Sharon Berry

Contingent Applications: The Role of Mathematics in Science

            In a previous section I addressed criticisms of my ‘practice-projection’ view of mathematics arising from the sense in which mathematics entails certain necessary (and apparently empirically verifiable) truths. I have argued that the role of logic within mathematics can account for the necessity of truths like ‘if you have three apples you have at least two apples’ and ‘if you have three apples and two oranges and no other fruit then you have five fruit’ and (in a different way) ‘if the car is accelerating at X rate from Y initial velocity for this amount of time then it will travel Z distance’.

Now I want to address an objection which arises from the role of mathematics in the expression of even contingent physical truths like the law of gravity. The objection goes like this: we have reason to believe that human practices of talking about numbers relate us to objective external entities (as opposed to being merely a matter of working out the full consequences of certain informally characterized scenarios) for the same reason that we have reason to think that our talk about electrons relates us to objective external entities. Our best scientific theories talk about these objects. These theories are well confirmed so we have reason to believe that they are true. And if they are true then numbers and electrons exist. Thus we have reason to believe that numbers and electrons exist.

The heart of this kind of argument for realism is the claim that theories whose truth requires the existence of a certain kind of object (realist theories) are better theories, scientifically speaking, than those whose truth doesn’t require the existence of that kind of object. Obviously, the plausibility of this claim depends on the kind of object in question, and the kind of evidence available to us. For this reason, I want to start by looking at two (toy) examples of the relevant kind of comparison between a theory which is ontologically committed to the existence of a certain kind of object and a nominalistic rival – one in which the argument pretty uncontroversially succeeds and one in which it equally uncontroversially fails.

To make things even simpler, I will restrict my attention to alternative theories of a certain kind: what we might call ‘the boring nominalization’ of the theory in question. If you have a theory to explain the observed behavior of Xs which says that Xs, Ys and Zs interact in a certain way w, then we can always, trivially, produce an empirically equivalent theory which is not committed to the existence of Ys. This is just the theory which says that Xs and Zs behave as they would if (there were Ys and) Xs Ys and Zs interacted in way w. Thus we will look at one case in which (its pretty uncontroversial that) the committed theory is better than its boring nominalization, and one in which it is not.  

I’ll start with a case where I think anyone but the most extreme of positivists would agree that an ontologically committed version of a theory is better than its boring nominalization. Suppose you know how rabbits typically behave and the foods they typically eat. When you start on a walk through a garden you notice that certain shrubs and vegetables are nibbled away, and you notice that these are exactly the ones which rabbits eat. You then form the theory that there was a rabbit in the garden and it has eaten away the damaged plants and as you continue to walk through the garden everything you observe fits with the theory: the missing plants are always ones that rabbits eat. Here is a case where, surely, the empirical confirmation of the theory gives us reason to believe in rabbits, and we can see why if we consider how you might respond to a (boring) rabbit-nominalist.

Suppose I am such a nominalist and I propose the following nominalized version of the theory: ‘Those plants are missing in the garden which would have been missing if there had been a rabbit’. We shouldn’t underestimate the merits of this theory. It unifies your past observations by fitting them into a simple pattern. It provides a kind of explanation for particular observations by fitting them into this pattern: ‘why is that lettuce nibbled?’ ‘lettuce is one of the foods that rabbits eat, and in this garden all the plants are nibbled which would be nibbled if a rabbit were let loose in it’. And it makes concrete predictions about which further shrubs will be missing – exactly the same ones which you made with your theory, and verified. Often, especially at the level of fundamental theories, this kind of explanation is the best we can hope to give: to unify observation, make accurate prediction and explain the particular by fitting it into a general descriptive pattern.

Nevertheless, few people would doubt that your theory is an improvement on mine – that nominalizing away the rabbit comes at a significant cost. We wonder what did cause the defoliation, and why (if there was no rabbit) there should be just this pattern of defoliation in the plants. In some cases, of course, no such explanation can be given: some regularities are just basic laws. But in this case, where we have a substantive explanation available (in the ontologically committed theory) for the pattern which the boring nominalization merely describes it seems clear that the merely descriptive nominalist theory looses out.

