This post assumes prior knowledge of philosophical vocabulary and concepts.



I'm a nominalist about all abstract objects.

This is a necessary feature of my overall metaphysics (elsewhere, I defend a version of existence monism, including in my dissertation).

So I'm motivated to justify a nominalist account of the philosophy of mathematics, too.

Doing so poses some unique and rather interesting challenges.

Nominalism

In general, aside from simply aligning with my preferred system of metaphysics, I reckon there are some good prima facie reasons to want to be a nominalist – to believe abstract objects don't exist.

First, it's ontologically parsimonious – that is to say, we don't have to posit any entities in order to be nominalists. It respects Occam's Razor (which I defended here), which suggests that one should not multiply entities beyond necessity. This means our ontology is leaner and more straightforward.

Second, it avoids epistemic problems around how we come to know about abstract entities. Nominalists don't have to put much work into explaining how we can come to know things about mathematics. Knowing something about maths is as simple as making some kind of discovery via empirical methods.

Third, we might think it aligns better with scientific practice (though there are a number of counterarguments available). For example, it's more compatible with the view that everything that exists is part of the natural, physical world; and is subject to empirical investigation (naturalism). We might also think nominalism is consistent with the idea of mathematics as a human-constructed tool or language used to describe patterns in the world.

Two competing intuitions

The core mission in the philosophy of mathematics is to find an adequate theory of mathematics. But what should that look like?

One of our most substantial challenges is to create a system that captures two competing intuitions.

The first intuition is that we want to be able to treat propositions about numbers in the same way as other propositions. For example, semantically, we want "Two is greater than one" to operate in a similar way to "France is next to Spain". We need a uniform system of semantics for mathematics.

The second is that we need to be able to explain how we come to have knowledge of mathematics. In other words, we need to develop a naturalistic epistemology for mathematics.

Paul Benacerraf explains this challenge in detail in Mathematical Truth – more on this here.

The problem for realists

Typically, people who argue that mathematical objects exist (realists) will have no issue with the challenge of creating a uniform semantics. Mathematical truths are just literal truths about abstract entities. But they'll struggle to explain how we can come to have that knowledge. How would we have access to entities outside of space and time?

The problem for nominalists

Conversely, mathematical nominalists will find the task of explaining how we come to have mathematical knowledge trivial, but struggle to give an account of uniform semantics. How would we go about explaining the seeming objectivity and incredible consistency of mathematical truths?

A method to resolve the tension: Fictionalism

Platonists about mathematical entities take mathematical statements to be true on the basis that there are mathematical entities. For Platonists, there's an objective mathematical reality. It exists independently of human thought or language.

There's a sharp contrast between the view that numbers exist and fictionalism.

Fictionalism is the view that:

Mathematical statements are strictly false, because there are no mathematical entities

Fictionalists broadly argue that mathematical statements can be true in some sense, in the same way that statements such as "Father Christmas delivers presents at Christmas" or "Harry Potter is a wizard" are true in some sense.

It is, if you like, a mathematical error theory. By that, I mean that when we mathematical statements to be true, we do so in error.

If we're fictionalists, we have to bite the bullet and say that the sentence "Two is greater than one" is strictly false.

By doing this, we have a way to resolve the tension we talked about earlier: creating both a uniform semantics, and a naturalistic epistemology.

"Two is greater than one" can be understood semantically in just the same way as "France is next to Spain" (it's just that the former is strictly false), while it's not hard to explain how we can come to have mathematical knowledge, either – we just need familiarity with the fiction.

The cost of fictionalism

How could a mere fiction have such incredible explanatory power in science?

This is the premise of extending the indespensibility argument to mathematical entities. It goes like this:

Premise 1: We ought to have ontological commitment to all of the entities which are indispensible for our best theories of science.

Premise 2: Mathematical entities are indispensible to our best theories of science.

C: We ought to have ontological commitment to mathematical entities.

The fictionalist needs to resolve this. We've got two choices: refute either Premise 1 or Premise 2. I'm going to start with the latter.

Refuting Premise 2

Are mathematical entities really indispensible to our best scientific theories?

Hartry Field has attempted to resolve the problem posted by the indispensability argument. His strategy involves two key components.

First, scientific theories can be reformulated without mathematics. Mathematics, he argues, can be rephrased in purely nominalistic terms, using only concrete objects and entities.

I've not read his original defence of this, which comes in his much-debated Science Without Numbers: A Defence of Nominalism (1980), but essentially he takes on the challenge of reformulating Newtonian gravity. The approach involves moves like replacing mathematical space with physical space, and functions with predicate logic. For example, we have to be substantivalists about spacetime (that's the view that spacetime really exists, or in other words that it is a real substance in and of itself – I discussed substantivalism here), and describe phenomena in terms of physical space – using locations and movements – rather than mathematical space.

