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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.

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My answer might be biased, so I hope someone else could complement it adding different viewpoints. Moreover, my answer is essentially philosophical, so it might sound incomplete in case you are searching for a technical answer.

I can underscore two threads in answering your question:

SDG and "Differential Geometry over general base field" are extremely flexible in creating the type of nilpotent infinitesimals you need for your problem. This is essentially because they are algebraic in nature. Like for Weil functors, these theories are wonderful generalizations of the same basic idea of the ring of dual numbers $\mathbb{R}[h]/(h^2=0)$. We can summarize this point of view, even too strongly, citing J. Conway who in The Math Forum Drexel of Feb. 17, 1999 said "SDG is just a formal algebraic technique for 'working up to any given order in some small variable $s$' - for instance if you want to work up to second order in $s$, you just declare that $s^3 = 0$". In my opinion, it is really difficult, or even impossible, to achieve the same formal power and beauty in other theories of nilpotent infinitesimals. On the other hand, Fermat reals are formally less powerful, but strongly intuitively clearer. This permits to gain intuition about possible developments of the theory of nilpotent infinitesimals which seem impossible following an algebraic approach. So, in the ring of Fermat reals, one can define a meaningful notion of $n$-th root of a nilpotent infinitesimal, with application to fractional calculus, there is a computer implementation of the whole ring, and we are working to define reciprocal of nilpotent infinitesimals and also stochastic nilpotent infinitesimals corresponding to Ito's calculus. This teaches us many things about nilpotent infinitesimals. If you want to write new papers (frequently simply reformulating SDG!), Fermat reals is the right framework. Therefore, the first thread to compare these theories could be: do you prefer more formal power but a lacking intuition or less formal power but a greater intuition?

The second thread can be summarized with the question: with whom do you want to talk? Very few people have a so strong formal control of the mathematics they are doing to have a sufficient sure sense that they are correctly working in intuitionistic logic. More generally, in my experience, for some mathematicians, a strong intuition is essential to like a theory. For this reason they found difficult to work in SDG or in a purely algebraic "formal" approach. Fermat reals have not this problem and are frequently well accepted by analysts, geometers and physicists.

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  • $\begingroup$ I'm looking at the issue as a user who is interested in applications, rather than a mathematician who wants to develop the theory. As a user, I do not feel encumbered by the use of constructive logic. In fact, as I'm reading "Synthetic geometry of manifolds", I'm barely aware of the fact that the constructive logic is being used to prove all theorems. It reads pretty much like any other math text. I'm curious about Bertram's approach as it seems very similar to SDG but without the extra baggage of category theory. Fermat's reals, on the other hand, look quite different. $\endgroup$ Jun 7, 2013 at 0:21
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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.

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  • $\begingroup$ Did you have any particular paper in mind at the link you posted? $\endgroup$ Oct 13, 2013 at 22:33
  • $\begingroup$ Well, Kostecki has an intro to topos theory and another to SDG, which contains SIA. He is also a colleague of C. Flori, at the same institute, who wrote a paper on topos and quantum theory and a book. By the way, have you decided which path to follow? I would like to find someone to team-study Bell's book. As Giordano argues, it is difficult to prove stuff respecting the intuitionistic logic. $\endgroup$
    – yann
    Oct 16, 2013 at 16:39

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