17 July, 2025

This three-part exchange is based on a debate hosted on March 10th by the Oxford University Philosophy Society called “Mathematical Fictionalism: Are Numbers Real?”.

Matti Eklund I

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 I

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 II

In my first contribution, I mainly intended to give a general introduction to the debate over fictionalism. But I also indicated my sympathy for a version of hermeneutic linguistic fictionalism, and my remarks hinted at what kind of view I am like. Picking up on that Daniel raises a number of reasonable objections. I will address those objections. But before I do so, I should say more about what I believe and why.

When writing about this earlier (see especially my “Fiction, Indifference and Ontology”, Philosophy and Phenomenological Research, 2005), I have called my own preferred view in the vicinity of fictionalist proposals indifferentism. We are indifferent to some of what is entailed by what is literally expressed by the sentences we use (whether this should be likened to fiction and figurative language).

Compare, as in my first contribution, talk of cups. Suppose I say, outside of philosophy, “The cup is on the table”. If you object, raising objections to the view that cups, these complex objects, exist, I will just find your remark irrelevant and I will be impatient with you. And I will react this way even if I myself actually share the doubts you raise. I happen to be a philosopher with views on things like this, but this would apply equally (and maybe more strongly) if I were a non-philosopher saying this.

Compare too a different kind of example, originally due to Keith Donnellan and used to make points about definite descriptions. Seeing a really happy man I want to indicate him to you. It is rude to point. I say “the man drinking water is really happy”. You object “that is not water; that is vodka”. One natural reaction for me is impatience. I don’t care what the transparent liquid in his glass is; I just want to indicate him to you. It is easy and convenient to refer to it as water. My utterance succeeds if you understand me to say that that guy is happy. This is so even if I actually believe that there is water in his glass. This belief, even if I have it, is not what is conversationally important, and I needn’t care about it at all.

Indifferentism says that our utterances of sentences implying the existence of mathematical objects are in relevant respects like what is illustrated in these two examples, regarding cups and regarding the happy man.

Daniel rightly notes that “commitment”, which I used when describing the fictionalist stance, threatens to be unclear. It is a technical term and one that needs to be explained. One relatively straightforward way to explain commitment talk would be in terms of what is entailed by what is asserted (even if there are complications there too). But it sounds strained to deny that, in the examples used, the speaker asserts that there is a cup on the table, or that the man drinking water is really happy. (Strained but not out of the question: “assertion” as used in these discussions is itself a technical term.)

The best general characterization I can give of how I speak of “commitment” is this. First, I claim that examples like the ones I have brought up shows how we have different attitudes towards different elements of what is conveyed through our assertions. Some we take less seriously than others; we are indifferent to them. Second, the fact that we take some elements less seriously is relevant for issues regarding how prima facie problematic it is that some ontological position is in tension with the way we speak. If it is only in tension with what we are indifferent to, it is much less of an issue than if it is in tension with other elements. It is this that suggests the label “commitment”. (Of course there is a bigger issue of how much of an issue it is anyway that an ontological view conflicts with something we firmly believe and take seriously in every way. But that is a different debate.)

Here is a general theoretical description that may help. The sentences we have available express propositions, excluding some possibilities and allowing for others. As speakers assertively uttering these sentences we intend to exclude some possibilities and allow for others. But the sentences we have at our disposal are often crude tools for the purpose. They don’t exclude exactly the possibilities that we intend to exclude. The cup case is an example. I don’t mean to exclude the possibility that there are only simples arranged cupwise. But unless I use a very convoluted formulaton, “the cup is the table” is the best sentence I have available.

Daniel also asks about my use of “ordinary”. My use of “ordinary” was there to emphasize what definitely is in the scope of my claim and not primarily to indicate what is not. I mean what I say to apply to some aspects of philosophical discourse. If I say, doing philosophy, “I have an objection to…” and someone else says “but are there such things as objections?”, this latter remark would often be beside the point in the way that “is it really water he is drinking?” is. And even in otherwise serious and non-ordinary discourse about numbers, whether in mathematics or financial analysis, the metaphysical status of numbers is often beside the point. Even if the financial analyst says something which on the face of it commits her to numbers, I bet she would regard doubts about whether numbers exist as relevant to what she is up to. Similarly for the practicing mathematician. I believe that cases where speakers mean all aspects of their utterances to be committing are extremely rare

Like hermeneutic fictionalism generally, my view is in the first instance a view on language and what we do when we use language. A different question concerns belief. Do we believe that there are numbers? Do we believe that there are cups? Etc. There are different ways for the indifferentist to go. One is to deny that we generally believe that there are numbers, despite what we happily assert. This is what may be called a misalignment version of the view, treating belief and assertion differently. Another is to say that we do have beliefs to the effect that there are numbers but have a don’t care attitude towards some aspects of these beliefs, just as we make the relevant assertions but have a don’t care attitude towards some aspects of these assertions. 

