same way. For example, they may give a paraphrase response to the standard argument for the existence of propositions but an error-theoretic response to the indispensability argument. Or a nominalist might divide up arguments for the existence of properties, taking a concretist response to some and an error-theoretic approach to others. (In fact, that is what Armstrong does, though in this section I will focus purely on the concretist part of his response.)

Let us examine the three types of response one by one, starting with the concretist response.

Concretism does not deny the existence of the objects in question. Rather, the idea is to dispute the claim that they are abstract. For example, one might concede that properties exist, but claim that they are concrete – or at least that it is unclear that they are abstract.

Armstrong’s ‘Aristotelian realism’ about properties is an example of this strategy. According to Armstrong, each property is causally active, and it is located wherever something has it (Reference EdwardsEdwards 2014: 28–46 is a helpful guide to Armstrong’s thinking on these matters). The property of being an electron is to be found just where the electrons are.

One might also accept the existence of numbers but argue that they are concrete objects. There is little plausibility to the thought that familiar concrete objects such as tables and chairs are numbers, which is probably why it receives almost no discussion. (If my chair is the number three, then why don’t mathematicians show any interest in it (see Reference HartHart 1991: 91)? And is it really within my power to destroy the number three?)

If we assume that relations are concrete, a more attractive form of concretism about numbers is open to us, one that sees numbers as relations; thus the number three could be identified with the relation that every mereological sum of three Fs bears to the property of being F. However, this view runs into difficulties with large infinite numbers, unless we are willing to countenance uninstantiated relations (I refer the reader to Reference ArmstrongArmstrong 2004: 116–7 for the details).

The concretist strategy returns us to the arguments above (Section 2.1) about which objects should be classed as abstract. When we consider a particular object, we may well have intuitions about whether it is spatially located and whether it is causally active. For instance, numbers seem not to be spatially located or causally active; in that sense, they seem to be abstract.

Such intuitions are not sacrosanct: they can be overturned if doing so brings sufficient theoretical benefit. Perhaps regarding numbers as causally active does enough theoretical work to make up for the strangeness of the belief. But where (2b) seems to be true, it is not enough for the opponent of the argument to object that there might be a sufficient case for overturning it. One can object to any premiss of any argument merely by pointing to the epistemic possibility of a refutation of it. To cause real trouble for the argument, the opponent needs to make a good case against (2b). In other words, they need to specify what would be gained by thinking of the objects in question as concrete. The gain might be ontological simplicity (see Section 4.1). Or it might be epistemological solvency (see Section 4.2).

Where (2b) is not intuitively obvious, the argument faces a more immediate challenge. To return to an example from Section 2.1: it is just not clear whether {Edinburgh} is spatially located. If it is spatially located, it is not abstract. In a case like this, the argument has no chance of working until the platonist gives a positive case for (2b). Without such a case, their opponent is entitled to reply that we have no reason to believe this premiss of the argument.

Whether or not (2b) is intuitive, an in-depth discussion of this premiss is going to go beyond intuitions, and examine the theoretical costs and benefits of incorporating it into one’s theory. One aspect of this is how well the claims made fit in with our best theories in other areas. Particularly relevant here are our general theories about what is spatially located and about what sorts of things take part in causal interactions.

3.4 Responses to the Arguments: Paraphrase

The paraphrase response is to resist (2a). The issue is what the sentences that play the role of (1) materially imply. Paraphrasers deny – or at least refuse to endorse – the claim that these sentences materially imply the existence of the objects in question. The ‘refuse to endorse’ version of the paraphrase response has not been popular, so I will focus entirely on the ‘deny’ version.

Let us return to this instance of the basic argument for the existence of numbers:

The paraphraser denies (2a). In their view, (1) does not materially imply the existence of numbers.

The obvious response is to argue that (2a) is true because (1) means what it does. This response is very plausible. ‘Five is a prime number but there are no numbers’ sounds self-contradictory. Reference Burgess and RosenJohn Burgess (1982: 7) compares Benacerraf’s claim that ‘if the truth be known, there are no such things as numbers; which is not to say that there are not at least two prime numbers between 15 and 20’ (Reference BenacerrafBenacerraf 1965: 73) to a statement attributed to George Santayana: ‘God does not exist, and Mary is His mother’. So resisting (2a) is likely to involve resisting a claim about the meaning of (1).

