This series is based on a debate hosted on March 10th by the Oxford University Philosophy Society called “Mathematical Fictionalism: Are Numbers Real?”. See here for Part II and here for Part III.

Matti Eklund

The topic of our debate is mathematical fictionalism. I will eventually defend a particular fictionalist thesis, or at least show myself to be a fellow traveler of sorts. But in these first remarks I will mainly present a map of the territory.

There are two importantly different kinds of “fictionalist” theses about numbers (or whatever purported entities you want to focus on). First, there is a thesis regarding metaphysics. Numbers are fictions – meaning here they have the metaphysical status of fictions. They don’t exist. (An awkwardness in discussing this is that some philosophers hold that fictional entities do exist. But let me pass over this difficulty. This metaphysical thesis taken by itself can also be error theory or eliminativism, or, in the special cases of abstract entities, nominalism. The view is opposed to “realism”. Second, there is at thesis regarding language – regarding number talk. This thesis says that in ordinary discourse regarding numbers we don’t commit to the existence of numbers, any more than a parent telling their kid a story about dragons commits to the existence of dragons.

In my remarks, I will focus on the linguistic thesis. It is this thesis that I want to defend a version of. We don’t commit. Specifically, as I will focus on what’s sometimes called a hermeneutic thesis regarding actual language use. (Some theorists focus on a revolutionary thesis, which is not about how we actually speak but about how we ought to speak.)

Let me therefore be brief about the metaphysical thesis and just indicate why one might accept it. There is first the general sense that there is something mysterious about these supposed entities, the numbers. More specifically, one can ask probing questions about them. It is generally agreed that numbers, if they exist, would have to be abstract. They don’t stand in causal relations, and they don’t exist in space and time. How then can we know about them? Or even manage to refer to them in the first place? In general, given their abstractness, how could it matter if numbers failed to exist?

The metaphysical and the linguistic theses are independent. I believe, with the hermeneutic fictionalist, that we don’t commit to numbers. But I also believe that numbers do exist, so I don’t accept metaphysical fictionalism. Whether or not my beliefs are true, the beliefs are certainly consistent. One can also be a metaphysical fictionalist, holding that numbers do not exist, while thinking, against the hermeneutic fictionalist, that we commit to numbers in ordinary number talk.

I will, again, focus on, and defend, hermeneutic linguistic fictionalism. Why focus on this thesis? Isn’t the philosophically interesting question whether numbers really exist, not how we talk about the existence of numbers? – Two responses. First, how we talk is often, whether for good reasons or not, treated as a guide to what exists. It seems that someone denying that numbers exist must thereby be saying that we often speak falsely, in a way that can seem like a problem. Second, questions about language are of interest in and of themselves. We may seem to commit when talking about numbers: number talk doesn’t seem like dragons talk. If we don’t, that reveals something interesting about language use.

Why find hermeneutic fictionalism attractive? Here are some things that can be said.

First, a general claim: even when we don’t speak of dragons, we often engage in loose talk  and don’t really mean to endorse what is literally expressed by the sentences we use. Hermeneutic fictionalism, in some forms, amounts to a way of spelling this out.

Loose talk doesn’t immediately have anything to do with fiction. But  compare fiction. Suppose I say “Once upon a time there was a dragon…”. If I say this in story-telling mode, there is no way in which I commit to the existence of dragons when uttering this. Rather I am putting forward something along the lines of: according to the story, once upon a time there was a dragon. If number talk functions similarly, then what I say when I say “there are infinitely many even numbers” is something along the lines of: according to the number fiction, there are infinitely many even numbers. This generic account invites an immediate, “phenomenological” objection: number talk just doesn’t feel like dragon talk.

There is a question of how seriously to take these differences in phenomenology. But there are also other things to stress. First, fictionalists do not always gloss what one really says in terms of fictions but instead use formulations like assuming that [numbers] in general exist, then... This doesn’t emphasize similarity to outright fiction. Second, Stephen Yablo, one of the main fictionalists, emphasizes an analogy with figurative speech. He says,

…figurative elements in our speech are very often unconscious, and resistant to being brought to consciousness. To hear ‘that wasn’t very smart’ (understatement) or ‘a fine friend she turned out to be’ (irony) or ‘spring is just around the corner’ (metaphor) as meaning what they literally say takes a surprising amount of effort. (“Apriority and Existence”, https://philpapers.org/rec/YABAP)

