There is an old tradition going back all the way to <PERSON> himself to think of differentials intuitively as "infinitesimal numbers". Through the course of history, big minds have criticized <PERSON> for this. Thus, <PERSON> accepted <PERSON>'s claim that infinitesimals are inconsistent and even reproduced it in his book Principles of Mathematics in 1903.
But then something profound happened in 1961: <PERSON>, one of the most profound thinkers of the foundations of mathematics, taught the world a rigorous construction of infinitesimals in the traditional framework of the Zermelo-Fraenkel set theory, expressed in terms of the theory of types. Among other things, this is a fully rigorous mathematical context in which the old intuition of <PERSON> and the intuition of plenty of naive physicists finds a full formal justification. In <PERSON>'s framework those differentials explicitly exist as infinitesimal elements of a suitable real closed field.
A detailed exposition of how this works is in <PERSON>'s 1966 book but simpler treatments have been developed since, such as the books by <PERSON> or by <PERSON>, including exposition of differentiation via infinitesimals.
Notice that this is not just a big machine to produce something you already know, as some will inevitably hasten to think. On the contrary, this leads the way to the more sophisticated places of modern physics, as developed in detail in the book by <PERSON> et al.:
<PERSON>, <PERSON>; <PERSON>, <PERSON>; <PERSON>, <PERSON>; <PERSON>, <PERSON>. 489
Nonstandard methods in stochastic analysis and mathematical physics. Pure and Applied Mathematics, 122. Academic Press, Inc., Orlando, FL, 1986. xii+514 pp.
Note 1. <PERSON>'s contribution in the framework of category theory dates from the 1970s.
Note 2. (In response to user <PERSON>'s question) <PERSON>'s framework is part of traditional analysis in the sense that it uses the traditional <PERSON>-Fraenkel foundations and classical logic (as opposed to <PERSON>'s approach which relies on intuitionistic logic in a break with classical mathematics). <PERSON>'s framework is an active research area today, featuring its own journal: Journal of Logic and Analysis (see http://logicandanalysis.org/) and an ever increasing number of monographs; most recently by <PERSON> and <PERSON> (see http://www.springer.com/us/book/9789401773263). 469