Statement on Infinitesimal calculus

-- Kurt Friedrich Gödel {Curtius Fridericus Gödel / കുര്‍ട് ഫ്രീഡ്രിഷ് ഗേഡല്‍ / குர்ட் ஃப்ரீட்ரிஷ் கேடல் / कुर्ट् फ्रीड्रिष् गेडल्}, Mar. 1973.

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 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 farther from the truth. Rather there are good reasons for believing 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 simplification facilitates discovery. Another, even more convincing reason, is the following: Arithmetic starts with 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.

Comments:

`The present writer holds to the view that the application of non-standard analysis to a particular mathematical discipline is a matter of choice and that it is natural for the actual decision of an individual to depend on his early training.

A more definite opinion has been expressed in a statement which was made by Kurt Gödel after a talk I gave in March 1973 at the Institute for Advanced Study, Princeton. The statement is reproduced here with Professor Gödel's kind permission.'

-- Abraham Robinson (Preface to the Second Edition of `Non-standard analysis')


© 2001. David C. Kandathil. All rights reserved.