Now for a case where I think it is pretty uncontroversial that a theory is equivalent, if not worse than its nominalization. Suppose two plumbers are studying the current at various points in an electric circuit with a certain configuration C and one of them puts forth the following theory: ‘The current in the circuit varies in the same way as the flow-rate of water varies as it goes through a pipe in configuration C’ where the resistance of the pipe at points in C’ corresponds to the resistance of the circuit at corresponding points in C’. And perhaps they test some of the consequences of the theory by measuring the current in the circuit and these predictions come out correct. Now consider the boring nominalization of this theory. This says that the current varies at it would if there were a pipe in configuration C’ and the current in the circuit varied in the same way as the flow of water through C’. It is not committed to the actual existence of a pipe in configuration C’, but rather it claims that the current in the circuit behaves in a way that corresponds to the way that water would flow through such a pipe.

Here I think its pretty uncontroversial that the original ontologically committed version of the theory has no advantage over its boring nominalization. Indeed it seems unwarranted to go from the usefulness of considering how water would flow through a pipe of configuration C’ in making correct predictions about the current at various points in the circuit to the conclusion that there is in fact some stretch of pipe somewhere in just this configuration. In order to explain the success of the theories which predict current by reference to a pipe in configuration C’ is not that such a pipe exist, but only that there be definite facts, which both plumbers know and agree on about how water would flow though such a pipe if it existed. To suppose that such a pipe actually exists gives rise to the same predictions about current and offers no explanation of why the current is what it is at various points in the circuit. Thus, I think its pretty uncontroversial that in this case there can be no argument from the superior theoretical virtues of the more ontologically committed theory to the real existence of the objects which it posits.

So the million dollar question is this. Generally: what makes the difference between the two kinds of cases? When is an ontologically committed theory to be preferred to its boring nominalization? And, more specifically: where do mathematical physics, biology, economics and the like fall with respect to the distinction?

a.       causal gaps

One theory would be that in order for an ontologically committed theory to earn its keep it needs to posit a causal role of some kind for the entities in question. This is certainly one difference between our two cases: according to first theory the rabbit causes the plant damage while even if the second one is right the water pipe plays no role in causing the current to be what it is at various points in the circuit. Supposing that there was in fact no rabbit leaves us with a mysterious causal gap: we wonder what did eat the plants and why exactly the plants which rabbits eat should have disappeared if there were no actual rabbits involved. But supposing that there is in fact no stretch of pipe in configuration C’ leaves no such mystery. Whether one believes the ontologically committed version of the theory or its nominalization one might wonder why the analogy holds between the behavior of the pipes and that of the circuit. But positing that a stretch of pipe in configuration C’ actually exists does nothing to explain this mystery. If the committed theory said e.g. that the increase or decrease in the flow of water through the pipes caused a corresponding change in current in the circuit then it would. And if this was the best theory it would give us reason to believe that some corresponding pipe existed to fill this causal role. But as it is, even the ontologically committed theory gives the pipe no causal role, so no gap is left when we nominalize it away.

On this view we can view scientific ontology as the creation of an expanding causal net. We start with some domain of objects and a theory of their default behavior and of how they interact with one another. This web is extended as new objects are posited to account for discrepancies between what the current theory predicts and what is observed. A rabbit is posited to explain why vegetables disappear since our present theory says that when there are no causes acting on the plants in a garden they remain. A planet is posited when all the causes which we know to be acting on a celestial body don’t account for its behavior. Then once these items are in the net, we try to account for their behavior in turn using the objects which we already believe in and perhaps positing others as required.