I'm not sure exactly how this would work, but as a student I imagined that it could be reduced to language that avoided referencing abstract objects something like this: instead of saying "one plus one is two", we might say "if you have one object and then add another similar object, you have a configuration of objects that is equivalent to what we typically call two objects". This may not be very accurate, but it helped me get an idea of what's going on.

Second, maths is useful, but doesn't contribute anything new. Field argues mathematics does not contribute any new factual content to scientific theories. The fictionalist can then say that the fact that mathamtical propositions are mere fictions doesn't get in the way of their empirical success.

As you can imagine, Field's approach has been met with plenty of criticism and debate, from worries about substantivalism to quantum mechanics.

Perhaps the most worrying criticism is simply that science done this way – without maths – is grossly impractical.

In fact, it looks to me that this way of doing things lacks almost all of the theoretical virtues of mathematics.

While those things might be true, a defence here might be that we're required to do science without maths – rather, maths is a convenient fiction that may used to represent – or that may be reduced – to genuinely true statements about the world. More on this later.

Refuting Premise 1

An alternative is to refute Premise 1 – that is, to deny that we ought to have ontological commitment to all of the entities which are indispensible for our best theories of science.

Particularly, we could argue we don't need have ontological commitment to mathematical entities.

To me, at least, it seems prima facie intuitive (not that I regard intuitiveness as a theoretical virtue – but you may!). After all, we typically think of the entities required to make scientific theories work to be physical entities located within spacetime. Do we really expect scientists to feel the need to posit the existence of abstract entities to make their science work?

Let's have a pick through some ideas from three contemporary philosophers.

Many have approached this in different ways, but most approaches involve the claim that mathematics is just a useful way of expressing what's required.

Penelope Maddy: Maddy's emphasis is on how mathematics is used in scientific practice rather than on the metaphysical status of mathematical entities. Originally fleshed out in Naturalism in Mathematics, they argue that the reason mathematics is so successful in science is because it provides an effective structure for organising and interpreting empirical data and experiences. This effectiveness need not imply the existence of abstract mathematical objects. More on naturalism in mathematics here.

Stephen Yablo: Yablo's approach, known as "figuralism", is to say that mathematics is a useful and powerful language with figurative truth. He likens mathematics to metaphor. Yablo, however, doesn't reduce mathematics to mere linguistic convenience – he's interested in how mathematical language can convey truths about the world in a non-literal way. Mathematical statements, he argues, can convey truths without committing to the existence of mathematical entities. More on Yablo's arguments in this (somewhat technical) collection of essays.

Jody Azzouni: Azzouni's approach – presented in Deflating Existential Consequence: A Case for Nominalism – is centred around something he terms "deflationary nominalism." Unlike Yablo, he views mathematical discourse as a literal language, but with truth not tied to the existence of its referents. He starts by arguing that linguistic reference shouldn't imply ontolgoical commitment. In this respect, it's a more direct challenge to traditional philosophical logic and language. Just because a term appears in a true statement, it doesn't necessarily mean that the term refers to an actual, existent entity. In mathematics, he points out that mathematical statements don't need to be able to be applied in science in order to be true. If that's the case, we don't need those ontological commitments based on their scientific indispensability.

Azzouni's challenge to traditional philosophical logic and language is interesting, because it takes us onto a new topic – truthmaking. It's far too big to tackle here, but I'm sympathetic to the viewpoint that, at least, insisting that mathematical statements should be strictly true need not imply abstract entities.



But, provisionally, I'm happy with a fictionalist system where mathematical statements need not be true, but could be true non-literally (after Yablo), or indeed, 'correct-but-not-true' as I've defended elsewhere. In either case, the application of mathematics doesn't imply the literal existence of mathematical entities – somewhat in the same way that lines on a map don't imply lines on the ground. Mathematical systems are "invented" (linguistically), rather than "discovered" (epistemically).

Yes, it's a very useful fiction. It provides a model for understanding the world.

I'm sympathetic to the intuitive pull of challenging nominalism based on what is commonly described as the "unreasonable effectiveness" of mathematics (Wigner's eponymous paper from 1960 is the original source for this term). Personally, I think we can acknowledge the success of mathematics – in things like explanatory power, predictive accuracy and so on – without committing to abstract entities. They're properties of the model or framework, not evidence of abstract mathematical entities.



Further Reading

An Introduction to the Philosophy of Mathematics – Mark Colyvan – A highly readable intro which covers this topic in detail in Chapter 4. He also discusses a couple of the philosophers I've mentioned in this blog in far more detail, including Yablo and Azzouni. Available from the Cambridge Book Shop.

Fictionalism – M. Ecklund. – An overview of fictionalism in metaphysics in general.

A Subject with No Object – J.P. Burgess & G. A. Rosen – A thorough and critical overview of a range of general nominalist approaches.