Since 1976 Hilary Putnam has drawn parallels between his ‘internal’, ‘pragmatic’, ‘natural’ or ‘common-sense’ realism and Kant’s transcendental idealism. Putnam reads Kant as rejecting the then current metaphysical picture with its in-built assumptions of a unique, mind-independent world, and truth understood as correspondence between the mind and that ready-made world. Putnam reads Kant as ove...

4. (A1 or B1 problem) Let C be a circle of radius 1, and let D be a diameter of C. Let P be the set of all points inside or on C such that p is closer to D than it is to the circumference of C. Find a rational number r such that the area of P is r. 5. Let n be a positive integer, let 0 ≤ j < n, and let fn(j) be the number of subsets S of the set {0, 1, . . . , n−1} such that the sum of the elem...

I discuss and try to evaluate the argument about constructible sets made by Putnam in “Models and Reality”, and some of the counterarguments directed against it in the literature. I shall conclude that Putnam’s argument, while correct in substance, nevertheless has no direct bearing on the philosophical question of unintended models of set theory. MSC2000: 00A30, 03A05, 03E45.

We extend the Putnam-Fuglede theorem and the second-degree Putnam-Fuglede theorem to the nonnormal operators and to an elementary operator under perturbation by quasinilpo-tents. Some asymptotic results are also given.

Bo J. Bernhard* David R. Dickens Paul D. Shapiro In his acclaimed 2000 book Bowling Alone. Robert Putnam documents a disturbing social trend of the broadest kind. Putnam cites a wide variety of data that indicate that over the past fifty years, Americans have become increasingly socially disengaged. In developing this theme, Putnam specifically cites the increase in casino gambling (and especia...