I won’t go into too much detail about this view because I want to focus on the more specific question of the status of mathematical objects, and this kind of theory has only a weak bearing on the subject. For, even if we accept that one way for new objects to earn their keep is for them to fill a causal role and that by hypothesis mathematical objects are supposed to be causally inert this doesn’t show that we should be nominalists about them. Perhaps there are other roles which an object can play in an ontologically committed theory that give us reason to prefer this theory to its boring nominalization. At least, someone with realist intuitions about mathematical objects will likely claim this. A careful survey of all the different kinds of objects uncontroversially posited by science might hope to show that all of these can be fit into the ‘expanding causal web’ picture mentioned above. And this would give some reason to suppose that filling a causal gap was not only a sufficient but also a necessary condition for an object to acquire ontological status. This would cut against the theory that mathematical objects’ role in scientific theories gives us reason to believe that they exist in an external (as opposed to merely practice-projecting) way. But I’m very far from being in a position to give such a theory.

b.      Causal explanation vs. subsumption under a general rule

Instead, I want to think about how we know that in the second kind of toy case we do not have reason to prefer the committed version of the theory – to believe there’s a pipe in configuration C’. This might lead us to a necessary condition for a theory’s giving us reason to believe in the existence of an object.

I will start by invoking an intuitive contrast between two (possibly not exhaustive) kinds of explanation: causal explanation on one hand and explanation by subsumption under a general rule on the other hand. In first case of the rabbit in the garden we had a theory which provided both kinds of explanations. It both gives us a general rule predicting which plants will be nibbled, and posts a cause for each nibbling. Contrast ‘this cabbage was nibbled because rabbits like cabbage and all the rabbit-favored plants were nibbled’ and ‘this cabbage was nibbled because a rabbit came and nibbled on it’. Then, when we turned to the nominalist version of the theory we saw that the latter causal explanation was removed while the explanation-as-subsumption-under-a-general-rule remained.

In contrast in the second case (I claim) we have only explanation as subsumption under a rule. As noted previously the plumber’s theory does not say what the cause for the different current flow in various parts of the circuit is. In principle there might be some third kind of explanation (neither giving a cause nor relating to a law) which this theory could give instead, but as far as I can tell there is not. Rather, this theory explains particular current measurements by showing them to be instances a simple rule which describes such measurements. Such explanations can be informative and even surprising (numerous applications of a few simple but general laws can lead to unexpected results) and in the most fundamental cases this is probably all the explanation that can be given.

            Nevertheless I think that this kind of explanation can never justify us in adding an object to our ontology. When a theory only offers subsumption style explanations it amounts to a mere description of how some known entities will behave. It gives us a sort of algorithm for working out correct predictions about what will happen, but it offers nothing more by way of explanation for why any particular observation turned out a certain way than a demonstration of how the algorithm in question actually gives this result. Thus the objects that are invoked to state such a merely descriptive theory are fungible. The plumber could have given the same description of what currents will be measured at what places by talking about orange juice flowing through an appropriately shaped tunnel or ectoplasm moving though spirits (assuming the plumbers shared some definite idea about how this was supposed to work) or –what is particularly relevant to our case– about what would happen if there were water flowing through an appropriately shaped tunnel.

            Thus the predictive success of a merely descriptive theory gives us no reason to believe in whatever objects are invoked in articulating it. The plumber’s successful use of his theory to make correct predictions about what current will be measured gives us no reason to think that his views about water are correct or even that there is such a stuff. All we need to suppose is that the plumber has a particular definite belief about how water moves through a C’ shaped pipe and that these supposed facts match the actual facts about current.

            If this is right then the following is a necessary condition for a theory to give us reason to believe in a certain kind of object. The theory must invoke these objects in a capacity that goes beyond merely computing what will occur. Frequently, this role is a causal one though there may be other possibilities. But in any case, the putative objects must offer more than an subsumption style explanation of the events which the theory describes.  

