48 citations · 200 across the 26 of their papers we have counts for
22 papers · 1 filter
Bridging Classical and Modern Approaches to Thales' Theorem
Piotr Błaszczyk, Anna Petiurenko
In this paper, we reconstruct Euclid's theory of similar triangles, as developed in Book VI of the \textit{Elements}, along with its 20th-century counterparts, formulated within th…
Euler's Series for Sine and Cosine. An Interpretation in Nonstandard Analysis
Piotr Błaszczyk, Anna Petiurenko
In chapter VIII of Introductio in analysin infinitorum, Euler derives a series for sine, cosine, and the formula His arguments employ infinitesimal and infi…
New model of non-Euclidean plane
Piotr Błaszczyk, Anna Petiurenko
We present a new model of a non-Euclidean plane, in which angles in a triangle sum up to . It is a subspace of the Cartesian plane over the field of hyperreal numbers $\mathbb{R…
On diagrams accompanying reductio ad absurdum proofs in Euclid's Elements book I. Reviewing Hartshorne and Manders
Piotr Błaszczyk, Anna Petiurenko
Exploring selected reductio ad absurdum proofs in Book 1 of the Elements, we show they include figures that are not constructed. It is squarely at odds with Hartshorne's claim that…
Procedures of Leibnizian infinitesimal calculus: An account in three modern frameworks
Jacques Bair, Piotr Blaszczyk, Robert Ely +2
Recent Leibniz scholarship has sought to gauge which foundational framework provides the most successful account of the procedures of the Leibnizian calculus (LC). While many schol…
Phenomenology of diagrams in Book II of the Elements
Piotr Błaszczyk
In this paper, we provide an interpretation of Book II of the Elements from the perspective of figures which are represented and not represented on the diagrams. We show that Eucli…
Continuity between Cauchy and Bolzano: Issues of antecedents and priority
Jacques Bair, Piotr Blaszczyk, Elias Fuentes Guillen +3
In a paper published in 1970, Grattan-Guinness argued that Cauchy, in his 1821 book Cours d'Analyse, may have plagiarized Bolzano's book Rein analytischer Beweis (RB), first publis…
Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimals
Jacques Bair, Piotr Blaszczyk, Peter Heinig +3
Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy app…
19th century real analysis, forward and backward
Jacques Bair, Piotr Blaszczyk, Peter Heinig +2
19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms…
Leibniz's well-founded fictions and their interpretations
Jacques Bair, Piotr Blaszczyk, Robert Ely +2
Leibniz used the term fiction in conjunction with infinitesimals. What kind of fictions they were exactly is a subject of scholarly dispute. The position of Bos and Mancosu contras…
Klein vs Mehrtens: restoring the reputation of a great modern
Jacques Bair, Piotr Błaszczyk, Peter Heinig +3
Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Kl…
Cauchy, infinitesimals and ghosts of departed quantifiers
Jacques Bair, Piotr Blaszczyk, Robert Ely +10
Procedures relying on infinitesimals in Leibniz, Euler and Cauchy have been interpreted in both a Weierstrassian and Robinson's frameworks. The latter provides closer proxies for t…
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Tiziana Bascelli, Piotr Blaszczyk, Alexandre Borovik +7
Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy's proof, and…
Gregory's sixth operation
Tiziana Bascelli, Piotr Blaszczyk, Vladimir Kanovei +6
In relation to a thesis put forward by Marx Wartofsky, we seek to show that a historiography of mathematics requires an analysis of the ontology of the part of mathematics under sc…
Toward a history of mathematics focused on procedures
Piotr Blaszczyk, Vladimir Kanovei, Karin U. Katz +3
Abraham Robinson's framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Thei…
A non-standard analysis of a cultural icon: The case of Paul Halmos
Piotr Blaszczyk, Alexandre Borovik, Vladimir Kanovei +4
We examine Paul Halmos' comments on category theory, Dedekind cuts, devil worship, logic, and Robinson's infinitesimals. Halmos' scepticism about category theory derives from his p…
Interpreting the infinitesimal mathematics of Leibniz and Euler
Jacques Bair, Piotr Blaszczyk, Robert Ely +10
We apply Benacerraf's distinction between mathematical ontology and mathematical practice (or the structures mathematicians use in practice) to examine contrasting interpretations…
Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania
Tiziana Bascelli, Piotr Blaszczyk, Vladimir Kanovei +4
Did Leibniz exploit infinitesimals and infinities `a la rigueur, or only as shorthand for quantified propositions that refer to ordinary Archimedean magnitudes? Chapter 5 in (Ishig…
Controversies in the foundations of analysis: Comments on Schubring's Conflicts
Piotr Blaszczyk, Vladimir Kanovei, Mikhail Katz +1
Foundations of Science recently published a rebuttal to a portion of our essay it published two years ago. The author, G. Schubring, argues that our 2013 text treated unfairly his…
A Purely Algebraic Proof of the Fundamental Theorem of Algebra
Piotr Błaszczyk
Proofs of the fundamental theorem of algebra can be divided up into three groups according to the techniques involved: proofs that rely on real or complex analysis, algebraic proof…
Is mathematical history written by the victors?
Jacques Bair, Piotr Blaszczyk, Robert Ely +9
We examine prevailing philosophical and historical views about the origin of infinitesimal mathematics in light of modern infinitesimal theories, and show the works of Fermat, Leib…
Ten Misconceptions from the History of Analysis and Their Debunking
Piotr Blaszczyk, Mikhail G. Katz, David Sherry
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 infinitesi…