In this post I'll quickly go over the Quine-Putnam indispensability argument in math, and describe two ways that I can think of applying this argument to ethics.
Here's a version of the indispensability argument (and it's sorta sloppy with the ideas, but hopefully clear)
(1) We should believe that something exists if, when we describe the world with our best scientific theories we are forced to speak as if that something exists.
(2) When we describe the world with our best scientific theories we are forced to speak as if mathematical entities exist.
Therefore:
(3) We ought to believe that mathematical entities exist.
Now, how could we apply this argument to ethics? In short, we can change premise (1) above, or we can change premise (2). Here's how.
Version #1:
(1) We should believe that something exists if, when we describe the world with our best scientific theories we are forced to speak as if that something exists.
(2) When we describe the world with our best scientific theories we are forced to speak as if ethical entities exist.
Therefore:
(3) We ought to believe that ethical entities exist.
Would it shock you to learn that such arguments have been made? I'm still at a very low level of familiarity with this kind of argument, but it's my impression that Sturgeon makes an argument very similar to this one. He argues that we should believe that ethics is objective, because if we don't believe that ethical statements are literally true and objective we are at an explanatory loss! I need to read more about this, and I also need to read about the relationship between the indispensability argument and inference to the best explanation.
Here's the second, less audacious version of the argument applied to ethics:
Version #2
(1) We should believe that something exists if, when we do something that we are really committed to taking seriously we are forced to speak as if that something exists.
(2) When we do that something that we are really committed to taking seriously we are forced to speak as if ethical entities exist.
Therefore:
(3) We ought to believe that ethical entities exist.
In short, this version can get off the ground by asking why, in the original Quine-Putnam version of the argument, we let science decide what we're committed to existing? Then you say, "the reason we give science such a privileged position is because X". The next step is to argue that there is something else that is X, and that something else makes us committed to ethical entities.
Just to make something up that sounds the tiniest bit plausible: Why do we say that science has the ability to determine what we're committed to? Because science helps us live our lives in safety. If we weren't to take science seriously then our lives would be unsafe! But that means that anything else that is indispensable for living our lives in safety should have the same status as science, with respect to its ability to determine what we believe in. Perhaps we believe that ethics is necessary for living our lives in safety--otherwise chaos in society would break out!--so we need to take ethics as seriously as science. And then you argue that you can't talk about ethics without believing in ethical entities, so you're justified in believing in ethical entities. This argument isn't sustainable, but it's the form of the argument, the move that's made, that I want to draw out.
As it happens, David Enoch takes an approach that resembles this one. I need to read him more carefully, but he talks aobut non-explanatory indispensabilities.
So we're in good shape! We've recognized, abstractly, two ways that the argument could go, and we've found some names who make such arguments. Plus (and I didn't tell you this yet) I found a couple articles that review both of these strategies. One of them is by Brian Leiter. So we're making some progress here.
Thursday, September 24, 2009
"What if we're ALL in the matrix?"
So, one of the stereotypes of philosophy is that it's a field that trades in justifying counter-intuitive things that nobody actually would seriously believe. Like, nobody believes that we live in a matrix or that experience is some dream. Right, yeah, but can you, you know, prove it? Well no, and I don't really care to, is the response that comment probably deserves.
But the truth is that the defense of counter-intuitive claims constitutes a lot of philosophy. I mean, that's predictable in a way. If something's intuitive, there's a good chance that it'll be said pretty early on in the discussion and won't be so interesting. And after a while, if you study a field long enough, you begin to lose your intuitions. This is a feeling familiar to philosophers of language and to linguists--you spend hours trying to explain why an analysis of language is bad because it runs counter to our intuitions about language, and by the time you're done you can't remember which one was the intuitive analysis and which was the non-intuitive one.
My introductory textbook in philosophy of math starts by saying that mathematics stands out because its a priori--that is, you don't bring empirical evidence to support the claims of math. And that's standard, and that's intuitive. After all, has anyone here ever looked outside to see if their math problem was correct? That having been said, Quine makes the claim that math is empirical, and there is no such thing as a priori knowledge at all. So math is not a priori, and you do figure out what math is correct by looking at the world. (As a matter of fact, Quine uses this method to throw out large areas of math as being fictional).
