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.
Monday, September 22, 2008
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)
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