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.

Gottfried Ploucquet, a teacher at the Tubingen seminary when Hegel was a student there, had been one of the few philosophers to take up Leibniz’s mathematized logic, including his project of reducing logic, and thought itself, to computational processes. In his Science of Logic, Hegel briefly discusses this project when expanding on his own “subjective” logic. The general tenor of the response is predictable. Computational logic seeks to mechanize conceptual processes, but conceptuality itself distinguishes free spiritual beings from machines. Beneath the surface, however, Hegel’s attitude to the relation of computation to conceptual reasoning is more complex. Here I argue that in Book I of his Logic, Hegel, following the approach of Plato in his late dialogues, treats a certain mathematical conception of number, the Neopythagorean triadic monad, as a model for the concept itself. In the section Quantity, Hegel focuses on the incommensurability between discrete and continuous quantities, the numbers of arithmetic and the lines, areas and volumes of geometry. This incommensurability had been discovered by the Pythagoreans and in his later writings, Plato had adopted a proposal for mediating it, attempting to generalize it to a solution of the conceptual incommensurability between the eternal realm of being and the transient realm of becoming. In line with Plato’s attempt, Hegel presents an account of the development of mathematical practices in which the concept of number from mere counting unit to a triadic form mediating numbers and geometric continua. This structure will in turn provide a model for his own later syllogism. This role for mathematics for Hegel is to be understood as in line with Plato’s later attempts to mediate being and becoming in ways in which eternal Ideas can be approximated in the form of worldly surrogates manifesting this triune structure. Conceptuality cannot be reduced to computation, but relations among computational processes nevertheless reveal much about the nature of conceptuality.

While in broad agreement with Rorty’s emphasis on the role of redescription in Hegel’s method, and with his wish to free Hegel from the constraints of traditional metaphysics, this chapter argues against Rorty’s account of Hegel’s redescriptive methodology. His account is, it suggests, tied to a misleading Sartrean interpretation of Hegel’s famous “master–slave” dialectic—an interpretation that is in fact closer to Fichte’s use of the notion of recognition than Hegel’s own. When Hegel’s concept of recognition is understood in relation to the logic of his concept of the will, a more nuanced account of recognition is achieved. This is one that coheres with a “redescription” of the task of metaphysics that portrays it as an inquiry into a modally conceived actual world. Unlike Rorty’s redescription, this is one that preserves the relevance of the value of truth and not merely that of freedom for metaphysics.

Parallels are often drawn between Hegel and the later Wittgenstein, but an examination of Hegel’s conception of judgement from the Science of Logic reveals curious parallels with central doctrines of Wittgenstein’s Tractatus. There Hegel appeals to a type of primitive linguistic structure or Satz consisting of the concatenation of two singular terms or names, and like Wittgenstein’s “Elementarsäzte”, these have the properties of being mutually independent, essentially positive, and with only one way of being true (or for Hegel, “correct”). For Hegel, to play a role in reasoning, such a Satz must have one of its terms re-determined as general such that it can now enter into inferential relations with other judgements. The immediate product of this type of transformation was a form of judgement, the judgement of existence (Dasein) that he exemplified by colour judgements, and such judgements had a logical form different to that of the Tractatus’ compound Sätze that Elementarsäzte were meant to constitute. When in the late 1920s Wittgenstein moved away from the doctrines of the Tractatus, considerations of the logic of colour judgements with similar features to those of Hegel’s judgements of Dasein played a role.

Stephen Gaukroger’s final book, The Failures of Philosophy: A Historical Essay, presents a compelling historical overview of a discipline threatened with irrelevance, a fate that he diagnoses as a consequence of its having attempted at its birth to establish itself entirely at the level of a “second-level” inquiry. Stephen presents his book itself as “a historical essay”, but read as exemplifying a historically influenced way of doing philosophy, the book itself testifies to the possibility of a healthy and relevant form of contemporary philosophical culture. Here I present Hans-Georg Gadamer as both exemplar and self-conscious theorist of such a mode of philosophy as enacted by Stephen’s book as well as his celebrated Descartes biography.

In this paper, I start from a criticism that John McDowell has made of the account of perception contained in Mind and World. In the essay Avoiding the Myth of the Given, he describes his earlier account as having been flawed by his having equated the idea of the conceptuality of perceptual experience with that of its propositionality. While agreeing with this criticism, I suggest that McDowell’s diagnosis of the earlier problem, as well as his suggestions for its solution, are obscured by his continuing to situate his account of perception within the predominantly epistemological framework of Mind and World. In contrast, and guided by Hegel’s account of the logic of perceptual judgment, I invoke a different function served by Kant’s idea of the “intuitive” content of perceptual experience—its modal function of indicating actual from merely possible states of affairs. Comparing Hegel’s metaphysical position to that of contemporary modal actualists, I argue that by transforming Kant’s concept–intuition distinction into a logical distinction between different types of judgment employing different notions of predication, Hegel was able to capture the modal function played by Kantian intuition without falling into the problematic “Myth of the Given.”

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.

Although Hegel is generally not known as a philosopher of mathematics, he maintained a deep interest in the history of mathematics, especially in its transformations between antiquity and the modern age. Charles S. Peirce, who was the son of a distinguished mathematician and was involved in developments in mathematics in the second half of the nineteenth century, was critical of what he perceived as Hegel’s lack of mathematical acumen. Nevertheless, he recognized in Hegel’s Science of Logic structural features of his own mathematically informed philosophy. In this article, I look to Hegel’s discussion of magnitude in The Science of Logic, and especially to his conception of the relation between continuous and discrete magnitudes, in order to articulate a solution he might have offered to difficulties encountered by Peirce in his opposition to Cantor’s set-theoretical analysis of the continuum. It is argued that Hegel’s interest in the ancient Platonic/Pythagorean tradition in mathematics provided him with crucial resources in this regard.

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.