For example, recall the Fregean case for platonism about numbers. Fregeans not only claim that (2a) is true: they also claim that its truth flows from the meaning of (1). To back up their claims about what mathematical sentences such as (1) imply, Fregeans argue that these sentences contain singular terms for numbers. For instance, they will aim to show that the first word of (1) is a singular term for a number. This is the point of developing theories about which expressions are singular terms, and making arguments to support them. Those who resist (2a) of the argument for numbers should have something to say about these Fregean arguments.

When they deny (2a), paraphrasers invite the following response: ‘I thought that (1) means that there’s an object, five, which is a prime number; and so (1) implies the existence of a number. But you deny that (1) implies the existence of a number, so you must think that (1) means something else. So what does it really mean, in your view?’.

When asked to clarify the meaning of a sentence, we commonly offer a paraphrase of it: another sentence that we take to mean the same as the original. (‘What does “C’est la vie” mean?’ ‘It means “That’s life”’.) This is the natural way to explain what (1) means, which is why this strategy for replying to the basic argument is called the ‘paraphrase’ response. Paraphrases are sometimes also known as ‘translations’, even if the original sentence and the paraphrase belong to the same language. The sort of meaning in question here is literal meaning. (Sometimes philosophers call regimentations ‘paraphrases’. But this usage is potentially confusing, because a regimentation is not expected to mean the same as the original sentence (Section 3.1). I will avoid it.)

For example, the paraphraser may say: ‘We misread (1) if we take it to imply the existence of a number. What it really means is the following:

And (1*) can be true even if there are no numbers’.

That claim is an example of the paraphrase response known as ‘modal structuralism’. The modal structuralist interprets sentences apparently about numbers as claims about what logically follows from having the natural number structure – that is, from having the structure characterised by the axioms of arithmetic. Modal structuralism is the most important paraphrase position in the philosophy of mathematics (Reference Burgess and RosenBurgess and Rosen 1997 include a wide-ranging survey of paraphrase positions). It is usually attributed to Reference HellmanHellman (1989) – though that attribution is actually wrong, as we will see later in this section.

Paraphrase responses are not confined to the philosophy of mathematics. Recall the argument for the existence of properties:

One response – a currently very unpopular one – is to deny (Q2a) and claim that (Q1) simply means ‘Patient people are virtuous’. So understood, (Q1) does not imply the existence of a property.

Recall also the argument for the existence of propositions:

One response is to claim that belief ascriptions express relations not to propositions but to sentences. This view is, naturally enough, called ‘sententialism’ (see e.g. Reference FelappiFelappi 2014). For the sententialist, (P1) really means ‘Justin believes “Snow is white”’, so (P2a) is false: (P1) really implies the existence of a sentence, not a proposition.

When a paraphrase response is under consideration, it is common to mention a classic paper by Reference AlstonAlston (1958). This is generally taken to contain a serious objection to the paraphrase response. But that is to misunderstand Alston’s article, as I will now explain, drawing on the excellent discussion of Alston’s article by Reference KellerKeller (2017).

The key passage of Alston’s paper reads as follows:

Here are several philosophically interesting translations … :

1. 1. There is a possibility that James will come.

2. 2. The statement that James will come is not certainly false.

3. 3. There is a meaning which can be given to his remarks.

4. 4. His remarks can be understood in a certain way.

5. 5. There are many virtues which he lacks.

6. 6. He might conceivably be much more virtuous than he is.

7. 7. There are facts which render your position untenable.

8. 8. Your position is untenable in the light of the evidence.

Alston points out that the relation of paraphrase is symmetrical: if sentence A means the same as sentence B, then sentence B means the same as sentence A. So if A implies the existence of Fs, then B implies the existence of Fs too. If only one of the sentences implied the existence of Fs, that would show that they did not mean the same thing after all. So either they both imply the existence of Fs, or neither of them does.

None of that is evidence against any paraphrase position. The paraphraser should claim that neither sentence implies the existence of the objects whose existence is being debated. For example, the modal structuralist should claim that neither (1) nor (1*) implies the existence of numbers. In this way, paraphrasers are perfectly capable of respecting the symmetry of the paraphrase relation.

Alston’s paper poses no objection to paraphrase positions. Rather, it points out that showing that A and B are paraphrases of each other does not establish that these sentences fail to imply the existence of the objects in question. It shows that the sentences have the same ontological implications as each other, but it leaves the question of what those implications are wide open. So merely establishing that two sentences are paraphrases of each other is not enough to vindicate a paraphrase position. Showing, for instance, that (1) and (1*) mean the same is not enough to vindicate modal structuralism, for (1) and (1*) might mean the same and both imply the existence of numbers.

