On Universal Algebra and the Whiteheadian Cosmology

“Ordinary algebra in its modern developments,” Whitehead observed in 1897, “is studied as being a large body of propositions, inter-related by deductive reasoning, and based upon conventional definitions which are generalizations of fundamental conceptions.” The use of ‘based upon’ here is perhaps too weak, for some “propositions” must of course be picked out as determinative of the kind of algebra in question by way of axioms. The definitions are then ancillary devices of notational abbreviation and may or may not be of terms or phrases previously introduced, and may or may not be of a “generalized” form. “Thus a science is being created [ordinary algebra in its modern developments],” Whitehead continues, “which by reason of its fundamental character has relation to almost every event, phenomenal or intellectual, which can occur…. Such algebras are mathematical sciences which are not essentially concerned with number or quantity; and this bold extension beyond the traditional domain of pure quantity forms their peculiar interest. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thought, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.”