Exploration Mathematics: The Rhetoric of Discovery and the Rise of Infinitesimal Methods

I. Exploration vs. Mathematics

In a famous passage of the New Organon, Francis Bacon challenged the natural philosophers of his time to live up to the example of geographical explorers: "It would be disgraceful," he wrote, "if, while the regions of the material globe--that is, of the earth, of the sea, of the stars--have been in our times laid widely open and revealed, the intellectual globe should remain shut up within the narrow limits of old discoveries." ¹ Bacon was not alone in this view: the great voyages of exploration were repeatedly cited as a model and an inspiration by early modern promoters of the new sciences. The image of the natural philosopher as a Columbus or Magellan, pushing forward the frontiers of knowledge, became a commonplace of scientific treatises and pamphlets of the period. The newly discovered lands and continents seemed both a proof of the inadequacy of the traditional canon and a promise of great troves of knowledge waiting to be unveiled. ²

The importance of the imagery of exploration in transforming the intellectual landscape of early modern Europe has been widely discussed in recent scholarship. Anthony Grafton has documented the intellectual crisis brought on by the geographical discoveries, when scholars began suggesting that other intellectual fields might harbor more undiscovered "Americas"; William Eamon and Paula Findlen have both recorded the widespread use of the rhetoric of exploration among natural philosophers in the sixteenth and seventeenth centuries; and Anthony Pagden has traced its fortunes in the hands of scholars from Bacon to Fontenelle to Alexander von Humboldt. ³

The exploration trope, however, did not promote all forms of knowledge equally: taking its cue from the explorers themselves, it tended to emphasize direct observation and experience over abstract reasoning. It was, after all, through actual travel and personal experience that the great voyagers forever undermined the authority of the academic geographers whose theories were based on abstract reasoning from ancient authority. Similarly, it was now argued, the hidden lands of nature could not be known by a priori reasoning, but must be discovered through unmediated experience. It is hardly surprising that Bacon and his fellow promoters of empiricism found the imagery of geographical discovery particularly appealing.

But while the experimental philosophers could easily imagine themselves as explorers of the secrets of nature, the case was more difficult for mathematicians ⁴ Mathematics, with its rigorous, formal, and deductive structure, appeared to be an ill-suited terrain for intellectual exploration. No mathematical object, after all, could ever be observed, experienced, or experimented upon. Mathematicians, it seemed, did not seek out new knowledge or uncover hidden truths in the manner of geographical explorers. Instead, taking Euclidean geometry as their model, they sought to draw true and necessary conclusions from a set of simple assumptions. The strength of mathematics lay in the certainty of its demonstrations and the incontrovertible truth of its claims, not in uncovering new and veiled secrets. Indeed, what could possibly be left "hidden" and "undiscovered" in a system where all truths were, in principle, implicit in the initial assumptions? ⁵

This view of mathematics was expressed most clearly by Christopher Clavius, the founder of the Jesuit mathematical tradition, in his tract In disciplinas mathematicas prolegomena, dating from the 1580s. Clavius's tract was his contribution to a debate about the truth value of mathematical propositions known as the "Quaestio de certitudine mathematicarum." He was here defending and promoting the value of mathematical learning, which was being challenged both from within the Jesuit Order and from without. ⁶ Using the common medieval classification, Clavius divided mathematics into "pure" and "mixed" domains. Pure mathematics included arithmetic and geometry, while mixed mathematics roughly corresponded to our notion of "mathematical sciences": it comprised such fields as astronomy and music, as well as engineering and geography. In these disciplines, according to Aristotle, physical objects are treated as...