In 1872, two definitions of the real numbers were introduced that are still used today: one based on Cauchy sequences, the other on Dedekind cuts. Before then, mathematicians spoke of "quantities" and offered definitions case by case: they defined algebraic irrational numbers (such as square roots and nth roots), searched for the best definition of π… But there was no precise general definition of irrational numbers. It was therefore difficult to know exactly what they were dealing with. Could these quantities be ordered? Could they be added, subtracted and multiplied? And if so, how were the order and operations to be defined?
Why was this a problem? Because some theorems of analysis—particularly those concerning the convergence of sequences—then rested on shaky foundations: what does a sequence of rational numbers tend to when its limit is itself irrational?
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Campaigning for greater rigor
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Charles Méray (1835–1911; pictured above), a French mathematician who spent most of his career at the Université de Dijon, was the first to propose a definition of irrational numbers. To him, the lack of rigor in analysis was a critical problem. He did not mince his words: "\[Analysis\] struck me as deplorable in its disjointedness, its methods and its utter lack of rigor, and my chief efforts have been directed towards making it natural, clear and rigorous." He criticized the discipline for resting on "precarious" principles and dubious arguments that relied too heavily on the notion of continuity. In his view, "this method has proved fertile in circular reasoning, fallacies, off-putting considerations and even errors". Mathematicians had managed to eliminate the errors only through "particular applications". Méray advocated relying on calculation instead.
