Handed into Nickel, 10/6

Indispensability Argument
I. Introduction
In this short essay I will analyze a version of the Quine-Putnam indispensability argument for the existence of mathematical entities. After quickly stating a version of this argument I will turn to analyzing its first premise, explaining what theses are required to make it plausible. Then I will motivate and present a variation on the indispensability argument offered by Michael Resnik. Finally, I will briefly describe how the two premises of the argument could be adapted for metaethics, and indicate that such projects have already been undertaken.
II. The Indispensability Argument
The indispensability argument concludes that mathematical entities exist, and that we know that they exist because mathematical entities form an indispensable part of our best overall scientific theories. This argument depends not only on the view that math is indispensable to our best scientific theories, but also on the truth of a general principle for deciding whether an entity exists or not. So there are two separate premises for the indispensability argument.
P1: (Methodological Claim) All the entities that are indispensable to our best scientific theories exist.
P2: Mathematical entities are indispensable to our best scientific theories.
From these two premises it is a simple, uncontroversial deduction to the conclusion that mathematical entities exist. The second premise is controversial; nominalistic philosophers, primarily Hartry Field, have argued that mathematical entities are actually dispensable to our best scientific theories. In this essay I will not engage in that discussion, however. Rather, I am interested in what theses, besides the two explicit premises, the indispensability argument is committed to. Since the indispensability requirement appears in P1, I think that these implicit theses can be overturned by analysis of P1 alone. I will now turn to analyzing the first premise.
III. The first premise
A. Inference to the Best Explanation
Why would one endorse a principle, such as P1, that advocates belief in entities without direct perceptual confirmation? Perhaps P1 should be rejected in favor of a statement that forbids us to believe that entities exist unless we have perceived them directly. P1 is plausible because it appears to describe a common and reliable form of reasoning that is employed in both everyday and scientific contexts. For example, suppose that water is pouring out of your kitchen wall. By far, the best explanation of the sudden appearance of water is that there is a leaky pipe behind the wall. So you come to the conclusion that there exists a leaky pipe, even though you have not directly perceived a leaky pipe (example due to Field, 15). Likewise, consider a physicist who observes “a vapor trail in a cloud chamber [and] he thinks “There goes a proton” (Harman 6). Though he hasn’t directly perceived the proton, the best explanation of his observation requires that protons exist. So the scientist, without qualms, concludes that he did perceive an entity, a proton.
Both of these examples—the everyday and the scientific—are instances of inferences to the best explanation. An inference to the best explanation provides a reason for belief in Q when our best explanation of some phenomenon would not be possible without Q (Field 15). Our premise, P1, is a principle for making certain, particular inferences to the best explanation. Specifically, P1 allows for making ontological inferences—inferences that certain entities exist—from scientific explanations. Indispensability should just be identified with the criterion we identified for inferences to the best explanation in general; that is, belief in an entity X is indispensable to a theory if our best scientific explanation is impossible without the truth of the sentence “Xs exist.”
There are some philosophers that have resisted inference to the best explanation, denying the validity of such a principle. They prefer a principle such as “inference to the best cause” (Cartwright, as presented in Colyvan 56). This is a total rejection of “inference to the best explanation” as a principle. This would limit the indispensability argument significantly; one would not be able to conclude that mathematical entities exist unless one could argue that mathematical entities can be causally efficacious, which is an untraditional view of mathematical entities. So P1 is committed to the validity of some kind of inference to the best explanation.
B. Naturalism
Our principle, P1, only looks to science to provide an explanation of the world. How do we know to restrict ourselves to science, though? Perhaps more than science should have a place in determining what our best explanation is. For example, perhaps the conclusions that philosophers reach independent of the scientific process should be considered part of our best explanation. Suppose that philosophical reasoning led philosophers to reach the consensus that one’s senses are unreliable; then one should not justify belief in a proposition with knowledge gleaned from one’s senses. So, if our best explanation of the world allowed philosophy to enter the conversation, then in this scenario P1 would be false. That is, our best explanation of the world involves the philosophical view that the senses are unreliable, and this undermines scientific epistemology. We would be unjustified in believing that an object exists only because science tells us it exists; we would first need philosophical approval.
This worry—the worry that philosophy or some other area could epistemologically precede science—is rejected by Quinean naturalism. This naturalism “rejects the view that philosophy”—or anything else.—“precedes science or oversees science” (Colyvan 23). That is, the only explanations are the scientific ones. P1 depends on this thesis, since P1 states that “all entities that are indispensable to our best scientific theories exist.” Without Quinean naturalism, it would be possible for philosophy to overturn the conclusions of science, preventing one from believing that entities exist even though they are indispensable to science. So commitment to P1 involves commitment to Quinean naturalism.
C. Confirmational Holism