One reason I like philosophy is because I like engaging in the practice of shifting my perspective. I think it's really cool to see a plausible version of something that I used to think was really crazy (like empiricism and mathematics, the claim that mathematics constitutes a posteriori knowledge). Moreover, it's important for figuring out where the stable resting points are in philosophy. If you restrict yourself to what seems intuitive, you'll be missing out on a lot of stable points in philosophy, a lot of viable positions.
So I don't mind when somebody says "Maybe we're all in the matrix" even though it's really counter-intuitive and there's very little chance that I'll believe it. OK, that's not true. I do mind. But not because I don't like dealing with possibilities that run counter to my intuitions. I do, and I think it's fruitful, but the matrix thing has just been overdone already. So, lesson: wild ridiculous claims=good. Talking about the matrix=bad.
But the truth is that the defense of counter-intuitive claims constitutes a lot of philosophy. I mean, that's predictable in a way. If something's intuitive, there's a good chance that it'll be said pretty early on in the discussion and won't be so interesting. And after a while, if you study a field long enough, you begin to lose your intuitions. This is a feeling familiar to philosophers of language and to linguists--you spend hours trying to explain why an analysis of language is bad because it runs counter to our intuitions about language, and by the time you're done you can't remember which one was the intuitive analysis and which was the non-intuitive one.
My introductory textbook in philosophy of math starts by saying that mathematics stands out because its a priori--that is, you don't bring empirical evidence to support the claims of math. And that's standard, and that's intuitive. After all, has anyone here ever looked outside to see if their math problem was correct? That having been said, Quine makes the claim that math is empirical, and there is no such thing as a priori knowledge at all. So math is not a priori, and you do figure out what math is correct by looking at the world. (As a matter of fact, Quine uses this method to throw out large areas of math as being fictional).
One reason I like philosophy is because I like engaging in the practice of shifting my perspective. I think it's really cool to see a plausible version of something that I used to think was really crazy (like empiricism and mathematics, the claim that mathematics constitutes a posteriori knowledge). Moreover, it's important for figuring out where the stable resting points are in philosophy. If you restrict yourself to what seems intuitive, you'll be missing out on a lot of stable points in philosophy, a lot of viable positions.
So I don't mind when somebody says "Maybe we're all in the matrix" even though it's really counter-intuitive and there's very little chance that I'll believe it. OK, that's not true. I do mind. But not because I don't like dealing with possibilities that run counter to my intuitions. I do, and I think it's fruitful, but the matrix thing has just been overdone already. So, lesson: wild ridiculous claims=good. Talking about the matrix=bad.
Tuesday, September 22, 2009
An indispensability argument for math
I just need to get this on the table, because it's basic to the topics that I'm currently learning more about. I'm going to quickly present the Quine-Putnam argument indispensability argument here, so that I can build off of it and delve deeper in future posts. Just to experiment with form a little bit here, I'm going to make it into a series of dialogues.
----
A: Zebras exist.
B: What makes you say that?
A: Well, I observed a bunch of zebras. I saw them. They're over there [points to zebras].
B: So what? That's inconclusive. You could be hallucinating.
A: That's hardly the best explanation of the fact that I observed those zebras. The best explanation is that I'm not hallucinating, and that there's a bunch of things over there, and those things are called zebras.
B: Fair point. You're always justified in believing things if belief in them is necessary for the best explanation you have of the evidence.
----
A: Electrons exist.
B: What makes you say that?
A: Well, I observed a bunch of stuff [note: I don't know physics], and the best explanation I have of it is that there is something that exists that we hadn't previously known of. I call it an electron.
B: Oh, fair point. After all, it's the best explanation that you have of the evidence, and that explanation requires you to believe that electrons exist.
----
A: Numbers don't exist.
B: Wait a second, that doesn't sound right.
A: Why not?
B: You believe in zebras and electrons because your best explanation of the world requires your belief in their existence, right?
A: Right.
B: And what does it mean for your explanation to "require you to believe in their existence"?
