Mathematical rigor, mathematical creativity, and the transgression of limits

Dignity and the highest certainty are the characteristics of mathematics guaranteed by rigorous demonstrations. The Ancients or – to be more precise – Archimedes was and remained the touchstone for the legitimacy of mathematical methods, objects, proofs up to the times of Euler. In order to demonstrate the equality of two quantities the double reductio ad absurdum was usual and necessary. Yet, thanks to Archimedes’ letter to Eratosthenes we know that he transgressed limits of legitimacy when he looked for the solution of problems. His demonstrations were criticized because of their obscurity. Hence authors tried to replace them with affirmative demonstrations. In 1615 Kepler explained Archimedean theorems in his New solid geometry of wine barrels using geometrical transformations and analogies thus being able to surpass the results of the Greek author. Was he entitled to do that? In 1641 Guldin spoke of Kepler’s new method of demonstrating. Guldin himself claimed to give clear and perspicuous demonstrations instead of the too obscure Archimedean demonstrations. When in 1676 Leibniz amply used infinitely small and infinite quantities in his Arithmetical quadrature of the circle etc. he admitted that this might appear obscure, but he emphasized that they provide abbreviations of speaking, thinking, discovering, and demonstrating and that his method differs from the Archimedean style only by the expressions. In 1755 Euler tried to justify the use of these quantities in another way. But he was convinced that the highest geometrical rigor was observed like in the books of the Ancients.
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