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.
Monday, October 6, 2008
Saturday, October 4, 2008
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...
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...
Monday, September 22, 2008
Notes on Colyvan, "The Indispensabilityof Mathematics" Chapter 2
Chapter 2: The Quinean backdrop
"I will argue that the two essential theses for our purposes--confirmational holism and naturalism--can be disentangled from the rest of the Quinean web."
Naturalism:
"Naturalism involves a certain respect for the scientific enterprise--that much is common ground--but exactly how this is cashed out is a matter of considerable debate. For instance, for David Armstrong naturalism is the doctrine that 'nothing but Nature, the single all-embracing spatio-temporal system exists', whereas for Quine naturalism is 'the abandonment of the goal of a first philosophy'." Quine rejects the view that philosophy precedes science or oversees science. This thesis has implications for the way we should answer metaphysical questions: we should determine our ontological commitments by looking to see which entities our best scientific theories are committed to.
So this means that we're defining science as the scientific practice? That doesn't seem very useful. Enter Cornell realism? "In the disccusion so far I've glossed over the question of what constitutes our best scientific theories. There is also the question of what constitutes a scientific theory as opposed to a non-scientific theory. I won't enter into that debate here: I'll assume that we have at least an intuitive idea of what a scientific theory is."
"It is worth bearing in mind that the primary targets of the indispensability argument are scientific realists disinclined to believe in mathematical entities. These scientific realists typically subscribe to some form of naturalism, so my accpetance of a broadly naturalistic perspective is not as serious an assumption as it may seem at first."
"Now, defences of such fundamental doctrines as naturalism are hard to come by. Typically such doctrines are justified by their fruits." Meaning, doctrines which eliminate in principle the possibility of philosophical justification only could have some kind of justification from the fruits of the theory. "So in order to defend Quinean naturalism over other versions of naturalism I'll examine some of the consequences of the Quinean position."
Good quote of Quine's: "From the point of view of Quine's naturalised epistemology there is no more secure vantage point than the vantage point of our best scientific theories. Thus, the naturalized epistemologist "no longer dreams of a first philosophy, firmer than science, on which science can be based; he is out to defend science from within, against its self doubts.""
Two Dogmas: "Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. Even a statement very close to the periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy or Einstein Newton or Darwin Aristotle?" The main point is that the history of science has taught us that what were once considered analytic truths, such as that Pythagoras's theorem holds in our wolrd or that any massive body can be accelerated without bound have been given up in order to cohere with new and better scientific theories. Thus by an inductive argument from such examples, we conclude that there are no analysitc truths.
First reaction: I'm not crazy about this book. It doesn't seem so careful as I would like.
"I will argue that the two essential theses for our purposes--confirmational holism and naturalism--can be disentangled from the rest of the Quinean web."
Naturalism:
"Naturalism involves a certain respect for the scientific enterprise--that much is common ground--but exactly how this is cashed out is a matter of considerable debate. For instance, for David Armstrong naturalism is the doctrine that 'nothing but Nature, the single all-embracing spatio-temporal system exists', whereas for Quine naturalism is 'the abandonment of the goal of a first philosophy'." Quine rejects the view that philosophy precedes science or oversees science. This thesis has implications for the way we should answer metaphysical questions: we should determine our ontological commitments by looking to see which entities our best scientific theories are committed to.
So this means that we're defining science as the scientific practice? That doesn't seem very useful. Enter Cornell realism? "In the disccusion so far I've glossed over the question of what constitutes our best scientific theories. There is also the question of what constitutes a scientific theory as opposed to a non-scientific theory. I won't enter into that debate here: I'll assume that we have at least an intuitive idea of what a scientific theory is."
"It is worth bearing in mind that the primary targets of the indispensability argument are scientific realists disinclined to believe in mathematical entities. These scientific realists typically subscribe to some form of naturalism, so my accpetance of a broadly naturalistic perspective is not as serious an assumption as it may seem at first."
"Now, defences of such fundamental doctrines as naturalism are hard to come by. Typically such doctrines are justified by their fruits." Meaning, doctrines which eliminate in principle the possibility of philosophical justification only could have some kind of justification from the fruits of the theory. "So in order to defend Quinean naturalism over other versions of naturalism I'll examine some of the consequences of the Quinean position."