Additional argument is needed to show that neither sentence implies the existence of the objects in question – for instance, an argument against the existence of Fs, together with an argument that we should respect the apparent truth-values of the disputed sentences. That is why ‘the point of the translation cannot be put in terms of some assertion or commitment from which it saves us’.

Alston’s objection, then, is not an objection to paraphrase positions, but to a type of argument one might offer in their favour. Perhaps that type of argument was popular at the time when Alston was writing. In places, Quine might seem to encourage this way of arguing. For example:

But Quine is not thinking of paraphrase positions: the paraphraser appeals to the notion of sameness of meaning – a notion which is not in good standing, according to Quine. Rather, Quine is thinking of regimentation.

Few if any contemporary paraphrasers support their view by giving arguments of the sort Alston criticised. So the value of Alston’s paper is to warn us against a tempting but inconclusive line of argument. It should do nothing to lower our credence in any paraphrase position.

By the way, nothing turns on what sorts of entities Fs are meant to be. Some philosophers have tried to ‘paraphrase away’ apparent reference to concrete objects such as tables, saying that ‘There is a table’ really means ‘There are some things arranged tablewise’. Alston’s paper is no threat to such positions either.

There are, however, many challenges facing the paraphraser.

It is not enough to paraphrase a few sentences that seem to imply the existence of the abstract objects in question. We want a scheme of paraphrase – that is, a recipe for generating a paraphrase of any sentence of the discourse that might play the role of (1).

A classic argument in this area illustrates the point. Suppose that ‘Red is a colour’ is paraphrased as ‘Everything red is coloured’. Then presumably we must paraphrase ‘Triangularity is a colour’ as ‘Everything triangular is coloured’. But that must be wrong, since even if everything triangular happened to be coloured, triangularity would still not be a colour. It is not clear to whom this argument should be attributed. Reference JacksonJackson (1977: 427) refers to it as a ‘standard objection’ and cites as an example an earlier statement by A. N. Prior. Its importance here is that it involves considering a scheme of paraphrase suggested by a paraphrase of a particular sentence. One might try to defend the original paraphrase of ‘Red is a colour’ as ‘Everything red is coloured’ by extending this to a different scheme than the one envisaged in the objection. And so the debate becomes one about schemes of paraphrase, not individual examples.

But giving a sentence-by-sentence scheme of paraphrase is not enough. As Reference BenacerrafBenacerraf (1973: 666–7) points out, we need to interpret mathematical language in a way which coheres with our interpretation of non-mathematical language. One of our goals (I assume) is to find a compositional semantic theory for English, and for other natural languages: a specification of sentences’ meanings based on the contributions made by their component words and phrases. From this perspective, the paraphraser owes us a compositional semantic theory for the sentences of the discourse which yields their scheme of paraphrase as a consequence. (See Reference Dever, Lepore and SmithDever 2008 for more on compositionality and the motivation for it.)

For an illustration of the difficulty of this, consider sentences which mix mathematical and non-mathematical language, such as:

(Some would brand these sentences as unintelligible on the ground that they involve ‘category mistakes’. See Reference MagidorMagidor 2013 for counter-arguments.). I take it that ‘large’ is clearly a predicate (if you think it is not, swap it for your favourite example of a one-place predicate). It seems that ‘large’ and ‘prime’ play a similar semantic role in (5); we could swap them around without loss of grammaticality. The theorist who denies that ‘prime’ is a predicate in (1) must therefore claim that the word functions differently in (1) and (5). But this will create trouble when the theorist comes to interpret (6). How should ‘prime’ be interpreted here? The same way as in (5) – in which case, it functions as a predicate? Or the same way as in (1) – in which case, it does not function as a predicate? Whichever option is chosen, it seems hard for to account for the logical relations between (6) and other sentences. For instance, (6) is entailed by

so presumably we should interpret ‘is prime’ the same way in (6) and (9). But (6) is also entailed by

so presumably we should interpret ‘is prime’ the same way in (6) and (10). But that would require us to interpret the expression in the same way in (9) and (10), which is impossible if the expression is a predicate in (10) but something else in (9).

