Talk:Non-standard analysis
From Academic Kids
Should you really be redirecting from nonstandard model? I was looking for a page on nonstandard models of first order versions of Peano's Postulates.
Re Connes' criticism, is this really a criticism of the hyperreals, or of one particular construction of the hyperreals? The construction using an ultrafilter is not very "constructive," since an ultrafilter can't be constructed. But there's another, more recent construction by Kanovei et al. that doesn't require an ultrafilter. I'd be grateful for a pointer to any further info on Connes' criticism, both because I'd like to understand it, and because I think this point in the article is not clear as currently written. --Bcrowell 17:52, 5 Mar 2005 (UTC)
- The point in the article is not very clear, because Connes is not very clear himself. The arguments he gives against NSA (in his now classic book Noncommutative Geometry) seem to suggest his criticism lies with its nonconstructive aspects. On the other hand, the Dixmier trace which plays an important role in his non-commutative theory of infinitesimals is non constructive, using a generalized limit in its definition. Fundamentally, Connes started his career in nonstandard analysis futzing around with logic, didn't get far with it, discovered Tomita-Takesaki, and has had a very low esteem of NSA ever since. This low regard is quite widespread among analysts, and is somewhat undeserved, but you do have to ask the question: "what has nonstandard analysis really done?" Not much, unfortunately. The one really unfortunate result of all this is that some very respected analysts (such as Edward Nelson) are now very unfairly regarded as somewhat marginal because they have used NSA or published articles in NSA. CSTAR 18:14, 5 Mar 2005 (UTC)
The author for the books at http://www.serve.com/herrmann/books.htm lloks rather questionable to me.
- Hi, 84.132.219.203, could you explain why you say that? He's a professor at the U.S. Naval Academy. I'm actually a physicist, myself, not a mathematician. Although I was initially a little put off by the title of his book "Einstein Corrected," it doesn't have any of the hallmarks of crank literature. I haven't sat down and read the whole thing, just browsed through it.--Bcrowell 14:37, 5 May 2005 (UTC)
