List of things to read

Justin Clarke-Doane: Disagreement in Mathematics
David Enoch: An Argument for Robust Metanormative Realism (at least part of it)
Soloman Feferman: The development of programs for the foundations of math
Peter Koellner: Truth in math: the question of pluralism

Need to find: paper on Mackie's argument from disagreement, and need to think more carefully about Mackie's argument from queerness and how that would apply to mathematical objects. Need to think more about the indispensability argument.

Here's one way that the proposal could look:

Introduction: similarities between the challenges to realism in math and ethics, and the similarity of approaches in defending realism, this thesis aims to show the fruitfulness of thinking this through more systematically by looking at arguments in math and ethics that haven't been applied to the other, thinking about whether they can be applied and what the major differences moving things around is.
Chapter 1: Argument from disagreement and queerness in Math
Chapter 2: Indispensability argument in ethics
Chapter 3: What are the significant differences between math and ethics that are exposed from this analysis?

Mackie's response to my last post

Mackie, in his article "The Subjectivity of Values" brings five arguments in support of his thesis that values are subjective, and not objective (and that people's common sense views are in error in this regard). The first one he calls "The argument from relativity", and it is essentially the view that I tried to criticize in my last post. The argument goes like this: the best explanation of the variance in moral belief across cultures is not that there is some objective value that people have more or less epistemic access to. Rather, the best explanation of this observed phenomenon is that people's moral beliefs are based on their ways of life and cultures. That is, that they're subjective.

I argued that this is an unfair argument, because no one denies that ancient history is objective even though there is a wide variance. Why not say that the disagreement in ancient history shows that the subject matter itself is subjective?

Mackie briefly raises this point and tries to counter it. He writes,

"Disagreement on questions in history or biology or cosmology does not show that there are no objective issues in these fields for investigators to disagree about. But such scientific disagreement results from speculative inferences or explanatory hypotheses based on inadequate evidence, and it is hardly plausible to interpret moral disagreement in the same way. Disagreement about moral codes seems to reflect people's adherence to and participation in different ways of life. The causal connection seems to be mainly that way round: it is that people approve of monogamy because they participate in a monogamous way of life rather than that they participate in a monogamous way of life because they approve of monogamy."

It seems to me that this argument is fishy. He's comparing the measured opinions of historians versus the beliefs of the masses. That seems to me unfair. I think that if you look at the way trained ethicists go about thinking about ethical problems it's much closer to making "speculative inferences or explanatory hypotheses based on inadequate evidence" than just assuming whatever their culture does.

It may be that people in general believe things based on their culture. But the real historical or scientific parallel to what he's observing in science would be the widespread belief in the American myth that varies with the French or British founding myth. Indeed, among the public we find wide variance in their opinions about the past. The best explanation is that these people aren't accessing some objective past, but rather are just believing what they are because of culture and background. But that's not the point! People aren't the ones that we should expect to be looking carefully at the evidence! There might very well be something objective about the past, but most people don't look for evidence about the past. Historians do. And ethicists might do so as well, even if there is widespread disagreement about what is good or right.

Countering a possible argument

Why is it that mathematical statements seem so darn capable of truth and falsity, while ethical ones seem so darn incapable of being absolutely true or false?

One observation that might seem relevant at first (and clearly is getting at something) is the observation that there is much disagreement about any particular ethical issue, while there is broad agreement about most things in math.

But is widespread disagreement a sign that something isn't objective? Sometimes it seems to be. For example, we all seem to think that taste isn't something true or false. It's not FALSE that chicken tastes bad. That's just one's opinion. And we know that it's an opinion because everybody has a different one. But other times when there is broad disagreement the subject still seems extremely objective. For example, ancient history or the nature of the universe. Both of these fields of study have a large degree of disagreement. Still, we're not in the realm of opinion in ancient history and science of the universe. Rather, the questions are very hard and we don't have so much evidence--that's why everybody who cares has their own opinion!

So, by itself, the observation that there is much disagreement in the field of ethics (and among people in the world as well) indicates nothing, because we find much disagreement both in matters of taste and in matters of history and science.

One reason why existence matters

Let's consider two statements: "Michael has 13 blue hats" and "Tom Sawyer has 13 blue hats."

The first one might be true, or it might be false. It would depend on whether I have 13 blue hats or not. There's a fact of the matter--it's either the case or it is not. It's objective, so to speak. And if you felt that the sentence was true and I believed it to be false, that disagreement would matter. After all, we can't both be right. And then we could begin to offer arguments and counter-arguments to attempt to establish the truth and falsity of that statement.

The second sentence is...messier, to say the least. For starters, it's not at all clear that the sentence is either true or false if Mark Twain didn't mention anything about it in his book. That is, if we don't know how many hats Tom Sawyer has from the author, then the disagreement about how many hats he has doesn't seem to amount to very much. We could disagree, but there's no reason for thinking that we'll get anywhere. The discourse about Tom Sawyer isn't objective.

What's the difference between these two sentences? It seems to be that the difference is just whether the subject of the sentence exists or not. Michael exists; Tom Sawyer doesn't exist.

This is true in math as well. Suppose that numbers do exist (and they are abstract objects). Now, it may seem that this is completely inconsequential. After all, we're talking about abstract objects here, and it's very unlikely that I'll trip over a number of a set anytime soon (the point: they can't cause stuff to happen). So does it matter whether they exist or not? It seems, from the example above, that if that which is being referred to in a sentence exists, then that sentence is capable of truth and falsity; otherwise, not. Then mathematical statements can easily said to be true as long as that which they discuss exists. It seems that mathematical statements talk about numbers and sets, so it would seem that numbers and sets need to exist in order for mathematical statements to be capable of truth and falsity.

Now, what other options are there? We could reanalyze the rest of language and see if there is some other factor that makes statements objective other than existence of what's discussed in the sentence. Further, we could accept the claim that existence is necessary for objectivity, but reject that what is being refered to are abstract obejcts such as numbers, and instead insist that what's being refered to are material objects or more acceptable abstract objects (and this is the nominalistic program).

The point is that whether numbers exist or not seems to be equivalent to asking whether mathematical statements are capable of truth and falsity, or if they are not. It seems that what gives sentences their objectivity is their "aboutness." That is, the sentences that are about the world seem to have right and wrong answers (and see Frege on this issue, and Goldfarb's insistence that the attempt to prserve the objectivity of discourse is what motivates Frege's work). So when we ask whether numbers exist as an important question, according to this line of thought what we're really asking are several questions: (1) Is math objective, capable of truth and falsity? (2) What gives it this objectivity? Is being "about" something real what makes it objective? (3) Are the "about" things numbers, or are the objectivity-granting things something besides numbers?

Why does existence matter?

Need to reflect on this, learn more about it.

Ethics and Indispensability Arguments

Right now I'm reading about indispensability arguments in math. On the agenda right now is:

(1) Hilary Putnam's elaboration on his version of it in "Philosophy of Logic"
(2) Hartry Field's criticisms and objections to the Putnam-Quine thesis
(3) Thinking what it would take for there to be a similar argument possible for ethical objects/concepts
(4) Understanding why such an argument has not been voiced

I'm on (1) right now, we'll see how it goes.

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.