How we learn mathematical language

Philosophical Review

Volumn 106, Issue 1, 1997, Pages 35-68

  McGee, Vann ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0012539398     PISSN: 00318108     EISSN: 15581470     Source Type: Journal    
DOI: 10.2307/2998341     Document Type: Article

References (59)

1. Cf. H. P. Grice, "Meaning," in Studies in the Way of Words (Cambridge: Harvard University Press, 1989), 213-223
2. "What Numbers Could Not Be," Philosophical Review 74 (1965): 47-73.
3. Reprinted in Philosophy of Mathematics, 2d ed., ed. Benacerraf and Hilary Putnam (Cambridge: Cambridge University Press, 1983), 272-294
4. Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," Malhematische Annalen 65 (1908): 261-281
5. English translation by Stefan BauerMengelberg, "Investigations in the Foundations of Set Theory I," in From Frege to Gödel, ed. Jean van Heijenoort (Cambridge: Harvard University Press, 1967), 199-223.
6. John von Neumann, "Zur Einführung der transfiniten Zahlen," Actalitterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae, Sectioscientiarum mathematicarum 1 (1923): 199-208.
7. English translation by van Heijenoort, "On the Introduction of Transfinite Numbers," in From Frege to Gödel, 346-54.
8. Donald Davidson, "The Inscrutability of Reference," Southwestern Journal of Philosophy 10 (1979): 7-19.
9. Reprinted in Davidson's Inquiries into Truth and Interpretation (Oxford: Oxford University Press, Clarendon Press, 1984), 227-41.
10. Hilary Putnam, Reason, Truth, and History (Cambridge: Cambridge University Press, 1981), chap. 2 and appendix.
11. For impure abstracta, like the set of justices of the U.S. Supreme Court, the inscrutability is mitigated, since, arguably, the set is located where the court sits.
12. See Penelope Maddy, Realism in Mathematics (Oxford: Oxford University Press, Clarendon Press, 1990).
13. Its status is discussed further in my "A Semantic Conception of Truth?" Philosophical Topics 21 (1993): 83-111.
14. The meaning of arithmetical terms, as we currently use them, ensures the truth of all conditionals with "The natural numbers exist" as antecedent and an axiom as consequent, but meaning is highly mutable. Should we discover that there is no structure that satisfies all the axioms, we might respond by saying diat we have learned that there are no natural numbers. But we might instead respond by changing the way we use number words, adopting a new conception according to which the natural numbers exist but some of the axioms we hitherto accepted are false. Which course we would choose is difficult to predict.
15. To connect numbers with popsicle sticks, we might use a rudimentary set theory, or we might employ higher-order logic. The latter counts as "non-arithmetical" only because the range of the individual variables includes things other than numbers.
16. See Harold Hodes, "Ontological Commitment, Thick and Thin," in Meaning and Method, ed. George Boolos (Cambridge: Cambridge University Press, 1990), 235-60.
17. To show that the theory is true, one must show that there is at least one structure that satisfies the axioms. Inasmuch as any two structures that satisfy the axiomsare isomorphic, an arithmetic sentence will count as true if and only if it is true inat least one structure that satisfies the axioms, or, equivalently, if and only if it is true in every structure that satisfies the axioms.
18. See Frank P. Ramsey, "Theories," in his Philosophical Papers (Cambridge: Cambridge University Press, 1990), 112-36.
19. Robinson's R is a simple system of extremely rudimentary arithmetical axioms, little more than extended addition and multiplication tables. See Alfred Tarski, Andrzej Mostowski, and Raphael M. Robinson, Undecidable Theories (Amsterdam: North-Holland, 1953), 53.
20. The general thesis that second-order logic is fundamental to our mathematical understanding is developed in Stewart Shapiro's highly insightful Foundations without Foundationalism (Oxford: Oxford University Press, Clarendon Press, 1991).
21. The second-order variables come in different sorts, taking the grammatical positions of one-place predicates, two-place predicates, three-place predicates, and so on. A careful notation would distinguish the sorts, but here I shall be sloppy, counting on context to indicate which sort is intended.