Good quote of Quine's: "From the point of view of Quine's naturalised epistemology there is no more secure vantage point than the vantage point of our best scientific theories. Thus, the naturalized epistemologist "no longer dreams of a first philosophy, firmer than science, on which science can be based; he is out to defend science from within, against its self doubts.""
Two Dogmas: "Any statement can be held true come what may, if we make drastic enough adjustments elsewhere in the system. Even a statement very close to the periphery can be held true in the face of recalcitrant experience by pleading hallucination or by amending certain statements of the kind called logical laws. Conversely, by the same token, no statement is immune to revision. Revision even of the logical law of excluded middle has been proposed as a means of simplifying quantum mechanics; and what difference is there in principle between such a shift and the shift whereby Kepler superseded Ptolemy or Einstein Newton or Darwin Aristotle?" The main point is that the history of science has taught us that what were once considered analytic truths, such as that Pythagoras's theorem holds in our wolrd or that any massive body can be accelerated without bound have been given up in order to cohere with new and better scientific theories. Thus by an inductive argument from such examples, we conclude that there are no analysitc truths.
First reaction: I'm not crazy about this book. It doesn't seem so careful as I would like.
Sunday, September 21, 2008
Some unclear explorations on indispensability
This post is exploratory, not expository. It makes less sense as it goes on.
I'm starting to dig a bit deeper into discussions of indispensability arguments. I'll do a longer exposition of these types of arguments at some point soon, but here's the short version: you need to refer to math and things like numbers in order to even express scientific theorems, and you should certainly believe in things that are necessary to express your best scientific theory. That's a LOT to unpack in such a short space, and all in good time.
The goal, at least in the short term (and hopefully in the long term as well) will be seeing how this argument can apply to ethics. Just to spell it out a bit: in the realm of philosophy of math this argument is believed by some to show that we should believe that numbers exist (and, correspondingly, that mathematical statements should be taken as true at face value). The question is, can the argument be fitted for ethics, to argue that we have a good reason to believe in the existence of ethical objects or concepts (such as "X is good")?
I just want to express two possible routes. There might be more, this is what occurs to me right now.
1) The easiest route is probably to analyze the math argument, and see what we "really" care about. That is, the argument says that because we care about scientific explanation we should believe the things that are necessary for scientific explanation. But maybe there are other things that we care about just as much as scientific explanation. Though I haven't read him carefully, DE makes this kind of argument. The idea is to find a parallel to scientific explanation in ethics that we can use to leverage an indispensability argument with.
2) You could argue that ethics is part of science. Now, this is probably impossible. But it's a fun idea for me to toy around with. After all, what is science? Is it just the investigation of the nature of the universe? Then we have to beg the question against ethics to get the discussion started, I could argue. After all, the form of argument for the existence of electrons has to be something like this: there is a fact of the matter, an objective truth about the nature of matter in the universe. The best explanation of the phenomenon of matter requires reference to electrons, so we should believe in the existence of electrons. So in order to exclude ethics we have to assume that ethical knowledge doesn't count as knowledge about the universe, such that the burden of proof is on ethics to show that it's objective. That is, if we start with the premise that ethical knowledge isn't very different from physical knowledge then ethical knowledge will be part of scientific explanation and the most common indispensability argument would have us believing in ethical concepts and objects.
Let me summarize this another way: standing behind the application of the indispensability argument seems to be an implicit acceptance that the other arguments for mathematical realism put mathematical realism in doubt. If mathematical realism weren't in doubt, then mathematical statements would be assumed to be true/false, just as scientific statements are. Then math would be part of the explanation and the leverage, not that which is dragged along by the certainty of physics. It's the same with ethics: you have to assume that ethics isn't part of proper knowledge in order to use the argument to who that it is part of proper knowledge.
Let me make the point one last way. Suppose that I presented you with the following argument: we should believe in the existence of those things, reference to which is necessary for our best scientific explanation of the world. Reference to trees is necessary for our best scientific explanation of the world, so we should believe in trees. This doesn't seem quite right to me--trees are part of the core scientific explanation, it's the secure rock that we can use to leverage other knowledge that is less sure along.
