Booss-Bavnbek, Bernhelm

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Roskilde Universitet

RoskildeUniversityBair, J.; Baszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K. U.; Katz, M. G.; Kudryk,T.; Kutateladze, S. S.; McGaffey, T.; Mormann, T.; Schaps, D. M.; Sherry, D. Cauchy,infinitesimals and ghosts of departed quantifiers. (English)Dealing with dissentBooss-Bavnbek, BernhelmPublished in:Zentralblatt MATH (Online)Publication date:2022Document VersionPublisher's PDF, also known as Version of recordCitation for published version (APA):Booss-Bavnbek, B. (2022). Bair, J.; Baszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K. U.; Katz, M. G.;Kudryk, T.; Kutateladze, S. S.; McGaffey, T.; Mormann, T.; Schaps, D. M.; Sherry, D. Cauchy, infinitesimals andghosts of departed quantifiers. (English): Dealing with dissent. Zentralblatt MATH (Online), 2022, [070537961].General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain.            • You may freely distribute the URL identifying the publication in the public portal.Take down policyIf you believe that this document breaches copyright please contact rucforsk@kb.dk providing details, and we will remove access to the workimmediately and investigate your claim.Download date: 05. May. 2022Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K. U.; Katz, M. G.; Kudryk,T.; Kutateladze, S. S.; McGaffey, T.; Mormann, T.; Schaps, D. M.; Sherry, D.Cauchy, infinitesimals and ghosts of departed quantifiers. (English)  Zbl 07053796Mat. Stud. 47, No. 2, 115-144 (2017)Dealing with dissent.This paper is about different readings of Cauchy’s work in analysis, particularly of his concept of con-vergence, continuity, and derivatives. More precisely, the paper is about a controversy whether Cauchy’sthinking about these topics may be considered, as the authors argue for, exclusively as a precursor of (A)non-standard analysis where, roughly speaking, the reals are expanded in field extension by infinitesimalswith their own rules, or whether Cauchy’s work shows traits of arguing in a way that can be read asprecursor of nowadays (B) standard analysis of limits, that is based on quantifiers. To the dismay of theauthors, Jesper Lützen, the highly respected historian of mathematics and physical sciences, had arguedfor the second view. As a service for the reader the authors present a short but insightful analysis of thedichotomy between practice and procedure versus ontology.That is not an esoteric topic which can be of interest for historians of mathematics alone: In undergraduateteaching, mathematicians must bridge a student’s intuitive understanding, based on a vague conceptof infinitely small changes (that, admittedly, can be made rigorous), with a, for some students oftenalienating, rigorous presentation of the abstract mathematical mainstream definitions based on quantifiers.Such bridges are useful, as well, when one works with a physicist who is trained to master even the mostelaborate chain rule and partial integration situations – without blinking, in an admirable way, and mostlycorrect and mostly without quantifiers. The late Richard Kadison, himself a genius in operator theory, abranch of modern analysis, used to tell the following story of a dissent between two geniuses, Marc Kacand Richard Feynman when they both were at Cornell: Kac was lecturing on his new ideas connectingpartial differential equations and stochastic processes, a hot topic in statistical mechanics. Feynman wassitting visible with his hight stature in the first row – and apparently increasingly disquiet, until he raised,interrupted Kac, went to the blackboard, took the chalk, and explained with only a few substitutionsand cancellations “here this, Marc, and there that” why Kac’s claim was obvious and didn’t need furtherelaboration. Then he returned to his seat. Kac smiled and said with his melodic Polish-American accent:“You are right, Dick. But look, I explain the audience how to argue when you are not around.”Since the rigorous foundation of non-standard analysis in the late 1950s, by Detlef Laugwitz and CurtSchmieden, and independently by Abraham Robinson, there is a consensus that approaches (A) and (B)are both valid and that preferences are a matter of taste and situation. This is perhaps best expressedby the inclusion of a separate chapter on nonstandard analysis in the representative Springer monograph“Zahlen” of 1983, also available in English translation. Then what is the controversy of the present paperabout?There is a widespread perception of the community of mathematicians as a brotherhood where conflictscannot endure due to the shared acceptance of the rules of logic, resulting in the final rejection andextinction of erroneous claims. However, as human artefacts, mathematical publications can arouse ad-versary feelings. Within the community, we cultivate adversary feelings in the elaborate system of peerreviewing. We distinguish between four types of objections, namely regarding I) meaning, II) clarity, III)correctness, and IV) priority. This distinction goes back to Gauss who is credited – and ironized – for the,logically somewhat inconsistent, constant grumbling: “This is meaningless”; “out of all reason, opaque,incomprehensible”; “utterly erroneous”; and “I did it before”.To me, the gist of the matter is the objection IV, the question of priority. Why so, is a short story that Ishall tell further below.Objection