Hermeneutic fictionalists about some discourse tend to provide accounts of what the speaker is really saying when engaging in the discourse. One simple model involves holding that when the speaker utters “p”, she is really saying that in the fiction, p. But go back to me, by assumption doubting the existence of cups, uttering “the cup is on the table”, reacting impatiently to someone raising concerns about whether cups exist. What am I really committing myself to? If I firmly believed some other ontological view – there are only simples, but some of them are arranged cupwise – maybe that is what I mean to commit myself to, despite speaking of “cups” to make myself understood by others. But I may well be agnostic. An open-ended disjunction may best capture my real aim: “there is a cup, or there are simples arranged cupwise, or the cosmos is cuppish, or there is cuppish stuff, or….”. “It is as if there is a cup….” or “Assuming there are cups…” do not actually quite cut it, for those utterances are compatible with my not ruling out that I am simply hallucinating, and that I may want to rule out. So here again, my indifferentist is less ambitious that the standard fictionalist. The fictionalist makes a definite claim about what the speaker asserts instead. I shy away from making such a claim. But the way I see it, this isn’t laziness on my part but a way of staying more true to the phenomena.

**

Daniel Kodsi II

In his second contribution, Matti explains that his own preferred view “in the vicinity of fictionalist proposals” is one that he calls “indifferentism”. He elaborates indifferentism as the claim that “we are indifferent to some of what is entailed by what is literally expressed by the sentences we use”. On that minimal statement, indifferentism is plausible. For instance, there is presumably at least one logical truth to which I am indifferent—if only an extremely complex one that I have never thought of. In classical logic, every logical truth is entailed by every sentence. Thus when I assert “it is sunny outside”, or indeed anything else, what I assert has some entailment to which I am indifferent. The example naturally generalizes to other finite beings. Perhaps it is enough to vindicate the bare doctrine of indifferentism. If so, then the bare doctrine of indifferentism is an inadequate and impoverished substitute for more standard fictionalist proposals. An analogy: radical philosophical sceptics, according to whom we know nothing, don’t get to claim credit for the common-sense observation that we don’t know everything. 

Of course, Matti could attempt to flesh out the doctrine of indifferentism to make it more interesting. His remarks provide no precise way of doing that. For instance, Matti further characterizes indifferentism as saying that “our utterances of sentences implying the existence of mathematical objects are in relevant respects” like a pair of informally described illustrative examples. But what exactly are the “relevant respects”? 

In the absence of a clear general statement of indifferentism, there is no way for me to challenge it further. Still, some of the more specific points and connections that Matti makes will be worth commenting on.

First, one feature that the illustrative examples that Matti presents have in common is that they involve a speaker (as it happens, Matti) being tempted to react with impatience to a challenge to what he has just said. But of course, people react with impatience to objections, doubts, and the like for all sorts of reasons—some good, others bad. For instance, a foreign policy expert may be impatient and dismissive towards an audience member who objects to their diagnosis of the Iraq War “but all of that neglects the possibility that 9/11 was an inside job”. That isn’t because the expert doesn’t carewhether 9/11 was an inside job; more likely, it’s because he knows that it wasn’t. Or again, a theologian who preaches the infallibility of the Bible may be impatient and dismissive when a non-believer points out some of its manifold small inaccuracies. That isn’t because the non-believer’s objection is irrelevant to the doctrine that it is infallible; more likely, it’s because it is relevant and so puts the theologian on the defensive. 

Again, without an explicit statement of what the “relevant” features of the original illustrative examples are supposed to be, it is hard to turn such reminders of the varied potential causes of impatience into a clear-cut objection to indifferentism. Still, they may be helpful for warding off the temptation to read too much into such impatience. 

