  Leibniz entertained various conceptions of infinitesimals, considering them
sometimes as ideal things and other times as fictions. But in both cases, he
compares infinitesimals favorably to imaginary roots. We agree with the
majority of commentators that Leibniz's infinitesimals are fictions rather than
ideal things. However, we dispute their opinion that Leibniz's infinitesimals
are best understood as logical fictions, eliminable by paraphrase. This
so-called syncategorematic conception of infinitesimals is present in Leibniz's
texts, but there is an alternative, formalist account of infinitesimals there
too. We argue that the formalist account makes better sense of the analogy with
imaginary roots and fits better with Leibniz's deepest philosophical
convictions. The formalist conception supports the claim of Robinson and others
that the philosophical foundations of nonstandard analysis and Leibniz's
calculus are cut from the same cloth.