We now turn our attention to another aspect of P1. Suppose that it is agreed that some entities are indispensable to our best scientific theory. Perhaps, though we can separate those parts of our theory that refer to these entities from the rest of the theory. If we were able to isolate the parts of our theory that refer to these entities, we could evaluate the evidence supporting these parts of our theory independently. Perhaps we’ll find that the evidence supporting the existence of these entities is weaker than the evidence supporting the rest of the theory. In this were possible, why would we agree to some principle such as P1? If it were possible to evaluate each claim of our theory on its own merits, we surely would rather than accept P1 and risk believing in entities that don’t exist.
Quine’s conformational holism tells us that it is impossible to separate our theory in such a way. As formulated by Resnik, “the evidence for a scientific theory bears directly upon its theoretical apparatus as a whole and not upon its individual hypotheses” (Resnik 166). With the thesis of conformational holism, P1 is reinforced, because any scientific evidence at all also counts for evidence in the existence of entities indispensable for the best scientific theory. So, if mathematical entities are indispensable to our best scientific theory, then every scientific observation supports the theory as a whole, including its mathematical commitments.
Some philosophers doubt the truth of conformational holism. Penelope Maddy argues that one can distinguish between parts of a scientific theory “that are true and parts that are merely useful” (Maddy 281). Once one can distinguish between the truth of parts of a theory, it is possible to argue that an entity being indispensable to a theory does not entail that the entity exists. In fact, Maddy does argue that mathematical entities are indispensable to science, yet only instrumentally useful. She believes that scientists are merely employing math, and not presupposing its truth.
In order to avoid this discussion about the truth of conformational holism, Michael Resnik developed a pragmatic version of the indispensability argument. In place of our premise P1 and P2, Resnik’s argument would have something similar to:
M1: We are justified in doing X, if doing X is the only way we know of doing science.
M2: The only way we know of doing science is by taking mathematics to be true.
(These premises are very compact, and in Resnik’s paper these two premises are expanded into eight). This argument depends crucially on naturalism, but is independent of conformational holism. It also does not seem to be a version of inference to the best explanation—after all, we’re not talking about justified belief, but rather justified activity. So there is a pragmatic parallel to inference to best explanation implicit in Resnik’s argument, a kind of principle that justifies activities necessary for activities whose justification has already been strongly secured.
IV. The Conclusion
Resnik’s pragmatic version of the indispensability argument concludes that “We are justified in taking mathematics to be true.” Is this conclusion significantly different from the conclusion that “mathematics is true”? Resnik argues that it would take a “kind of incoherence” (172) to say of oneself “I am justified in believing in p, but not p.” So he thinks that this incoherence means that it must be valid for a person to conclude that “mathematics is true” from the belief that “I am justified in taking mathematics to be true.”
The conclusion of Colyvan’s preferred version of the indispensability argument has a gap similar to Resnik’s. That is, Colyvan’s argument concludes that “We ought to have ontological commitment to mathematical entities” (Colyvan 11); he doesn’t conclude with an actual ontological commitment. Is there a way to fill the gap between “We ought to believe that mathematical entities exist” and “Mathematical entities exist”? One could rehearse Resnik’s argument in this context; it would be incoherent for an individual to believe that he ought to believe that mathematical entities exist while denying that such entities exist.
V. In conclusion; application to ethics
What if we wanted to apply the Quine-Putnam argument to metaethics? In this short conclusion I want to indicate (too briefly and compactly) how such a project could proceed, and then I want to identify these possibilities with actual philosophical projects.
We began by introducing two premises on which the indispensability argument depends.
P1: All the entities that are indispensable to our best scientific theories exist.
P2: Mathematical entities are indispensable to our best scientific theories.
One can translate this argument into metaethics by altering either P1 or P2. The most direct way would be to defend the following premise:
ME2: Ethical entities are indispensable to our best scientific theories.
Nicholas Sturgeon and the Cornell Realists defend the view that our best explanation is impossible without reference to ethical entities. They attempt to do so within a naturalistic framework, and so their project is quite close to the defense of ME2. Of course, in order to properly defend ME2 one would either have to discuss whether our best scientific theory depends on actual scientific practice (and if that’s what “best scientific explanation” means then one has to argue that ethicists are scientists, or one has to give up). One could also try to fine-tune the indispensability argument in mathematics to come up with a plausible version that is more easily adaptable into ethics.
Another way would be to adapt P1 for metaethical use. We noted that P1 is dependent on some kind of inference to the best explanation. But why should we believe in those entities that are necessary for achieving the best explanation? Perhaps there is something special about explanation, some explanation for our deep commitment to having the best explanation. But once we know why we are deeply committed to having the best explanation, we can ask whether there are any purposes/ends that are in the same boat at explanation. Is there anything else that we are so deeply committed to, that we may be justified in believing in that which is indispensable to that end? This is the approach that David Enoch takes (Enoch 34). He argues that believing in X because it’s indispensable to our best explanation is only valid if believing in X because it’s indispensable to our deliberative project (the task of deliberating over our actions) is. On that basis he gets a modified version of P1,
ME1: All the entities that are indispensable to our deliberative project exist.
He then argues that belief in ethical entities is indispensable to the deliberative project.
So, our analysis of the indispensability argument for mathematics yields at least two routes for applying a form of the argument to metaethics. Further, there are two projects that take these routes.

