SCHIZOPHRENIA IN CONTEMPORARY MATHEMATICS 29 possible to develop a theory that seems to accommodate the known examples and to have certain pleasing functorial qualities, but the theory is somehow not convincing -- for one thing, it is too involved. For another, there is a certain sort of space - - let us call it a ball space - - that does not fit well into the theory. DEFINITION. A ball space is a set X, together with a function that to each r 2 0 and point x of X associate a subset B(x, r ) of X (to be thought of as the closed ball of radius r about x) satisfying the following axioms. (a) B(x, r ) c B(x, s) if r s s . (b) B(x, 0) = Cx 3 . (c) B(x,r) = ~ { B ( x , s ) : r 3 . (d) If y E B(x, r), then x E B(y,r) . (e) If y E B(x, r ) and z E B(y, s), then z E B(x, r + s). (f) U CB(X,~): r 2 0 1 = X . Duals of Banach spaces are particular instances of ball spaces, as are various other function spaces. Algebraic topology, at least at the elementary level, should not be too difficult to constructivize. There is a problem with defining singular co- homology constructively, as pointed out in [2] . Richman [25] points out that the classical Vietoris bmology theory is not satisfactory constructively, and he gives a new version that constructively (and also classically) has certain features that are more desirable. I would like to conclude these lectures by discussing some of the tasks that face constructive mathematics. Of primary importance is the systematic constructive development of enough of algebra for a pattern to begin to emerge. Of course, it may be that much of the classical theory is inherently unconetructivizable, and that con- structive algebra will go its own way. It is too early to tell. Less critical, but also of interest, is the problem of a convincing con- structive foundation for general topology, to replace the ad hoc definitions in current use. It would also be good to see a constructivization of algebraic topology actually carried through, although I suspect this would not pose the critical difficulties that seem to be arising in algebra. To sum up, the first task is to constructivize as much of existing classical mathematics as is suitable for constructivization. As this is being
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