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Borges, Gödel and Creativity

Jorge Luis Borges’ Library of Babel (which contains every book that can be generated with a given alphabet and line/page count) is, in many respects, a self-referential structure, inasmuch as many of its books will inevitably refer to the library itself. For any merely generative algorithm that is capable of generating all possible books in the library will also generate all possible books whose topic is the library (including Borges’ own story The Library of Babel). Many of these referential works will purport to be alternate catalogues of the library. But for each possible catalogue, a multitude of other books will sing its praises, while yet another multitude will offer the most damning critiques of its gaps and inaccuracies. In fact, these books will in turn be criticized or praised by other books in the library, and so on, ad nauseum. Any of these reference works might offer an invaluable guide to the library and its contents, if only we knew which ones were trustworthy. However, the library offers no reliable means of identifying which books are to be trusted, even as they concern the library itself.

Though the generative principle of Borges’ Library of Babel  is a remarkably simple one, the same tantalizing dilemma also plagues real generative systems of much greater complexity in mathematics and computer science. In fact, this dilemma was formally demonstrated just a few years before Borges published his story, in a proof by a young Austrian mathematician named Kurt Gödel. Mathematicians had long sought to place mathematics on a sound systematic footing, and in the early part of the 20th century, mathematicians set themselves a bold goal: to find a generative proof procedure capable of generating all theorems that can be formally derived from a given set of axioms. This procedure could thus be used to crank out proofs for the truth or falsity of any conjecture that had previously eluded their best efforts, in much the same way as a Chomskyan grammar can churn out every valid sentence, or Arthur C. Clarke’s computer might crank out all the names of God (in his short story The Nine Billion Names of God). However, Gödel was to derail this enterprise with his fascinating proof. Just as Borges’ library contains a potentially perfect index whose veracity cannot be established within the library itself, Gödel showed that every formal system of a certain generative power must also contain a Gödel statement whose truth cannot likewise be proven within the system. Like Borges’ index, a Gödel statement refers both to itself and to the system in which it is constructed. It is just the kind of statement a system should be able to prove, yet Gödel showed it cannot do so without compromising either its consistency or its completeness.

Gödel showed -- with his famous incompleteness theorem -- that formal generative systems can suffer from blind spots, insofar as they cannot ascertain the veracity of everything they can generate. But from the perspective of a creative system, this seems like a minor setback at most: after all, we don’t expect great writers to be capable of writing every great book, or good poets to be capable of generating every good poem. Indeed, talented creators often produce works that they later disown, as we earlier saw with Picasso’s admission that he sometimes painted fakes. However, some theorists have argued that Gödel’s proof irrevocably undermines the generative powers of purely formal systems, and thus of computers and their algorithms. His proof is cited as evidence that formal systems are inherently limited, and cannot therefore reach the bar established by human intelligence. If we believe that human-like creativity requires human-like intelligence, then this argument may also prove fatal to the enterprise of computational creativity. Though Gödel himself did not quite go so far, interpreters of his work such as Ernest Nagel and James Newman have used Gödelian arguments to argue against the possibility of artificial intelligence. But in a revised edition of their classic work Gödel’s Proof, Douglas Hofstadter offers the following editorial rebuttal of their negative view:

Although computers, as their name implies, are built of rigidly arithmetic-respecting hardware, nothing in their design links them inseparably to mathematical truth. It is no harder to get a computer to print out scads of false calculations (“2 + 2 = 5; 0/0 = 43”, etc.) than to print out theorems in a formal system. A subtler challenge would be to  devise ‘a fixed set of directives’ by which a computer might explore the world of mathematical ideas (not just strings of mathematical symbols), guided by visual imagery, the associative patterns linking concepts, and the intuitive processes of guesswork, analogy, and aesthetic choice that every mathematician uses.”

While computers are engineered using mathematical principles, computational pragmatism and mathematical truth are two quite different notions. As Hofstadter notes, computers are just as capable of telling lies (with respect to their model of the world) as telling the truth. The challenge of computational creativity is to go beyond the cliché of computers as machines that can “only give you (true) answers”, to create the kind of machine that can formulate, in the words of Picasso, artful “lies that tell the truth”.

But are we opening the door to dishonest computers by imbuing our machines with a creative capacity? Does the latter necessarily presuppose the former?

 

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