When Einstein Walked with Gödel: Excursions to the Edge of Thought

by Jim Holt

During this golden age of discovery, scientists treated the infinitesimal as they would any other number, until it became convenient in their calculations to set it to zero (as Newton somewhat fishily did in cases like that of the falling rock, above). This cavalier attitude toward the infinitely small is captured by the advice of the French mathematician Jean Le Rond d’Alembert: “Allez en avant, et la foi vous viendra” (Go forward, and the faith will come to you). Still, there remained those who felt it a scandal that the edifice of modern science was being erected on such metaphysically shaky foundations. Throughout the eighteenth century, there were many efforts to answer the charges against the infinitesimal put by critics like Berkeley and to find a logical set of rules for its use. None were successful; some were simply fatuous. (Well into the next century, Karl Marx would try his hand at this problem, leaving nearly a thousand unpublished pages devoted to it.) One of the more philosophically appealing attempts was that of Bernard de Fontenelle, who tried to rationalize the infinitesimal by characterizing it as the reciprocal of the infinitely large. Though Fontenelle was ultimately defeated by formal difficulties, he was prescient in arguing that the reality of objects like the infinitesimal rested ultimately on their logical coherence, not on their existence in the natural world.