" 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. The root is his mathematical realism (often confused with Platonism), which dictates that there is one "true" mathematics, working like physics of the ideal: After Cohen proved it independent of ZFC Gödel suggested that new axioms will be "discovered" to resolve it. The NSA prediction comes from the same place as another Gödel's prediction, about the continuum hypothesis. 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." continuous mathematics is an approximation of the discrete one in contraposition to the traditional point of view. technically it is usually much more difficult. While discrete analysis is conceptually simpler. " Continuous analysis and geometry are just degenerate approximations to the discrete world. In a short note About Perspectives of Nonstandard Analysis Gordon quotes Zeilberger suggesting a particular way of how Gödel's prediction might come true: 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." 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. 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. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. 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. One reason is the just mentioned simplification of proofs, since simplifications facilitates discovery. Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future. 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.
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. " 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.
PHYSICS STACKING INFINITESIMALS FULL
Here is the full text of the remark (boldface mine):
It is reproduced in the preface to the second edition of Robinson’s Non-Standard Analysis (1974). The quote is from the remark Gödel made after Robinson's talk at the Institute for Advanced Study in Princeton in March 1973.
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