Sentences (7) and (8) raise similar problems. If ‘There are tables’ is an existential quantification, as it seems to be, but ‘There are prime numbers’ is not an existential quantification, then how are we to interpret (7)? What reading should we give to ‘There are’ as it appears in (7)? If ‘Edinburgh’ is a name but ‘forty-two’ is not, then how are we to read ‘loves’ in (8), bearing in mind that ‘Bo loves Edinburgh’ and ‘Bo loves forty-two’ collectively entail (8)?

In short, philosophers of mathematics who adopt an alternative to the default interpretation have a difficult task integrating their interpretation of mathematical sentences with their interpretation of the rest of the language. (Note that proponents of the default interpretation avoid these problems by taking mathematical language at face value.) For all I have said, perhaps a sufficiently ingenious philosopher can successfully complete the task: but the result will be a complicated semantics.

Paraphrasers often claim not just that the abstract objects in question do not exist, but that there are no abstract objects whatsoever. For example, paraphrasers in the philosophy of mathematics are often nominalists across the board. These philosophers face an additional challenge: they must show that their paraphrases are consistent with nominalism.

Modal structuralism in the philosophy of mathematics provides a good example. For the modal structuralist, true arithmetical sentences such as ‘Five is a prime number’ express truths about the logical consequences of having a certain structure. This raises the question: what is the metaphysics of logical consequence? Many philosophers believe that logical consequence is best understood in model-theoretic terms: for B to be a logical consequence of A is for B to come out as true on every model on which A comes out as true (Reference Tarski and CorcoranTarski 1983). This approach to logical consequence presupposes the existence of models. But models are sets. These sets are different to the example of {Edinburgh} discussed in Section 2.1, in that they do not contain physical objects at any level. They are what are called ‘pure sets’. These are presumably abstract mathematical objects. So the modal structuralist who wishes to be a nominalist therefore needs either to explain how model theory does not violate nominalism, or provide an alternative nominalist account of logical consequence. They are more likely to take the latter option.

The importance of these problems should not be overstated. Every nominalist who wants to use the notion of logical consequence in their theorising needs to provide an account of logical consequence, whether or not they are a paraphraser. We will re-encounter this point in connection with Field’s nominalism, which is an alternative to paraphrase nominalism (Section 3.5).

Sententialists face a similar problem. For the sententialist, belief ascriptions express relations to sentences. Some beliefs have contents that have never been tokened, so the sententialist ought to say that ascriptions of these beliefs express relations to sentence types, not sentence tokens. But sentence types, like other types, seem to be abstract objects (see Section 2.3).

The main problem for the paraphraser is making their claims about the meanings of the sentences in question plausible. Often, the sentences in question simply do not seem to mean what the paraphraser says they do. For example, ‘Patience is a virtue’ and ‘Patient people are virtuous’ seem to mean different things. (See also Reference FieldField 1989: 113–5.)

Reference HellmanHellman’s (1989) Mathematics without Numbers is usually regarded as the classic expression of modal structuralism. However, Hellman has claimed that paraphrase positions are

Hellman goes on to explain that in his Reference Hellman1989 he was offering a ‘rational reconstruction’ of mathematics rather than an interpretation of mathematical language (Reference Hellman1998: 342). So Mathematics without Numbers was not intended to present a paraphrase position. At the time he was writing it, Hellman rejected modal structuralism because he found its claims about linguistic meaning implausible. (More recent work, written jointly with Stewart Shapiro, indicates that Reference PettigrewPettigrew 2008 has persuaded Hellman to change his mind on this; see Reference Hellman and ShapiroHellman and Shapiro 2019: 67–8.)

Defenders of paraphrase will probably say that these costs are worth paying. One potential benefit is especially worth mentioning. Where concretism is not a serious contender, the only options will be the paraphrase response or the error-theoretic response. In contrast to error theories, paraphrase views enable us to uphold the truth of the sentences that play the role of (1). Paraphrasers tend to regard that as one of the main benefits of their approach. In the next section, I will argue that error theories are worthy of serious attention and cannot be easily dismissed.

Another objection to paraphrasers’ claims about linguistic meaning is that they violate an important methodological principle. On whether paraphrasers in the philosophy of mathematics are correct in their claims about the meaning of mathematical sentences, Burgess and Rosen write:

In other words, paraphrasers are indulging in armchair linguistics: they should leave questions about the meaning of mathematical language to the experts. No doubt this argument could be extended to paraphrase responses in other domains. It threatens to show that philosophers are unjustified in believing paraphrasers’ semantic claims.

