Recent zbMATH articles in MSC 01Ahttps://zbmath.org/atom/cc/01A2023-11-13T18:48:18.785376ZUnknown authorWerkzeugMathematics as metaphor. Selected essays. Translated from the English by Claire Vajou. With a preface by Freeman J. Dyson. With an afterword by Pierre Lochakhttps://zbmath.org/1521.000032023-11-13T18:48:18.785376Z"Manin, Yuri I."https://zbmath.org/authors/?q=ai:manin.yurii-ivanovichPublisher's description: Les Mathématiques comme Métaphore est, au même titre que La Science et l'Hypothèse de Poincaré, un témoignage accessible et rigoureux de la beauté mathématique. Sa première partie constitue une méditation sur l'expérience intime de la pensée algébrique, la vocation d'un chercheur et la fonction sociale de la science. Yuri Manin y livre les clefs de son propre destin en dévoilant les mathématiques comme une métaphore de l'existence. Dans sa seconde partie, l'ouvrage aborde l'épineuse et récurrente question des relations entre constructions mathématiques, spéculations physiques et algorithmes informatiques. Refusant les positions unilatérales, la réflexion s'installe dans un va-et-vient connectant motifs et figures, évoquant en particulier celle d'Alexandre Grothendieck. Exposant les puissances de l'autre hémisphère du cerveau, la troisième partie élabore une série de conjectures sur le Trickster, les mythes, le langage, la poésie, etc. Celles-ci complètent les analyses précédentes et s'y réverbèrent. Traversé par le fantôme de la dialectique, Les Mathématiques comme Métaphore offre une magistrale leçon de philosophie mathématique pour non-mathématiciens.
Cette édition réunit les textes traduits en anglais, une sélection complémentaire issue de l'édition russe, des textes postérieurs choisis par l'auteur ainsi qu'une postface inédite de Pierre Lochak.
See the review of the original English edition in [Zbl 1172.00003].F things you (probably) didn't know about hexadecimalhttps://zbmath.org/1521.000042023-11-13T18:48:18.785376Z"Strickland, Lloyd"https://zbmath.org/authors/?q=ai:strickland.lloyd"Jones, Owain Daniel"https://zbmath.org/authors/?q=ai:jones.owain-daniel(no abstract)Book review of: E. Giusti (ed.) and P. d'Alessandro (ed.), Leonardi Bigolli Pisani vulgo Fibonacci. \textit{Liber abbaci}https://zbmath.org/1521.000052023-11-13T18:48:18.785376Z"Folkerts, Menso"https://zbmath.org/authors/?q=ai:folkerts.mensoThe \textit{Liber abaci} (Book of calculation) is a treasure written by Leonardo of Pisa in 1202 and revised by him in 1228. It consists of fifteen chapters, some of them very long, and represents one of the most extensive mathematical works written in Latin in the Middle Ages. Nineteen manuscripts of the \textit{Liber abaci} are known, nine of them contain essentially the entire work and the remaining ten only the last chapters (or parts thereof). In addition, there is one privately owned manuscript but its current location is unknown. Only since 1857 a printed edition has been existing, published by Baldassare Boncompagni, but it is based only on a single manuscript. This edition is faithful to the manuscript, unfortunately also in terms of its mistakes. In 2002, an English translation by L. E. Sigler was published posthumously [Fibonacci's \textit{Liber abaci}. A translation into modern English of Leonardo Pisano's Book of calculation. New York, NY: Springer (2002; Zbl 1032.01046)]. However, this translation is based on Boncompagni's edition.
A new edition of this important Latin text, that takes into account all known manuscripts, has long been desired. Now such an edition [Zbl 1457.01028] is available thanks to Enrico Giusti, a mathematician with broad experience in the history of medieval and early modern mathematics, and Paolo d'Alessandro, philologist and linguist. Both editors have gone through all accessible manuscripts of the \textit{Liber abaci} and collated the most important ones. They chose six manuscripts for Chapters 1--11, five for Chapter 12 and six for Chapters 13--15. Thus, they succeeded in providing a reliable edition of this central mathematical work that can be used as a basis for further research on mathematics in the Western Middle Ages and Renaissance.
