The Babel Algorithm

A BEYOND BORDERS column by David Krakauer, President of the Santa Fe Institute.

Max Brückner’s Collection of Polyhedral Models (1900)

In his 1901 essay, “Mathematics and the Metaphysicians,” Bertrand Russell suggests that George Boole, the autodidactic son of a cobbler, was the discoverer of pure mathematics in his work, An Investigation of the Laws of Thought (1854). Russell argues that Boole was “mistaken in supposing he was dealing with the laws of thought” — that in fact the book is concerned with “formal logic, and this is the same thing as mathematics.” In case one supposes that it was Aristotle who invented logic, Russell dispatches such quaint historicism, observing that “in each decade since 1850 more has been done to advance the subject than in the whole period from Aristotle to Leibniz.”

What Russell’s essay is really about is infinity, and through the work of Georg Cantor and others, the anchoring of mathematics in logic and logic in infinity. Although Russell did not yet see it, whereas ironically Boole already had, infinity is ultimately the constructive principle supporting knowledge and thought. As Boole puts it in “Constitution of the Intellect,” the final chapter of An Investigation of the Laws of Thought: “When from a large number of observations on the planet Mars, Kepler inferred that it revolved in an ellipse, the conclusion was larger than his premises, or indeed than any premises which mere observation could give. What other element, then, is necessary to give even a prospective validity to such generalizations as this? It is the ability inherent in our nature to appreciate Order . . .”

It is order that empowers the archiving mind in finite time to operate in an infinite universe. And this permits in the words of Boole, “The necessary sequence of states and conditions in the inorganic world, and the necessary connexion of premises and conclusion in the processes of exact demonstration thereto applied, seem to be co-ordinate.” Knowledge and reality converge through the right choice of system.

Perhaps too much has been written about Jorge Luis Borges’ story, “The Library of Babel.” But to my knowledge, much of this commentary has been aimed at scholarly futility and the impossibility of originality. The great power of infinity — as obtained through the ordering principles of Boole, Cantor, and Russell — is that like alchemy, it seems to transform discovery into creativity, or search into invention. When Borges writes that “The universe (which others call the Library) is composed of an indefinite and perhaps infinite number of hexagonal galleries” and that within these hexagons are all the books that could ever be written, he is suggesting that if there is a catalog or index to the library (a source of order), then knowledge is identical to intellection — to know where something is equates to its discovery. To know it all is to appear genius.

This last year has been full of know-it-alls. The artists have names like NightCafe, DALL•E 2, and Stable Diffusion, and the writers, GPT-3 and Bloom. They are Babel Algorithms, accumulating limitless hexagons of online information, where every artist born and unborn is somewhere in the endless permutation of what came before. And the search index seems to provide the necessary order to turn mere facts into fancy. It is a challenge for a dawning science to think through the implications of solving problems by exploiting practically searchable “effective infinities.”

Boole had foreseen this utilitarian dilemma in a bygone configuration as well as its dire limits for the future: “In the extreme case it is not difficult to see that the continued operation of such motives, uncontrolled by any higher principles of action, uncorrected by the personal influence of superior minds, must tend to lower the standard of thought in reference to the objects of knowledge, and to render void and ineffectual whatsoever elements of a noble faith may still survive.”

— David Krakauer,
President, Santa Fe Institute

From the Winter 2022–2023 edition of the SFI Parallax newsletter. Subscribe here for the monthly email version, or email “news at” to request quarterly home delivery in print.



The Santa Fe Institute is an independent research center exploring the frontiers of complex systems science.

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Santa Fe Institute

The Santa Fe Institute is an independent research center exploring the frontiers of complex systems science.