Timeline for Salvaging Leibnizian formalism?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 13, 2014 at 7:33 | comment | added | Mikhail Katz | Steve, the problem with this approach to infinitesimals is that it does not capture the full strength of Robinson's transfer principle. This apparently applies to the other answer as well since the reliance on intuitionistic logic undermines certain arguments that Robinson's transfer principle does apply to. | |
| Aug 12, 2014 at 13:40 | comment | added | Steven Gubkin | the algebra of infinitesimals in your manifold. | |
| Aug 12, 2014 at 13:40 | comment | added | Steven Gubkin | A Weil algebra is an $\mathbb{R}$ algebra of the form $\mathbb{R} \oplus N$ with $N$ a nilpotent ideal. This ideal gives the "form" for the "infinitesimals" you want to consider. A major point in the book is that to each such algebra, there is a product preserving functor from the category of manifolds to the category of bundles over manifolds. For example, the tangent bundle corresponds to $\mathbb{R}[x]/(x^2)$, the tangent bundle to the tangent bundle to $\mathbb{R}[x,y]/(x^2,y^2)$, the space of $k$-jets to $\mathbb{R}[x]/(x^k)$. Essentially the algebra is keeping track of | |
| Aug 12, 2014 at 12:11 | comment | added | Mikhail Katz | I finally got a chance to look up Kolar et al. This seems to focus on naturality in differential geometry and it is hard to see how this relates to developing a framework for infinitesimals. Perhaps @Peter Michor could comment? | |
| Aug 11, 2014 at 13:26 | comment | added | Mikhail Katz | Differential forms are fine, but if you consult Spivak's book on differential geometry you will find him struggling to translate infinitesimal arguments in Gauss and Riemann into modern terminology and usually comes up with a more cumbersome result. | |
| Aug 11, 2014 at 13:17 | comment | added | Mikhail Katz | "as far as you know" is not correct. Leibniz knew perfectly well how to work with second differentials. See for example link.springer.com/article/10.1007/BF00327456 | |
| Aug 11, 2014 at 13:13 | history | answered | Steven Gubkin | CC BY-SA 3.0 |