Hegel: Mathematics

This project retraces activations of Kierkegaard in the development of polit­ical theology. It suggests alternative modes of states of exception than those attributed to him by Schmitt, Taubes and Agamben. Several Kierkegaardian themes open themselves to 'something like pure potential' in Agamben, namely: living death, animality, criminality, auto-constitution, modification, liturgy, love and certain articulations of improbabilities. Attention is drawn to a modal ontology and auto-constitution at work in Kierkegaard's writings, as well as a complicated and indissociable operation between killing and (...) letting-live in legalist exceptionalism, comparable to similar functions found in Foucault regarding the biopowers and necropolitics of territorial and governmental apparatuses. It closes in consideration of Kierkegaard's critique of enumeration, large numbers, and statistical probability alongside contemporary tele-technoscientific social controls via the online datafication of people by surveillance or platform capitalisms. After Kierkegaard, such apparatuses are perhaps suspect as calculated to tranquilize humanity into more docile subhumans as it fools folk into becoming part of its numbers. (*Accompanying file includes only front matter, abstract, and endnotes*). (shrink)

We attribute three major insights to Hegel: first, an understanding of the real numbers as the paradigmatic kind of number ; second, a recognition that a quantitative relation has three elements, 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. 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. (shrink)

It is fair to say that Georg Wilhelm Friedrich Hegel's philosophy of mathematics and his interpretation of the calculus in particular have not been popular topics of conversation since the early part of the twentieth century. Changes in mathematics in the late nineteenth century, the new set-theoretical approach to understanding its foundations, and the rise of a sympathetic philosophical logic have all conspired to give prior philosophies of mathematics (including Hegel's) the untimely appearance of naïveté. The common view was expressed (...) by Bertrand Russell: -/- The great [mathematicians] of the seventeenth and eighteenth centuries were so much impressed by the results of their new methods that they did not trouble to examine their foundations. Although their arguments were fallacious, a special Providence saw to it that their conclusions were more or less true. Hegel fastened upon the obscurities in the foundations of mathematics, turned them into dialectical contradictions, and resolved them by nonsensical syntheses. . . .The resulting puzzles [of mathematics] were all cleared up during the nineteenth century, not by heroic philosophical doctrines such as that of Kant or that of Hegel, but by patient attention to detail (1956, 368–69). (shrink)

This essay explores Hegel’s treatment of Carl Friedrich Gauss’s mathematical discoveries as examples of “Analytic Cognition.” Unfortunately, Hegel’s main point has been virtually lost due to an editorial blunder tracing back almost a century, an error that has been perpetuated in many subsequent editions and translations.The paper accordingly has three sections. In the first, I expose the mistake and trace its pervasive influence in multiple languages and editions of the Wissenschaft der Logik. In the second section, I undertake to explain (...) the mathematical significance of Gauss’s discoveries. In the third section, I take a look at the deeper implications of Hegel’s treatment of Gauss’s work as a window onto the nature and limitations of analytic cognition. In conclusion, I seek to explain how the linear method embodied in deductive reason leads by its own inner principle, according to Hegel, to its dialectical Aufhebung (sublation). The result is a kind of deliberately circular reasoning that he describes as “the Absolute Idea.”. (shrink)

This paper argues that Hegel has much to say to modern mathematical philosophy, although the Hegelian perspective needs to be substantially developed to incorporate within it the extensive advances in post-Hegelian mathematics and its logic. Key to that perspective is the self-referential character of the fundamental concepts of philosophy. The Hegelian approach provides a framework for answering the philosophical problems, discussed by Kurt Gödel in his paper on Bertrand Russell, which arise out of the existence in mathematics of self-referential, non-constructive (...) concepts (such as class). (shrink)

There is at present no intelligible account of what the statements of pure mathematics are about. The philosophy of mathematics is in a mess! Marvin J. Greenberg.