A: I can clarify that. In order to give my best explanation of the world, I have to speak as if zebras and electrons exist. So if I'm committed to my explanation, I can't really avoid being committing to talking as if zebras and electrons exist. I think that this justifies my belief that zebras and electrons exist.
B: Is the law of gravity an important part of your best explanation of the world?
A: Sure--as long as it's up to date with current research...
B: Now, here's the important question for my argument: when you talk about the law of gravitation in your best explanation, is there any way that you can avoid talking as if numbers exist.
A: Well, errrm, no. How else can I talk about distances?
B: That's my point, then. Your best explanation of the world requires you to talk as if numbers exist. But that's the exact same justification you gave for why you believe that zebras and electrons exist. So if you're justified in believing that zebras exist, then you're justified in believing that numbers exist too.
------
----
A: Zebras exist.
B: What makes you say that?
A: Well, I observed a bunch of zebras. I saw them. They're over there [points to zebras].
B: So what? That's inconclusive. You could be hallucinating.
A: That's hardly the best explanation of the fact that I observed those zebras. The best explanation is that I'm not hallucinating, and that there's a bunch of things over there, and those things are called zebras.
B: Fair point. You're always justified in believing things if belief in them is necessary for the best explanation you have of the evidence.
----
A: Electrons exist.
B: What makes you say that?
A: Well, I observed a bunch of stuff [note: I don't know physics], and the best explanation I have of it is that there is something that exists that we hadn't previously known of. I call it an electron.
B: Oh, fair point. After all, it's the best explanation that you have of the evidence, and that explanation requires you to believe that electrons exist.
----
A: Numbers don't exist.
B: Wait a second, that doesn't sound right.
A: Why not?
B: You believe in zebras and electrons because your best explanation of the world requires your belief in their existence, right?
A: Right.
B: And what does it mean for your explanation to "require you to believe in their existence"?
A: I can clarify that. In order to give my best explanation of the world, I have to speak as if zebras and electrons exist. So if I'm committed to my explanation, I can't really avoid being committing to talking as if zebras and electrons exist. I think that this justifies my belief that zebras and electrons exist.
B: Is the law of gravity an important part of your best explanation of the world?
A: Sure--as long as it's up to date with current research...
B: Now, here's the important question for my argument: when you talk about the law of gravitation in your best explanation, is there any way that you can avoid talking as if numbers exist.
A: Well, errrm, no. How else can I talk about distances?
B: That's my point, then. Your best explanation of the world requires you to talk as if numbers exist. But that's the exact same justification you gave for why you believe that zebras and electrons exist. So if you're justified in believing that zebras exist, then you're justified in believing that numbers exist too.
------
Monday, September 21, 2009
Philosophy is a mess
The point of this post is to show how a discussion that would seem to be restricted to the world of ethics quickly becomes entangled with very different parts of philosophy. (And I'll be paraphrasing Harman throughout). And I take the argument to show that a good philosopher can't study only (say) ethics or only (say) math. The nature of philosophy is such that you need to know your way away around a bunch of different areas.
Is ethics objective--is there a matter of fact that is independent of any one of us, with our biases, whether an action is wrong? Or maybe ethics is (at least) subjective? As I've argued before, this question matters largely because we want to know if we can disagree and agree with what other people think about ethics.
A plausible first attempt at answering this question could sound like this: ethics isn't objective, because the way we justify ethics isn't up to the standards of the rest of our objective knowledge. Take, for example, our knowledge that zebras exist. We know that zebras existing is an objective truth because we can observe zebras with our senses (our eyes see 'em at the zoo or in pictures). But ethics--feh! how do you observe an ethical truth? Now, it's true that some people form ethical beliefs after observing (seeing) something. For example, someone can decide that it's wrong to burn cats after observing a cat burning. But that's not a pure observation, like seeing a zebra. Nope. Rather, what you see and observe is a cat burning, but what you decide is that it's bad, after you observe a cat on fire. But you haven't SEEN bad. There was no "badness" that you saw at the cat burning. What would "badness" look like anyway?!
