Other Formulations of the Quine-Putnam Indispensability Argument
In the previous post, I formulated the Quine-Putnam Indispensability argument as follows.
In contrast, here is Mark Colyvan's much more recent formulation of the argument (from his Stanford Encyclopedia entry, "Indispensability Arguments in the Philosophy of Mathematics"):
First, Quine and Putnam were not primarily advocating a view about what "we ought to have ontological commitment to". For Quine and Putnam, ontological commitment is a semantic property of sentences and theories, not a normative epistemic property of human beings. Whether a theory $T$ implies that there are $F$s can be established by regimenting this theory into some precise canonical notation, as $T^{\ast}$, say. And seeing if, thus regimented, $T^{\ast}$ logically implies $\exists x Fx$. Whether one "accepts" this theory or not is immaterial, as Quine emphasized in 1948. Admittedly, Quine and Putnam do sometimes talk loosely, about "accepting abstract entities into our ontology", but that really is loose talk, elliptical for "accepting some theory which implies that there are abstract entities". Here, acceptance of the theory is an epistemic matter, separable from the semantic matter, concerning which sentences of the form $\exists x Fx$ the theory implies.
Second, the notion of "we" (i.e., human cognition) having "ontological commitment" to an entity is problematic. All parties to the debate up to, say, the mid-1990s took the notion of ontological commitment to involve a property of sentences and theories; we may accept, or may not accept, sentences and theories (representations, if you prefer; or even propositions, if you prefer). Those sentences, theories, etc., may, or may not, imply that there are $F$s. The notion of cognition bearing "ontological commitment" to an entity is a bit mysterious to me. Is the relation semantical? All knowledge of the world is mediated by cognitive representations, and the whole idea of direct contact with objects is something I'm a rather sceptical about. (In other words, I am defending Lockean indirect realism.)
Third, there is an epistemological side of things, for Quine and Putnam. This comes from a background acceptance of science. For both, we are to accept science roughly as is. Quine's long-held view might be called a kind of realistic pragmatism; and Putnam's view (as of 1971, Philosophy of Logic, and 1975, "What is Mathematical Truth?"), was classic scientific realism. As philosophers, we then subject science---that we have already accepted---to analysis. This position is defended by Russell too, in his 1950 article "Logical Positivism":
Finally, it seems misleading to me to say that the Quine-Putnam indispensability argument is meant to provide reasons for accepting the existence of mathematical entities. Rather, it argues that our working scientific theories presuppose the existence of mathematical entities, and are the best theories around. Consequently, we are being "intellectually dishonest" if we accept science, while feigning to reject those entities.
The Quine-Putnam Indispensability Argument (JK)I believe that this formulation is quite faithful to the intentions of Quine and Putnam, as well as to the intentions those of those involved in the early phase of the debate: Field, Burgess, Shapiro, Chihara and Hellman.
(1) Mathematicized theories are inconsistent with nominalism.
(2) Our best scientific theories are mathematicized.
(C) So, if one accepts our best scientific theories, one must reject nominalism.
In contrast, here is Mark Colyvan's much more recent formulation of the argument (from his Stanford Encyclopedia entry, "Indispensability Arguments in the Philosophy of Mathematics"):
The Quine-Putnam Indispensability Argument (MC):This kind of formulation has become quite widely cited. But I think it is mistaken as a formulation of what Quine and Putnam had in mind. Here are the reasons.
(P1) We ought to have ontological commitment to all and only the entities that are indispensable to our best scientific theories.
(P2) Mathematical entities are indispensable to our best scientific theories.
(C) We ought to have ontological commitment to mathematical entities.
First, Quine and Putnam were not primarily advocating a view about what "we ought to have ontological commitment to". For Quine and Putnam, ontological commitment is a semantic property of sentences and theories, not a normative epistemic property of human beings. Whether a theory $T$ implies that there are $F$s can be established by regimenting this theory into some precise canonical notation, as $T^{\ast}$, say. And seeing if, thus regimented, $T^{\ast}$ logically implies $\exists x Fx$. Whether one "accepts" this theory or not is immaterial, as Quine emphasized in 1948. Admittedly, Quine and Putnam do sometimes talk loosely, about "accepting abstract entities into our ontology", but that really is loose talk, elliptical for "accepting some theory which implies that there are abstract entities". Here, acceptance of the theory is an epistemic matter, separable from the semantic matter, concerning which sentences of the form $\exists x Fx$ the theory implies.
Second, the notion of "we" (i.e., human cognition) having "ontological commitment" to an entity is problematic. All parties to the debate up to, say, the mid-1990s took the notion of ontological commitment to involve a property of sentences and theories; we may accept, or may not accept, sentences and theories (representations, if you prefer; or even propositions, if you prefer). Those sentences, theories, etc., may, or may not, imply that there are $F$s. The notion of cognition bearing "ontological commitment" to an entity is a bit mysterious to me. Is the relation semantical? All knowledge of the world is mediated by cognitive representations, and the whole idea of direct contact with objects is something I'm a rather sceptical about. (In other words, I am defending Lockean indirect realism.)
Third, there is an epistemological side of things, for Quine and Putnam. This comes from a background acceptance of science. For both, we are to accept science roughly as is. Quine's long-held view might be called a kind of realistic pragmatism; and Putnam's view (as of 1971, Philosophy of Logic, and 1975, "What is Mathematical Truth?"), was classic scientific realism. As philosophers, we then subject science---that we have already accepted---to analysis. This position is defended by Russell too, in his 1950 article "Logical Positivism":
For my part, I assume that science is broadly speaking true, and arrive at the necessary postulates by analysis. But against the thoroughgoing sceptic I can advance no argument except that I do not believe him to be sincere. (Russell, 1950)Both Quine and Putnam reject a certain kind of First Philosophy. They are not arguing from the perspective of the armchair First Philosopher or Cartesian, who has purged her mind of all "commitments" and who is wondering what to "accept" from a baseline of noble ontological innocence. On the contrary, one already accepts science, as is. One is subjecting science itself to analysis.
Finally, it seems misleading to me to say that the Quine-Putnam indispensability argument is meant to provide reasons for accepting the existence of mathematical entities. Rather, it argues that our working scientific theories presuppose the existence of mathematical entities, and are the best theories around. Consequently, we are being "intellectually dishonest" if we accept science, while feigning to reject those entities.
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