Burgess and Rosen’s position is arguably too extreme. One can acknowledge that work in linguistics is relevant to philosophical debates about the existence of abstract objects while denying that linguistics on its own can settle the question of what (for instance) mathematical sentences mean. Philosophers take into account pieces of evidence that play no role in linguistics: for example, the arguments for and against the existence of abstract objects. It would be irresponsible to ignore such evidence when deciding what sentences mean, because that would be to ignore some of the available evidence. (See Reference Daly and LigginsDaly and Liggins 2011 for further discussion of philosophical deference to linguistics and other disciplines.)

However, Burgess and Rosen’s comments on paraphrase do highlight a striking feature of contemporary debate over the existence of abstract objects: a lack of contact with linguistics (Reference van Elswyk, Tillman and Murrayvan Elswyk 2022 is a refreshing exception). Closer engagement promises to be fruitful, so this is an avenue for future work.

(On paraphrase responses, see Reference von Solodkoffvon Soldokoff 2014 and Reference Szabó, Loux and ZimmermanSzabó 2003: section 3.)

3.5 Responses to the Arguments: Error Theory

4.1 Ontological Motivations for Nominalism

Simplicity is also known as ‘parsimony’ or ‘economy’, and it comes in several varieties. The sort that is relevant here is ontological simplicity. It seems that the nominalist has a simpler ontology than the platonist, because the platonist posits abstract objects whereas the nominalist does not.

But simplicity is more complicated than that. Reference LewisLewis (1973: 87) distinguishes different sorts of ontological simplicity. A theory is ‘quantitatively’ simple if it posits fewer entities; it is ‘qualitatively’ simple if it posits fewer types of entities.

It is impossible to judge the quantitative simplicity of a theory without knowing its details, but it is fair to say nominalist theories tend to be quantitatively simple compared with platonist ones. Most platonist accounts of properties, propositions, and mathematical entities posit infinitely many abstract objects. Mathematics teaches us that some infinite numbers are greater than others, but set theory posits so many objects that mathematics lacks a number to count them.

However, it is controversial whether quantitative simplicity is a theoretical virtue. Reference LewisLewis (1973: 87) denied that it is (see Reference NolanNolan 1997 for discussion). Some would also deny that qualitative simplicity is a virtue (see e.g. Reference HuemerHuemer 2009), but, like many metaphysicians, I will assume that it is, and examine the significance of qualitative simplicity to the abstract objects debate.

Let ‘global’ nominalism be the claim that there are no abstract objects whatsoever. And let ‘nominalism about Fs’ be the claim that there are no abstract objects that are Fs. For example, a nominalist about properties claims that either there are no properties or they are concrete objects. A nominalist about properties need not be a global nominalist: they might think there are some abstract objects that are not properties (propositions, perhaps).

Nominalism about a specific kind of abstract object that falls short of global nominalism is hard to defend by appeal to simplicity. This nominalist does posit abstract objects of one sort, so it is hard to see their theory as qualitatively simple compared with other platonist theories. Moreover, such a position faces challenges over its motivation. For instance, consider the view that there are no abstract mathematical objects, but there are propositions, and these are abstract objects. Anyone who holds this view faces the challenge: why do your reasons for avoiding abstract mathematical objects not carry over to propositions? If that avoidance is motivated by epistemological considerations (see Section 4.2) then why do those considerations not apply to propositions as well? It may be possible to overcome such challenges, but they certainly give the non-global nominalist more work to do.

Let us turn now to global nominalism. It seems that any form of global nominalism is bound to be qualitatively simpler than a theory that posits both abstract and concrete objects: the latter posits objects of both those types, the former only posits concrete objects. An under-discussed option for platonists here is to deny the existence of concrete objects, and say that abstract objects give rise to the perceptual experiences that convince us concrete objects exist (see Reference Szabó, Loux and ZimmermanSzabó 2003: 29).

So far I have assumed that when we assess qualitative simplicity, it is appropriate to take ‘abstract object’ and ‘concrete object’ as kinds. Actually, that is not so clear. There is a wider methodological problem concerning what to count as a kind when judging the qualitative simplicity of a theory. At one extreme, for each object o posited by the theory, we could take ‘thing that is identical to o’ as a kind: then there would be as many kinds as objects posited. At the other extreme, we could say that the only thing to count as a kind is ‘self-identical object’: then each theory that posits at least one thing would posit just one kind. In between these extremes there are many different alternatives. Our evaluation of a theory depends on what we count as a kind, but which alternative should we pick, and why? (See Reference OliverOliver 1996: 7. Reference LewisLewis 1973: 87 speaks of ‘fundamentally different kinds of entity’, which may help, but does not completely resolve the problem.)

