Nominalism, Quine, and Burgess’ Dilemma

1. Ontology and certainty

You and I and working mathematicians also are all quite sure that there are an infinite number of primes. But I don’t think this means that we’re sure that the answer to the philosopher’s question ‘are there numbers?’ is yes. Or, for that matter, that we are sure of some nominalist reconstruction of ‘there are infinitely many primes’ which allows that the answer is no.

Certainty is very often (maybe always) really certainty-with-respect-to a range of alternatives which a person has in mind or conversational context supplies to them. We are sure that there are infinitely many primes rather then, say, there being only finitely many primes, or every number being composite. There’s a mathematical practice which determines right or wrong answers, and when people ask mathematical questions this is normally what they are asking you to engage in. And we’re certain that this practice determines that the right answer to the question ‘are there infinitely many primes?’ is yes. But when the philosopher asks whether numbers exist, they are (trying) to ask something else. They want to ask something like, ‘are numbers merely a manner of speaking?’ or ‘is mathematical practice more like metaphorical talk or like talk about physical objects?’

Now, this question may or may not make sense. But in either case (whether the philosophical question about existence makes sense or not), a yes answer to it can’t be part of what we’re all certain of when we are certain that there are infinitely many primes. If ‘do numbers *really* exist?’ doesn’t make sense, then ‘yes numbers do *really* exist’ also doesn’t make sense so it can’t be entailed by what we are sure of when we say that ‘there are infinitely many primes’ for surely this latter comment does make sense. And, on the other hand if the question ‘do numbers *really* exist?’ makes sense, then the fact that it is a correct part of mathematical practice to say that they exist is neither here nor there. For it’s also a correct part of metaphorical practice to say that people have chips on their shoulders. The question the metaphysical philosopher wants to ask is not whether mathematical practice includes existential talk but whether this talk has some further feature or being real or literal.

So, I don’t think that our mathematical certainty that there are numbers with certain properties cuts any philosophical ice. For, what we are certain of when doing mathematics doesn’t determine an answer to philosophical questions about the existence of numbers. We can get further evidence for this view by considering what happens when we ask actual mathematicians this pair of questions, and by considering how certainty figures in talk of objects which are generally agreed not to be literal.

A philosophical questioning about chips on shoulders might go something like this. A: “Jon might have a chip on his shoulder but Simon certainly does” B: “ So you really think that some people have chips on their shoulders?” A: “Yes, certainly. It’s a sad truth that some people go through life with an attitude of unjustified and unprovoked resentment” B: “So you think that there really are chips which sometimes reside on people’s shoulders” A: “What a strange way of rephrasing your question…. Oh, wait, I see what you mean now. You are asking whether some people really literally have chips on their shoulders or whether it’s just a manner of speaking, right? Well, the answer is that it’s just a manner of speaking, or course there aren’t really chips on people’s shoulders”. We can be certain about statements like ‘there are numbers’, as opposed to uncertain about whether (mathematical or metaphorical) practice entitles us to make it, without being certain that this existence claim has some kind of ontological import, or even while being equally certain that they don’t.

And, in my experience if you ask a mathematician a similar chain of questions about numbers (“are there infinitely many primes” “so you think there are infinitely many numbers” “so you think numbers exist?”) the conversation will go similarly - up to the last point. At the last question some mathematicians will say yes numbers do really exist, others will say no and tell you about some kind of formalism or that math is really just about deducing consequences etc. and most of them will finally ‘see’ that rather then asking a well defined mathematical question you are asking about some kind of philosophical nonsense (what would it mean for a numbers to “really exist”) and abuse you for wasting grant money on it.[1]

 

2. Burgess’ Dilemma

Burgess supports Quinean realism about numbers by posing the following cunning dilemma for the anti-realist. Mathematicians seem to say that numbers exist, and you say they numbers don’t exist. So, what is your attitude towards what working mathematicians believe? Are you a revolutionary who says that all of existing mathematics is wrong and should be replaced with some nominalist substitute? Or are you a revisionary who claims that your hygienic nominalistic substitute (math without numbers) is what mathematicians have believed all along? The former sounds rather hubristic, and philosophy has such a bad track record in its meddling with empirical science that there’s little reason to think it will do well in attempting to correct math. And the latter option sounds implausible. There’s plenty of well-documented evidence that many brilliant mathematicians (like Godel) were rabid Platonists.

I claim that this question presents a false dilemma. What mathematicians learn by example, and agree on, and do in journals is a practice which, as I have argued in the previous essay, determines nothing about whether numbers *really* exist, in the stricter sense which the nominalist and the realist have in mind.

Now, I realize that this kind of answer depends crucially on a fashionable and perhaps suspect focus on the ‘practice’ of mathematics rather than the meaning of mathematical terms and the purpose of the endeavor. So I will try to dispel any suspicions by making the point more explicitly here.

