Instrumental nominalism about set-theoretic structuralism Richard Pettigrew Department of Philosophy University of Bristol 7 th August 2015 Richard Pettigrew (Bristol) INSTS 7 th August 2015 1 / 16
Interpreting mathematics The first question in the philosophy of mathematics is this: How should we interpret mathematical language? Some examples of mathematical language: R contains transcendental numbers. V 4 is the smallest non-cyclic group. where p n is the n th prime. lim inf n (p n+1 p n ) < 7 10 7 Richard Pettigrew (Bristol) INSTS 7 th August 2015 2 / 16
Mathematics studies collections Mathematical language includes terms for collections: group, field, R, V 4, the natural numbers, etc. And it includes terms for their members: the identity element of V 4, 1, i, etc. The sentences of mathematics seem to be concerned with these collections, their members, operations on them and relations between them. Richard Pettigrew (Bristol) INSTS 7 th August 2015 3 / 16
Three questions This raises three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? No Richard Pettigrew (Bristol) INSTS 7 th August 2015 4 / 16
Three questions This raises three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? No Richard Pettigrew (Bristol) INSTS 7 th August 2015 4 / 16
Three questions This raises three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? No Richard Pettigrew (Bristol) INSTS 7 th August 2015 4 / 16
Three questions This raises three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? No Richard Pettigrew (Bristol) INSTS 7 th August 2015 4 / 16
I Aristotelianism about mathematical language Grammatically, numerals seem to function as singular terms, and according to ante rem structuralism, numerals are singular terms. (Shapiro (2006) Structure and Identity) Do numerals really seem to function as singular terms? Not necessarily (Pettigrew (2008) Platonism and Aristotelianism in Mathematics). Richard Pettigrew (Bristol) INSTS 7 th August 2015 5 / 16
I Aristotelianism about mathematical language Consider the following phrases: 1 H is stable. i 2 + 1 = 0. In each case, the expression in boldface is a dedicated parameter. A parameter: Let r be a real number. A dedicated parameter functions like a parameter. But there is no need to state the introductory stipulation because the expression is always used in the same way. A dedicated parameter is not a singular term. It does not refer at all, let alone uniquely. It is akin to a free variable. Richard Pettigrew (Bristol) INSTS 7 th August 2015 6 / 16
I Aristotelianism about mathematical language The conclusion of (Shapiro (2008) Identity, Indiscernibility, and ante rem structuralism): i is a dedicated parameter. The claim of (Pettigrew (2008) Platonism and Aristotelianism in Mathematics): We have no reason to think that N, R, C, 0, 1, π, e, Z 2, V 4 are singular terms rather than dedicated parameters. Richard Pettigrew (Bristol) INSTS 7 th August 2015 7 / 16
I Aristotelianism about mathematical language Objections to ante rem structuralism: Burgess (1999): It is impossible to state the required condition of property deficiency. Hellman (2005): There can be no unique property-deficient ante rem structure. Richard Pettigrew (Bristol) INSTS 7 th August 2015 8 / 16
Three questions Our three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? (III) Is the use of these terms in assertions ontologically committing? Richard Pettigrew (Bristol) INSTS 7 th August 2015 9 / 16
II Set-theoretic structuralism Consider the mathematical sentence: There is no greatest prime. We interpret this as the following conjunction: (i) For all collections N, successor functions s on N, and zero element 0 in N that satisfy the Dedekind-Peano axioms, there is no element of N that is the greatest N prime N. (ii) There is a collection N, successor function s on N, and zero element 0 in N that satisfies the Dedekind-Peano axioms. Richard Pettigrew (Bristol) INSTS 7 th August 2015 10 / 16
II Set-theoretic structuralism What are the collections over which we quantify? Reasons to prefer set theory: The freedom with which we can construct new collections from old. Non-naturalist justification of postulates. (The posits of set theory are posits also of our everyday business; the posits of category theory are not.) Richard Pettigrew (Bristol) INSTS 7 th August 2015 11 / 16
Three questions Our three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? Richard Pettigrew (Bristol) INSTS 7 th August 2015 12 / 16
III Instrumental nominalism As interpreted by the set-theoretic structuralist, mathematical statements look ontologically committing. This threatens the rationality of having a very high degree of confidence in mathematical theorems, even if it doesn t threaten belief in them or knowledge of them. Thus, instead, we opt for an instrumental nominalist interpretation... Richard Pettigrew (Bristol) INSTS 7 th August 2015 13 / 16
III Instrumental nominalism Let s say the concrete core of a world w is the largest wholly concrete part of w: the aggregate of all of the concrete objects that exist in w. [... ] [Then] S is nominalistically adequate iff the concrete core of the actual world is an exact intrinsic duplicate of the concrete core of some world at which S is true that is, just in case things are in all concrete respects as if S were true. (Rosen (2001) Nominalism, Naturalism, and Epistemic Relativism) Richard Pettigrew (Bristol) INSTS 7 th August 2015 14 / 16
III Instrumental nominalism Making this precise using a primitive notion of modality requires: (i) A clear distinction between concrete and abstract objects and properties. (ii) Modal operators and that allow us to shift the locus of evaluation of formulae. These are over and above and. (Pettigrew (2012) Indispensability arguments and instrumental nominalism) Richard Pettigrew (Bristol) INSTS 7 th August 2015 15 / 16
Three questions Our three questions: (I) Are the terms that refer to collections singular or general? General (II) Are the collections best understood using set theory or category theory or type theory or plural logic or SOL/HOL? Set theory (III) Is the use of these terms in assertions ontologically committing? These answers characterize Instrumental Nominalism about Set-Theoretic Structuralism. No Richard Pettigrew (Bristol) INSTS 7 th August 2015 16 / 16