But maybe this entire analysis is wrong. I've been looking at the indispensability argument in the wrong way, perhaps. I've been looking at is as if it can take objects whose existence was previously in doubt, and then through the argument, makes belief in those objects' existence more secure. But maybe I shouldn't look at it this way. Maybe the better way to look at the indispensability argument is just as an observation of the theory, to see what is referred to in our explanation of the world.
So does that explain anything for ethics? I'm totally lost now. Need to try this again tomorrow or later. Ech.
I'm starting to dig a bit deeper into discussions of indispensability arguments. I'll do a longer exposition of these types of arguments at some point soon, but here's the short version: you need to refer to math and things like numbers in order to even express scientific theorems, and you should certainly believe in things that are necessary to express your best scientific theory. That's a LOT to unpack in such a short space, and all in good time.
The goal, at least in the short term (and hopefully in the long term as well) will be seeing how this argument can apply to ethics. Just to spell it out a bit: in the realm of philosophy of math this argument is believed by some to show that we should believe that numbers exist (and, correspondingly, that mathematical statements should be taken as true at face value). The question is, can the argument be fitted for ethics, to argue that we have a good reason to believe in the existence of ethical objects or concepts (such as "X is good")?
I just want to express two possible routes. There might be more, this is what occurs to me right now.
1) The easiest route is probably to analyze the math argument, and see what we "really" care about. That is, the argument says that because we care about scientific explanation we should believe the things that are necessary for scientific explanation. But maybe there are other things that we care about just as much as scientific explanation. Though I haven't read him carefully, DE makes this kind of argument. The idea is to find a parallel to scientific explanation in ethics that we can use to leverage an indispensability argument with.
2) You could argue that ethics is part of science. Now, this is probably impossible. But it's a fun idea for me to toy around with. After all, what is science? Is it just the investigation of the nature of the universe? Then we have to beg the question against ethics to get the discussion started, I could argue. After all, the form of argument for the existence of electrons has to be something like this: there is a fact of the matter, an objective truth about the nature of matter in the universe. The best explanation of the phenomenon of matter requires reference to electrons, so we should believe in the existence of electrons. So in order to exclude ethics we have to assume that ethical knowledge doesn't count as knowledge about the universe, such that the burden of proof is on ethics to show that it's objective. That is, if we start with the premise that ethical knowledge isn't very different from physical knowledge then ethical knowledge will be part of scientific explanation and the most common indispensability argument would have us believing in ethical concepts and objects.
Let me summarize this another way: standing behind the application of the indispensability argument seems to be an implicit acceptance that the other arguments for mathematical realism put mathematical realism in doubt. If mathematical realism weren't in doubt, then mathematical statements would be assumed to be true/false, just as scientific statements are. Then math would be part of the explanation and the leverage, not that which is dragged along by the certainty of physics. It's the same with ethics: you have to assume that ethics isn't part of proper knowledge in order to use the argument to who that it is part of proper knowledge.
Let me make the point one last way. Suppose that I presented you with the following argument: we should believe in the existence of those things, reference to which is necessary for our best scientific explanation of the world. Reference to trees is necessary for our best scientific explanation of the world, so we should believe in trees. This doesn't seem quite right to me--trees are part of the core scientific explanation, it's the secure rock that we can use to leverage other knowledge that is less sure along.
But maybe this entire analysis is wrong. I've been looking at the indispensability argument in the wrong way, perhaps. I've been looking at is as if it can take objects whose existence was previously in doubt, and then through the argument, makes belief in those objects' existence more secure. But maybe I shouldn't look at it this way. Maybe the better way to look at the indispensability argument is just as an observation of the theory, to see what is referred to in our explanation of the world.
So does that explain anything for ethics? I'm totally lost now. Need to try this again tomorrow or later. Ech.
Notes on Colyvan, "The Indispensabilityof Mathematics" Chapter 1
This post is just going to be random thoughts and passages that I want to write down in the course of my reading. Not necessarily meant for making sense.
