The Foundations of Mathematics

Dr. Paul Carus's The Foundations of Mathematics is a profound philosophical inquiry dedicated to the philosophy of geometry. The book begins by addressing the historical "flaw" in Euclidean geometry: the axiom of parallels. Carus provides a historical sketch of "metageometry" (non-Euclidean geometry), tracing the efforts of giants like Gauss, Riemann, Lobachevsky, and Bolyai who challenged Euclid's absolute certainty. This historical search sets the stage for the book's central philosophical problem: Are the foundations of mathematics a priori (transcendental) or derived from experience (empirical)? Carus rejects the extremes of both, arguing against Kant's subjectivism and Mill's total empiricism, and proposes his own "New Positivism". He redefines the a priori not as innate knowledge, but as the "purely formal", derived from abstraction. He argues that geometry is not "rigidly a priori" like logic, but a "purely a priori" construction based on the concept of "anyness" and "pure motility," or the potential for motion. This approach allows him to defend the unique, classical status of Euclidean geometry—not as the only tactual space, but as the indispensable system of measurement based on "even boundaries" like the straight line. He explores tridimensionality and the concept of "The Superreal", concluding with a profound epilogue that links the eternal norms of mathematics to the superpersonal nature of God.