Dummett contra Wittgenstein on Counting

In his article on Wittgenstein’s philosophy of mathematics, Dummett points out that if you count 5 boys and 6 girls and 10 students all together, you will always find that you have made a mistake. It’s not just that you will always say when you finish counting all the students together that you must have made a mistake somewhere. That behavior might be explained by a determination to hold on to certain sentences at all cost by declaring any observations which seem to falsify it to be wrong. Rather, the fact is that if you now scrutinize how you did the counting you will always find that at some point you made *what you would previously have counted as a mistake*. To put things (I hope) even more clearly: Suppose we have sets of tapes which show a person counting the girls in the room then the boys then all the children together. And suppose we show you parts of these tapes separately, so you only see the counting of the girls or the counting of the boys or of the schoolchildren all together, and mark down the segments under which you say a mistake was made. Then the fact is, that all of the tapes which show a person coming to the end result that there are 5 boys and 6 girls and 10 students will turn out to contain stretches which you had previously, not knowing that their results would conflict with the laws of addition, have marked as containing a counting error. At least, this must be true if we suppose that the tapes are sufficiently detailed.

This simple, and almost undisputable point poses a serious challenge to (my version of) Wittgenstein’s philosophy of mathematics. In this section I’m going to explain what the problem is and then say how I think we can get around it.

Let’s start with the intriguing but maddeningly hard to formulate idea from the Tractatus that necessary truths show patterns in our use of language rather than describing very general truths about the world. Not only is the idea immediately plausible and beautiful when we see the examples of it in the Tractuatus (the fact that this cup isn’t full and not full certainly seems to be more an illustration of something about predication and negation, than an articulation of a substantive fact about the cup) but it brings huge epistemological benefits to the field of modal epistemology (i.e. to the philosophical task of explaining our knowledge of possibility and necessity). The previously bewildering question of how we can know things about wildly different possible worlds which, even if they do exist, are completely causally isolated from us is reduced to the more tractable question of how we can know that, whatever experience presents us with, our terms will always function in such a way as to make certain sentences true.

I’m not going to argue that this idea of necessity can be made out and reconciled with the insights of the Investigations here, in order to focus on Dummett’s objection. But if it is the right view of necessity then it should be possible to account for any truth in terms of some combination of substantive, contingent truths discovered empirically, and a priori truths which are mere artifacts of our use of language. In particular, it should be possible to separate mathematics, pure and applied into an empirical, substantive component and a necessary, trivial one component arising from our conventional tendencies to use words.

Now I claim that applied arithmetic throws a spanner into the works, as far as this project is concerned. On the one hand, we have the substantive empirical prediction pointed out by Dummett: if you get 5 boys and 6 girls and 10 children all together it will turn out that you have made (what you would have recognized even prior to getting this surprising result as) a mistake. On the other hand we have the necessity of both arithmetic ‘5+6=11’ and applied arithmetic ‘whenever there are 5 things and 6 other things you have 11 things all together’.

In order to bring out what I think the true shape of this problem is, I want to contrast the case we have here with that of applied geometry. Applied geometry, I claim, factors nicely into a substantive empirically discovered component and a necessary linguistic one. First (not to say first historically) we have geometries as pure mathematical structures each of which might be e.g. a set of points together with a function that assigns a non-negative rational “distance” to each pair of points and gives 0 whenever a point is paired by itself. There are then necessary truths, with no substantive empirical consequences, about what features each of these geometries have. Mathematics provides us with a sort of store house of different geometrical structures, and works out truths like ‘any geometry which counts as having feature F1 also counts as having feature F2’. Then on the empirical side we have physical theories which are connect up empirical measuring procedures (or, alternately the physical properties which these procedures measure) with one or the other of these geometries. We note a kind of sameness of structure between the readings of rulers when we measure from one point to another and the distances assigned by the distance function in a certain geometry, and use this similarity to state a theory about what further measurements will tell us. The latter theory which matches up measurements and physical properties measured with a given geometry is highly contingent. Mathematics on its own entails nothing about how distances between points in the physical world should be related, as we can see from the fact that empirical research is required to determine which of the many geometrical structures in mathematics can be used to discover distance relations in our world.

