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).