Writes that "Hilary Putnam...points out that it is possible to be a mathematical realist without being commited to matheatical objects--realism is about objectivity not objects" In What is mathematical truth on pages 69-70. The point is not that you can completely avoid meaningful reference to anything, but that you just elimiante reference to objects that are peculiarly mathematical. For Putnam in one point in his life, that means reinterpreting mathematical statements into statements about possibility and necessity. Colyvan that semantic realism isn't the only reason why you would care about metaphysical, ontological realism; he argues that the question is interesting in of itself (true, but is it important? well, important to whom? fair point, I guess, maybe?).
What should I do? I tend to think that semantic realism is what matters, and in this regard I'm influenced by Goldfarb's lectures on Frege. But the indispensability argument is based on metaphysical realism? I think that I just deal with metaphysical realism, but I argue in addition that semantic realism is plausibly dependent on metaphysical realism (though not necessarily so in either domain).
Colyvan endorses a view where mathematical objects exist contingently. Is that possible before Quine? Hardly, I'd think. Let's find out.
Fictionalism is an error theory for math? Are mathematicians saying false things? Prob, in as much as they think that they're refering to non-fictional objects. The footnotes and hedges you'd expect on page 5.
Man, NYU gets Parfit, Field and Nagel. Sheesh. They basically rock Harvard (though Scanlon and Korsgaard are pretty awesome, as is Goldfarb and the other teachers).
He's going to set aside an account of the applicability of math. That's an important point--applicability is not the same as indispensability for expression.
"An idispensability argument, as Field (1989) points out is "an argument that we should believe a certain claim..because doing so is indispensable for certain purposes (which the argment then details)". Colyvan continues, "Clearly the strength of the argument depends curciallyon what the as-yet unspecific purpose is. For instance, few would ifnd the following argument persuasive: We shoul believe that whites are morally superior to blacks becuase doing so is indispensable for the purpose of justifying black slavery. Similarly, few would be convinced by the argument that we ought to believe that God exists because to do so is indispensable to the purpose of enjoying a healthy religoius life."
He continues: "This raises the very interesting question: Which purpose are the right sort for cogent arguments? I know of no easy answer to this question, but fortunately an answer is not required for a defence of the class of indispensability arguments with which i am concerned. I will restrict my attention largely to arguments that address indispensability to our best scientific theories. I will argue that this is the right sort of purpose for cogent indispensability arguments."
"COnsider the argument that takes providing explanations of empirical facts as its purpose. I'll call such an argument an explanatory indispensability argument."
Page 7: Inference to the best explanation is a kind of indispensability argument. Ex: Reference to God is necessary to explain the creation of species, or the best explanation of the facts involves reference to dark matter.
Controversy about inference to the best explanation: Bas van Fraassen and Nancy Cartwright reject unrestricted usage of this style of inference. Typically, rejectionof infernce to the best explanation results in some form of anti-realism. ("I'm not claiming here that the ia for math entities is an instance of inference to the best explanation; I'm just noting that inference to the best explanation is a kind of indispensability argument.")
The use of indispensability arguments for defending mathematical realism is usually associated with Quine and putnam, but it's important to realise that the argument goes back much further. Frege considers the difference between chess and arithmetic and concludes "it's applicability alone which elevates arith from a game to teh rank of a science". Godel also appeals to some ort of indispensability argumentin "What is Cantor's Continuum Problem?" Colyvan says something important: "these arguments are not simply artifacts of the Quinean worldview. Although the form of the argument that I favor is esssentially Quinean, part of my task is to disentangle this argument from the rest of the Quinean web. I will argue that while the argument does depend on a couple of Quinean doctrines (namely, confirmational holism and naturalism) it doesn't depend on acceptance of all of Quine's views on science and language."
"Both Quine and Putnam, in these passages, stress the indispensability of mathematics to science. It thus seems reasonable to take science, or at least whatever the goals of science are, as the purpose for which mathematical entites are indispensable. But as Putnam also points out (1971) it is doubtful that ther eis a single unified goal of sicnece. Thus we see that we may construct a variety of indispensablity arguments all based on the various goals of science.
