User rachel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T18:14:18Z http://mathoverflow.net/feeds/user/12629 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/57072/a-remark-of-connes/57113#57113 Answer by Rachel for A remark of Connes Rachel 2011-03-02T14:09:36Z 2011-03-02T15:04:58Z <p>A Google search turns up a page on nonstandard analysis on WorldLingo that cites a quote from Connes in the section on criticisms. This may give you more understanding of what Connes is referring to, given that this seems to be a re-statement of what you have quoted above:</p> <blockquote> <p>The answer given by nonstandard analysis, a so-called nonstandard real, is equally deceiving. From every nonstandard real number one can construct canonically a subset of the interval [0, 1], which is not Lebesgue measurable. No such set can be exhibited (Stern, 1985). This implies that not a single nonstandard real number can actually be exhibited.</p> </blockquote> <p>The next remark is:</p> <blockquote> <p>A. Connes Noncommutative Geometry and Space-Time, Page 55 in The Geometric Universe, Huggett et al. The point of Connes' criticism is that nonstandard hyperreals are as fictitious as non-measurable sets. These sets can be shown to exist, assuming the axiom of choice of set theory, but are not constructible. Non-measurable sets are usually considered pathological, a sort of irritant that must be tolerated in order to have the axiom of choice available.</p> </blockquote> <p>As there are sources listed, you may be able to get some additional insight in reading some of the references.</p> <p><a href="http://www.worldlingo.com/ma/enwiki/en/Non-standard_analysis#Criticisms" rel="nofollow">http://www.worldlingo.com/ma/enwiki/en/Non-standard_analysis#Criticisms</a></p>