 An interesting alternative to nonstandard analysis, which allows for a development of substantive parts of mathematics, goes under the name of (smooth) infinitesimal analysis. This differs from both ordinary and nonstandard analysis by allowing nilpotent infinitesimals, namely ‘linelets’ \(dx\) such that \(dx\neq 0\) but \(dx\cdot dx = 0\). The consistency of such a theory is proved using toposes in category theory. The best exposition of the topic is Bell (1998b) (see also Bell 1988a, 2019); philosophical aspects of the theory are discussed in Hellman and Shapiro (2018). Arthur (2013) discusses infinitesimal analysis in connection to Leibniz and makes points similar to those made by Bos (1974) on Leibniz and non-standard analysis. Constructive interpretations of non-standard analysis are developed in Salanskis (1999), which includes discussions of Nelson’s approach as well as the French school of non-standard analysis (Reeb, Harthong). For further discussion of infinitesimals, see Davis (1977), Thomason (1982), Bell (2005), and the entry on   continuity and infinitesimals.