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  • The Emergence of the Cyborg and the End of the Classical Tradition:The Crisis of Alfred North Whitehead's Process and Reality
  • Don Byrd (bio)

Logic fell into a Hegelian funk in the nineteenth century. Even the most fundamental notions could not be consistently distinguished: truth and falsity and life and death flipped like the rabbit and the duck in the optical illusion. Nihilism was the inevitable product of this theoretical subroutine, which Nietzsche brought to a brilliant conclusion. Toward the end of the century, however, Georg Cantor and Gottlob Frege led a brilliant recovery of formal logic. They constructed a logic that took abstract collectivities or sets, rather than simple distinctions, as its primitive form. Thus, logic addressed not the unit of the natural-language sentence but the unit of the logical or mathematical function. Frege's Begriffsschrift or concept writing was not only clearer, it was more abstract. William and Martha Kneale, in their magisterial Development of Logic, noted:

Frege says that the relation of the script to ordinary speech is like that of the microscope to the eye and claims for it the merits which had been predicted by Leibniz and others in the seventeenth century for a calculus philosophicus et ratiocinator. It is one of the tasks of philosophy, he tells us, to break the dominion of the word over the human mind, and his invention has already done something towards this by freeing logic from too close attachment to the grammar of ordinary language.1

The Begriffsschrift set forth clearly the axioms of set theory. For the first time, it was possible to specify the deep forms of post-Cartesian [End Page 95] logic. The clarity of the symbolism, however, also made it possible for Bertrand Russell to see a fatal flaw in the system. It had been known since the Greeks that certain abstractions or essences—such as the essence of the consistent liar—were paradoxical, but they had seemed only logical curiosities. When, in the seventeenth century, the essences were emptied out, however, and signs took their values not from concrete instances of distinguished objects but from relations with other signs in the formal calculus, the self-referential objects covertly appeared in the very foundations of logic. Russell noted a difference between ordinary collections, such as the collection of all lions, and a relatively rare but, as it turned out, crucial order of collections, such as the collection of all collections. The collection of all lions is unproblematic. The collection of all collections, however, is perplexing: the collection is itself a collection. It is not possible to determine whether it includes itself: if it does, it does not, and so forth. On June 16, 1902, Russell wrote to Frege just as he was about to complete his major work, pointing out that the set of all sets, which was fundamental to Frege's formalism, was paradoxical. The Greek liar had his revenge at last.

Despite the setback, it was assumed that mathematics could still be successfully formalized and completed: the Russell paradox was a mere linguistic statement that belonged not to mathematics but to metamathematics; it seemed to be a problem because it interpreted the formal signs in relation to an extraneous grammar. Russell and Alfred North Whitehead, on the one hand (in their jointly authored Principia Mathematica), and David Hilbert, on the other (in a series of important texts, culminating in Grundlagen der Mathematik), developed philosophically distinct ways to address the problem and, they believed, to inoculate the formalism against viciously self-referential propositions.

Russell and Whitehead's Principia Mathematica2 and Whitehead's Process and Reality3 were profoundly conservative works—the former a powerful, if somewhat ungainly, reconstruction of the Platonic formal universe, using a version of Frege's concept writing; the latter an allegorical interpretation of the abstract system of Principia by way of a patchwork of concepts and phrases from Plato, Aristotle, and the pre-Kantian modern philosophers, most notably Locke. Although [End Page 96] Hilbert's formalism finally fell to the same argument as Russell and Whitehead's Platonism, his proposal of mechanical proof procedures proved in the long run to be more useful.

At the outset...

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