Ideals In Partially Ordered Sets

Abstract

is. The study of partially ordered sets is a kind of abstract mathematics of the most general sort, compared with the theory of algebraic integers which gave rise to ideal theory. I should like to say a few words about the distinction which is often made between abstract mathematics and the other kind, which may be called classical mathematics. Classical mathematics is based on the real number system, or equivalently on the system of positive integers or on the complex number system. Abstract mathematics is based on set theory. It studies the consequences of quite arbitrary postulate systems. From the point of view of the abstract mathematician, the real number system is a particular ordered field, or a particular locally compact topological ring; in other words, one of a class of similar abstract systems, and not necessarily more interesting than the others. From this same point of view, classical mathematics is a special case of a more general abstract theory. Actually, all mathematics is equally abstract. The real number system is no more concrete than, say, the theory of groups. Neither deals with the physical universe. To be sure, classical mathematics has seemed to be the kind most useful in applications to the sciences. However, perhaps this is an illusion, due to the time lag between the development of a theory and its application. The classical mathematics of the last century is being applied now. The abstract mathematics now being developed may turn out to be just as useful in the next century. Perhaps it will be found that you can kill just as many people just as efficiently using the theory of quasi-groups or p-adic fields as you can with the real number system. Of course many mathematicians will disagree with this view. But it is true now, just as it has been found to be true in the past, that mathematical results arrived at with no thought of their possible use are being applied to situations not imagined by their originators.