Document Type

Article

Comments

Published in Cardozo Law Review, vol. 29, no. 5 (April 2008).

Abstract

This Article, transcribed from a symposium talk given by the author, examines two critical junctures at which foundational decisions must be made in three areas of theoretical inquiry - mathematics, law, and economics. The first such juncture is that which the Article labels the "arbitrary versus criterial choice" juncture. This is the decision point at which one must select between what is typically called an "algorithmic," "principled," "law-like," or "intensionalist" understanding of those concepts which figure foundationally in the discipline in question on the one hand, and a "randomized," "combinatorial," or "extensionalist" such understanding on the other hand.

The second decision point concerns how to respond to certain paradoxes and/or indeterminacies that typically attend recursive, reflexive, or self-referential capacities in the discipline in question. Many practitioners attempt to circumvent or head-off such difficulties merely by fiat: They impose, in the form of axioms, ad hoc restrictions that simply rule out self-reference itself. Other practitioners, by contrast, face the difficulties occasioned by self-reference head-on. They then endeavor to ascertain what these tell us about the underlying structures of the subjects treated of in the disciplines in question.

The Article argues that the most important lesson that has emerged in foundational mathematics since the time of Cantor is that neither a fully intensionalist nor a fully extensionalist understanding of the foundational concepts upon which the discipline is built - whether these be categories, classes, or sets - is sustainable. What the author calls a "thin" form of intensionalism has proved to be the most graceful and intuitively plausible means of accommodating the puzzles raised at the full intensionalist and extensionalist extremes. These means have been discovered, moreover, precisely by reflecting with care upon what the puzzles occasioned by self-reference reveal about cognition as engaged in by self-conscious, freely creative yet norm-observant creatures such as ourselves.

The Article shows that we find the "thin intensionalist" accommodation that it advocates at work in both of the best known nonclassical logics upon which workable, non-paradox-ducking foundational mathematics programs have come to be based: Those are so-called "epistemic," or "intuitionist," logics in the one case, and one or another of the best known "paraconsistent" logics in the other. In effect, the Article shows, these logics compensate for the fully extensional treatments of classes or cognate foundational objects necessary to found Peano arithmetic, by reintensionalizing certain erstwhile extensional, truth-functional logical operators. The latter include negation in the intuitionist case, and either or both of negation and the material conditional in the most attractive paraconsistent cases.

It is no accident, the Article argues, that its thin intensionalist accommodation, within foundational mathematics, of the puzzles arising at the intensional versus extensional choice divide, is discovered precisely upon confronting the puzzles that arise at the self-reference decision-point. For reflection upon our forms of cognition.

Date of Authorship for this Version

4-2008

Keywords

Mathematics