Second, in response to my challenge to clarify the reading of “commitment” relevant to the no-commitment hypothesis, Matti says that he takes the examples to “show how we have different attitudes towards different elements of what is conveyed through our assertions”. In fact, however, the examples show no such thing, because the tests for the attitudes in question—like taking seriously—do not clearly distinguish between some elements of what is conveyed and others. For instance, about the first example, Matti in effect points out that a partygoer who asserts “the guy drinking water is happy” may be satisfied if his friend takes away the message “that guy is happy”. But wouldn’t the partygoer be similarly likely to be satisfied if his friend takes away the message “the guy drinking water is in a good mood”? (Perhaps being happy requires being disposed to be in a good mood.) Again, it is independently plausible that we have different attitudes towards different consequences of our assertions—but without further elaboration, Matti’s examples don’t show that. 

Third, restoring the connection with mathematical fictionalism, Matti continues by suggesting that “if [some ontological position] is only in tension with what we are indifferent to, it is much less of an issue than if it is in tension with other elements”. But compare the case of indifference to some theorem of propositional logic—say, (p → (q → r)) → ((p → q) → (p → r)). At first, the average philosophically naïve subject will probably be quite indifferent as to whether that specific sentence is true. Nevertheless, it is a very serious issue if a position—ontological or otherwise—is inconsistent with it, since then the position is inconsistent simpliciter. 

Of course, with enough time and energy, and good will and intelligence on the part of the subject, a philosopher might be able to convince them that it is in fact a serious issue whether a position is inconsistent with even the most boring-looking theorem of propositional logic (then again, they might not). But that is no disanalogy with the case of indifference to whether there are numbers. Through a very similar process of reflection and instruction, philosophically naïve subjects may become convinced that though denying “there are numbers” looks harmless, its failure would ramify throughout our web of beliefs. In any case, the more general point is that indifference—even widespread indifference—as to whether something is so hardly implies that it doesn’t matter whether it is so. It is sometimes simply an artefact of failing to appreciate the stakes.

Fourth, in response to my challenge to clarify his use of “ordinary”, Matti says that he included the restriction to ordinary discourse “to emphasize what definitely is in the scope of [his] claim and not primary to indicate what is not”. That is fine, but it does not engage with the reason I had assumed that the restriction was there to restrict the no-commitment hypothesis. Namely, it is clear that the unqualified claim “speakers never commit to the existence of numbers” has counterexamples: for instance, philosophers of mathematics who say things like “I hereby unequivocally commit to there being numbers”. Some restriction is needed to restore accuracy. But how exactly is that restriction to be spelt out? In particular, can it be spelt out in a way which is not tantamount to “obvious cases of commitment to numbers aside…”?

Finally, Matti briefly alludes to an attractive model of conversation on which a central function of asserting a sentence is to rule out possibilities. He suggests that “the sentences we have at our disposal … don’t exclude exactly the possibilities that we intend to exclude”. For instance, someone who say that there is a cup on the table need not intend to “exclude the possibility that there are only simples arranged cupwise”. Matti gives the impression that such a model of conversation supports his indifferentism. However, it is quite independent of it. For suppose that someone asserts that it is raining outside. The simplest and most natural answer to the question “which possibilities did they intend to exclude?” is “possibilities in which it is not raining outside”. 

More generally, the simplest and most natural answer to “which possibilities did S intend to exclude in asserting that p?” is “possibilities in which not p”. If Matti wants to make plausible that the simple and natural answer is in general wrong—someone who says that there is a cup on the table does not, in fact, intend to rule out that there is no cup on the table, and so on—he owes some non-trivial account of what possibilities speakers do intend to exclude when making assertions. As a corollary, it would be interesting to know what possibilities he himself intends to exclude in asserting indifferentism, if not simply those in which indifferentism isn’t true. 

**

Matti Eklund III

In my first contribution to this exchange, I described the theoretical map. In the second, I briefly outlined by favored view regarding talk of numbers, indifferentism. In this third contribution I will mostly play defense. For further remarks spelling out the positive view, see the work of mine I have referred to earlier. Also, Daniel Hoek develops a position much like what I call indifferentism in his “Conversational Exculpature” (Philosophical Review, 2018). And both Hoek and I in different ways pick up on ideas from Stephen Yablo. For a relatively recent statement of Yablo’s views, see ch. 12 of Aboutness (Oxford University Press, 2014).

Maybe unsurprisingly, I find some of Daniel’s concerns more significant than others; and there are so many of them that I may not cover them all. I will work my way up to what I see as more significant.

In his first contribution, Daniel asked some probing questions about the Oracle argument. I think the best way to defend the Oracle argument would involve being mor careful about which dispositions to act and believe are at issue. But let me not get into that. I referred to the Oracle argument and I find it an attractive thought experiment. But even independently of Daniel’s specific concerns it is unclear how much argumentative weight it can carry.

