On the Use of the Equivalence Symbol and Parentheses Symbols in Associative Distributive Algebra

We have already treated the foundations of associative algebra, which include the foundations of the theory of integers, in four papers published in this magazine, each under the title "A Development of Associative Algebra and an Algebraic Theory of Numbers," appearing as follows: (I)-Vandiver, vol. 25, 1952, pp. 233-250. (II) -Vandiver, vol. 27, 1953, pp. 1-18. (III) -Vandiver and Weaver, vol. 29, 1956, pp. 135-149. (IV)-Vandiver and Weaver, vol. 30, 1956, pp. 1-8; Errata, vol. 30, 1957, p. 219. Part of this paper may, in effect, be regarded as a supplement to one or more of these papers. In fact, we shall pursue much further some of the ideas expressed in the above papers concerning use of the equivalence symbols (= or =), and also the use of parentheses. In connection with the first topic mentioned in the title, we note that two advances have been made during the history of mathematics that some mathematicians refer to as the greatest advances made, in that period, in mathematical thought, if not all abstract thought. These are, first, the recognition by the ancient Greeks, in particular Thales, of the desirability of beginning the discussion of geometry by setting up a system of postulates, that is, postulates or axioms presumably self-evident. The second step of this character was made by Lobachewsky, who, in his invention of non-Euclidean geometry, in effect discarded the notion that postulates or axioms should be self-evident. We think it is quite possible that historians and critics of mathematics in the future may refer to the development of the idea that we can, for example, use algebraic formulas and transform them according to a set of axioms which do not depend on defining any particular symbol in itself, but only on the manipulation of the symbols in the formulas. When we follow this scheme we seem to be getting closer and closer to Hilbert's idea that mathematics is the art of manipulating meaningless symbols. Related to this is the fact that algebraists at the present time are particularly active in reducing other parts of mathematics to algebraic patterns, such as the subject which Artin calls geometric algebra. It seems to me that in spite of the fact that considerable advances