Mikhail G. Katz

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

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.

TEN MISCONCEPTIONS FROM THE HISTORY OF ANALYSIS AND THEIR DEBUNKING

The widespread idea that infinitesimals were “eliminated” by the “great triumvirate” of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d’Alembert, Cauchy, and others.

Entropy of systolically extremal surfaces and asymptotic bounds

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with A. Katok’s lower bound in terms of the volume, to obtain a simpler alternative proof of M. Gromov’s asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov’s theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.

Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.

Boundary case of equality in optimal Loewner-type inequalities

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, G), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with respect to suitable norms. These norms are the stable norm of G, the conformally invariant norm, as well as other L p -norms. Here we exploit L P -minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our optimal inequalities, in terms of the criticality of the lattice of deck transformations of T b , while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.

Stevin Numbers and Reality

We explore the potential of Simon Stevin’s numbers, obscured by shifting foundational biases and by 19th century developments in the arithmetisation of analysis.

Leibniz's laws of continuity and homogeneity

We explore Leibnizs understanding of the differential calculus, and argue that his methods were more coherent than is generally recognized. The foundations of the historical infinitesimal calculus of Newton and Leibniz have been a target of numerous criticisms. Some of the critics believed to have found logical fallacies in its foundations. We present a detailed textual analysis of Leibnizs seminal text Cum Prodiisset, and argue that Leibnizs system for differential calculus was free of contradictions.

Cauchy's Continuum

Cauchys 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, Cauchys hypothesis is formulated in terms of a single variable x, rather than a pair of variables, and requires the error term rn = rn(x) to go to zero at all values of x, including the infinitesimal value generated by explicitly specified by Cauchy. If one wishes to understand Cauchys modification/clarification of the hypothesis of the sum theorem in 1853, one has to jettison the automatic translation-to-limits.

Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

We analyze some of the main approaches in the literature to the method of ‘adequality’ with which Fermat approached the problems of the calculus, as well as its source in the παρισότης of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermats method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermats collected works (62, pp. 133–172). We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, variation e. Fermats treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary definitions of παρισότης and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermats texts by Carcavy. We argue that Fermat relied on Bachets reading of Diophantus. Diophantus coined the term παρισότης for mathematical purposes and used it to refer to the way in which 1321/711 is approximately equal to 11/6. Bachet performed a semantic calque in passing from parisoo to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat (1896) and Leibniz (1858) on the problem of maxima and minima.

Hyperelliptic surfaces are Loewner

We prove that C. Loewners inequality for the torus is satisfied by conformal metrics on hyperelliptic surfaces X as well. In genus 2, we first construct the Loewner loops on the (mildly singular) companion tori, locally isometric to X away from Weierstrass points. The loops are then transplanted to X, and surgered to obtain a Loewner loop on X. In higher genus, we exploit M. Gromovs area estimates for e-regular metrics on X.