4 ERRETT A. BISHOP that it is important to have some content, but don't especially care to find out what it is. Still others, for whom GGdel (see for example [16]) seems to be a leading spokesman, do their best to develop content within the accepted frarne- work of platonic idealism. One suspects that the majority of pure mathe- maticians, who belong to the union of the first two groups, ignore as much content as they possibly can. If this suspicion seems unjust, pause to consid- er the modern theory of probability. The probability of an event is commonly taken to be a real number between 0 and 1. One might naively expect that the probabilists would concern themselves with the computation of such real num- bers. If so, a quick look at any one of a number of modern texts, for instance the excellent book of Doob [14], should suffice to disabuse him of that expec- tation. Fragmentation ensues, because much if not most of the theory is use- less to someone who is concerned with actually finding probabilities. He will either develop his own semi-independek theories, or else work with ad hoc techniques and rules of thumb. I do not claim that reinvolvement of the prob- abilists with the basic questions of meaning would of itself reverse the process of fragmentation of their discipline, only that it is a necessary first step. In recent years a small number of constructivists (see [3], [9], [lo], [ll] , [12], [23], and [24] ) have been trying to help the probabilists take that step. Whether their efforts will ultimately be appreciated remains to be seen. When I attempt to express in positive terms that quality in which con- temporary mathematics is deficient, the absence of which I have characterized as "schizophrenia, " I keep coming back to the term "integrity. " Not the in- tegrity of an isolated formalism that prides itself on the maintenance of its own standards of excellence, but an integrity that seeks common ground in the researches of pure mathematics, applied mathematics, and such mathemati- cally oriented disciplines as physics that seeks to extract the maximum mean- ing from each new development that is guided primarily by considerations of content rather than elegance and formal attractiveness that sees to it that the mathematical representation of reality does not degenerate into a game that seeks to understand the place of mathematics in contemporary society. This integrity may not be possible of realization, but that is not important. I like to think of constructivism as one attempt to realize at least certain aspects of this idealized integrity. This presumption at least has the possible merit of preventing constructivism from becoming another game, as some construc- tivism~ have tended to do in the past.
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