Cauchy's Continuum

Abstract

Cauchy's sum theorem of 1821 has been the subject of rival interpretations ever since Robinson proposed a novel reading in the 1960s. Some claim that Cauchy modified the hypothesis of his theorem in 1853 by introducing uniform convergence, whose traditional formulation requires a pair of independent variables. Meanwhile, Cauchy's hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term r n = r n (x) to go to zero at all values of x, including the infinitesimal value generated by 1 n , explicitly specified by Cauchy. If one wishes to understand Cauchy's modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits. Contents 1. Sifting the chaff from the grain in Lagrange 2 2. Cauchy's continuity 3 3. Bråting's close reading 5 4. Cauchy's 1853 text 8 5. Is the traditional reading, coherent? 9 6. Conclusion 12 Appendix A. Spalts Kontinuum 12 Appendix B. Fermat, Wallis, and an "amazingly reckless" use of infinity 17 Appendix C. Rival continua 21 Acknowledgments 24 References 24

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

“…Sections 6 through 8 Y. Sergeyev [152], and D. Spalt [156]. These were dealt with respectively in KatzSherry [102], Bascelli et al [15], Kanovei-Katz-Sherry [87], Bair et al [7], BorovikKatz [28], B laszczyk et al [25], B laszczyk et al [23], Bascelli et al [16], B laszczyk et al [26], Gutman et al [58], Katz-Katz [94]. 6 See e.g., a quotation in Delfini-Lobry [35] from Berkeley Physics Course, Crawford [33] and Section 6.2. explore additional aspects of Robinson's framework.…”

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

“…Cantor's interest in eliminating infinitesimals is paralleled nearly a century later by Bishop's interest in eliminating LEM, and by a traditional nominalist's interest in eliminating Platonic counting numbers. The automatic infinitesimalto-limit translation as applied to Cauchy by Boyer and others is not only reductionist, but also self-contradictory, see [86].…”

“…The stronger condition bars Abel's counterexample. See our text [86] for more details. whose logical weakness was criticized by Berkeley, was modified by A. Robinson by exploiting a map called the standard part, denoted "st", from the finite part of a B-continuum (for "Bernoullian"), to the Acontinuum (for "Archimedean"), as illustrated in Figure 2.…”

Intellectual Property and Development

“…242 There were several complementary factors that helped China in its economic development, such as market-oriented reforms of the economic system, the recognition of private property rights, the transition and optimisation of the National Innovation System (NIS), and persistent and comprehensive policies opening it up to the world. 244 China is an interesting case for study by IP scholars in developing countries because although IP may not be a prerequisite for economic development, it is an integral and essential part of the country's innovation system, which is important for progress. 238 tremendous pressures on China to change its IP systems and provide strong protection for American corporations working in the country.…”