There's the issue of justified in belief or ontological commitment versus having a reason to believe that it's true that mathematical objects exist. Colyvan on page 11: "If you try to turn the above argument into a descriptive argument so that thte conclusion is that math entities exist, you find you must have something like "all and only those entities that are indispensaleo our bet theories exist" as the crucial first primse." I need to work on this more, I don't quite get the jump, or the lack of jump.
Important for further research: Other indispensability arguments
1) Resnik's pragmatic indispensability argument: (response to problems with Q/P raised by Maddy and Sober discussed in chapters 5 and 6). Resnik "The argument is similar to the confirmational argument except that instead of claiming that the evidence for science is also evidence for its mathematical components, it claims that the justification for doing science also justifies our accepting as true such math as science uses." It puts things in terms of acts instead of body of statements.
2) Semantic Indispensability argument: "A theory of truth for the language we speak, argue in, theorize in, mathematize in, should provide similar truth conditoins for similar sentences. The truth conditions assigned to two sentences containing quantifiers should reflect in relatively similar ways the contribution made by the quantifiers. Any departure from a theory thus hmogenous woul d have to be strongly motivated to be worth considering. (Benecerraf 1973).
Writes that "Hilary Putnam...points out that it is possible to be a mathematical realist without being commited to matheatical objects--realism is about objectivity not objects" In What is mathematical truth on pages 69-70. The point is not that you can completely avoid meaningful reference to anything, but that you just elimiante reference to objects that are peculiarly mathematical. For Putnam in one point in his life, that means reinterpreting mathematical statements into statements about possibility and necessity. Colyvan that semantic realism isn't the only reason why you would care about metaphysical, ontological realism; he argues that the question is interesting in of itself (true, but is it important? well, important to whom? fair point, I guess, maybe?).
What should I do? I tend to think that semantic realism is what matters, and in this regard I'm influenced by Goldfarb's lectures on Frege. But the indispensability argument is based on metaphysical realism? I think that I just deal with metaphysical realism, but I argue in addition that semantic realism is plausibly dependent on metaphysical realism (though not necessarily so in either domain).
Colyvan endorses a view where mathematical objects exist contingently. Is that possible before Quine? Hardly, I'd think. Let's find out.
Fictionalism is an error theory for math? Are mathematicians saying false things? Prob, in as much as they think that they're refering to non-fictional objects. The footnotes and hedges you'd expect on page 5.
Man, NYU gets Parfit, Field and Nagel. Sheesh. They basically rock Harvard (though Scanlon and Korsgaard are pretty awesome, as is Goldfarb and the other teachers).
He's going to set aside an account of the applicability of math. That's an important point--applicability is not the same as indispensability for expression.
"An idispensability argument, as Field (1989) points out is "an argument that we should believe a certain claim..because doing so is indispensable for certain purposes (which the argment then details)". Colyvan continues, "Clearly the strength of the argument depends curciallyon what the as-yet unspecific purpose is. For instance, few would ifnd the following argument persuasive: We shoul believe that whites are morally superior to blacks becuase doing so is indispensable for the purpose of justifying black slavery. Similarly, few would be convinced by the argument that we ought to believe that God exists because to do so is indispensable to the purpose of enjoying a healthy religoius life."
He continues: "This raises the very interesting question: Which purpose are the right sort for cogent arguments? I know of no easy answer to this question, but fortunately an answer is not required for a defence of the class of indispensability arguments with which i am concerned. I will restrict my attention largely to arguments that address indispensability to our best scientific theories. I will argue that this is the right sort of purpose for cogent indispensability arguments."
"COnsider the argument that takes providing explanations of empirical facts as its purpose. I'll call such an argument an explanatory indispensability argument."
Page 7: Inference to the best explanation is a kind of indispensability argument. Ex: Reference to God is necessary to explain the creation of species, or the best explanation of the facts involves reference to dark matter.
Controversy about inference to the best explanation: Bas van Fraassen and Nancy Cartwright reject unrestricted usage of this style of inference. Typically, rejectionof infernce to the best explanation results in some form of anti-realism. ("I'm not claiming here that the ia for math entities is an instance of inference to the best explanation; I'm just noting that inference to the best explanation is a kind of indispensability argument.")