But, as Hanson points out, this idea runs into trouble. Because you're not just making a claim about ethics here; you're also making a claim about the way we observe zebras.
Is it true that when you see a zebra you've made something like a pure observation, that has nothing to do with how you think? Some people (most people, actually) argue that this is not the case. After all, you know what a zebra is before you look into the world and observe one. You have some kind of theory and understanding about the world in order for you to be able to conceptualize what a zebra is, and that has to come before having an observation of it. To realize this, imagine that I ask you whether bliggles exist. I say that they do, because I've observed them directly. You've never observed them directly, but that's because you have no idea what they are. Would it be right to say that you've observed bliggles when you have no idea what they are? (It turns out the bliggles are what I call the combination you get when you duct-tape a parrot to an elephant, a strange object indeed. I'm not just renaming some normal object.)
The point is: in order to make your ethical argument, you also have to make a completely different argument about the way we observe things in general, and how that counts as justification for our empirical beliefs.
And if things weren't complicated enough, we also have this weird thing called "math." Do we observe all mathematical truths the same way we observe empirical truths? Yes? Then with which of our senses? Our eyes? Sure, we can see that 2 stones and 2 more stones make 4 stones, but have you ever seen 2 million stones be added to 2 million stones, and counted to make sure that we have 4 million stones? No, you haven't. So should we doubt that 2 mil+ 2 mil=4 mil? Well, that makes no sense. So maybe we don't observe mathematical truths. Well, maybe we can be sure about things without observational evidence. But then what about ethics? Maybe ethics gets off the hook, and you can be objective without having observational evidence? Or maybe math gets on the hook, and it isn't objective, just like ethics isn't objective.
So now we have three problems entangled: how do we justify our belief in empirical truths, how do we justify our belief in ethical truths, and how do we justify our belief in mathematical truths? Dissolving this tangle isn't really possible; a good philosopher would need answer to all three of these questions.
So maybe philosophy needs to be done as a whole. This is more the way that philosophy was done in the past than the way that it is sometimes done now. And I think the philosophy that tends to get remembered now, the good stuff, doesn't restrict itself to just one domain. After all, if all of philosophy is really tangled up like this, a good idea in one domain is probably going to be a good important idea in the other domains too.
Is ethics objective--is there a matter of fact that is independent of any one of us, with our biases, whether an action is wrong? Or maybe ethics is (at least) subjective? As I've argued before, this question matters largely because we want to know if we can disagree and agree with what other people think about ethics.
A plausible first attempt at answering this question could sound like this: ethics isn't objective, because the way we justify ethics isn't up to the standards of the rest of our objective knowledge. Take, for example, our knowledge that zebras exist. We know that zebras existing is an objective truth because we can observe zebras with our senses (our eyes see 'em at the zoo or in pictures). But ethics--feh! how do you observe an ethical truth? Now, it's true that some people form ethical beliefs after observing (seeing) something. For example, someone can decide that it's wrong to burn cats after observing a cat burning. But that's not a pure observation, like seeing a zebra. Nope. Rather, what you see and observe is a cat burning, but what you decide is that it's bad, after you observe a cat on fire. But you haven't SEEN bad. There was no "badness" that you saw at the cat burning. What would "badness" look like anyway?!
But, as Hanson points out, this idea runs into trouble. Because you're not just making a claim about ethics here; you're also making a claim about the way we observe zebras.
Is it true that when you see a zebra you've made something like a pure observation, that has nothing to do with how you think? Some people (most people, actually) argue that this is not the case. After all, you know what a zebra is before you look into the world and observe one. You have some kind of theory and understanding about the world in order for you to be able to conceptualize what a zebra is, and that has to come before having an observation of it. To realize this, imagine that I ask you whether bliggles exist. I say that they do, because I've observed them directly. You've never observed them directly, but that's because you have no idea what they are. Would it be right to say that you've observed bliggles when you have no idea what they are? (It turns out the bliggles are what I call the combination you get when you duct-tape a parrot to an elephant, a strange object indeed. I'm not just renaming some normal object.)
The point is: in order to make your ethical argument, you also have to make a completely different argument about the way we observe things in general, and how that counts as justification for our empirical beliefs.
