The Problem of the Existence of Mathematical Entities

of Paper read on 2nd March, 1951. The problem of the existence of mathematical entities takes its origin from the fact that, while the truths of mathematics belong to those elements in human knowledge to which we ascribe the highest degree of certainty, we search in vain, in the world of human experience, for entities which present the properties described in these truths. The first attempts to solve i t : platonism, aristotelianism, and constructivism (Plotinus, Nicolaus Cusanus, Kepler, Hobbes), belong to speculative philosophy. The crisis in the foundations of mathematics, however, forced the mathematicians to consider the problem from their own point of view. At first, they borrowed their solutions from speculative philosophy. Later, the development of axiomatics in geometry led Poincare and Hilbert to accept the formal consistency of a system of axioms as a necessary and sufficient condition for the existence of a model. I t was observed by L. Lowenheim (1915) that, for axiom systems formulated within the Peirce-Schroder calculus, the existence of a model can be discussed in a rigorous manner. The development of his ideas led finally to the Lowenheim-Skolem-Godel theorem : an axiom system formulated within elementary logic has a model, if and only if it is formally consistent. Recently, this result has been extended to various other logical systems by A. Mostowski, L. Henkin, and Helena Rasiowa. Far from leading to a final settlement of the problem, however, this result was shown by Th. Skolem to give rise to a complete relativisation of mathematical existence. So it seems reasonable to reconsider the problem from the point of view of speculative philosophy ; of course, we cannot expect to obtain, on this basis, a deductive theory of mathematical existence. Nevertheless, it is felt that a realistic and pluralistic conception of human knowledge, which can borrow precious elements from older doctrines, may contribute to a clarification of the situation. terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0950563600001913 Downloaded from https://www.cambridge.org/core. IP address: 207.241.231.83, on 30 Oct 2018 at 04:32:48, subject to the Cambridge Core