Math by Pure Thinking: R First and the Divergence of Measures in Hegel's Philosophy of Mathematics

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number (which also accords with their role in physical measurement); second, a recognition that a quantitative relation has three elements (the two things being related and the relation itself), which is embedded in his conception of measure; and third, a recognition of the phenomenon of divergence of measures such as in second‐order or continuous phase transitions in which correlation length diverges (e.g., the critical point of water at which the reciprocal size of the droplets diverges). For ease of exposition, we will refer to these three insights as the R First Theory, Tripartite Relations, and Divergence of Measures. Given the constraints of space, we emphasize the first and the third in this paper.