Given that applied geometry factors this way we can explain how it gives rise to the kind of empirical predictions which Dummett’s objection points out. We know that if you measure the three sides of a right triangle to be 3, 4 and 6 you will turn out to have made a mistake, not because it is a necessary truth that the distances between those three points could not be so related but because we know an empirical law which connects observable properties with geometric truths in a way that forbids it. The prediction is substantive and empirical but not necessary, so we can fit it in with the Tractarian theory of necessity.

But, unfortunately for our theory, the case of algebra is importantly different. I’ll start with the similarities. Pure mathematics gives us different algebras as well as different geometries (take any set of numbers and come up with ‘plus’ and ‘times’ functions that associate each pair of numbers with a single number in the set in a way that’s consistent with certain axioms depending on whether you want a group or a field). We can take Z/6Z, the structure which you get when you do all your adding and multiplying mod 6, as a concrete example of such an alternative algebra. And in both cases we have theories which associate the results of empirical measurements/ the properties thus measured with items in one of these mathematical structures. But is this association contingent and empirically discovered? One feels a strong inclination to say that it is not. We can imagine discovering that a spatial right triangle with sides 3 billion miles by 4 billion miles does not have a hypotenuse of exactly 5 billion miles. But we cannot imagine discovering that there were actually 3 billion male fleas and 4 billion female fleas but only 6 billion fleas all together. The geometry of the world seems to be an empirical and contingent matter, but its algebra seems to be necessary.

For this reason applied arithmetic presents a serious challenge to the Tractarian theory of necessity advocated above. Dummett’s point is an apparent counterexample to the account of necessity as a mere artifact of language, for here is a statement which is both necessary and of substantive empirical import: ‘if you got 5 boys and 6 girls but only 10 children then you will find that you have made a mistake at some point in your counting.’

I propose to answer this objection by analyzing our inclination to say that the world has its algebra necessarily. Specifically, I claim that what makes the difference between the necessity of applied arithmetic and the contingency of applied geometry is the fact that we count things other than physical objects (numbers, quantifiers, reasons) but we don’t measure such things. Applied geometry only has to match with the results of empirical measuring (what will happen if we put a ruler against the third side of the triangle?). But applied algebra is supposed to match with both empirical results (what will happen if we pile these two bunches of fruit together and then count them?) and with certain linguistic and logical applications which swing completely free of any empirical observation (if there are two positive arguments for the view and three negative, critical arguments then there are five arguments all together).

In particular we use numbers to articulate a body of logical, non-substantive, necessary truths of the form ‘If ExP(x)^AwP(w)->w=x and EyEz Q(y) Q(z) ~y=z ^ Aw Q(w)->w=y v w=z and Aw ~P(w)^Q(w) then ExEyEz ~x=y^~y=z^~x=z’. In empirical physical counting we associate numbers with things (under a concept) by applying certain counting procedures like pointing to apples 1 by 1 or stacking up quarters to a certain height, and use the operation plus to predict what will happen when we collect together these items and apply the same tests to the resulting bunch. But in logical applications we associate numbers with combinations of existence and identity statements like the above, and associate the operation of addition with disjunction.

