Who Gave You the Cauchy–Weierstrass Tale? The Dual History of Rigorous Calculus

Foundations of Science 17 (3):245-276 (2012)
Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered
Keywords Archimedean axiom  Bernoulli  Cauchy  Continuity  Continuum  du Bois-Reymond  Epsilontics  Felix Klein  Hyperreals  Infinitesimal  Stolz  Sum theorem  Transfer principle  Ultraproduct  Weierstrass
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DOI 10.1007/s10699-011-9235-x
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