Rather than pressing this problem, platonists are more likely to point out that appeals to simplicity are inconclusive. Nominalist theories may have the virtue of simplicity, but that is just one virtue among many: their simplicity and other virtues may well be outweighed by their theoretical vices. A well-known formulation of Occam’s Razor states that ‘Entities are not to be multiplied without necessity’. The last two words are important here: although a shortage of simplicity may well count against platonist theories, we cannot establish nominalism merely on the basis of simplicity. Rather, we have to compare the overall theoretical benefit of positing abstract objects with the overall theoretical benefit of not. So the appeal to simplicity advances debate by issuing a challenge to platonists. Whether it motivates nominalism depends on whether platonists can meet the challenge. Abstract objects might be complications we cannot do without.

Because explanatory power is an important theoretical virtue, one way for platonists to meet the challenge is to show that we need abstract objects to explain the phenomena that need explaining. So considerations of simplicity lead us back to indispensability arguments (see Sections 3.1 and 3.2).

4.2 Epistemological Motivations for Nominalism

Epistemological motivations for nominalism are developed in most depth in the debate over the existence of abstract mathematical objects, so I will focus on that debate. Much of it carries over straightforwardly to nominalism about other sorts of abstract objects. (On epistemic challenges to belief in properties, see Reference SwoyerSwoyer 1996.)

We saw in Section 3.5 that Reference BenacerrafBenacerraf (1973) posed a dilemma for all philosophers of mathematics. Part of this was to argue that if we take mathematical claims to be accurate descriptions of abstract mathematical objects, then we will be unable to explain how people acquire mathematical knowledge. For example, ‘2+2=4’ will be interpreted as a claim about two abstract objects, the number two and the number four – but because these objects are abstract, we are unable to know how they are related, so the platonist is at a loss to explain how we come to know that 2+2=4.

Benacerraf’s argument assumes that the platonist will not want to deny that people have such knowledge. This assumption is reasonable. Since it forms part of a dilemma, Benacerraf does not endorse this argument: he simply adds it to the discussion. It potentially generalises to other platonist theories. For example, many want to say that we know that the proposition Fido barks entails the proposition something barks: but if propositions are abstract objects, that makes it impossible to explain how we gain this knowledge.

Benacerraf’s argument uses a non-obvious epistemological assumption: it relies on the claim that we know about objects only if we are causally related to them. Reference BenacerrafBenacerraf (1973: 671) justifies it by appealing to causal theories of knowledge, which were popular at the time.

According to Reference LewisLewis (1986: 109), Benacerraf’s whole strategy is flawed, because there is no way of using an epistemological assumption to refute platonism: ‘Our knowledge of mathematics … is ever so much more secure than our knowledge of the epistemology that seeks to cast doubt on mathematics. … Causal accounts of knowledge are all very well in their place, but if they are put forward as general theories, then mathematics refutes them’. Perhaps Lewis is alluding here to the fact that causal theories were not always intended as general theories (for instance, Reference GoldmanGoldman (1967: 357) says he intends to account only for knowledge of ‘empirical propositions’). Lewis’s argument presupposes that we begin philosophical reflection knowing some truths about abstract objects. But in Section 2.3 I cast doubt on that assumption. If nominalists have the burden of proof, then perhaps Lewis is right; but if they do not, it is hard to see how our knowledge of mathematics can have the force Lewis takes it to have.

Causal theories of knowledge have not worn well. Contemporary epistemologists reject them for reasons that have nothing to do with abstract objects (see Swain 1998). So Benacerraf’s argument has few, if any, adherents nowadays. Yet the sense remains that abstract objects raise an epistemological problem.

Reference FieldField (1989: 25–30, 68, 230–9; Reference Field2016: section 5) offers a different way of articulating this problem. Unlike Benacerraf’s argument, Field’s does not rely on any assumptions about necessary conditions on knowledge (Reference FieldField 1989: 232–3). It is epistemological only in the broad sense of concerning true belief.

