Mathematical Confusions Behind a Common Misunderstanding of Idealism

The paper starts by questioning the highly influential but extremely misleading characterizations of Plato and Hegel by Bertrand Russell and Karl Popper. It is argued that mathematical assumptions concerning the ancient problem of the incommensurability of continuous and discrete quantities underlie the ways in which Russell and Popper portray the metaphysics of Plato and Hegel—Popper explicitly, and Russell implicitly, presupposing a particular response to this problem by broadening the concept of number to include irrational numbers. Recent work on Plato, however, suggests a different strategy for responding to this ancient conundrum, one that involves a mediated “duality” of the continuous and discrete that Hegel would later generalize to a duality of determinate and indeterminate aspects of cognition more generally. This Platonic alternative had originated with the Pythagorean natural philosopher Philolaus of Croton and would later be expressed in modern mathematics in a non-Cartesian way of applying numerical metrics to geometric figures in disciplines such as projective geometry. Such an alternative approach to both quantitative and conceptual incommensurability, I claim, had influenced Plato’s later conception of philosophical method that would be adopted by Hegel via the intermediary of Leibniz, the first modern “idealist”. Understanding the actual mathematics modeling philosophical concepts for Plato and Hegel becomes crucial for understanding the philosophical claims of modern idealism.