And if things weren't complicated enough, we also have this weird thing called "math." Do we observe all mathematical truths the same way we observe empirical truths? Yes? Then with which of our senses? Our eyes? Sure, we can see that 2 stones and 2 more stones make 4 stones, but have you ever seen 2 million stones be added to 2 million stones, and counted to make sure that we have 4 million stones? No, you haven't. So should we doubt that 2 mil+ 2 mil=4 mil? Well, that makes no sense. So maybe we don't observe mathematical truths. Well, maybe we can be sure about things without observational evidence. But then what about ethics? Maybe ethics gets off the hook, and you can be objective without having observational evidence? Or maybe math gets on the hook, and it isn't objective, just like ethics isn't objective.
So now we have three problems entangled: how do we justify our belief in empirical truths, how do we justify our belief in ethical truths, and how do we justify our belief in mathematical truths? Dissolving this tangle isn't really possible; a good philosopher would need answer to all three of these questions.
So maybe philosophy needs to be done as a whole. This is more the way that philosophy was done in the past than the way that it is sometimes done now. And I think the philosophy that tends to get remembered now, the good stuff, doesn't restrict itself to just one domain. After all, if all of philosophy is really tangled up like this, a good idea in one domain is probably going to be a good important idea in the other domains too.
Thursday, September 17, 2009
Ethics and observation
"Ethics and observation" is the title of the first chapter in Harman's (very readable) introduction to ethics. He is trying to explain what makes ethics problematic in a way that science or math is not. Harman is a particularly good person for me to be reading, because he was a student of Quine's. Quine was the author of the "indispensability argument" in mathematics, which is something that I'm focusing research on.
Quine was an empiricist--a position that I can't think of any good way to characterize, so I'll just say that it's the view that the justification of any knowledge has to come from observation. Empiricism makes a lot of good sense. After all, how do I know that I'm justified in believing that my computer is sitting on a table? Isn't it just that I can observe that my computer is on the table! So just generalize from there, and suppose that all knowledge is like my knowledge about where my computer is. But accounting for math was a sore spot for empiricists right from the beginning. Cuz math doesn't seem to be true because of any empirical observation! You don't look at anything to try to prove that 2+5=7, you just figure it out. Empirical observation seems to be worthless when it comes to math. (I'll post a stronger argument for that at some point; Frege made some good ones, and they were directed against J.S. Mill).
So math is difficult for empiricists. If you're an empiricist, you owe us an account of how math fits into your empiricist framework. Take the logical empiricists. They said that all math is logic, and all logic is empty of content (and here they were inspired by Wittgenstein). So since it's devoid of content, you don't have to worry about how we know things about math; it's empty anyway (does that make sense? def not the way I wrote it). But everything else needs empirical justification.
Quine, who was originally an advocate of this view, destroyed it in his landmark paper "Two Dogmas of Empiricism." What he did with math was, actually way cooler and more radical. He argued that math was actually empirical, and so it's just like all other knowledge. But he didn't do this in a kinda stupid way. Rather, he argued that lots of math is absolutely necessary for science; and that therefore we should believe that math is true. That's a one sentence version of the "indispensability argument for the existence of mathematical objects". Another way of putting it is that inference to the best explanation is the way we justify scientific beliefs, and the truths of math are part of the best explanation of the world in a deep, unavoidable way. This is the argument that I'm going to be looking at more closely in the thesis, if all goes according to plan.
The question is, is ethics different from scientific and mathematical knowledge in this regard? At first glance, obviously it's different. Ethics doesn't seem to be tied to our best explanation of the world, it doesn't seem necessary for scientific practice. In this first chapter Harman makes this argument a bit more carefully, and tomorrow I'll present his argument as a first shot way of distinguishing between ethics and math in an empiricist framework.
Well, I'm really tired and this post didn't really go anywhere. But tomorrow I'm going to be looking at Harman's chapter more carefully.
