TITLE: Nilpotent infinitesimals comparison
QUESTION [6 upvotes]: I'd like to understand better the advantages and disadvantages of various approaches to nilpotent infinitesimal numbers and their application to differential geometry in the context of physics and engineering problems. I'm aware of the following three systems
1) Synthetic differential geometry (SDG), which is described in a number of publications. Most recently, in "Synthetic geometry of manifold" by Anders Kock, available at home.imf.au.dk/kock/SGM-final.pdf‎
2) A proposal by Wolfgang Bertram to use dual numbers to represent tangent spaces, which he describes in his "Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings" (http://fr.arxiv.org/abs/math.DG/0502168)
3) An approach based on Fermat reals promoted by Paolo Giordano in a number of publications, e.g. http://arxiv.org/abs/1104.1492
The first approach appears to be the most significant of the three, with a long list of publication by various people, but it uses constructive logic. The last two rely on classical logic but are much less developed than SDG. What are the relative merits of these three approaches? What do they teach us about nilpotent infinitesimals? Particularly, when applied to differential geometry.

REPLY [2 votes]: Perhaps the disadvantage pointed out by Paolo Giordano is actually a real advantage. I mean, if SIA and SDG lead you to learn intuitionistic logic and topos theory (or more generally category theory), as it did for me, then instead of just allowing you to remain within the bounds of your existing knowledge, it expands your mind. Furthermore, learning the latter enables you to understand how they can be used in novel foundations of quantum theory.