Digital 18003

Summary

WRITING AT THE dawn of the computer age, the famous logician Kurt Gödel looked back to the period around 1800 in order to describe the developments in mathematics around 1900. Gödel's fame rested on his incompleteness theorems of the early 1930s, which showed that any formalized system of mathematics will contain statements that cannot be proven true or false by the axioms that define that system. And, yet, the constraints implied by his work regarding the solvability of every mathematical problem disappear, Gödel wrote in 1961, if one continues to add new axioms to the system—an idea that “agrees in principle with the Kantian conception of mathematics.” Indeed, Kant's claims regarding mathematics, made most notably at the outset of The Critique of Pure Reason, are “incorrect if taken literally, since Kant asserts that in the derivation of geometrical theorems we always need new geometrical intuitions,” Gödel continues, and “therefore a purely logical derivation from a finite number of axioms is impossible. That is demonstrably false. However, if in this proposition we replace the term ‘geometrical’—by ‘mathematical’ or ‘set-theoretical,’ then it becomes a demonstrably true proposition.” For Gödel, it was “a general feature of many of Kant's assertions that literally understood they are false but in a broader sense contain deep truths.” Although Gödel himself was not involved in the birth of the digital computer, these passages suggest that the mathematicallogical developments of the early twentieth century that led to the first computers are essentially an extension of the Kantian project. In fact, there exists an oblique yet significant relationship of um 1800 to a group of thinkers who created the digital universe and, hence, to our conception of the digital itself.

The relationship between digital computing and um 1800 is reciprocal, perhaps uniquely so, and this reciprocity flips the questions of this forum on its head. It is widely recognized, for instance, that mass digitization has changed not only the methods of researching and analyzing literature, but also cultural histories of both the eighteenth and nineteenth centuries (as their materials often evade copyright).