Reviewer: Marita Blankenhagel (Berlin)History of mathematics through collaboration: toward a composite portrait of Oswald Veblen. Abstracts from the workshop held December 4--10, 2022https://zbmath.org/1521.000132023-11-13T18:48:18.785376ZSummary: Oswald Veblen played a pivotal role in the history of American mathematics in the twentieth century. His life, however, remains largely unstudied. This conference was designed to redress this issue by exploring Oswald Veblen and his contributions to the history of American and international mathematics in an interactive workshop that used the Veblen Papers from the US Library of Congress as a foundational and shared resource. With this frame, the conference raised queries and discussed issues related to Veblen, his mathematical contributions, and his collaborative initiatives, including his critical work aiding refugee mathematicians in WWII that helped establish long standing programs at American institutions that continue to advance mathematics at the highest level. The workshop echoed Veblen's collaborative focus and brought together historians of mathematics and mathematicians to work alongside one another during the conference. This content and collaborative approach combined to advance our understanding of Veblen's collaborations and the history of twentieth-century mathematics more broadly.Georg Cantor. Infinity and the subconscious. With a preface by Pierre Cartier. With an afterword by Éric Laurenthttps://zbmath.org/1521.010012023-11-13T18:48:18.785376Z"Charraud, Nathalie"https://zbmath.org/authors/?q=ai:charraud.nathaliePublisher's description: Georg Cantor (1845--1918) est le père de la théorie des ensembles qui est devenu aujourd'hui le cadre universel de présentation des mathématiques. Mais son apport le plus révolutionnaire est la découverte de la non-dénombrabilité des nombres réels qui implique qu'il n'y a pas qu'un seul infini, contrairement à l'opinion répandue. Ce saut vertigineux vers une infinité d'infinis ne concerne pas seulement les mathématiques, mais aussi bien la philosophie et la théologie, et participe de l'avènement de la modernité.
Une nouvelle qualité s'affirme avec Cantor dans le domaine des mathématiques, celle de la liberté qu'il revendique face à ses détracteurs. Ce livre suit, au fil de ses travaux, l'équilibre qu'il lui fallut maintenir entre vérité logique et liberté dont la conjonction peut paraître paradoxale.
Ces difficultés réelles sont-elles la cause de la psychose qui sourd au cours des années ?
Mathématicienne et psychanalyste, Nathalie Charraud suit, à travers une étude des documents historiques, les aspects intérieurs de la création cantorienne. Elle réussit à décortiquer son activité mathématique en termes lacaniens, sans pour autant ôter aux réflexions mathématiques leur signification scientifique.
See the review of the first edition in [Zbl 0809.01015].Another look at the two Egyptian pyramid volume `formulas' of 1850 BCEhttps://zbmath.org/1521.010022023-11-13T18:48:18.785376Z"Siegmund-Schultze, Reinhard"https://zbmath.org/authors/?q=ai:siegmund-schultze.reinhardAuthor's abstract: ``This paper provides some methodological, didactical, and historiographical reflections on Egyptian pyramid volume formulas, responding to suggestions by \textit{P. M. E. Shutler} [Int. J. Math. Educ. Sci. Technol. 40, No. 3, 341--352 (2009; \url{doi:10.1080/00207390802641692})]. These suggestions partly reiterate a historically documented proof by the Chinese Liu Hui (third century CE), although Lui Hui's contribution was apparently unknown to Shutler. The latter came forward, in addition, with intuitive arguments which might have been used by the Egyptians to convince themselves of the correctness of their formula for the volume of the full pyramid. In a broad sense, the reflections in this paper may contribute to the use of history in the mathematical classroom. As a cautionary note: The paper is an abridged version of a longer manuscript that contains detailed explanations and discussions of historical secondary sources. Since the paper is somewhat outside the usual canon of mathematics historiography, I have deposited the longer manuscript on [``Intuitive, didactically useful, and historically possible proofs for the two Egyptian pyramid volume formulas (1850 BCE). Thoughts on the border between history and didactics of mathematics'', Preprint, \url{arXiv:2207.04427}].''
Reviewer's comments: The author has established in full his intentions as summarized in his summary. It was a pleasure to read in the paper that dealed with questions of volume calculations/determinations arising of about four thousand years ago.
As to recovering methods of measurements in classical and pre-classical historic times, the paper of \textit{W. M. F. Greenhill} [``How Plato designed Atlantis'', Symmetry Art Sci. 13, No. 1--4, 163--202 (2002)] might also give rise to new ideas and interpretations; let us say, who knows?
Anyway, the author of the paper under review did produce a welcome addition to older (but even more important) recent literature. Highly recommended!
Reviewer: Robert W. van der Waall (Huizen)The \textit{Arithmetica} of Diophantus. A complete translation and commentaryhttps://zbmath.org/1521.010032023-11-13T18:48:18.785376Z"Christianidis, Jean"https://zbmath.org/authors/?q=ai:christianidis.jean-p"Oaks, Jeffrey"https://zbmath.org/authors/?q=ai:oaks.jeffrey-aIf I had to select the two most influential books in algebra and number theory, I would choose Diophantus' \textit{Arithmetica} and Gauss's \textit{Disquisitiones arithmeticae}. Both books contain parts that are next to impossible to read for modern mathematicians; for this reason, all modern editions of Diophantus have ample comments explaining the ideas behind the calculations.
