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    $\begingroup$ Well, Arnold famously dislikes axiomatics not grounded in concrete examples, especially coming from abstract algebra. The paragraph preceding the one which appears above is funnier (although not related to local rings): during a written exam, in Paris, a student told Arnold that he forgot his calculator and could Arnold tell him if 4/7 was greater or less than 1. (Convergence of an integral depended on this.) Arnold generally compared this knowledge-without-examples in higher math to Feynman's discussion of "Brazilian" physics in his Surely You're Joking book. $\endgroup$
    – KConrad
    Commented May 25, 2010 at 17:08
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    $\begingroup$ Victor, as you know, though related, the notion of localization from the question is not the same as the notion of a local ring. Typical localization used for Zariski open subsets look geometrically and intuitively radically different from the localization at a point leading to a local ring; the latter are about the formal/infinitesimal neighborhood -- in my opinion much more abstract notion than the Zariski open sets. Your preference to localization at point seems based on liking the personality of Arnol'd who was systematically rude to any misgiving of French students and mathematicians. $\endgroup$
    – Zoran Skoda
    Commented May 15, 2011 at 16:55