People see examples of mathematical calculations, terms and proofs and learn to produce these items themselves. If you think that this process works by way of examples allowing us to grasp the meaning of mathematical statements, the notion of consequence and the like, then the question ‘is math about mathematical objects or about what can be deduced from certain hypotheses?’ must have a definite answer – at least for each individual. The question is just this: by ‘there are infinitely many primes’ do mathematicians mean that there are so many abstract objects, or that certain formulae could be deduced or that if there were something which had the structure supposed by certain axioms then there would be infinitely many primes? Which of these do mathematicians have in mind? Which of these describes the meaning which we grasp in learning to do math? So, from this point of view, Burgess’ question poses a genuine dilemma: there’s something which mathematicians mean by ‘there are infinitely many primes’ and this is either a nominalist surrogate or a claim about abstract objects. The actual practice of math is compatible with either of these ways of ascribing meaning to mathematical sentences (that’s what the nominalist surrogate is rigged up to ensure). But underneath this there is a question of which of these options describes what mathematicians really mean.

This idea, that underneath every practice there are meaning facts which determine the answer to every unforeseen and exotic question (e.g. do numbers literally exist), is what makes Burgess’ argument against nominalism work. If the nominalist says the real meaning of math is some nominalist thing about what’s deducible or what would have to be the case if something were true, then their claim is historically and psychologically implausible. If they say that this is not what mathematical statements mean but we should adopt new ways of talking which *do* mean the nominalist surrogate then they seem to be advocating an under-motivated change in mathematics for the sake of philosophical hygiene. Neither option is appealing.

But why should we believe the initial hypothesis? Why think that people learn the practice of math by grasping this kind of fully determinate meanings? There are other reasons to think that people are able to detect and continue patterns without grasping any kind of conscious picture or rule, or only grasping a very vague one. So why not say that people just learn how to continue the practice which they see in examples without supposing that there’s any special item which they need to grasp in order to do this? When children learn to add, they don’t see examples, then grasp that the recursive definition of addition is what the teachers have in mind by ‘+’ and then add accordingly. Rather, they first learn how to add in the sense of getting the right answers to sums, and only later see the recursive definition, think things over, and then agree that this gives the meaning of ‘+’. Similarly it may be that mathematicians see proofs, and learn to construct proofs like the ones which they have seen either without grasping any detailed picture of the meaning of mathematical primitives, or with each person having a different picture in mind.

If this is the case then the mathematical practice won’t determine an answer one way or another to the philosophical question of whether numbers exist. Thus the nominalist won’t be embarrassed by Burgess’ dilemma, for what the nominalist claims will neither contradict mathematics as we now know it nor require a revision in it. Instead, the nominalist aims to answer questions which present mathematical practice leaves indeterminate.

 

3. What exactly is Quine trying to do?

It would make a lot of sense for a philosopher to be skeptical about whether the question ‘do mathematical objects *really* exist?’ makes sense. If we take metaphorical practice of talking about chips on people’s shoulders on one hand and scientific practice of talking about bacteria on the other hand and then ask which of these talk of numbers is closest to, it would be very reasonably reject the question as under-defined, and perhaps nonsensical. If this kind of thing were Quine’s point a) he probably would have said so, being the clear fellow he was and b) he wouldn’t have distinguished between metaphorical/eliminable talk and literal scientific discourse, but would have allowed that all the kinds of things which we are inclined to believe in exist. Someone who held this kind of anti-metaphysical view might say “Sure, there are numbers and atoms and tables and chips on some unfortunate people’s shoulders. Of course, chips on shoulders are very different from woodchips, and numbers are very different from atoms, but all of these exist. Any further scruples about real existence are metaphysical nonsense.” I think this would be a respectable position, but I’m sure we can all agree that it wasn’t Quine’s.

On the other hand, one might think that underneath some amount of philosophical confusion there is a real question, or at least an important distinction associated with anxieties about *real, literal* existence. In this case one would be interested in analyzing (and perhaps sharpening up or correcting along the way) our actual inclinations to distinguish between objects that really exist and verbal fictions, what we are inclined to take as evidence for real existence and the like. But if we suppose that Quine is involved in this project then his recipe for checking whether to accord something ontological status looks like a failure.

If you actually ask a physicist whether their reasons for believing that 12 has an odd factor are of the same kind as their reasons for believing that there is a black hole somewhere (that these claims form parts of well-confirmed scientific theories) most of them will say no. And if we think about what convinces us that new physical objects like fields really exist or that there is an additional planet orbiting the sun, will the answer have more to do with the difficulty in writing empirically equivalent theories which don’t refer to, or with causal or explanatory gaps in these instrumentalist theories (why are the orbits of planets perturbed around that region of space? wouldn’t we need spooky action at a temporal distance to explain how it can take time for one object to feel attraction towards another if we don’t say there is really a field that permeates?). And does our conviction that metaphorical talk like ‘during the meeting Jones behaved like he had hyperactive ants in his pants’ doesn’t carry an ontological commitment to hyperactive ants come from certainty that this talk could be rephrased in the vocabulary of fundamental physics (the difficulty of giving analyses for natural language terms is notorious) or from the fact that the activities of these ants are being used merely to describe Jones’ uncomfortable fluttering rather than to explain it?

So, while the question Quine introduces, ‘what objects do we need to quantify over in stating our best literal scientific theories of everything?’ is intrinsically interesting, I don’t see what relation it is supposed to bear to the original question ‘what is there?’. It would seem that Quine means neither to do away with the ontological question as metaphysical nonsense nor to offer a plausible philosophical analysis of it.