The present edition of ``The \textit{Arithmetica} of Diophantus'' is going to be the definitive edition of this masterpiece. The introduction deals with the problem of dating Diophantus, lists his mathematical works, and gives an overview of the existing editions and translations of the Arithmetica. Chapter 2 discusses Diophantus' notation for numbers and polynomials, and Chapter 3 presents the history of algebra from Diophantus to Viète with a careful investigation which problems e.g. Arabic authors copied from Diophantus. Chapter 4 deals with the mathematical language in the Greek and Arabic books.
The actual translation of Diophantus' \textit{Arithmetica} in Part II begins on p. 273; it contains a faithful translation of the first three Greek books, the four Arabic books IV--VII, and the remaining Greek books IV\(^G\)--VI\(^G\). Part III presents comments for each of the problems.
The appendices contain a reconstitution by E. Stamatis of four missing problems in the fifth Greek book, techniques for solving indeterminate equations, a short dictionary of Greek and Arabic expressions, and a conspectus of the problems in modern algebraic notation. This is followed by an extensive bibliography and an index.
I recommend this book to everyone who is interested in the history of algebra and number theory.
Reviewer: Franz Lemmermeyer (Jagstzell)Ptolemy's Sun and Moonhttps://zbmath.org/1521.010042023-11-13T18:48:18.785376Z"Girstmair, Kurt"https://zbmath.org/authors/?q=ai:girstmair.kurtThe aim of this paper is to present in the simplest possible terms, without getting sidetracked by the astronomical raw data, the three models of the motion of the Moon and the Sun that can be found in Ptolemy's \textit{Almagest}. The idea is to reduce at a necessary minimum the reliance on astronomical data and to emphasize the geometrical nature of the models, in the hope that this presentation will be easier to understand for a contemporary reader with a mathematical background than those in [\textit{V. M. Petersen}, Centaurus 14, 142--171 (1969; Zbl 0196.00304); \textit{O. Pedersen}, A survey of the Almagest. Odense, Denmark: Odense University Press (1974; Zbl 0343.01003); \textit{O. Neugebauer}, A history of ancient mathematical astronomy. In 3 parts. Vol. 1. Berlin etc.: Springer-Verlag (1975; Zbl 0323.01002)], which the author credits for having provided inspiration for the current work.
Reviewer: Victor V. Pambuccian (Glendale)Ptolemy's treatise on the meteoroscope recoveredhttps://zbmath.org/1521.010052023-11-13T18:48:18.785376Z"Gysembergh, Victor"https://zbmath.org/authors/?q=ai:gysembergh.victor"Jones, Alexander"https://zbmath.org/authors/?q=ai:jones.alexander|jones.alexander-daniel"Zingg, Emanuel"https://zbmath.org/authors/?q=ai:zingg.emanuel"Cotte, Pascal"https://zbmath.org/authors/?q=ai:cotte.pascal"Apicella, Salvatore"https://zbmath.org/authors/?q=ai:apicella.salvatoreThis paper reports an ongoing project of studying the contexts of the manuscript Milan, Veneranda Biblioteca Ambrosiana L 99 Sup., involving and based upon modern imaging technologies. While the principle text in this manuscript is an 8th-century copy of Isidore of Seville's \textit{Etymologies} -- a Latin compilation of late-ancient knowledge, arranged by subject matter -- it contains 15 palimpsested leaves written over fragments of three mathematical texts in Greek, each coming from a single bifolium, which had themselves been copied in the 6th or 7th century. Two of the underlying texts are well known -- the so-called \textit{Fragmentum mathematicum Bobiense}, dealing with mathematical optics, and Ptolemy's \textit{Analemma}, dealing with geometrical constructions pertaining to the geometry of a sphere as used for sundial theory -- while J. L. Heiberg argued that the third text, which he could not read enough of to edit, was related to the technical material of the \textit{Analemma}, on the basis of certain similarities of terminology. More recently, the research effort reported in this paper has produced new images of the palimpsested pages and shown that the third mathematical text is, in fact, a treatise on an instrument known as the \textit{meteoroscope}.
The meteoroscope was an articulated armillary sphere, consisting of nine rings, which could be used both to make observations, as well as to perform analog calculations for problems in spherical geometry and spherical astronomy. This instrument is described in the \textit{Geography} of Ptolemy, who claims it as his own, as well as by Pappus, and Theon of Alexandria, who wrote a treatise on the device, which only survives in medieval Arabic translations and summaries.
