Seminar 5: Diagrammatization of written mathematical practices

Iterated exponentials, powers and iterates: Which diagrammatization? Which symbolism?

Ivahn Smadja (Nantes Université & Institut Universitaire de France)

This exploratory contribution aims at analyzing a significant shift that occurred around 1800 in the way (some) diagrammatic dimensions of (some) mathematical symbolism were addressed, conceived, and exploited in mathematical practice. As a guiding thread into the thick of it, we will first focus on iterated exponentials, tracing the history of the topic from Euler to the British mathematicians of the Analytical Society, and beyond. A broader purpose, however, is to show how this perplexing issue connects with new ways to think about diagrammatization and symbolism being increasingly brought into the open in the early decades of the nineteenth century.

While commenting on the striking analogy between the Bodhisattva’s calculus in the Lalitavistara Sūtra and Archimedes’ Sand Reckoner, in his Mémoire sur la propagation des chiffres indiens (1863), historian of mathematics and mathematician Franz Woepcke made an interesting observation, suggesting a watershed in the history of symbolism. Whereas the numbers attested in these ancient sources, being analogously expressed by means of extended scales of numerical names, rightly appear huge, it would nevertheless be mistaken, he argued, to believe them to be “the limit of the possible in terms of notation”. As a token of a broadened perspective, Woepcke pointed to his own mathematical work on iterated exponentials, taking up and clarifying issues that arose decades earlier. These debates hinged on the role of nested parentheses in defining iterated operations, the status of such symbolic expressions as

and further the possibility of constructing an open-ended hierarchy of levels of operations. Among the mathematicians involved were Carl Friedrich Gauss, Charles Babbage and Gotthold Eisenstein. Unfolding this sequence, though, prompt us, more generally, to look deeper into the meaning and scope of Arbogast’s so-called “method of separation of symbols”, and his use of detachable “scales of operation”, as appropriated by the British Analysts, insofar as these shed light on the interplay between diagrammatization and symbolism.