Notes on Field, "math realism and modality"

p.14 "An indispensability argument is an argument that we should believe a certain claim (for instance, a claim asserting the existence of a certain kind of eneity) because doing so is indispensable for certain purposes (which the argument then details). In this section I will focus on one special kind of ind argument: one involving indispensability for explnations. (There is some discussion of other kinds of indispensability argument in several of the essays in this volume, especially essays 3 and 7). To rely on this special kind of ind argument is to rely on a principle of inference to the best explanation. Some such principle seems to underlie much of our knowledge of the physical world.

It seems to me that most of us accept the principle of "inference to the best explanation" in the sense that this principle (or something pretty close to it) governs our ordinary inductive methodology.

The principle of inference to the best explanation makes no discrimination among these three cases: if a belief plays an ineliminable role in explanations of our observations, then other things being equal we should believe it, regardless of whether that beleif is itself observational, and regardless of whether the entitires it is about are observable. That I think is the metholdogly we (nealry) all employ and I think it would be unwise to change it. The fact tat the principle doesn't discriinate over whether the explanation is obesrvational (or whether it postulates unobservable entities) stands up well to reflection: intuitively, the observational nature of the explanation shoul dmake no difference in an inference to the best explanation. After all, in any case where we rely on inference to the best explanation our belief goes beyond what we have observed; the fact that one belief could be fairly directly tested by observation while the other couldn't seems to have no relevance to their evidential status when such an independent test has not been made. (When the independent test has been made--when th eleak behind the wall has been directly observed--then we need no longer rely on inference to the best explanation. When we do rely on inference to the best explanation our belief s go beyond th observations we have made, and my point is that the difference with respect to possible observations that haven't been made is irrlevant to our actual evidential situation.

I think that this sort of argument for the existence of mathematical entities (the Q-P argument, I'll call it) is an extremely powerful one, at least prima facie. It should be noted that the 1p argument is not merely that just as there are good explanations in which the posulation of unobservables is essential, so too are there good explanations in which the posulation of math entitie siessneital, so that if inference to the best explanation licnesess one is licenses the other. The argument is stronger in that it says that the very same explanations in whichthe posulation of unobseravbles is essential are explanations in which the posultaton of math enties is essential: math enters essentially into our theory of (say) electrons.

"At present of course we do not know in detail how to eliminate mathematical entities from every scientific explanation we accept; consequently I think that our inductive methodology does at present give us some justification for believing in math entities. But this brings me to my second point, which is that justification is not an all or nothing affair. The belief in math entities raises some problems which I and many other s believe to be fairly serious (I will briefly discuss two of these problems in the next section ...It seems to me that the most satisfying explanations are usually intrinsic ones that don't invoke entities that are causally irrelevant to what is being explained. Extrinsic explanations ar e acceptable but it is natural to think that for any good extrinsic explanation there is an intrinsic explanation that underlies it. This principle seems plausible indpenddently of anti-platonist scrulples...