III) doesn’t seem relevant here: Lützen’s claim, on one side, doesn’t call the authors’ enthusiasmfor nonstandard analysis in question and for their various suggestions for precursors of their ideas. Neither,on the other side, do the authors cast doubt on Lützen’s correct quotes. The description of the past byan interpreter who uses the full knowledge of developments of subsequent importance is a typical case ofso-called ‘present-centred history (PCH)’. Recognizing the descriptive value of PCH, Philip J. Davis andI once raised the question: “But can the interpreter do better than that and really get into the minds ofthe past creators?” – “Only imperfectly.” That was the best answer we found by Eleanor Robson, notedEdited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities© 2022 FIZ Karlsruhe GmbH Page 1for her studies of ancient Babylonian mathematics.Let’s turn now to objections of type I) and II). Each period of the history of mathematics had its owncontroversies of these types. I shall give three examples.(1) The irrationality of the length of a unit square’s diagonal was easily established in the sense,that√2 cannot be written as the quotient of two natural numbers. The lack of commensurabilitybetween the lengths of the sides and the diagonal of a square, however, invites to logical quarrels:Kronecker vehemently rejected the concept of irrational numbers as opaque long before Gödel’sIncompleteness Theorem. He demonstrated that in number theory deep results can be obtained byworking solely with a working language of ideals in polynomial rings and of equivalence classes ofrecursive formulae to determine a wanted quantity with arbitrary precision, without appealing toan ontological concept of, e.g., algebraic or transcendental irrational numbers. Though it is knownthat Kronecker acknowledged the beauty and ingenuity of Lindemann’s proof of the transcendencyof π, “at least if one could assert a meaning to the opaque concept of irrationality”.(2) Similar controversies arouse about the stunning usefulness of calculating with imaginary units in theabsence of a definite clarification of the concept of complex numbers. The controversies began withthe study of the cubic and quartic equations in the 16th century and culminated with d’Alembert’sproof of the Fundamental Theorem of Algebra and Gauss’ criticism of it.(3) More recent are controversies about the prehistory of the theory of distributions, illuminated byLützen in a widely read monograph. Long before the rigorous introduction of distributions asfunctionals, physicists demonstrated that, working with ‘functions’ that are constant zero withan infinity at one point and having finite nonvanishing integral helps finding ‘true’ solutions ofimportant partial differential equations which otherwise were hard to win. Clearly, lifting thatworking knowledge to a rigorous concept of distributions was a breakthrough in the theory of systemsof linear partial differential equations, e.g., when guaranteeing distributional (‘weak’) solutions bysimple inversion formulae, leaving the hard work to the researcher, to establish whether these weaksolutions can be written as hard solutions, i.e., as conventional functions, or proving a intricate resultby approximating distributions by sequences of conventional step functions and other piecewiselinear functions.In light of these three examples, we must agree with the authors that one has to distinguish between theworking practice and the development of what they call ontological concepts. Rightly they emphasize thataspects of Cauchy’s working practice do not injure or devaluate the common perception that elements ofmodern non-standard analysis are present in Cauchy’s writing and thoughts.Personally, I’m grateful to the authors that in their criticism of Lützen’s observations and claims theyhave clarified the relation – and difference between working practice and carrying ideas.For a general mathematical reader, however, the emotional atmosphere of the present article is disconcert-ing. What they discover and emphasize is the difference between working practice and mature concepts.That is an old hat for most mathematicians. Therefore, I stick to an objection of type IV. The priorityof their alleged new discovery belongs to 2500 years of history of mathematics and physical sciences.Reviewer: Bernhelm Booß-Bavnbek (Roskilde)MSC:01A45 History of mathematics in the 17th century01A55 History of mathematics in the 19th century01A85 HistoriographyCited in 3 DocumentsKeywords:historiography; infinitesimal analysis; Latin model; butterfly model; continuity; convergence; derivativesBiographic references:Cauchy, Augustin-LouisFull Text: DOIEdited by FIZ Karlsruhe, the European Mathematical Society and the Heidelberg Academy of Sciences and Humanities© 2022 FIZ Karlsruhe GmbH Page 2

Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K. U.; Katz, M. G.; Kudryk, T.; Kutateladze, S. S.; McGaffey, T.; Mormann, T.; Schaps, D. M.; Sherry, D. Cauchy, infinitesimals and ghosts of departed quantifiers. (English):Dealing with dissent

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Bair, J.; Błaszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K. U.; Katz, M. G.; Kudryk, T.; Kutateladze, S. S.; McGaffey, T.; Mormann, T.; Schaps, D. M.; Sherry, D. Cauchy, infinitesimals and ghosts of departed quantifiers. (English):Dealing with dissent

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