22. Within purely arithmetical investigations, the restriction of one's quantifiers to natural numbers is typically tacit. For applications, however, one needs to make the restriction explicit.
23. Theorem 132 of Was sind und was sollen die Zahlen (Brunswick: 1888).
24. English translation by Wooster Woodruff Beman, "The Nature and Meaning of Numbers," in Dedekind's Essays on the Theory of Numbers (New York: Dover, 1963), 29-115.
25. That is, every two models of the theory are isomorphic.
26. Intuitionists would dispute this, of course. Affirming the traditional view that logic is universal (which goes back to Metaphysics I) is part of our presumptive commitment to realism.
27. Philosophy of Logic, 2d ed. (Cambridge: Harvard University Press, 1986), 66.
28. These so-called laws of second-order logic are, on the mooted view, really first-order comprehension axioms, in a disguised notation.
29. "A System of Axioms for Geometry," Transactions of the American Mathematical Society 5 (1904): 343-84.
30. ZFCU is often formulated with two axiom schemata, Replacement and Separation, but the latter is redundant.
31. Notice that since we have Urelemente, we cannot follow the practice, customary among set theories, of taking our quantifiers tacitly to be restricted to sets; we require the predicate 'Set(x)'.
32. "Über den Begriff der logischen Folgerung," Actes du Congrès International de Philosophie Scientifique 7 (1936): 1-11.
33. J. H. Woodger, "On the Concept of Logical Consequence," in Tarski's Logic, Semantics, Metamathematics, 2d ed. (Indianapolis: Hackett, 1983), 409-20.
34. Making no provision for varying the universe of discourse as we go from one model to another, Tarski's paper defines a model for a formula φ (Set, ε) to be a variable assignment for the formula φ (X,Y) that we obtain from φ (Set, ε) by replacing its nonlogical terms with variables of appropriate types. Making no provision for varying the universe of discourse as we go from one model to another, Tarski's paper defines a model for a formula φ (Set, ε) to be a variable assignment for the formula φ (X,Y) that we obtain from φ (Set, ε) by replacing its nonlogical terms with variables of appropriate types. However, as the discussion in John Etchemendy's The Concept of Logical Consequence (Cambridge: Harvard University Press, 1990) makes abundandy clear, we need to make allowance for varying the universe of discourse, else the notion of "logical consequence" we define will diverge wildly from standard mathematical usage.
35. As found, for example, in C. C. Chang and H. J. Keisler, Model Theory, 3d ed. (Amsterdam: North Holland, 1990).
36. The kinds of nonlogical terms available to us when we use second-order logic are the same as when we use first-order logic: individual constants, signs for functions from individuals to individuals, and predicates designating properties and relations among individuals. Such resources will not suffice to enable us to define truth in a model, for we naturally define truth in a model in terms of satisfaction, and to represent the satisfaction relation for second-order formulas, we require a predicate that has a second-order variable in one of its argument places.
37. See Shapiro, Foundations without Foundationalism, 134-37.
38. 2>)).
39. α:: α an ordinal >, only it will be a sequence of proper classes, rather than a sequence of sets. This way of talking is only credible if it isn't taken literally.
40. On one way of thinking about classes, there aren't any proper classes; the only classes there are are the sets. This is the point of view that's being developed here. Talk about proper classes is figurative, a vivid way of expressing ideas that can be expressed more precisely either by employing higher-order logic or by metalinguistic ascent. Far be it from me to object to a figure of speech.
41. 0 will be a proper class that includes all proper classes. But according to the standard conception of class theory, which requires classes to be well founded, such a thing isn't possible.
42. αs as proper classes, we need to take the Urelemente to be the non-classes, rather than the non-sets. As we shall see below, such an account, which takes the theory of classes, rather than the theory of sets, as its mathematical foundation, still requires a new axiom, in addition to the standard second-order theory of classes, in order to ensure mat the non-classes form a class.
