Conflicts Between Generalization, Rigor, and Intuition: by Gert Schubring

By Gert Schubring

This e-book bargains with the advance of the phrases of study within the 18th and nineteenth centuries, the 2 major options being unfavourable numbers and infinitesimals. Schubring experiences frequently missed texts, particularly German and French textbooks, and divulges a far richer background than formerly proposal whereas throwing new gentle on significant figures, corresponding to Cauchy.

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Extra info for Conflicts Between Generalization, Rigor, and Intuition: Number Concepts Underlying the Development of Analysis in 17th-19th Century

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Hitherto, just a small part of the surviving texts of Arabic mathematicians have been evaluated. Nevertheless, the texts already studied give no indication that mathematicians within this cultural context considered negative solutions acceptable. The six classical types of algebraic equations of first and second degree were always conducive to positive solutions. Mathematicians who treated problems of indeterminate algebra adhered to Diophantus’ model, selecting coefficients so as to obtain positive solutions (Sesiano 1985, 108).

Newton and Leibniz, however, extended their investigations to transcendental functions. The problems of how to develop transcendental functions into series and the latter’s impact on the meaning of the foundational concepts were to become the principal focus for research into analysis. In his Method of fluxions, Newton introduced the distinction between independent and dependent (variable) quantities, as quantitas correlata and quantitas relata. In Leibniz’s manuscripts, the term function is found for the first time in 1673, for the relation between ordinates and abscissae of a curve given by an equation.

A further impulse was provided by the problems raised by multidimensional integration; this is where presenting the integral as a sum becomes necessary. Potential theory necessitated the calculation of definite integrals. 11 Cauchy, however, was the first to raise the concept of the definite integral to the rank of a privileged fundamental notion, and the first to comprehensively make the concept and the existence of the integral a subject proper of mathematical research in his textbook of 1823.

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