In his last contribution. Daniel asks for more precision regarding the scope of the thesis. Indifferentism is a kind of fictionalism. Just as fictionalist theses are often presented as theses regarding specific discourses – “I am a fictionalist about mathematical discourse” – indifferentist theses should be understood as being about certain circumscribed parts of language use, even if the idea of language as neatly divided into “discourses” is unduly simple. And a sensible hermeneutic fictionalist about “mathematical discourse” may well want to make exceptions for utterances about mathematics in philosophy of mathematics settings, inviting reasonable questions about exactly how far her thesis extends. I think the claim is clear enough to be sensibly discussed even absent further specification. This isn’t to say that there isn’t more to be said than what I have said here.

Regarding the “ordinary”, I see Daniel’s concern about informativeness. But I despair about being more precise. Daniel’s own example “I hereby unequivocally commit to…” serves to make the point. I can easily see someone saying “I hereby unequivocally commit to a number of objections, namely…”, while later disavowing any commitment to such entities as objections. This simply is messy.

Daniel complains about my “in relevant respects”. I thought it was clear what was meant by this. Just as the truth of what the speaker is concerned with in the “happy man” utterance is independent of what’s in the man’s glass, the truth of what a speaker is concerned with in utterances of sentences whose literal truth demands the existence of mathematical objects is independent of whether there really are mathematical objects. 

Daniel notes that speakers can be impatient with interlocutors for all sorts of reasons. They may simply think that what the interlocutor says is beyond the pale. This is right and it is a complication. But it does seem to be very surmountable. One way to deal with it is by considering cases where the speaker in fact agrees that the interlocutor’s doubts are justified, and asking whether impatience is still the likely reaction. This was for example how I already described the cup case.

I don’t think I get Daniel’s final remark. What I talked about were cases in which the speaker assertively utters a sentence that semantically expresses that p, but the possibilities inconsistent with the proposition that p aren’t exactly the ones the speaker means to exclude. In his response, Daniel seems to just insist that when a speaker assertively utters a sentence semantically expressing that p, she means to exclude the possibility that not-p. That is just what I was disputing. The examples used illustrate my point.

Regarding my Donnellan-inspired case, Daniel notes that the partygoer is “likely to be satisfied if his friend takes away the message ‘the guy drinking water is in a good mood’”. He indicates that this is a problem for me. I rather take Daniel’s point, generalized, to indicate that the phenomenon I point to permeates language use. The partygoer says “water” because that is convenient – he is not really concerned with what’s in the guy’s glass. And somewhat similarly, maybe the partygoer says “happy” because that is a convenient way of gesturing towards the mental state in question. Maybe something else – “ecstatic”? “blissful”? “content”? – would be more accurate, and better convey the partygoer’s exact impression of the guy’s mood. But “happy” is simple and works neatly.

The same general phenomenon is nicely illustrated by an ordinary person’s attempt to describe what some wine tastes like, or the aesthetic qualities in a painting. One finds oneself using general, unhelpful descriptions like “delicious” and “beautiful”, all the while being painfully aware that this doesn’t really get at what one really thinks the wine or the painting is like. If one gets the reply, “well, the Guernica isn’t exactly beautiful” one may simply be inclined to agree.

What to me is the most theoretically interesting point Daniel makes concerns examples where the speaker’s first instinct is to regard something with indifference, but the speaker can be rationally convinced that this something is worth caring about. However, there are different issues in the vicinity, and distinctions to make.

First, one must distinguish between whether some claim is worth caring about full stop and whether it is worth caring about given conversational purposes here and now. Daniel gestures towards how the falsity of “there are numbers” could reverberate throughout our web of beliefs. I imagine he has in mind that science requires the existence of mathematical objects. Maybe it would thus reverberate. But that says little about conversational purposes here and now. Indifferentism might be true of ordinary utterances of mathematical sentences even so. Compare a drastic analogy. “There is a climate crisis” is a truth that, in some sense, we ought to care about, always. Even so, uttering that sentence may be completely conversationally irrelevant in many contexts. 

So the mere fact that something has some kind of more general importance is not immediately germane. Other examples, though, do problematize even conversational relevance. A point made by Daniel in our debate at the Oxford Philosophy Society serves to show this. “That isn’t water in the glass…it is vodka” may turn out be relevant in the Donnellan case, for it might serve to explain the happy guy’s mood. This is a more interesting case, for it indicates how issues the speaker may initially take to be irrelevant may turn out to be relevant. Daniel’s point about truths of logic may make a similar point.

