“The Death of Socrates,” by Jacques-Louis David (1787)
The following essay will be published in three installments; this is the second.
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Congratulations on Doing the Impossible: On the Death of Philosophy, #2
From our perspective, in the 21st Century, Frege’s work looks naïve because of the Russell paradox—and, in fact, has been mothballed under the name “naïve set theory.” Russell and Whitehead’s work looks, from a certain ruthless perspective, equally naïve. As Hanna writes,
Despite its triumph and triumphalism, in fact classical Analytic philosophy was seriously theoretically hobbled in the 1930s and 40s by Kurt Gödel’s profoundly important first and second incompleteness theorems…which, when they’re taken together with Alfred Tarski’s semantic conception of truth, bearing further witness to the categorical distinction between truth and logical proof, collectively amount to a logico-mathematical 1-2 punch that permanently KO’d the classical Frege-Whitehead-Russell logicist project for reducing mathematics to logic. (Hanna, 2022: pp. 52-53)
Hanna’s point is that “[an] organicist wave crashed upon the rocky shores of early 20th century Analytic philosophy and was lost” (Hanna, 202: p. 54), because organicists like Henri Bergson were bitterly decried by Bertrand Russell, who had built his institutional power and celebrity megaphone on an (apparently) successful masterstroke: the Principia. From Hanna’s point-of-view, Russell should not have been given the acclaim he received, considering the dubiously ambitious nature of his effort to limit all truth-claims to provable, sequential mathematical operations—an effort that was finally smashed to bits.
But by whom? By Gödel, Tarski, and Quine, all of whom responded specifically to the work of the Analytic logicians, and had been counted among the faithful (see Hanna, 2022) until their own work convinced them of the flaws in the Principia. There is no question in my mind that Russell could be, at times, a repressive force who prevented other, more “speculative” philosophers (including the organicists) from reaching wider audiences. But I would strongly object to the idea, popularized by Richard Rorty and backed by Hanna, that the project of Analytic philosophy was based on an impossible, dogmatic logicism that took an entire body of philosophical work down with it when it was finally disproven.
First of all, Russell’s paradox already contains the seeds of the self-referential statements Gödel later used to disprove Russell and Whitehead, as is obvious from the paradox’s linguistic form, “This set (I) is (am) the set of all sets that do not contain themselves.” For this reason, and because of Russell and Whitehead’s stronger formulation of the rules Frege invented for manipulating sets, the Incompleteness Theorems never come into existence without the precedent set by the Principia.[i] It may have been wrong, but it was a necessary error to make en route to the truth.
Second, the Analytic philosophers were, with a couple unfortunate exceptions, mostly not logicists for their whole lives. In 1929, about a decade after a definitive break with Russell, Whitehead published Process and Reality, a work that expanded upon both organicism and phenomenology. Scratch him off the list. Ditto Gödel and Quine, for reasons discussed above. But even Ludwig Wittgenstein was not limited by the mathematical constraints of number theory, a fact articulated beautifully by none other than Bertrand Russell. Here’s Russell on Wittgenstein:
Mr. Wittgenstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages […] The totality of possible values of x which might seem to be involved in the totality of propositions of the form fx is not admitted by Mr. Wittgenstein among the things that can be spoken of, for this is no other than the totality of things in the world, and thus involves the attempt to conceive the world as a whole. (Russell in Wittgenstein, 1922/1981: p. 22)
Russell is referring, of course, to the series of propositions that lead Wittgenstein to his final, famous conclusion: “What we cannot speak about, we must pass over in silence” (Wittgenstein, 1922/1981: prop. 7, p. 189). Not only does Russell’s “loophole” obviously foreshadow Tarski’s work, but it also potentially allows a set to describe its own unresolvable uncertainty, a potentiality that would be reached and fully described when Gödel invented “Gödel numbers,” which are computable sets represented as integers. As we will see, this had unbelievably important implications in a wide range of fields, including computer science, neurobiology, and systems theory.