Field’s argument centres on the mathematical beliefs held by mathematicians. As an error theorist, Field thinks that these are largely untrue, because the objects that are required for their truth do not exist. He points out that platonists will accept that most of the mathematical beliefs held by mathematicians are true; they will think that, although mathematicians make the occasional mathematical mistake, their mathematical beliefs are largely accurate. This phenomenon would be ‘so striking as to demand explanation’ (Reference Field1989: 26); the problem for the platonist is about how they can explain it.

There are two strategies open to them, Field argues. They can give a causal explanation – but the acausal nature of the entities they posit rules this out. Or they can give a non-causal explanation – but for Field

So it seems that the platonist can give no explanation of mathematicians’ mathematical accuracy. In other cases of non-accidentally true belief, it is possible to give an explanation: for instance, we do not know all the details, but we have the makings of a scientific account of how our perceptual faculties provide us with true beliefs about our environment. Field’s point is that it is hard to imagine what shape a convincing platonist explanation of mathematicians’ mathematical accuracy would take.

Contemporary mathematical theories have an axiomatic structure: mathematicians’ mathematical beliefs are deduced from the axioms of the area of mathematics in question. Since true axioms only have truths as logical consequences, explaining why mathematicians’ beliefs in their axioms tend to be true would go a long way towards explaining mathematicians’ mathematical accuracy, as Reference FieldField (1989: 231–2) acknowledges. But it is hard to see what form a satisfactory explanation could take – or so Field claims.

At this point, exposition of Field is complicated by the fact that that different groups of philosophers interpret his argument in different ways. Reference Sjölin WirlingSjölin Wirling (forthcoming) calls the two camps ‘Team Explanatory Power’ and ‘Team Undercutting Defeat’. The dispute is about what we should conclude from being unable to envisage a satisfactory platonist explanation of mathematicians’ mathematical accuracy.

According to Team Explanatory Power, the significance of this is that we should lower our credence in platonism. Platonist accounts of mathematics fail to explain a phenomenon they ought to explain, and that counts against them. Team Undercutting Defeat, on the other hand, take the significance to be that it removes the justification for mathematicians’ mathematical beliefs (see e.g. Reference Clarke-Doane and PatautClarke-Doane 2017).

Here, I will avoid the debate over what Field really intended, and focus on my preferred interpretation, that of Team Explanatory Power, because I think this argument offers a more promising case for nominalism.

I say ‘promising’ rather than ‘successful’, because it is important to appreciate the dialectical status of Field’s contribution. If we treat it as an attempt to refute platonism about mathematical objects, then I think it has little prospect of succeeding. The weak spot is the passage quoted above: ‘it is very hard to see what this supposed non-causal explanation could be’. Here Field suggests that it will not be possible for the platonist to give a convincing non-causal explanation of the phenomenon in question, but he does not provide a strong reason for thinking so, merely that it doesn’t seem to him that this is possible. Considered as an objection to platonism, this is flimsy (see Reference LigginsLiggins 2010a: 74).

When he introduces his argument, Reference FieldField (1989: 25) uses the word ‘challenge’, and that seems a better way of understanding its significance. At the moment, platonist theories do not explain why mathematicians’ mathematical beliefs are accurate. That counts against them. But we cannot rule out that a sufficiently ingenious platonist will be able to explain the phenomenon.

Field’s argument involves the idea that it is a theoretical cost to leave some phenomena unexplained – they ‘call for explanation’. This methodologically important idea is poorly understood, as Dan Baras has emphasised; his Reference Baras2022 is a book-length discussion of the notion. (The book discusses Field’s argument briefly (170–1), though it seems to me that this discussion does not focus on the argument’s core.) Better theories of ‘calling for explanation’ can only help us understand Field’s argument more deeply. But the absence of such theories does nothing to diminish its force. (Parallel: failing to find the correct philosophical theory of the nature of justice should not stop us from trying to promote justice. See Reference Nolan, Cappelen, Gendler and HawthorneNolan 2016: 169 for relevant discussion.) Exploring connections between Field’s contribution and epistemological challenges elsewhere in philosophy can also bring illumination (see, for instance, Reference EnochEnoch 2010).

If we accept that Field’s argument does not refute platonism, but simply advances the debate by challenging the platonist to explain a particular phenomenon, then we would expect the responses to Field to consist of discussion of particular platonist proposals. This is not the case: there is a voluminous contemporary literature on Field’s argument, but it mostly circles round the question of what (if anything) the argument shows, and how it relates to the modal epistemic conditions of safety and sensitivity (see Reference TopeyTopey 2021 for an illuminating discussion). I think this is because of the prominence of the Undercutting Defeat interpretation of the argument. A more fruitful way forward is to see Field as Team Explanatory Power propose – as offering a relatively straightforward challenge – and to discuss various ways of trying to meet it.

