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 II.
Matti Eklund
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.
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Daniel Kodsi
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.