A quick way of dealing with this complication is to say that the focus should not simply be on what the speaker would regard as conversationally relevant and not, but rather what the speaker upon reflection should regard as relevant given actual conversational aims and not.

Recall, finally, the distinction between hermeneutic and revolutionary theses that I drew in an earlier contribution to this debate. Hermeneutic theses concern how we do speak; revolutionary theses concern how we ought to speak. Indifferentism as I defend it is a hermeneutic thesis. One reason to care about hermeneutic claims is because of what they show regarding language, regardless of the metaphysical interest. But as I also stressed, they are also held to be of metaphysical interest. What commitments speakers regularly take on is treated as evidence regarding metaphysical theses. Specifically, it is held to be theoretically costly to hold that there are no Fs if speakers commit to Fs.

I will end by making two remarks related to this. First, Daniel’s point about reverberation can be seen as a claim about why the existence of mathematical objects is important and there is good reason to believe in mathematical objects, regardless of actual aims and practices. Insofar as that is so, my response is that this is perfectly compatible with everything I wish to argue. Maybe the relevance both of hermeneutic fictionalism and indifferentism to whether we ought to believe in mathematical objects is extremely limited. Second, insofar as indifferentism and similar theses are metaphysically relevant, some weaker theses than indifferentism arguably are so too. Consider the thesis that it is indeterminate whether we commit, and also the thesis that the evidence at hand doesn’t settle whether we commit. These theses are weaker than indifferentism, but share the kinds of consequences (again: if any!) that indifferentism has for disputes over the metaphysical question of the existence of mathematical objects. If I needed to retreat to either of these weaker theses, that would be fine as far as consequences for metaphysics go.

Needless to say, there is more that can be said about indifferentism, both about its motivation and about objections that critics like Daniel are apt to raise.

**

Daniel Kodsi III

In his final contribution, Matti diligently works his way through various objections that I raised to his preferred brand of fictionalism in my first two responses. At this point, readers may feel that the dialectic has become rather intricate. Accordingly, in this final contribution, I will zoom out and explain the general motivation for my objections to Matti, before making a representative application to something he says in response to me. 

In philosophy as in the natural and social sciences, hypotheses are appropriately assessed on several dimensions. The most obvious dimension is that of fit with the evidence. It is a (decisive) problem with a philosophical hypothesis if it is inconsistent with what we know. In particular, that is the problem with some eye-catchingly radical metaphysical claims, like nihilism (“there is nothing”) and monism (“everything is one”). Whatever else nihilism and monism have going for them, they are obviously false. We know that there are many things—for instance, that there are more than eight billion people. 

Faced with this objection, some nihilists or monists may simply double down. They will make no qualifications to their doctrine at all and unhesitatingly dismiss the evidence against it. As unsatisfying as it sounds, it is not clear how much more there is to be said in response to such hardliners. For better or worse, philosophical methodology provides no foolproof recipe for disabusing people of their philosophical errors.

In any case, many philosophers initially sympathetic to such extremist metaphysical positions eventually recognize that some concessions to reality are needed. In one way or another, they weaken or attenuate their original commitments. For instance, instead of asserting the unqualified “there is nothing”, they assert “for all we know, there is nothing” or “fundamentally speaking, there is nothing”. The trouble is that such attempts to restore fit with the evidence often either fail to do so or succeed in doing so by depriving the original radical claim of any philosophical interest. “For all we know, there is nothing” exemplifies the first horn of this dilemma: we don’t just know that there are things, we know that we know that there are things. “Fundamentally speaking, there is nothing”, by contrast, arguably exemplifies the second. For what is supposed to follow from the claim that “fundamentally speaking, there is nothing”? What does it imply in independent terms? Methodologically, the point is that although “fundamentally speaking, there is nothing” may do fine on the score of fit with the evidence, unless supplemented with a rich theory of fundamentality, it does badly on the score of informativeness, another crucial dimension of hypothesis-comparison.  

My objections to Matti have been driven by a version of this dilemma for theorists who espouse radical-sounding metaphysical doctrines. As Matti has been at pains to emphasize, there are more and less radical versions of mathematical fictionalism. Radical versions of mathematical fictionalism—on which numbers and sets are no more real than unicorns or witches—clearly face the objection that they are inconsistent with what we know. By non-skeptical standards, that there are numbers and sets is a direct consequence of our evidence, of what we know. And it isn’t just that we have some evidence, sourced from a single unreliable channel, which entails that there are such things. We have lots and lots of it, sourced from many channels, including some of the most reliable. We know about numbers by means ranging from sense-perception to formal proof. In fact, it is contemporary orthodoxy in cognitive science that we have a number sense, which enables us to learn through multiple modalities about the number of things, just as our visual system enables us to learn about the color of things. 