Quine was an empiricist--a position that I can't think of any good way to characterize, so I'll just say that it's the view that the justification of any knowledge has to come from observation. Empiricism makes a lot of good sense. After all, how do I know that I'm justified in believing that my computer is sitting on a table? Isn't it just that I can observe that my computer is on the table! So just generalize from there, and suppose that all knowledge is like my knowledge about where my computer is. But accounting for math was a sore spot for empiricists right from the beginning. Cuz math doesn't seem to be true because of any empirical observation! You don't look at anything to try to prove that 2+5=7, you just figure it out. Empirical observation seems to be worthless when it comes to math. (I'll post a stronger argument for that at some point; Frege made some good ones, and they were directed against J.S. Mill).
So math is difficult for empiricists. If you're an empiricist, you owe us an account of how math fits into your empiricist framework. Take the logical empiricists. They said that all math is logic, and all logic is empty of content (and here they were inspired by Wittgenstein). So since it's devoid of content, you don't have to worry about how we know things about math; it's empty anyway (does that make sense? def not the way I wrote it). But everything else needs empirical justification.
Quine, who was originally an advocate of this view, destroyed it in his landmark paper "Two Dogmas of Empiricism." What he did with math was, actually way cooler and more radical. He argued that math was actually empirical, and so it's just like all other knowledge. But he didn't do this in a kinda stupid way. Rather, he argued that lots of math is absolutely necessary for science; and that therefore we should believe that math is true. That's a one sentence version of the "indispensability argument for the existence of mathematical objects". Another way of putting it is that inference to the best explanation is the way we justify scientific beliefs, and the truths of math are part of the best explanation of the world in a deep, unavoidable way. This is the argument that I'm going to be looking at more closely in the thesis, if all goes according to plan.
The question is, is ethics different from scientific and mathematical knowledge in this regard? At first glance, obviously it's different. Ethics doesn't seem to be tied to our best explanation of the world, it doesn't seem necessary for scientific practice. In this first chapter Harman makes this argument a bit more carefully, and tomorrow I'll present his argument as a first shot way of distinguishing between ethics and math in an empiricist framework.
Well, I'm really tired and this post didn't really go anywhere. But tomorrow I'm going to be looking at Harman's chapter more carefully.
Sunday, September 13, 2009
Thesis Proposal
Debates about realism in ontology and realism in truth-value are central to both metaethics and philosophy of mathematics. In both fields the existence of objects and concepts central to the field’s practice are debated. But more seems to be at stake when it is questioned whether ethical or mathematical statements have truth-value at all. After all, if ethical statements are neither true nor false, then it would seem that I’m unjustified in criticizing someone else’s moral practice. Likewise, if mathematical statements cannot be false, how can I be justified in criticizing someone for embracing the law of non-contradiction, or other seemingly false claims?
So philosophers of math and ethics share similar concerns; the connections between the two subjects run deeper, though. It is plausible to suppose, for example, that the shared concern of these philosophers stems from a shared set problems facing both math and ethics. For example, it has seemed for many that neither math nor ethics is able to bring empirical justification for their claims. This seems to be a challenge to the objectivity of mathematical and ethical discourse. How can a philosopher of math or ethics respond to such a challenge? A defender of realism could suggest that humans have some way of mathematical or ethical intuition that gives them access to the truth, in a way that is analogous to sensual perception. Another choice is to challenge the premise itself, arguing that math and ethics really can be justified empirically. On the other hand, perhaps the lack of empirical justification suggests that our discourse is actually subjective, not objective. Then either math and ethics are false, or they lack truth-value (then the challenge is to explain why our discourse sounds so objective, and projectivist accounts begin to enter the picture).
What is fascinating, yet predictable, is that all of these positions have been staked out in both philosophy of math and in metaethics. Despite this broad similarity in the landscape of both subjects, there are still some arguments that are particular to either math or ethics. Consider, for example, one of J.L. Mackie’s arguments from “The Subjectivity of Values”: the “argument from relativity.” Stated sloppily, this argument states that the best explanation of widespread, deep and irresolvable disagreements in ethics is that ethical statements are subjective rather than objective. This is a powerful challenge in metaethics, but is not often brought in philosophy of mathematics. Likewise, in philosophy of math a powerful argument for realism is the Quine-Putnam indispensability argument, which argues that we are justified in believing in the existence of some mathematical objects (and therefore given a better chance at maintaining truth-value realism in math) due to their indispensability in expressing our best scientific theory. This is an argument that seems peculiar to math and versions of it are less often brought in ethics.
