POSTULATES, or axioms, are sets of propositions belonging to mathematical dis ciplines from which the remaining propositions can be deduced. Thus the axioms of Euclid constitute a tentative set of postulates for ordi nary geometry. The old-fashioned view was that these axioms constitute a set of self evident truths, referring only to one system, that of the space in which we live. The work of Bolyai, Lobachevski and Riemann, however (see GEOMETRY, Nos-Eucutie.Alt), cast grave doubts on the self-evidence of these proposi tions, and led to the more modern attitude, according to which the axioms are merely hy potheses concerning space. As far as the mathe matician makes use of the axioms it is a matter of entire indifference whether they concern space or not; in other words, strictly speaking, postulates are not propositions concerning a definite subject matter at all, but are merely forms of such propositions. Once this view is accepted it becomes easy to see that sets of postulates can be formulated for systems as unlike ordinary geometry or any ordinary mathematical system as you please, and that these sets will have just as much intrinsic, un derived right to exist as that of Euclid. This has led to a large number of sets of postulates for various mathematical systems, such as non Euclidean geometry, line geometry, analysis sites, algebra, the theory of functions of a real variable, quaternions, symbolic logic, etc. (qq.v.).
Furthermore, various sets of Postulates can be formed for the same system. Thus, one can describe all the entities of geometry in terms of points and their distance-relations, or in terms of spheres and their relations of inclu sion. Accordingly there are at least two possi ble sets of propositions characterizing the properties of geometrical entities, or, in other words, two sets of postulates for geometry. Furthermore, it is possible to define all geo metrical properties in terms of the same entities by different sets of postulates. As a conse quence of these facts it becomes a desideratum to determine those sets of postulates which are in some sense the most practical or simple. TEe
criteria of simplicity usually employed are (a) the independence of the postulates—no set of the postulates must imply another; (b) their fewness in number. Independence of the pos tulates is demonstrated by the exhibition of systems which fail to satisfy but one of the postulates, this postulate being each in turn. E. K. Moore of Chicago discusses a strength ened form of independence, in which not only are the postulates independent of one another, but no postulate or its denial is dependent on any combination of the other postulates or their denials. Many mathematicians, however, con sider this a rather unnecessary refinement. As to the fewness of the postulates, for some pur poses of analysis the opposite extreme seems desirable as indicating a greater degree of analysis. However, no really unequivocal definition of the number of distinct postulates in a set has yet been given.
Besides these properties of postulates that are more or less independent of the systems they define, there are others which are bound up with the latter. In the first place, though a set of postulates always defines a class of mathematical systems, this may be the null class (see Loom, SYMBOLIC). In that case the postulates are called inconsistent, otherwise they are consistent. The consistency of a set of postulates is demonstrated by the exhibition of a system which satisfies them. For all ordi nary purposes the exhibition of a system formed from the natural numbers is regarded as sufficient. The policy of producing such a system in the case of all mathematical systems is called the arithmetization of mathematics.
Another desideratum of a set of postulates is that the systems •they define be one from the mathematical standpoint — that is, that the set be categorical. A categorical set has the prop erty that a one-one correspondence can be set up between any two systems which it defines; and that this correspondence will leave their structure invariant.