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 I and here for Part III.
Matti Eklund
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
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 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.