Russell is also able to foresee, albeit with rather abject terror, the possibility of infinitely recursive hierarchies:
These difficulties suggest to my mind some such possibility as this: that every language has, as Mr. Wittgenstein says, a structure concerning which in the language, nothing can be said, but that there may be another language dealing with the structure of the first language, and having itself a new structure, and that to this hierarchy of languages there may be no limit.[ii] Mr. Wittgenstein would of course reply that his whole theory is applicable unchanged to the totality of such languages. The only retort would be to deny that there is any such totality. The totalities concerning which Mr. Wittgenstein holds that it is impossible to speak logically are nevertheless thought by him to exist, and are the subject-matter of his mysticism…I do not see how any easier hypothesis can escape from Mr. Wittgenstein’s conclusions. (Russell in Wittgenstein, 1922/1981: p. 23)
That means that in 1922, when Russell wrote his introduction to the Tractatus, Russell was fully aware of Wittgenstein’s mysticism, which clearly moves beyond logicism towards a mystical reckoning with totality that recalls Immanuel Kant’s category of the “noumenal” and Kant’s definitions of the “supersensible.” He was also aware of the implications for his own work: “The totality resulting from our hierarchy would be not merely logically inexpressible, but a fiction, a mere delusion” (Russell, in Wittgenstein, 1922/1981: p. 23). Russell is using “our” abstractly here—he is not deliberately referring to himself and Whitehead—but he is also inadvertently announcing the impending death of the mathematical totality he had attempted to adduce in the Principia.
The same problem that plagues any attempt to create a “totality” of languages, and higher-level metalanguages to verify those languages, is equally fatal to any attempt to create a totality of sets. In other words, Russell was one step away from using Wittgenstein to demonstrate the interchangeable nature of linguistic and mathematical logic (which Tarski would go on to codify), and then using the infinite regress of metalinguistic truth-claims to prove that no set can express a totality (which Gödel would prove nine years later). Consider this: the resurgence of interest in Kant that has enlivened recent decades of philosophical research is already foreshadowed by the Tractatus itself, which abounds with Kantian language and concepts.[iii]
Furthermore, the path to this renaissance of interest, in which this blog itself participates, does not primarily lead through the “outsider” history of Continental philosophy, which continued the projects of organicism and transcendental critique during the reign of logicist analysis. Continental philosophy in the first half of the 20th Century was largely willing to replace Kant with one of his successors, G.W.F. Hegel. Hegel did not preserve in his own work very many features of Kant’s transcendental critiques, although he draws heavily on Kant’s system of perceptual boundaries and modalities to describe their dialectical “sublation.”
Instead, it was Kant’s relevance to 20th century mathematics and natural science that brought him back into vogue. Kant’s writings helped expand our ability to apply recent scientific discoveries more broadly. The long list includes discoveries made by structural linguists about the logic of syntax and language acquisition. Kant made it easier to knit together the logical structure of language with the fact that language could not be acquired by children if it were not present a priori as a substratal, innate capacity (and limit) of the human brain.[iv] Even more crucial to this Kantian renaissance was the work of physicists Albert Einstein and Werner Heisenberg. Einstein’s Theory of General Relativity and Heisenberg’s Uncertainty Principle both suggest limitations on our ability to arrive at any empirical description of the universe as a totality, limitations that later provoked key mathematicians and scientists to return to Kant in a mood of wonder and admiration.