I suggest that Field’s challenge can be widened along various dimensions. According to platonists, plenty of non-mathematicians form mostly true mathematical beliefs as well, so it is reasonable to ask platonists to explain how these people manage to do that. I see no particular reason for avoiding concepts such as justification and knowledge here, so long as it’s understood that the challenge does not appeal to any theories about justification or knowledge. A satisfying form of mathematical platonism would explain how people manage to form justified beliefs about abstract mathematical objects, and gain mathematical knowledge.

Quite a lot of work in philosophy of mathematics can be seen as a response to these challenges, even though it is not always explicitly framed as such. But it is notable that the literature often does not meet Field’s challenges head on.

The Neo-Fregean philosophy of mathematics developed by Hale and Wright is an example. Here, the goal is to show a way in which, by making appropriate stipulations, we could come to acquire mathematical knowledge (see Reference Wright, Hale and WrightWright 2001: 279–80). It is not clearly explained how to turn that into an account of the source of our actual mathematical knowledge, or even of our actual mathematical true belief.

Another example is work based on the Quinean idea that science is the fundamental source of mathematical knowledge. Quinean philosophy of mathematics tends to focus on the indispensability argument for the existence of abstract objects rather than on filling in the details of an epistemology for them.

Let me put those points in a slightly different way. There is a theoretical virtue which, I suggest, has been neglected in the recent debate over the epistemology of abstract objects. That virtue is strength. Other things being equal, a theory is more worthy of belief the more informative it is: that is, the more it rules out. As Williamson puts it: ‘strength is a strength’ (Reference Williamson and Armour-GarbWilliamson 2017: 337; see also Reference HuberHuber 2008). Once we remember the virtue of strength, we see that the discussion is currently out of balance: the epistemology of abstract objects tends to focus on relatively thinly specified theories. Giving the virtue of strength greater prominence would result in a diminished emphasis on Field’s challenge itself and a greater emphasis on theory construction.

What sort of platonist epistemology might succeed in meeting Field’s challenge? Let us work back from the challenge and see where we get. We want to explain the correlation between mathematicians’ mathematical beliefs and the mathematical facts as the platonist conceives them. To do this, we could say that the beliefs and the facts are correlated because they both depend on some third thing. But it is simpler to say either that the beliefs depend on the facts or the facts depend on the beliefs. Since the latter threatens to compromise the objectivity of mathematics, the more natural step is to explore the former, and seek to explain how the beliefs could depend on the facts. Abstract objects are acausal, so the dependence must be acausal. Working back from Field’s challenge, then, the obvious avenue to explore is whether the mathematical facts non-causally influence mathematical beliefs.

Metaphysicians of abstract objects are lucky: in recent years, the notion of non-causal influence has been studied extensively, often under the name of ‘grounding’. (On grounding, Reference Rosen, Hale and HoffmanRosen 2010 is a good place to start. It is controversial whether the notion of grounding is in good standing: see Reference RavenRaven 2022, Section ‘Skepticism and Anti-Skepticism’.) Elijah Chudnoff’s work on intuition is a good example of how appeal to grounding can advance the debate. Reference ChudnoffChudnoff (2013) uses grounding to build an explanation of how intuitive experiences might make us aware of abstract objects. He argues that, just as our sense experiences can depend on concrete objects in our surroundings, so intuitive experiences can depend on abstract objects. These ideas merit much discussion. And we should also explore in detail other ways of using grounding to respond to Field.

It is natural to assume that non-causal explanations work by locating the explanandum in a network of non-causal dependence, as Chudnoff’s explanation does. But that picture has been challenged: perhaps there are non-causal explanations that work in other ways, for example, Marc Lange’s ‘explanations by constraint’ (Reference LangeLange 2017). Understanding the different types of non-causal explanation can only help platonists trying to meet Field’s challenge: the more sorts of non-causal explanation we can identify, the more types of explanation we might hope to offer of our mathematical accuracy. Each type provides a possible sort of answer for the platonist.

In short: Field’s challenge does not refute platonism, but presents platonists with work to do. Whether they can meet the challenge remains to be seen.