E. I. Gordon : Will the Nonstandard Analysis be the Analysis of the Future? (17.2.2017)

v piatok 17.2.2017 o 14:00 hod. v miestnosti M/126


08. 02. 2017 12.06 hod.
Od: Jaroslav Guričan

Prednášajúci: E. I. Gordon (Eastern Illinois University, Charleston) 

Názov: Will the Nonstandard Analysis be the Analysis of the Future?

Termín: 17. 2. 2017, 14.00, M 126


Abstrakt: 
In 1973 Abraham Robinson gave a talk about the nonstandard analysis (NSA) at the Institute for Advanced Study in Princeton. After his talk Kurt Godel made a comment, in which he predicted that "...there are good reasons to believe that Non-Standard Analysis, in some version or other, will be the analysis of the future." One has to admit that during the fty years since this prediction did not come true. Although the NSA simpli ed proofs of many deep results in standard mathematics and even allowed to obtain new standard results, among which there were some long standing open problems, it did not become the working tool for any more numerous part of mathematicians. When they are interested in some result obtained with the help of the NSA, they prefer to reprove it in standard terms. One of the reasons of rejecting the NSA is that, as a rule, the job of standard reproving is not too difficult. Another reason stems from the fact the Transfer Principle, one of the main principles of the NSA, which is crucial for deduction of standard results from nonstandard ones and vice versa, relies heavily on the formalization of mathematics within the the framework of superstructures or of the Axiomatic Set Theory. For mathematicians working, e.g., in ODEs, PDEs or other areas oriented toward applications, using mostly the naïve set theory, these formal languages may appear difficult and irrelevant, and some objects to which they refer, like, e.g., in nitesimal or in nite numbers, even suspicious, so they may not feel con dent in nonstandard proofs.

In my talk I will present a new version of nonstandard set theory, that is formulated on the same level of formalization as the naïve set theory. I will try to justify my opinion that this is perhaps the version, within which formulated, the nonstandard analysis may become the "analysis of the future". I will discuss some examples of NSA theorems about interaction between some classical mathematical notions and theorems on one hand and their computer simulations on the other, that are rigorous theorems in the NSA but which cannot be formulated in terms of classical (standard) mathematics. These theorems have clear intuitive meaning and the phenomena they describe can even be monitored in computer experiments. Nowadays many applied mathematicians share the point of view that the continuous mathematics is an approximation of the discrete one, but not vice versa. This point of view can easily be formalized in the naïve nonstandard set theory mentioned above. Once we are not interested in proving classical theorems with the help of the NSA, we don't need the standardness predicate and the Transfer Principle of the NSA in full. This allows us to avoid an excessive logical formalism of the usual versions of NSA.