Consequently, there is significant pressure on mathematical fictionalists not to be radical mathematical fictionalists. In particular, again as Matt emphasizes in his discussion, rather than deny that there are numbers outright, many mathematical fictionalists instead opt for the more qualified claim that we are not committed to there being such objects. The trouble is that, as I have for my part emphasized in my responses to Matti, the implications of the claim “we are not committed to there being numbers” are unclear. In particular, it is unclear what consequences it has for what we believe and assert under the guise of sentences like “there are four prime numbers less than 10”. Some mathematical fictionalists treat it as having quite striking consequences for the contents of our assertions and beliefs, such as that—a few exceptional cases aside—no one believes or asserts anything which entails that there are numbers. Those consequences look false. For instance, someone who sincerely asserts “there are four prime numbers less than 10” asserts and believes that there are four prime numbers less than 10, which straightforwardly entails that there are numbers. By contrast, other theorists, like Matti, are more cautious in drawing consequences from it. Like caution about practical matters, such theoretical caution may seem like a sensibly safe policy. However, take it too far and you end up with a doctrine that fails to meet reasonable minimal standards for informativeness. 

Independently of the ideology of commitment, the general dilemma for mathematical fictionalists is this. Either they deny that we know, believe, state or act on information which entails that there are numbers and sets, or they do not. If they do, they must explain how to reconcile fictionalism with our evidence about what we know, believe and say, as well as with the exceptional success of a wide variety of activities apparently based on mathematical knowledge. If they do not, they must explain what independent interest fictionalism has. If it has no clear implications for what we know, believe, say or act on, then it starts to look irrelevant to the philosophical action. 

Of course, that dilemma is schematic. Fictionalists have some room for maneuver. They can pick and choose exactly which attitudes or relations they want to say we don’t have towards facts about numbers, and under exactly which conditions we don’t have them. For instance, they could claim that ordinary people never believe or assert anything about numbers, though mathematicians and philosophers sometimes do. However, such a pick-and-choose approach runs the danger of being ad hoc, of drawing distinctions that are unstable under further scrutiny. The toy example I just gave illustrates that problem: mathematicians and philosophers are ordinary people. Though the pick-and-choose fictionalist can help themselves on the spot to yet more distinctions, multiplying distinctions simply to avoid counterexamples is bad philosophy. 

Naturally, I do not expect Matti to agree that his indifferentism is impaled on one horn or other of the dilemma just sketched. In particular, though his responses to my objections acknowledge significant vagueness and unclarity in his statement of indifferentism, it is clear that he does not think they show that his view falls, as I have implied, beneath reasonable minimal standards of informativeness. For my part, having explained the general methodological concern driving my objections, I am happy on the whole to leave it to readers to judge the adequacy of his responses for themselves. 

One point deserves elaboration, however, in line with the preceding remarks. In response to the final objection in my second contribution, Matti says: “Daniel seems to just insist that when a speaker assertively utters a sentence semantically expressing that p, she means to exclude the possibility that not-p. That is just what I was disputing.” But I did not “just insist” that when a speaker asserts, for instance, the sentence “there is a cup on the table”, she means to exclude the possibility that there is no cup on the table. Rather, I pointed out that the simplest and most natural answer to the question of what possibilities a speaker who asserts “there is a cup on the table” intends to exclude is “possibilities in which there is no cup on the table”. Having noted this readily available answer to the question, I challenged Matti to provide some non-trivial alternative to it. In effect, he declined the opportunity to do so, though in his second contribution, he did say: “An open-ended disjunction may best capture my real aim: ‘there is a cup, or there are simples arranged cupwise, or the cosmos is cuppish, or there is cuppish stuff, or….’.” 

Taken at face value, this suggestion neatly illustrates the dilemma for Matti’s indifferentism. Does the claim “a speaker who asserts ‘there is a cup on the table’ intends to exclude an open-ended disjunction” have consequences for what that speaker asserts or believes, or, perhaps, for what is going through her head? If so, those consequences can be assessed independently—and must be plausible by normal standards on attitude-ascription. If not, the interest of this claim is no clearer than that of the other quasi-theoretical formulations by which Matti has attempted to articulate his indifferentism. 

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.