This essay asks what it means to ascribe a halting problem to modernism, and how modernist literature affords us the opportunity to conceptualize such a claim. A precedent for modernism’s halting problem emerges in 1922, in Ludwig Wittgenstein’s inimitable
Tractatus Logico-Philosophicus—a text that weakens the foundations of formal logic even as it seeks to edify them. As the infectious lure of the
Tractatus infiltrates the modern bloodstream, analogous responses begin to appear in both literature and the sciences. Not quite so deconstructive as Derrida’s
parergon, modernist literature simultaneously exhibits a faith in, and skepticism toward, the stability of the aesthetic frame. A parallel skepticism emerges in mathematics and computer sciences as their practitioners begin to distinguish, in a manner similar to Wittgenstein’s
Tractatus, between expressing isolated facts in the form of individual theorems and algorithmically determining the total formal system in which those theorems appear: “To be able to represent the logical form,” Wittgenstein writes, “we should have to be able to put ourselves with the propositions outside logic, that is outside the world.” With the publication of the Tractatus, modernist writers and scientists alike find themselves confronted with the implacable presence of the Outside. For mathematicians, this implacable presence rears its head in the figure of the halting problem; for modernist writers, it emerges in their ambivalence toward aesthetic form.
The halting problem initiated a fascination with how exactly to handle the question of determining solvability without actually doing any solving. The operation demands a language in which one can talk about proofs, and a set of formal statements about the provability of statements. The recursion of this demand opens the door to Gödel’s famous incompleteness theorems, which establish that within any given formal systems there are true statements that cannot be proven true.
[ii] Douglas R. Hofstadter refers to this as “Gödel’s trick,” and describes it as “like trying to quote an entire sentence inside itself.”
[iii] The Church-Turing hypothesis refined Gödel’s conceptual logic about five years later, in 1936, essentially proving that no program can be written in a language that can perform solvability tests on all other programs also written in that language.
[iv] These programs cannot define their limits in their own vernacular. Even a program that attempted to make statements about programs in a lower-level language cannot guarantee that all its statements about those lower-level programs would be true, since in order to do so it would it need to address its own language, thereby necessitating another level of metalinguistic discourse.
[v]
This is all well and good for mathematics, but what of modernism? Logical form is not aesthetic form, and we would not do well to confuse the two. Works of art are not arguments; they do not persuade us so much as seduce us. The emphasis in this paper lies not in conceiving of modernism as a logical system, but in conceiving of modernism as an aesthetic expression of the logical upsets in the works of Gödel, Turing, and Wittgenstein. Modernism’s halting problem emerges in the corollary between the epistemological drive for complete knowledge and the ontological drive for complete meaning—the full wealth of occupying our human experience.
Wittgenstein, Tractatus Logico-Philosophicus, 1922, trans. C.K. Ogden, New York: Barnes & Noble, 2003, 4.12. All citations to the Tractatus refer to aphorisms.
[ii] Fortnow gives an example in the form of an adaptation of the liar’s paradox: “There is no proof that this sentence is true” (111). If the sentence is false, then there is a proof that it is true, in which case the sentence would be true, but then would have no proof of being true. The insidious effect of this equation is that it persuades its readers to equate provability with truth: “Gödel also shows that we cannot prove that ‘everything we can prove is true is true’ unless we can also prove false things” (111). Put another way, Gödel’s incompleteness theorems reveal a glaring aporia in the recursive functions of formal systems. When looking for proofs of solvability, practitioners will always inevitably encounter true statements whose solvability cannot be guaranteed by the formal language available to them.
[iii] Hofstadter, Gödel, Escher, Bach: An Eternal Golden Braid, 1979, New York: Basic Books, 1999, 426.
[iv] Hofstadter, 429. See also C.A.R. Hoare and D.C.S. Allison, “Incomputability,” Computing Surveys 4.3 (1972), 178: “Any language containing conditionals and recursive functions definitions which is powerful enough to program its own interpreter cannot be used to program its own ‘terminates’ function.”
[v] The suggested overlap between Wittgenstein and Turing is not accidental, nor is it original. For an exceptional and detailed account of Wittgenstein’s influence on Turing (and vice versa), see Juliet Floyd, “Chains of Life: Turing, Lebensform, and the Emergence of Wittgenstein’s Later Style,” Nordic Wittgenstein Review 5.2 (2016): 7-89. Floyd suggests that Turing’s and Wittgenstein’s mature works challenge the formalization of metalevel systems that can account for the complexity of all possible formal statements, phrasing this challenge as an embrace of infinite recombination—that is, as a halting problem: “with Turing’s analysis in hand, [Wittgenstein] now realized that he could—or should—continually detach, move, rearrange, amalgamate and reconfigure motifs and pieces of procedure and thought and conversation (and its ending) within one another without end” (17).
