Mathematics is an iceberg on which the Titanic of modern empiricism founders. It is good now and then to remind ourselves why, and Gottlob Frege’s famous critique of John Stuart Mill in is a useful starting point. Whether Frege is entirely fair to Mill is a matter of debate. Still, the fallacies he attributes to Mill are often committed by others. For example, occasionally a student will suggest that the proposition that 2 + 2 = 4 is really just a generalization from our experience of finding four things present after we put one pair next to another – and that if somehow a fifth thing regularly appeared whenever we did so, then 2 and 2 would make 5.
A comparable thesis from Mill that is criticized by Frege is the claim that the proposition that 1 + 2 = 3 is made true by our experience of finding that a group of objects that looks like OOO can be separated into two groups that look like O and OO. Frege jokes that in that case it is a good thing that all the objects in the world aren’t nailed down, otherwise we wouldn’t be able to separate them and thus it wouldn’t be true that 1 + 2 = 3. In fact, of course, 1 + 2 = 3 would still be true even in this scenario, in which case Mill’s account is wrong.
Someone might accuse Frege of begging the question here, but he is not. Consider again my imagined student’s suggestion that if whenever we put two pairs of things together we regularly found that this left us with five things, we would judge that 2 + 2 = 5. A little reflection shows that this is not necessarily the case. For there are at least two ways we might describe such a scenario. We might, as the student proposes, characterize it as a world in which 2 + 2 = 5. But we might instead characterize it as a world in which 2 + 2 = 4 but where there is a strange causal law operating that ensures that bringing two pairs of things together to make a collection of four will immediately generate a fifth thing.
Now, which of these is the correct description of the student’s scenario? Experience itself cannot tell us, because any set of experiences is consistent with either description. I would say, and Frege would say, that we can know a priori that the first description is wrong, because the proposition that 2 + 2 = 5 is just nonsense. But put that aside for present purposes. What matters is the fact that we can make sense of the difference between these two alternative descriptions of the scenario despite the fact that experience cannot tell in favor of one rather than the other. And that entails that there is more to the content of the proposition that 2 + 2 = 4 than a mere description of what we happen to experience.
That, I submit, is Frege’s point. The proposition that 2 + 2 = 4 says more than merely that putting two pairs of things together will regularly give you four things, because we can describe a scenario in which it is still true that 2 + 2 = 4 even when putting two pairs of things together will regularly give you five things. That scenario would not by itself entail that 2 + 2 = 5, as opposed to merely entailing the operation of a weird causal law. And by the same token (and to return to Frege’s own example), the proposition that 1 + 2 = 3 says more than merely that a collection that looks like OOO can be separated into parts that look like O and OO, because we can describe a scenario in which no such separation is physically possible and yet it is still the case that 1 + 2 = 3. Mill’s account fails to capture even the meaning of the proposition that 1 + 2 = 3, let alone the grounds for judging it true.
A second and related problem with Mill’s view, notes Frege, is that it cannot account for examples that don’t involve collections of physical objects which we might know via sensory perception and separate into smaller parts. He gives the example of there being three methods of solving a certain equation. Methods of solving an equation are not physical objects that we might literally perceive to be lumped together as a collection, or which we might physically separate into parts (one part consisting of two of the methods, with the other part being the remaining method sitting off by itself).
A third problem is that Mill’s account is a non-starter when applied to facts concerning large numbers like 777,864. Obviously, it is absurd to suppose that our grasp of the proposition that 770,001 + 7,863 = 777,864 is grounded in experience of finding that whenever groups of 770,001 things are lumped together with groups of 7,863 things, we find that the resulting collection has 777,864 things in it.
Mill says several other things which Frege shows cannot be right. For example, there is the claim that number is merely a property that a bundle of things has, alongside its color, shape, or the like. Hence a pile of ten red pens can be said to be ten, just as it can be said to be red. As Frege would ask, why say that the pile has the property of being ten, as opposed to being twenty (on the grounds that if we distinguish the pens from their pen caps, we get twenty things)? Or why not say that it has the property of being in the billions (on the grounds that we get such a number when we distinguish the particles out of which the pens are composed)?
Furthermore, Frege points out, 1,000 grains of wheat remain 1,000 grains even after they are sown far and wide and no longer form a bundle. Nor would Mill’s account explain what the number 0 is, since it obviously can’t be a property that a pile of pens or a bundle of grains of wheat has, the way Mill thinks being ten and being a thousand are properties of such collections. And then there is the fact that we can apply number to things that are not physical objects that might be lumped together into a bundle (for example, the number of ways to prove a theorem, the number of concepts one entertained Wednesday morning, or the number of events occurring right now).
Mill also says that even 1 = 1 can be false, insofar as one one-pound weight does not always weigh exactly the same as another. This is like the fallacy committed by the student who thinks that 2 + 2 = 4 is merely a generalization of the observation that when we put pairs of things together we typically find that the result is a collection of four things. As Frege says, it simply gets wrong what a claim like 1 = 1 is asserting. It isn’t an empirical claim to the effect that, as a matter of contingent fact, any item that we happen to characterize as weighing one pound will always be exactly equal in weight to any other item we happen to characterize as weighing one pound. One reason why this is a mistake is, of course, that typically we are only approximating when we characterize something as weighing a pound. But the deeper point is that, even if we were not speaking merely approximately, the claim would not be a mere description of the empirical facts. If it turned out that no two things were ever exactly of the same weight, that would not entail that it is false that 1 = 1. It would entail only that this arithmetical truth does not strictly describe anything in the empirical world.
As Frege says, Mill’s error is to suppose that arithmetical claims are inductive generalizations from particular cases, and to confuse what are in fact applications of arithmetical truths with empirical evidence for those truths. When we stick one pair of apples next to another to yield four apples, we are not assembling one further bit of empirical evidence in a way that gives additional inductive support for a contingent general claim to the effect that 2 + 2 = 4. Rather, we are applying a necessary truth, and one that is already known a priori, to a specific case. And the same thing is true of our application of the proposition that 1 = 1 to the comparison of two weights and the like.
Again, it would be a mistake to accuse Frege of begging the question against Mill. He isn’t stomping his foot and refusing to listen to empirical evidence against a contingent generalization to the effect that 1 = 1. Indeed, it would beg the question against Frege to characterize the situation that way, because his point is precisely that the proposition that 1 = 1 is not a contingent empirical generalization in the first place. His point is that when Mill characterizes arithmetical statements that way, he is changing the subject. He is no longer talking about the proposition usually expressed by “1 = 1,” but rather about some empiricist-friendly ersatz. Mill is really just ignoringthe arithmetical truth that 1 = 1 and talking instead about a very different sort of claim while using the same symbols.
The actual situation, for Frege, is that it is only because we already have an independent grasp of the meaning and truth of arithmetical propositions that we know how to apply them to concrete empirical cases. Frege gives the example of pouring 2 unit volumes of liquid into 5 unit volumes of liquid. We judge that this will yield 7 unit volumes of liquid only given the absence of some chemical reaction or other causal factor that might alter the volume. We don’t “work up” from the specific empirical cases to the general arithmetical proposition but rather “work down” from the arithmetical proposition to a description of what is really going on in the specific empirical cases.