Wednesday, December 17, 2008

Parsons's Mathematical Thought: Sec. 47, Induction and the concept of natural number (continued)

To continue. Parsons now takes up three more issues about his self-styled “justification” of induction.

1. His first question is: “What is the range of the first-order variables?” over which we can apply the rules which ground his “justification” of induction? Some domain of entities, presumably, that can be given to us prior to our specifying its “numbers”, i.e. the zero and its successors. “However,” says Parsons, “this is . . . to assume that some infinite structure is given to us independently of our knowledge of the kind of structure the natural numbers instantiate.”

But I’m not sure why Parsons says this. Take any domain which contains a zero element 0 and for which a function S is defined. Then, whether the function S is injective or otherwise, whether the domain is finite or infinite, we’ll be able to similarly define N -- meaning ‘is 0 or one of its successors’ -- and the induction rule will hold for the Ns. We need, of course, further rules governing S to ensure that the Ns form an infinite progression: but Parsons’s “justification of induction” seems to work equally well whether they do and whether they don’t. If he thinks that there is something special about the infinite case, then he doesn’t bring the point out clearly here.

2. Second, Parsons comments on “the schematic character of the induction rule. . . . the applicability of the rule is not limited to predicates defined in some particular first-order language such as that of first-order arithmetic. But we must not take it as implying the unavoidability or even the legitimacy of full second-order logic.” The target here, I suppose, is Kreisel’s well known contrary claim that we accept instances of a schematic form of the induction rule because we already accept the full second-order induction axiom -- though Parsons doesn’t mention Kreisel here. I take it that the argument is that the reasoning that led us to accept the induction rule was silent on the particular character of the filling for φ – that, it seems, was left entirely open ended (the permitted fillings will be whatever we can make sense of, as wide or as narrow a class as that is): but silence doesn’t mean agreeing to the coherence of the full second-order notion of quantifying over arbitrary properties, where these are conceived of as being in effect arbitrary subsets of the domain of the first-order variables (when that domain is infinite). I agree.

3. “A third question is whether and in what sense is induction an analytic or conceptual rule or truth.” Parsons’s line is that “The explanation of the number concept by rules makes induction follow from an explanation of that concept: it is certainly in some sense ‘conceptual’.” But then what of someone who does not accept induction across the board -- say, a finitist who doesn’t countenance Σ1 induction? Is he then guilty of failing to acknowledge a conceptual truth? No, says Parsons, and surely rightly. We should take the finitist objection to be not to the schematic rule but rather to the admission of certain [say, Σ1] predicates as fully kosher.

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