Conceptual Harmonies by Paul Redding presents an innovative examination of G. W. F. Hegel’s Science of Logic by tracing the fine interconnections between philosophical and mathematical thought throughout history. Redding posits that Hegel’s work is deeply embedded within the evolution of European mathematics, a perspective that offers fresh insights into the often bewildering landscape of Hegelian logic.
At the core of Redding’s interpretation is the idea that the dialogue between philosophy and mathematics has been mutually influential since antiquity. He emphasizes how Hegel’s thinking was shaped by Pythagorean harmonics, Platonic reasoning, and the geometric logic of Aristotle. By doing so, Redding situates Hegel within a lineage that extends from classical antiquity through the mathematical developments of the early modern period, culminating in the sophisticated algebraic geometries of the 19th century.
Redding carefully traces the historical trajectory of these intellectual currents, detailing how the mathematical innovations of figures like Leibniz, with his work on calculus and combinatorial logic, and later developments in algebraic geometry, were pivotal in shaping Hegel’s conceptual framework. This historical contextualization allows Redding to illuminate Hegel’s logic in a new light, challenging the prevailing interpretations that often overlook the mathematical dimensions of Hegel’s thought.
In Conceptual Harmonies, Redding extends his analysis beyond the confines of Hegelian scholarship. He places Hegel’s ideas in a broader philosophical and mathematical context, engaging with the works of pivotal figures such as Kepler, Newton, Frege, Boole, and Peirce. This broad approach not only enriches our understanding of Hegel’s Science of Logic but also demonstrates its relevance to contemporary conceptions of logic and mathematics. Redding’s work contests Robert Brandom’s interpretation, which frames Hegel within the traditions of analytic philosophy, proposing instead that Hegel’s logical framework anticipates many modern developments in mathematics and logic.
Paul Redding, an emeritus professor of philosophy at the University of Sydney, brings his considerable erudition to this project, drawing on his extensive knowledge of both Hegelian philosophy and the history of mathematics. His previous works, including Continental Idealism: Leibniz to Nietzsche and Analytic Philosophy and the Return of Hegelian Thought, have established him as a leading figure in the study of Hegel. In this latest endeavour, Redding delivers a compelling and thorough reassessment of Hegel’s logical project.
The book cites Robert Pippin’s observation that Hegel’s Science of Logic has yet to receive the full contemporary engagement it merits. Redding argues that this neglect is partly due to a widespread misunderstanding of Hegel’s logic, particularly its last part, known as “Subjective Logic,” which addresses forms of judgment and inference in a manner that aligns more closely with traditional logic than is often acknowledged.
Redding’s explores the ancient roots of Hegel’s thought, examining the influence of Greek geometry and Pythagorean harmonics on Hegel’s conception of the syllogism. He highlights Hegel’s fascination with the “most beautiful bond” described in Plato’s Timaeus, which Hegel interpreted as a geometric ratio symbolizing unity and harmony. This Platonic influence is traced through Hegel’s early philosophical development, including his lost fragment on the Analogie from Plato’s Timaeus, which employed a fractal-like “triangle of triangles” to represent conceptual relations.
Redding revisits Hegel’s infamous dissertation, On the Orbits of the Planets, often criticized for its perceived reliance on number mysticism. Redding argues that Hegel’s intention was not to supplant empirical science with mystical numerology, but to defend the role of observation against the purely empirical methods attributed to Newton. By situating Hegel’s work within the broader context of Pythagorean and Platonic mathematical traditions, Redding reveals a more nuanced understanding of Hegel’s engagement with the natural sciences.
Redding’s detailed historical analysis extends to the later influences on Hegel’s logic, such as the developments in algebraic geometry by Leibniz and others. He shows how these mathematical advancements were integral to Hegel’s conceptual innovations, highlighting Hegel’s anticipation of later logical theories and his relevance to contemporary philosophical discourse.
Conceptual Harmonies is a significant contribution to Hegel studies and the history of logic. Redding’s exhaustive research and original insights provide a comprehensive and compelling reinterpretation of Hegel’s Science of Logic, bridging the gap between ancient mathematical traditions and modern logical theory. This work is essential reading for anyone interested in Hegel, the history of mathematics, and the interplay between philosophical and mathematical thought.
Simon Gros 
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