43. There will be many intended models, not just one, since the best we can hope for is that our intentions should pick out the preferred models uniquely up to isomorphism.
44. A collection C of pure sets is said to be an initial segment of the pure sets if and only if every member of a member of C is a member of C. It is called a proper initial segment if there are some pure sets that aren't in C.
45. That is, there isn't any model < U,S,E > of second-order ZFCU such that there is a one-one function I, taking pure sets to members of S, that meets the following conditions: every member of the range of I satisfies "x is a pure set" in the model, but not everything that satisfies "x is a pure set" in the model is a member of the range of I; if y is in the range of I and Exy, then x is in the range of I; for any pure sets x and y, we have x ε y if and only if EI(x) I(y).
46. The Foundations of Geometry, 2d ed., trans. E. J. Townsend (Chicago: Open Court, 1910), 25.
47. Sometimes a specification of a mathematical system is regarded as categorical only if it characterizes the system, uniquely up to isomorphism, by its internal structure. Our axioms are not categorical is this strict sense, since we characterize the pure sets by reference to things that aren't pure sets, namely, the Urelemente. Usage here is not uniform. Hilbert's axioms for geometry are often referred to as "categorical," even though they refer to things outside the given system of points, lines, and planes.
48. The Gödel sentence for the enlarged language, "This sentence is not provable wiuhin the version of PA that permits substituents from the language gotten from the language of arithmetic by adjoining a truth predicate," is not provable within the version of PA that permits substituents from the language gotten from the language of arithmetic by adjoining a truth predicate. But it is provable in the version of PA gotten by allowing substituents from the language gotten from the language of arithmetic by first adjoining a trudi predicate for the language of arithmetic, then adding another truth predicate for the enlarged language. And so on, up the Tarski hierarchy.
49. Jeff Paris and Leo Harrington, "A Mathematical Incompleteness in Peano Arithmetic," in Handbook of Mathematical Lope, ed. Jon Barwise (Amsterdam: North Holland, 1977), 1133-42.
50. The advantages of using schemata as a substitute for universally quantified second-order sentences are discussed by Shapiro, Foundations without Foundationalism, 246-50.
51. Lavine, Understanding the Infinite (Cambridge: Harvard University Press, 1994), 228-34.
52. If we describe the force of the rule by saying that every Induction Axiom holds in every logically permissible extension of our language, we commit ourselves, at least prima facie, to the dubious category of logically permissible languages. But we by no means require diis metalinguistic description in order to employ the rule.
53. Talk of the cardinality of the universe is figurative, to be cashed out in terms of second-order quantification.
54. That is, we take second-order ZFCU + the Urelement Set Axiom as premises and prove the conclusion, as opposed to giving a model-theoretic argument that the conclusion holds in every model of second-order ZFCU + the Urelement Set Axiom.
55. κ, which is κ.
56. Cantor's Theorem has both first- and second-order formulations, both theorems of second-order ZFCU. Here we are using the latter.
57. The derivation of the schema mimics the second-order proof. To play the role of H, the one-one function mapping the universe into the pure sets, find a formula γ(g,x,y) such uiat it is provable that, for any g, if g is a bijection from the set of Urelemente onto an ordinal, then the ordered pairs such that γ(g,x,y) describe a one-one map from the entire universe into the pure sets. Then prove, using the Urelement Set Axiom and the Axiom of Choice, that such a bijection g exists.
58. "Fraenkel's Addition to the Axioms of Zermelo," in Essays on the Foundations of Mathematics, ed. Yehoshua Bar-Hillel, E. I. J. Poznanski, and Abraham Robinson (Jerusalem: Magnes Press, 1961), 91-114.
59. "Axiom Schemata of Strong Infinity in Axiomatic Set Theory," Pacific Journal of Mathematics 10 (1960): 223-38.