To the extent that the fictionalist emphasizes other things than fiction when arguing for the “we don’t commit” thesis, the “fictionalism” label may not seem apt. But that label is dubiously apt also as used for the metaphysical thesis. Non-existence is not peculiar to fictional entities. While the fictionalism label is apt to attract attention, the underlying thesis can, in both the metaphysics case and the semantics case, be stated without invoking talk of fiction.. Metaphysical fictionalism about Fs is simply the thesis that Fs do not exist. Linguistic fictionalism about Fs is simply the thesis that when engaging in F-talk we do not commit to the existence of Fs. In each case fiction and fictional entities may be a useful case to consider. Dragons don’t exist. Moreover, when engaging in dragon talk we don’t commit to dragons. But the link to fiction need not be stronger than that: fiction is just one analogy among many.

A different consideration Yablo adduces is this. Suppose you get to ask the all-knowing Oracle whether there are numbers. She says no. Would this really matter to how you go on to talk about numbers (in ordinary – non-philosophical – settings)? Hardly! You would utter the same sentences as before, and, moreover, it would not really feel or seem any different.

My aim in this first contribution to the debate is just expository. But let me end by stressing a notable point about these fictionalist considerations: don’t they generalize beyond abstract objets such as numbers? Consider a different ontological debate than the one about numbers. Some people hold that there are not really any complex material objects; all that exist are the fundamental things that make everything else up (“simples”). Where we say that there are cups, they hold that the real truth nearby is that there are simples arranged cupwise. Can’t one be a hermeneutic fictionalist about cup talk just as about number talk? Might not the hypothesis that cup talk proceeds under a silent “assuming there are complex objects…” be as plausible as the hypothesis that number talk proceeds under a silent “assuming that there are numbers…”? Wouldn’t your reaction to the Oracle be the same in the cup case as in the number case? I am as tempted by something like fictionalism in the case of cup talk as in the case of number talk. But of course one can agree that the cup case and the number case are analogous, but hold that this simply casts doubt on the fictionalist case regarding number talk.

**

Daniel Kodsi

Matti is scrupulous in distinguishing between different theses associated with the label “mathematical fictionalism”. He disavows many of the associated claims. He does not believe that mathematical objects—like Matti, I will primarily focus on the special case of numbers—are a mere fiction, similar in metaphysical status to dragons or demons. Indeed, he himself accepts that there are numbers. Matti also does not believe that mathematical discourse is especially similar to paradigms of fictional discourse. He raises a phenomenological objection to the assimilation of the former to the latter: “number talk just doesn’t feel like dragon talk”. Fictionalist doctrine is therefore best elaborated without the help of the “dubiously apt” analogy to fiction. 

In brief, Matti’s fictionalist sympathies are subtle and qualified. Still, he thinks that the “hermeneutic linguistic” strain of mathematical fictionalism is on to something. In particular, Matti says he agrees with the hermeneutic fictionalist hypothesis “regarding actual language use” that “in ordinary discourse regarding numbers we are not committed to the existence of numbers”. Call that the no-commitment thesis. Is it any more plausible than the various fictionalist theses that Matti rejects? 

The question is premature. The most pressing problem for the no-commitment thesis is not that it is implausible but that it is unclear, both in content and motivation.

An initial unclarity in the no-commitment thesis is this. In elaborating it, Matti helps himself to a distinction between “ordinary number talk” and non-ordinary number talk, explicitly restricting the claim that “we don’t commit to numbers” to ordinary number talk. The restriction enables Matti to accept the no-commitment thesis without thereby committing himself to the implausible-sounding claim that speakers never say things which commit them to the existence of numbers. The trouble is that without clarification about what counts as “ordinary number talk”, the effect of the restriction is to leave it quite unclear in what cases the no-commitment thesis does apply. For instance, does it apply to the highly specialized discourse of number theorists? The highly specialized discourse of financial analysts? Of Dungeons and Dragons players?

