Cauchy's contribution

Question

Sometime, I believe perhaps 2 years, ago I asked a question about breakthroughs, such as those within mathematics and physics which may lead a whole discipline forwards in many ways. One example from physics (perhaps also mathematics) is Newton’s Principia.

I considered the problem of infinitesimals, used by Newton and criticized by idealist bishop Berkeley as “ghosts of departed quantities”, being “neither finite of infinitely small” (by which I assume he meant =0). We consider this contradiction to be solved by the modern concept of the mathematical limit, by a relation between e.g. epsilon and delta.

My question was the following:

If this breakthrough (which I attribute to Cauchy) was a significant step forward, why did we not see a number of significant advances as a consequence such as for Newton and Leibniz regarding celeste mechanics?

I now take the view that the question was never properly answered. It was also put on hold being considered unclear. I also believe that I now have a clear answer: the concept can be used to define the real numbers. I am not sure but I guess that this was the first of several methods of defining the set of real numbers. That amounts to a major consequence, does it not?

Answer 1

This breakthrough enabled the real numbers to be put on a firm foundation. It also enabled the different types of continuity (ordinary and uniform, for example) to be used, which allowed the errors of the past to be explained and corrected.

It also lead (as far as I know, which is not much) to filters and other generalizations.

I know that I do not know much about advanced math, so any corrections and additions would be appreciated.

Answer 2

Cauchy did have a fairly satisfactory notion of number, but it is not the one you outlined in your question. Cauchy was relying on a notion of number via decimal notation that was developed already by Simon Stevin in his work L'Arithmetique (not in La Thiende) at the end of the 16th century, before luminaries like Fermat, Barrow, Newton, and Leibniz developed the calculus. Stevin's insight was a declaration of equality among all numbers, rationals, surds, irrationals, etc. He proposed to represent them all by their unending decimal. This leads to a satisfactory notion of number (modulo identifying the tails of 9s). In particular, Cauchy's proof of the intermediate value theorem is satisfactory in this context. Isaac Newton was inspired by unending decimals to invent the theory of infinite series, as pointed out by the historian Victor J. Katz.

The epsilon-delta machinery was introduced by Weierstrass and was a major advance because it allowed mathematicians to formulate their results in analysis in a framework they felt was satisfactory. Analysis practiced at the time by the best practitioners including Cauchy relied on infinitesimals which were not thought to be grounded satisfactorily.

To respond specifically to your question concerning significant advances as a consequence such as for Newton and Leibniz regarding celeste mechanics, I would mention that ironically the framework based on real numbers and set theory eventually led to grounding the Leibniz-Euler-Cauchy infinitesimals in a satisfactory fashion. This was certainly a significant advance.