Foundations of Science

, Volume 17, Issue 3, pp 245-276

First online:

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

  • Alexandre BorovikAffiliated withSchool of Mathematics, University of Manchester
  • , Mikhail G. KatzAffiliated withDepartment of Mathematics, Bar Ilan University Email author 

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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.


Archimedean axiom Bernoulli Cauchy Continuity Continuum du Bois-Reymond Epsilontics Felix Klein Hyperreals Infinitesimal Stolz Sum theorem Transfer principle Ultraproduct Weierstrass