Paul Redding

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.

Aristotle is often regarded as providing a potentially appropriate model for a naturalistic human psychology that is able to reconcile the commonly opposed normative or “manifest” and factual or “scientific” images of the world and restore to the world the qualities that constitute its value. Such Aristotelian features were taken up after Newton by Goethe in his Theory of Color in his attempt to restore the actual color to the world that had seemingly been drained of it by Newtonian science. Here, I argue that beneath the “modificationalist” elements that Goethe took from Aristotle lies a mathematical approach to color originating in Plato that exploits similarities between color and tonal consonances and dissonances. The logical structure of Goethe’s color theory has recently been investigated by proponents of “universal logic”, but only when this theory is viewed against the background of Plato’s appropriation of Pythagorean harmonic theory does its full explanatory potential become apparent.

While Kepler is regarded as a major figure in standard historical accounts of the scientific revolution of early modern Europe, he is typically seen as having one foot in the new scientific culture and one in the old. In some of his work, Kepler appears, along with Galileo, to be on a trajectory towards Newton’s celestial mechanics. In addition to his advocacy of Copernicus’s heliocentrism, he appealed to physical causes in his explanations of the movements of celestial bodies. But other work appears to express a neo-Platonic “metaphysics” or “mysticism”, as most obvious in his embrace of the ancient tradition of the “music of the spheres”. Here I problematize this distinction. The musical features of Kepler’s purported neo-Platonic “metaphysics”, I argue, was also tied to Platonic and neo-Platonic features of the methodology of a tradition of mathematical astronomy that would remain largely untouched by his shift to heliocentrism and that would be essential to his actual scientific practice. Importantly, certain features of the geometric practices he inherited—ones later formalized as “projective geometry”—would also carry those “harmonic” structures expressed in the thesis of the music of the spheres.

While Kepler is regarded as a major figure in standard historical accounts of the scientific revolution of early modern Europe, he is typically seen as having one foot in the new scientific culture and one in the old. In some of his work Kepler appears, along with Galileo, to be on a trajectory towards Newton’s celestial mechanics. Besides his advocacy of Copernicus’s heliocentrism, he appeals to physical causes in his explanations of the movements of celestial bodies. But other work expresses a neo-Platonic “metaphysics” or “mysticism”, as most obvious in his self-alignment with the ancient tradition of the “music of the spheres”. Here I problematize this distinction. Kepler’s purported neo-Platonic “metaphysics”, I argue, had been tied to Platonic and neo-Platonic features of the methodology of a tradition of mathematical astronomy that would remain largely untouched by his shift to heliocentrism and that would be essential to his actual scientific practice. Importantly, certain inherited geometric practices—one’s later formalized as “projective geometry”—would carry those “harmonic” structures expressed in the thesis of the music of the spheres.

Aristotle is often looked to as providing a potentially appropriate model for a naturalistic human psychology that is able to reconcile the commonly opposed normative or “manifest” and factual or “scientific” images of the world. In contrast, this paper argues for the greater relevance of Plato’s comparatively neglected approach, exploring this in the context of a psychophysics of colour perception—a topic still resistant to formalization in modern science. While Plato’s natural philosophy is often dismissed as caught up in a fanciful pre-scientific approach based on the “harmonies” of Pythagorean music theory (the so-called “music of the spheres”), it is argued that such Pythagorean harmony theory had actually provided Plato with the rudiments of contemporary mathematical tools useful for the study of colour phenomena, tools such as projective geometry, linear algebra, algebraic topology, and graph theory. It is argued that Plato’s approach to the psychophysics of colour underlies the colour phenomena discovered by Goethe in the nineteenth century, phenomena that are consistent with modern mathematical analyses of order. It is Plato, not Aristotle, whose work is suggestive of a successful psychophysical approach to colour.

In this paper, I approach Hegel’s philosophy under the banner of a “Keplerian Revolution”, the implicit reference being, of course, to Kant’s supposed Copernican philosophical revolution. Kepler had been an early supporter of the Copernican paradigm in astronomy, but went well beyond his predecessor, and so is invoked here in an attempt to capture some of the important ways in which Hegel attempted to go beyond the philosophy of Kant. To make these issues more determinate, however, Hegel’s Keplerian orientation will not be presented in its contrast to Kant’s “Copernicanism” as such, but as contrasted with that of another early follower of Copernicus, Giordano Bruno, and this Brunian orientation will be used to characterize Kant’s philosophy as seen from Hegel’s rival Keplerian point of view. Interpreting Hegel as a philosophical Keplerian will require that we broach those worrisome aspects of Kepler’s astronomy, namely his support for Plato’s cosmology and the tradition of the “music of the spheres”, but this will be shown to have connections to Hegel’s own approach to logic. This in turn will help shed light on the meaning of Hegel’s form of idealism and, in particular, on its usually unacknowledged Platonic dimensions.

Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper, I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here, I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.

Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.