Given the great deal of overlap between the two subjects, arguments that seem peculiar to either math or ethics provide an important opportunity to explore the relationship between math and ethics. First, is it possible to map one argument into the other’s field? For example, can a plausible version of Mackie’s argument from relativity be made with regards to math? Can some kind indispensability argument be made in ethics? If the answer is “yes,” then we have made some progress to showing some stronger connection between math and ethics. If the answer is “no” then we also have an important opportunity: to investigate closely to determine why the argument failed to be plausible when translated into the other subject. For example, if study shows that the argument from relativity fails to be plausible in math, we should be able to isolate why the argument failed. Whatever the reason for this failure is, it should be a very important difference between the two subjects. In other words, failure for the argument to map between the subjects should isolate an important difference between math and ethics.
My initial investigations have shown me that there are some philosophers engaged in this sort of project. For example, there is a dissertation in NYU being written on ethics and mathematics by Justin Clarke-Doane (Hartry Field, Thomas Nagel, Derek Parfit and Stephen Schiffer are on his committee). Clarke-Doane has argued that Mackie’s argument from relativity applies just as strongly to mathematics than it does to ethics. Another NYU-educated philosopher, David Enoch, wrote his dissertation defending a version of the indispensability argument for metaethics (again, Field, Nagel and Parfit were on the committee).
I propose to write a thesis that continues this line of thought. In order to focus the project, I will pick one argument for or against realism in truth-value that is made either solely in philosophy of math or metaethics. Then I will, following the procedure that I laid out above, see how plausibly the argument translates into the other subject. After I determine how plausible translation is I will be able to make (only modestly, given the scope of the project) some observations concerning the similarities and differences between math and ethics that make translation either plausible or implausible. My continued research on this topic will determine which argument I choose to focus on—my research so far has looked at Mackie’s argument from relativity and the Quine-Putnam indispensability thesis (my interest in these arguments led to the discovery that Clarke-Doane and Enoch had also done work on them, not the other way around), but as I continue this project better candidates for arguments to focus on will emerge (analyzing the argument from queerness or the attempt to identify mathematical with logical truths might yield interesting insights).
Preliminary Bibliography:
Introductory textbooks:
“Thinking about Mathematics” by Stewart Shapiro
“An Introduction to Contemporary Metaethics” by Andrew Miller
Collections:
“Philosophy of Mathematics: Selected readings” edited by Paul Benacerraf and Hilary Putnam
“Essays on Moral Realism” edited by Geoffrey Sayre-McCord
Articles and books:
“Philosophy of Logic” by Hilary Putnam
“Realsim, Mathematics and Modality” by Hartry Field
“Mathematics in Philosophy” by Charles Parsons
“Essays in Quasi-Realism” by Simon Blackburn
“Moral Realism and the Argument from Disagreement” by D. Loeb
“How is Moral Disagreement a problem for realism” by David Enoch
“Moral Realism and the Foundation of Ethics” by David O. Brink
So philosophers of math and ethics share similar concerns; the connections between the two subjects run deeper, though. It is plausible to suppose, for example, that the shared concern of these philosophers stems from a shared set problems facing both math and ethics. For example, it has seemed for many that neither math nor ethics is able to bring empirical justification for their claims. This seems to be a challenge to the objectivity of mathematical and ethical discourse. How can a philosopher of math or ethics respond to such a challenge? A defender of realism could suggest that humans have some way of mathematical or ethical intuition that gives them access to the truth, in a way that is analogous to sensual perception. Another choice is to challenge the premise itself, arguing that math and ethics really can be justified empirically. On the other hand, perhaps the lack of empirical justification suggests that our discourse is actually subjective, not objective. Then either math and ethics are false, or they lack truth-value (then the challenge is to explain why our discourse sounds so objective, and projectivist accounts begin to enter the picture).
