Abstract

Already as far back as 1934, Peter Hoenen complained that the Scholastics, hardly loyal to Aristotle, had almost entirely neglected the philosophy of mathematics, while several modern schools (logicism, intuitionism, and formalism) vigorously cultivated it. What Hoenen did not know is that the moderns, too, would almost completely abandon this important branch of knowledge in the aftermath of the foundational crisis that was brewing due to the introduction of set theory. Alas, the question is yet to be settled: almost a hundred years after the catastrophe, mathematicians are unable to tell us, with any certainty, what the subject-matter of their discipline is, or what method should be used. We propose in this dissertation that the metaphysics of Saint Thomas Aquinas is quite capable not only of accounting for the principles of ancient mathematics, but also of pointing out the principles of modern set theory—and even of supporting a thorough and realistic alternative to this problematic theory. In the first part, we bring to light, in order to dispel entrenched modern fictions, what ancient Greek mathematicians had to say about the principles of their own discipline. Faithfully following Saint Thomas, we devote the second and major part of this work to establishing what a principle is; what kinds of principles there are; what is required for something to be a first principle; and what should be the first principles of any science or discipline, and specifically those of mathematics. In the third part we determine, in the light of Saint Thomas’s theory, the elusive essence of algebra—the discipline that would later be identified by Descartes with universal mathematics or analysis. We follow its development until the introduction of set theory, whose principles we clarify. Finally, we propose ancient foundations for modern so-called mathematics.