To make the force of the question vivid, it may be helpful to approach it from the other end. What is the non-ordinary discourse to which the no-commitment thesis is supposed not to apply? Although he does not say so explicitly, a natural suspicion is that Matti has in mind the discourse of professional philosophers, who intentionally assert that there are numbers, in the contexts of debates like this one. However, there are many respects in which the discourse of professional philosophers debating whether there are numbers is far from distinctive. In particular, it is no more reflective, explicit, theoretical or specialized than a great deal of other discourse, like that of number theorists, financial analysts or Dungeons-and-Dragons players. If there is any respect in which explicit assertions that there are numbers by professional philosophers engaged in debating whether there are numbers are distinctive, it is simply that they are an especially clear case of commitment to numbers. But it would be grossly ad hoc to exclude such assertions from the scope of the no-commitment thesis on that basis alone, akin to restricting some universal generalization H by the proviso “blatant counterexamples to H aside”. 

There is a second, perhaps more basic unclarity in the no-commitment thesis. It is that “committed” itself is somewhat theoretical terminology. As it were, the claim that in ordinary number talk speakers are not “committed” to there being numbers calls out for elaboration in independently understood terms. For instance, what consequences does it have for what speakers say or for what they believe? 

The question is pressing because it corresponds to a dilemma for proponents of the no-commitment thesis. On the one hand, they can posit close connections between what speakers are committed to and what they say or believe. For instance, they can treat “S is committed to p” as equivalent to “something S said implies p”: in effect, speakers are committed to the consequences of what they say. However, given that equivalence, the no-commitment thesis has the radical consequence that in ordinary number talk, speakers don’t say things which imply that there are numbers. By non-skeptical standards, that is clearly false: after all, the claim that there are prime numbers occurs in ordinary number talk and implies that there are numbers. On the other hand, if proponents of the no-commitment thesis sever the connections between the theoretical relation of being committed to and more familiar cognitive relations like saying and believing, the interest of the no-commitment thesis is left quite unclear. 

For now, I will leave the issue there, to give Matti the chance to explain the commitments of commitment-talk as he sees fit.

A final unclarity in Matti’s discussion worth highlighting concerns his motivation for the no-commitment thesis. Matti does not attempt to motivate the no-commitment thesis in detail. But he does suggest that it gains support from what we can call the Oracle thought experiment, originally due to Stephen Yablo. In the thought experiment, “you”—Matti couches his claims in the second-person—are told by an all-knowing Oracle that there are no numbers. Matti confidently predicts that you wouldn’t do anything differently in response to such testimony: you would say the same things and feel the same way when saying them (no guilty intellectual conscience). He takes this piece of reflection on the Oracle thought experiment to provide a consideration in favor of hermeneutic fictionalism. But how exactly does it do so?

More specifically, it is quite unclear what the argument from the Oracle thought experiment to the no-commitment thesis is supposed to be. What are its premises? Perhaps one of them is supposed to be this, where S is an “ordinary” subject:

(Premise 1)    If S were to come to believe that were no numbers, S would not act or think any differently. 

But if so, then there’s a problem: (Premise 1) isn’t true. For part of what it is to believe something is to be disposed to act on it. Such a disposition need not be very robust. It might involve little more than an inclination to say the thing that one believes: for instance, to answer the question “Does God exist?” by saying what one believes about whether God exists. Such an inclination will be defeasible. Some people prefer to demur about religious questions, and so on. But if you aren’t even tempted to assent to the claim that God exists, that is a clear sign that you don’t believe it. Similarly, if S really has been convinced by the Oracle that there are no numbers, then S would act and think at least somewhat differently, again if only in virtue of now being disposed to say “No” if asked whether there are numbers, rather than “Yes” or “I don’t know”. 

Of course, Matti could take that point on board by regimenting the argument from the Oracle thought experiment using a weaker version of (Premise 1):

(Premise 1*)   If S were to come to believe that were no numbers, S would not act or think very differently. 

But why think that one is only committed to something if one would act or think very differently if one were to come to believe that it was false? Certainly, it isn’t necessary for believing something that one would act or think very differently upon coming to believe that one was mistaken. Right now, I am confident that the Prime Minister of Australia has never murdered anyone; learning otherwise would be amusing but make little difference to my day-to-day dispositions. However, taking the issue further would require getting clearer on what “committed” means as it occurs in the no-commitment thesis. Again, I leave that task of clarification to Matti. 

Matti Eklund is Chair Professor of Theoretical Philosophy at Uppsala University and author of the books Choosing Normative Concepts and Alien Structure: Language and Reality (2024). Daniel Kodsi is a lecturer in philosophy at Magdalen College and editor-in-chief of The Philosophers’ Magazine.