Editing Differential (mathematics) (section)

=== Synthetic differential geometry ===

A fifth approach to infinitesimals is the method of [[synthetic differential geometry]]<ref>See {{Harvnb|Kock|2006}} and {{Harvnb|Lawvere|1968}}.</ref> or [[smooth infinitesimal analysis]].<ref>See {{Harvnb|Moerdijk|Reyes|1991}} and {{Harvnb|Bell|1998}}.</ref> This is closely related to the algebraic-geometric approach, except that the infinitesimals are more implicit and intuitive. The main idea of this approach is to replace the [[category of sets]] with another [[category (mathematics)|category]] of ''smoothly varying sets'' which is a [[topos]]. In this category, one can define the real numbers, smooth functions, and so on, but the real numbers ''automatically'' contain nilpotent infinitesimals, so these do not need to be introduced by hand as in the algebraic geometric approach. However the [[logic]] in this new category is not identical to the familiar logic of the category of sets: in particular, the [[law of the excluded middle]] does not hold. This means that set-theoretic mathematical arguments only extend to smooth infinitesimal analysis if they are ''[[constructive mathematics|constructive]]'' (e.g., do not use [[proof by contradiction]]). [[Constructivism_(philosophy_of_mathematics)|Constructivists]] regard this disadvantage as a positive thing, since it forces one to find constructive arguments wherever they are available.