Origins of Formal Structure

Abstract

Mathematics, at the beginning, is sometimes described as the science of Number and Space—better, of Number, Time, Space, and Motion. The need for such a science arises with the most primitive human activities. These activities presently involve counting, timing, measuring, and moving, using numbers, intervals, distances, and shapes. Facts about these operations and ideas are gradually assembled, calculations are made, until finally there develops an extensive body of knowledge, based on a few central ideas and providing formal rules for calculation. Eventually this body of knowledge is organized by a formal system of concepts, axioms, definitions, and proofs. Thus Euclid provided an axiomatization of geometry, with careful demonstrations of the theorems from the axioms; this axiomatization was perfected by Hilbert about 1900, as we will indicate in Chapter III. Similarly the natural numbers arise from counting, with notation which provides to every number the next one—its successor, and with formal rules for calculating sums and products of numbers. It then turns out that all these formal rules can be deduced from a short list of axioms (Peano-Dedekind) on the successor function (Chapter II). Finally, the measurements of time and space eventually are codified in the axioms (Chapter IV) for the real numbers.