What is fascinating, yet predictable, is that all of these positions have been staked out in both philosophy of math and in metaethics. Despite this broad similarity in the landscape of both subjects, there are still some arguments that are particular to either math or ethics. Consider, for example, one of J.L. Mackie’s arguments from “The Subjectivity of Values”: the “argument from relativity.” Stated sloppily, this argument states that the best explanation of widespread, deep and irresolvable disagreements in ethics is that ethical statements are subjective rather than objective. This is a powerful challenge in metaethics, but is not often brought in philosophy of mathematics. Likewise, in philosophy of math a powerful argument for realism is the Quine-Putnam indispensability argument, which argues that we are justified in believing in the existence of some mathematical objects (and therefore given a better chance at maintaining truth-value realism in math) due to their indispensability in expressing our best scientific theory. This is an argument that seems peculiar to math and versions of it are less often brought in ethics.
Given the great deal of overlap between the two subjects, arguments that seem peculiar to either math or ethics provide an important opportunity to explore the relationship between math and ethics. First, is it possible to map one argument into the other’s field? For example, can a plausible version of Mackie’s argument from relativity be made with regards to math? Can some kind indispensability argument be made in ethics? If the answer is “yes,” then we have made some progress to showing some stronger connection between math and ethics. If the answer is “no” then we also have an important opportunity: to investigate closely to determine why the argument failed to be plausible when translated into the other subject. For example, if study shows that the argument from relativity fails to be plausible in math, we should be able to isolate why the argument failed. Whatever the reason for this failure is, it should be a very important difference between the two subjects. In other words, failure for the argument to map between the subjects should isolate an important difference between math and ethics.
My initial investigations have shown me that there are some philosophers engaged in this sort of project. For example, there is a dissertation in NYU being written on ethics and mathematics by Justin Clarke-Doane (Hartry Field, Thomas Nagel, Derek Parfit and Stephen Schiffer are on his committee). Clarke-Doane has argued that Mackie’s argument from relativity applies just as strongly to mathematics than it does to ethics. Another NYU-educated philosopher, David Enoch, wrote his dissertation defending a version of the indispensability argument for metaethics (again, Field, Nagel and Parfit were on the committee).
I propose to write a thesis that continues this line of thought. In order to focus the project, I will pick one argument for or against realism in truth-value that is made either solely in philosophy of math or metaethics. Then I will, following the procedure that I laid out above, see how plausibly the argument translates into the other subject. After I determine how plausible translation is I will be able to make (only modestly, given the scope of the project) some observations concerning the similarities and differences between math and ethics that make translation either plausible or implausible. My continued research on this topic will determine which argument I choose to focus on—my research so far has looked at Mackie’s argument from relativity and the Quine-Putnam indispensability thesis (my interest in these arguments led to the discovery that Clarke-Doane and Enoch had also done work on them, not the other way around), but as I continue this project better candidates for arguments to focus on will emerge (analyzing the argument from queerness or the attempt to identify mathematical with logical truths might yield interesting insights).
Preliminary Bibliography:
Introductory textbooks:
“Thinking about Mathematics” by Stewart Shapiro
“An Introduction to Contemporary Metaethics” by Andrew Miller
Collections:
“Philosophy of Mathematics: Selected readings” edited by Paul Benacerraf and Hilary Putnam
“Essays on Moral Realism” edited by Geoffrey Sayre-McCord
Articles and books:
“Philosophy of Logic” by Hilary Putnam
“Realsim, Mathematics and Modality” by Hartry Field
“Mathematics in Philosophy” by Charles Parsons
“Essays in Quasi-Realism” by Simon Blackburn
“Moral Realism and the Argument from Disagreement” by D. Loeb
“How is Moral Disagreement a problem for realism” by David Enoch
“Moral Realism and the Foundation of Ethics” by David O. Brink
Thursday, September 10, 2009
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?
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?
Monday, September 7, 2009
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,
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.
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.
Saturday, September 5, 2009
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 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?
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?
Thursday, September 3, 2009
Wednesday, September 2, 2009
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.
(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.
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