The use of indispensability arguments for defending mathematical realism is usually associated with Quine and putnam, but it's important to realise that the argument goes back much further. Frege considers the difference between chess and arithmetic and concludes "it's applicability alone which elevates arith from a game to teh rank of a science". Godel also appeals to some ort of indispensability argumentin "What is Cantor's Continuum Problem?" Colyvan says something important: "these arguments are not simply artifacts of the Quinean worldview. Although the form of the argument that I favor is esssentially Quinean, part of my task is to disentangle this argument from the rest of the Quinean web. I will argue that while the argument does depend on a couple of Quinean doctrines (namely, confirmational holism and naturalism) it doesn't depend on acceptance of all of Quine's views on science and language."
"Both Quine and Putnam, in these passages, stress the indispensability of mathematics to science. It thus seems reasonable to take science, or at least whatever the goals of science are, as the purpose for which mathematical entites are indispensable. But as Putnam also points out (1971) it is doubtful that ther eis a single unified goal of sicnece. Thus we see that we may construct a variety of indispensablity arguments all based on the various goals of science.
There's the issue of justified in belief or ontological commitment versus having a reason to believe that it's true that mathematical objects exist. Colyvan on page 11: "If you try to turn the above argument into a descriptive argument so that thte conclusion is that math entities exist, you find you must have something like "all and only those entities that are indispensaleo our bet theories exist" as the crucial first primse." I need to work on this more, I don't quite get the jump, or the lack of jump.
Important for further research: Other indispensability arguments
1) Resnik's pragmatic indispensability argument: (response to problems with Q/P raised by Maddy and Sober discussed in chapters 5 and 6). Resnik "The argument is similar to the confirmational argument except that instead of claiming that the evidence for science is also evidence for its mathematical components, it claims that the justification for doing science also justifies our accepting as true such math as science uses." It puts things in terms of acts instead of body of statements.
2) Semantic Indispensability argument: "A theory of truth for the language we speak, argue in, theorize in, mathematize in, should provide similar truth conditoins for similar sentences. The truth conditions assigned to two sentences containing quantifiers should reflect in relatively similar ways the contribution made by the quantifiers. Any departure from a theory thus hmogenous woul d have to be strongly motivated to be worth considering. (Benecerraf 1973).
Wednesday, September 17, 2008
Bibliography for Ethics and Mathematics
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:
Math
“Philosophy of Logic” by Hilary Putnam
“Realsim, Mathematics and Modality” by Hartry Field
“Mathematics in Philosophy” by Charles Parsons
"The Indispensability of Mathematics" by Mark Colyvan
Ethics
“Essays in Quasi-Realism” by Simon Blackburn
“Moral Realism and the Argument from Disagreement” by D. Loeb
"How to be a moral realist" by Richard Boyd
"The Nature of Morality" by Gilbert Harman, especially the first two chapters.
Ethics and Math
"Ethics, Mathematics and Relativism" by Jonathan Lear
The work of Justin Clarke-Doane
"What We Owe to Each Other" (first chapter, scattered)
"An Outline of an argument for Robust Metanormative Realism" by David Enoch (though it provides an indispensability argument, the discussion is largely divorced from mathematical concerns)
“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:
Math
“Philosophy of Logic” by Hilary Putnam
“Realsim, Mathematics and Modality” by Hartry Field
“Mathematics in Philosophy” by Charles Parsons
"The Indispensability of Mathematics" by Mark Colyvan
Ethics
“Essays in Quasi-Realism” by Simon Blackburn
“Moral Realism and the Argument from Disagreement” by D. Loeb
"How to be a moral realist" by Richard Boyd
"The Nature of Morality" by Gilbert Harman, especially the first two chapters.
Ethics and Math
"Ethics, Mathematics and Relativism" by Jonathan Lear
The work of Justin Clarke-Doane
"What We Owe to Each Other" (first chapter, scattered)
"An Outline of an argument for Robust Metanormative Realism" by David Enoch (though it provides an indispensability argument, the discussion is largely divorced from mathematical concerns)
Subscribe to:
Posts (Atom)