Now let’s think about the role of mathematical objects in physics, biology and the like. Again, I can’t claim to have done an exhaustive survey of the uses of mathematics in science. But in many standard cases its hard to see how mathematical objects could be playing more than a merely descriptive role. Take, as we philosophers love to do, the Newtonian laws about gravity. These laws use mathematics to great effect, and provide many explanations. But are mathematical objects ever invoked to give more than explanation-by-subsumption? The laws of Newtonian physics can tell us why a cannon ball landed where it did by invoking a bunch of mathematical entities like numbers, functions and functions from physical properties like velocity and mass to numbers. And these laws (give rise to notions of causal power and default behavior which) can certainly justify belief in new objects, from planets to tiny particles in the air. But what role do mathematical objects play beyond their merely descriptive use in stating the theory which talk about supposed numbers, or what mathematical structures would have to be like if they existed does equally well? None that I can see.

c.       what does need explanation

I have just argued that I don’t think that the role of mathematics in the formulation of well confirmed empirical laws gives us the same kind of reason to believe in mathematical objects which we have to believe in physical objects. This is not to say that the role of mathematics in science imposes no constraints on philosophy of math. Firstly, as noted above, in order for math to play the striking role which it does in scientific theorizing there must be definite, agreed on, facts about what status is assigned to different mathematical propositions. Consider, for example, the case of the plumbers mentioned above. We don’t need to suppose that there is actually a stretch of pipe in configuration C’ in order to explain the predictive success of their theory about electrical currents. But we do need to suppose that the plumber has some definite idea (it needn’t be a true one) about how such a pipe would behave. Otherwise how would he get the definite, confirmed, predictions about current in the circuit from his theory? Similarly, without some definite shared idea about what is supposed to be true in the realm of mathematical objects, a physical theory couched in mathematical terms would be unable to communicate a definite description from one person to another. So, even if one considers mathematics to be more like communal make-believe than like scientific discovery of the objects around us, there must be definite facts about what can and can’t be said about this make-believe and different people, who aren’t in communication with one another must all be inclined to extend it in the same way.

Secondly there is the question of why it is that the particular kinds of structures found in mathematics prove to be so useful in scientific investigation. Part of this usefulness of mathematics, this echoing of natural structures in mathematical ones can, on my view, be explained by the role of scientific descriptive considerations (among many others) in determining which informally characterized structures mathematics investigates. But the striking usefulness of mathematics goes beyond this. Mathematical structures which gained interest for entirely different reasons, of beauty or intra-mathematical usefulness have often proved to have unexpected physical applications. This deep match between science and mathematics has even led some philosophers to believe in a divine designer (cf. Mark Steiner ‘The Applicability of Mathematics as a Philosophical Problem).

Personally I think that a more natural explanation can be given if we focus on the relationship between two notions of ‘sameness’ or ‘continuing on in the same way’. On the one hand we have the idea of sameness given to us by induction (and arguably originally determined by evolution), the sense in which humans can’t but expect things to behave in the future in the ‘same way’ as the have in the past. On the other we have the sameness of rules and language: training in applying a certain rule gives people an idea of what would count as continuing on in the same way (and it gives most people largely the same idea). I suspect that the way humans do induction determines that the worlds which finite experience could ever lead us to believe to be actual form a small subset of the worlds which are (and we think are) metaphysically possible, and that these worlds tend out to be either the same, or a subset of, those worlds which can be economically described in the language of mathematics. But I will give this argument in full in a later section.

 

            In this section I have presented the question of whether we have empirical, scientific reason to believe in mathematical objects and offered two styles of arguments that we don’t. The more ambitions one claimed that an ontologically committed theory is only to be preferred over its boring nominalization when it posits objects that help causally explain the regularities which it and its nominalization agree in predicting or fills some other causal gap. If this is right than the causal inertness of mathematical objects will mean that our use of them in empirical scientific theories could never give us reason to believe in them: being a causal they could never fill a causal gap. The other, more modest, argument consists in a sort of challenge to the mathematical realist. We can easily think of cases where putative facts about a certain domain of objects are merely invoked to state a predictive theory so that the confirmation of the theory gives us no reason to believe in the actual existence of these objects. The challenge then, for the person who wants to think of mathematical objects as part of the ontology given to us by physics, is to say what further role mathematical objects play.

The most obvious answer to this challenge arises from Quine’s radically different way of thinking about the question of what objects science gives us reason to believe in. This will be the topic of the next section.