So applied addition performs the mixed job of predicting what will happen if you pile stuff up and articulating necessary and logical truths with no empirical consequences like ‘if there is one F and two Gs and no Fs are Gs then there are three F or Gs’. To demonstrate the difference (and the potential gap) between these roles of applied algebra, I want to consider what our intuitions would be about a physical world worked very differently. Imagine a world with discrete units both in space and time populated with a single kind of tiny moving object. As time passes the objects blip from one place to another, either staying in the same place or disappearing and reappearing in an adjacent block of space. Now suppose that whenever a bunch of these objects head over to the same unit of space the number of objects we find in this place is equal to the original number mod 131. So, for example if 70 objects move in to a given spot from one direction and 70 move in from the opposite direction at the same time we will find that there are 9 in that spot in the next instant of time. We could formulate a physical law stating this and describing the behavior of the objects using Z/131Z. And we could make Dummett-style predictions like ‘if you count 3 objects in one place and 2 in the other and you don’t get 5 when you heap them together you’ll find that counted wrong or some object has anomalously appeared or disappeared’ and the same for combining 100 objects and 40 objects and getting anything other than 9. Now would this be a world where the objects in question ‘only had number mod 131’? I think not. We can imagine that all practical purposes for combining the objects depend on their being in the same unit of space so that both scientific and practical theories used Z/131Z rather than Z. Certainly in this world the norm in contrast to which we speak of objects ‘appearing’ and ‘disappearing’ would be given by Z/131Z’s version of addition. But this would not count as discovering that the world had a different algebra for we would still say that insofar as there were at a given time 70 things in spot A and 70 in spot B there were 140 things in spot A or in B, even if one could never have 140 things in the same place. The logical application has, in a sense, more weight than any empirical application. We think of addition both in terms of counting procedures and non-identity statements, in terms of heaping things in the same spot and disjunction. But our allegiance is ultimately to the latter, logical application of algebra, and since this body of logical truths is necessary the right algebraic structure for counting is necessarily fixed.

Given this idea of the primacy of logical applications of arithmetic I want to explain the apparent combination of necessity and empirical prediction in Dummett’s point as follows. We have an allegiance to applying numbers to logical statements in a certain way (plus for disjunction etc.) and given this logic determines that Z is the right structure to use. Then we decide which activities qualify as procedures for counting a collection of objects on the basis of this combined logic and arithmetic. In the first stage we fixed counting for abstracta so that e.g. 2 gets assigned to a property Q iff EyEz Q(y) Q(z) ~y=z ^ Aw Q(w)->w=y v w=z does. Then we ‘learn how to count’ by looking at the facts about our empirical world and finding out which processes yield the number 2 in exactly the circumstances where we would apply EyEz Q(y) Q(z) ~y=z ^ Aw Q(w)->w=y v w=z. Thus a person’s idea of what’ s involved in counting correctly actually embodies an empirical theory about which physical processes yield give results which match the association of logical statements about existence with numbers, and what sorts of things can go awry to prevent them from doing so. For example, in our world middle-sized objects tend to persist in such a way that you count the number of objects there were at a given time by doing something to them five minutes later. And they tend to be pretty insensitive to being piled together (unlike in the fanciful example above) so that you can often count the A-or-Bs by pushing the As and the Bs together.

Thus the fact that we began with, ‘if you count 5 boys and 6 girls and 10 students all together, you will always find that you have made a mistake’ expresses a necessary truth for the following curious reason. Our allegiance to applying numbers to logical statements in a certain way determines that 2 goes with a logical statement like EyEz Q(y) Q(z) ~y=z ^ Aw Q(w)->w=y v w=z and the like. And this combines with empirical facts about our world to determine just what we consider to be counting correctly or and making a mistake. Let L(5, A) stand for the logical version of ‘there are five As’ Then it’s a fact about language that whenever we are willing to apply L(5, A) and L(6, B) we will apply L(11, A or B). And our counting procedures are empirically, sometimes laboriously, arrived at so that they associate n with P iff we are willing to assert L(n, P).

So, I think that a subtler version of the idea which Dummett originally wants to fend off is correct. It’s not that we insist on denying 5+6=10 at all cost so we call a given counting procedure mistaken just in virtue of it giving that result. Rather we can correctly predict that further investigation will reveal a mistake because it’s a linguistic fact that we would never assent to L(5, A) and L(6, B) and L(10, A or B) at the same time and because our counting procedures are exactly those empirical processes which match our tendencies to apply statements of the form L(n, P) when performed correctly. It’s an empirical fact that you can count by doing such-and-such. But it’s a necessary truth that, given that you can count by doing such and such, if you get the wrong answer you have made a mistake.