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I found this : "There are good reasons to believe that nonstandard analysis, in some version or other, will be the analysis of the future."

What did Kurt Gödel mean exactly ?

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The quote is from the remark Gödel made after Robinson's talk at the Institute for Advanced Study in Princeton in March 1973. It is reproduced in the preface to the second edition of Robinson’s Non-Standard Analysis (1974). Here is the full text of the remark (boldface mine):

"I would like to point out a fact that was not explicitly mentioned by Professor Robinson, but seems quite important to me; namely that non-standard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be further from the truth. Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.

One reason is the just mentioned simplification of proofs, since simplifications facilitates discovery. Another, even more convincing reason, is the following: Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. I am inclined to believe that this oddity has something to do with another oddity relating to the same span of time, namely the fact that such problems as Fermat’s, which can be written down in ten symbols of elementary arithmetic, are still unsolved 300 years after they have been posed. Perhaps the omission mentioned is largely responsible for the fact that, compared to the enormous development of abstract mathematics, the solution of concrete numerical problems was left far behind."

In a short note About Perspectives of Nonstandard Analysis Gordon quotes Zeilberger suggesting a particular way of how Gödel's prediction might come true:

"Continuous analysis and geometry are just degenerate approximations to the discrete world... While discrete analysis is conceptually simpler... technically it is usually much more difficult... continuous mathematics is an approximation of the discrete one in contraposition to the traditional point of view. Under this approach the notion of a very big finite set is very important and the definition of a hyperfinite set in NSA is an appropriate formalization of this notion that satisfies the modern requirements to mathematical rigor."

The NSA prediction comes from the same place as another Gödel's prediction, about the continuum hypothesis. After Cohen proved it independent of ZFC Gödel suggested that new axioms will be "discovered" to resolve it. The root is his mathematical realism (often confused with Platonism), which dictates that there is one "true" mathematics, working like physics of the ideal:

"Despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception".

See da Costa's Gödel Incompleteness Theorems and Physics.

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In the 1970s Gödel recommended Robinson for membership in the British Academy of Sciences. His recommendation was based in part on Robinson's framework for analysis with infinitesimals. From the viewpoint of Gödel's realism, the significance of Robinson's work was to extend further the ordered number system, beyond the natural, integer, rational, and real numbers, so as to include the hyperreal numbers (e.g., infinitesimals). Robinson for his part did not subscribe to realist views about set theory, and wrote explicitly that he did not view set theory as a "foundation" of mathematics. In private correspondence, Gödel chastised Robinson for expressing views contrary to Gödel's realist ideas about number systems! To Gödel, the significance of Robinson's work was to be able to give precise mathematical formulation in currently acceptable foundational frameworks to Leibniz's ideas about infinitesimal and infinite numbers. Robinson and Gödel agreed in their admiration for Leibniz.

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