In my recent post on
difficulties with how we are taught to characterize induction, I noted favorably (although without commitment to it) the existence of the position that mathematical induction is inductive, not deductive, contrary to what is often claimed; and in the discussion that followed I defended it as being at least as intelligible as any other position on the subject, given the terms in which we generally characterize induction. The position was, of course, most famously insisted upon, or put in its most famous form, by Poincaré in the opening
Science and Hypothesis. Given some of the discussion of that point, I found it interesting to read Duhem's response to Poincaré's thesis. Duhem's basic argument is similar to the one that Ocham proposed in the comments; although he allows that Poincaré's characterization does show how we have a sense of the generality of our conclusion prior to actually showing the generality, he thinks Poincaré's account confuses this vague sense with the actual reasoning, and fails to recognize that mathematical induction as presented in mathematical treatises is often presented in an abbreviated form that can easily be expanded into a finite number of syllogisms. But the interesting part of the review is that Duhem recognizes that this leaves open the broader problem Poincaré is considering, namely, how to understand generalization in general in mathematics. If mathematics is purely deductive, it is not easy to see how we can build an account of mathematical generalization that actually does justice to progress in mathematics. Poincaré's claim about mathematical induction or recursion is simply the key part of his argument that the best way to handle this problem is simply to deny that mathematics is wholly or even primarily deductive. Deduction, for Poincaré is not the means of mathematical progress; it is merely that whereby mathematicians sometimes neatly present what they have discovered after they have discovered it. Before it reaches that point, what is being put in that form has been proven non-deductively. Duhem insists that mathematics is, in fact, primarily deductive, and that all mathematical demonstration is deductive. But he recognizes that this leaves him with the question of how mathematical generalization is possible, and so he proposes an answer: generalization is not a matter of the
form of reasoning used by mathematics, but of
what they are reasoning about. Mathematics does not deduce conclusions from axioms alone; it deduces them from axioms
and definitions, and while axioms are rare and hard to come by, definitions are not. Because of this, "it is always possible for us to add to the previously defined and studied mathematical concepts new mathematical notions obtained by combination, modification, and generalization of the previously defined and studied notions". It is the power of the intellect to come up with new definitions that is the basis for the fertility of mathematics, not, as Poincaré would have it, its ability to perform inductions involving infinite series when that series is perfectly regular.
It is now easy to understand what one means by generalization in mathematics.
We have demonstrated a proposition A, which states a certain property of a mathematical notion a. We compose a mathematical notion b, which includes the notion a as a particular case. Finally, with respect to this notion b, we demonstrate a proposition B that restores proposition A when we substitute for notion b its particular determination a. Theorem B is a generalization of theorem A.
[Pierre Duhem, "The Nature of Mathematical Reasoning," in
Essays in the History and Philosophy of Science, Ariew and Barker, eds. Hackett (Indianapolis: 1996) p. 231.]