So much for Rorty’s idea that Analytic “scientism,” and the adjacent positions of the logicists, were poisonous to the 20th Century reputations of philosophers from bygone centuries. What about Russell’s hostility to organicism? Well, first of all, it is quite possible to overstate Russell’s success. Organicism did not fail to thrive, as Hanna’s own psychological analysis of Russell’s motives incidentally shows: “it’s also clear from Russell’s correspondence and other biographical evidence that he was jealously annoyed by Bergson’s great fame during the first three decades of the 20th century” (Hanna, 2022: p. 54). Also, Russell was right to be suspicious of Bergson. Bergson was not some kind of intuitive savant, or even a purely speculative bootstrapper like Rene Descartes. He was trying to be scientific, which is why he attributed many features of consciousness and organic life to an élan vital, a “life force” posited by 19th century scientists who did not have the tools to describe life as a system with emergent properties. There is no such thing, however, as the élan vital, and Bergson faded into obscurity when vitalism was abandoned by the scientific community at large. He is mostly known today thanks to the enthusiastic reception he received from Marcel Proust, who incorporated his theories into a great novel, In Search of Lost Time. This is probably a shame, since Bergson’s work speaks eloquently to unsolved problems raised, within neuroscience, by researchers like Mihaly Csikszentmihalyi (Flow) and Daniel Kahneman (Thinking, Fast and Slow). Nonetheless, surely organicism’s embarrassing debts to the muddy sidetrack of vitalism counts as a mistake equal to, or greater than, the Gödel-shaped hole in the side of the Principia Mathematica.
Moreover, organicism’s great comeback would arrive in the most unexpected way: through the intervention of an American mathematician, Norbert Wiener, who invented cybernetics in the 1940s, and the work of Austrian biologist Ludwig von Bertalanffy. Today, cybernetics is a subdivision of “systems theory,” a gigantic edifice that expanded Bertalanffy’s preliminary “General Systems Theory.” Systems theory became the foundation for ecological science, communications theory, information theory, chaos theory, complexity theory, computer science, and artificial intelligence. Today, dozens of cutting-edge theorists have built bridges connecting quantum theory, the discoveries of Einstein and Isaac Newton, systems theory, and the modern technologies indebted to them (the transistor, the microchip, etc.). That’s how, in 2024, we can see that a version of Tarski’s semantics of truth (for example) enables us to create the error correction algorithms, and hierarchized processing of information, that detects and corrects quantum noise in electric currents and quantum computers.
Even more exciting, advances in cybernetics and (organicist) systems theories have made possible promising speculations about the nature of consciousness, especially the work of Douglas Hofstadter, who synthesized systems theory with Gödel’s incompleteness theorem (in Gödel, Escher, Bach) to explain how living organisms can use the logic of self-reference to become “aware,” figuratively or literally, of their own existence, limits, and entropy. Even the élan vital may eventually return, transfigured, as the “allostatic responses” that enable living systems to subsist, reproduce, and shield themselves over time. If it does, however, it will bear the imprint of analytic (and post-analytic) theorists like Kurt Gödel and Alfred Tarski, none of whom could have achieved what they did without standing on the shoulders of “idiots” like Bertrand Russell. If you’ll pardon the joke, they did more than just a supreme job of error correction on Russell and Whitehead. Their work suggested that uncertainty, noise, and incompleteness could be predicted and transformed into technologies that help us understand and preserve life, information, and the global environment. Putting such impossibilities to work has already transformed the world, and modern researchers and thinkers are still using them to expand the frontiers of knowledge today.
Fig. 1: Gary Larson’s anthropocentric take on allostasis: vitalism’s last, great hope.
NOTES
[i] The same thing is obviously true of Tarski’s method of creating multi-level sets for verifying truth-claims within sets, which was really just a refinement of the typological hierarchies that Russell and Whitehead established to prevent logical impossibilities from arising within their system.
[ii] Boldfacing mine.
[iii] See, for example, “Each thing is, as it were, in a space […] I cannot imagine the thing without the space” (Wittgenstein, 1922/1981: prop. 2.013, p. 33), which succinctly re-states Kant’s “transcendental exposition of the concept of space,” in The Critique of Pure Reason.
[iv] I am thinking of work by structural linguists like Ferdinand de Saussure and Claude Levi-Strauss, as well as Noam Chomsky’s famously groundbreaking work on language acquisition.
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