After (1) an introduction setting out the principle scholarship on the palimpsested leaves of Ambrosiana L 99 Sup., the paper provides (2) an overview of the manuscript, (3) a discussion of all ancient sources discussing Ptolemy's meteoroscope, (4) a summary of the contents of the newly determined \textit{Meteoroscope}, descriptions of the technical terminology of the instrument and the instrument itself (including two diagrams), which correct the previously standard scholarship in this regard, (5) a critical edition and translation of select passages of the text of the \textit{Meteoroscope}, along with a convincing argument that this newly identified text was written by Ptolemy himself.
Reviewer: Nathan Camillo Sidoli (Tokyo)Who proved Pythagoras's theorem?https://zbmath.org/1521.010062023-11-13T18:48:18.785376Z"Lučić, Zoran"https://zbmath.org/authors/?q=ai:lucic.zoranThis is a well-argued speculative reconstruction of early Greek proofs of the Pythagorean theorem. Given that, in \textit{A commentary on the first book of Euclid's Elements} Proclus writes that he admired ``those who first became acquainted with the truth of this theorem'', and that he ``marvel[s] more at the writer of the \textit{Elements}, not only because he established it by a most lucid demonstration, but because he insisted on the more general theorem by the irrefutable arguments of science in the sixth book'', the author concludes that the proofs in I.47 and VI.31 of Euclid's \textit{Elements} not only could not have been the original ones (VI. 31 needed Eudoxus' theory of proportion), but that they must have been somehow ``refutable''. He concludes, by analysing Hippocrates' quadrature of the lunes and Aristotle's definition of proportionality in \textit{Topics}, 158b, that the original proof must have been one using the concepts of length and area, by way of ``geometric algebra''. Pre-Eudoxian geometry being less rigorous, lacking proper definitions of the notions of length and area and based on Aristotle's problematic definition of proportion, the pre-Eudoxian proof could have been looked upon, in light of Eudoxus' achievement, as ``refutable''.
Reviewer: Victor V. Pambuccian (Glendale)Eudoxus' simultaneous risings and settingshttps://zbmath.org/1521.010072023-11-13T18:48:18.785376Z"Schironi, Francesca"https://zbmath.org/authors/?q=ai:schironi.francescaThe works of fourth-century BCE astronomer Eudoxus of Cnidus included two treatises -- the \textit{Phaenomena} and the \textit{Enoptron} -- that identified constellations that rise above, or set below, the horizon when certain other constellations rise. Reconstructing the precise meanings of Eudoxus's reports is difficult, partly because the fragments of Eudoxus's writings that we have come from works by Aratus and by Hipparchus of Rhodes a couple of centuries later. The author clears up some confusion about these fragments by postulating that Eudoxus used two different reporting systems: (i) the constellations that rose/set when a zodiacal constellation was beginning to rise; and (ii) the constellations that rose/set during the entire time it takes a zodiacal constellation to rise.
Reviewer: Glen Van Brummelen (Langley)Fifteenth-century Italian symbolic algebraic calculation with four and five unknownshttps://zbmath.org/1521.010082023-11-13T18:48:18.785376Z"Høyrup, Jens"https://zbmath.org/authors/?q=ai:hoyrup.jensThis striking article discusses a 1463 algebraic manuscript by Florentine abbacist author Benedetto da Firenze. The author convincingly argues that Benedetto's work exhibits symbolic algebraic calculation with four and five distinct unknowns, nearly a century before the earliest generally recognised examples of such algebraic calculation. Through formal considerations of the manuscript presentation and close textual analysis, the author establishes that Benedetto used symbolic variables as a primary means of solving algebraic problems that he then described rhetorically, in contrast to the familiar practice from the history of rhetorical algebra of using symbols only as after-the-fact shorthands for rhetorical or geometrical reckoning.
The article includes extensive critical translations and transcriptions of the manuscript source, exhibiting a masterful handling of an early text to support a profound argument hinging on subtle interpretations. The article and its earlier companion in the same journal [the author, Gaṇita-Bhāratī 41, No. 1--2, 23--67 (2019; Zbl 1476.01005)] together significantly revise how historians may understand the relationship between the Italian abbacist tradition associated with Fibonacci and the adaptation of Sanskrit and Arabic symbolic mathematics in southern Europe. Far from the epochal breakthrough that symbolic calculation is often taken to be in histories of European algebra, the author shows how such calculation could appear marginal and ambivalent to its authors. This underscores the contingency of knowns and unknowns in the early history of algebra.
Reviewer: Michael J. Barany (Edinburgh)Commercializing arithmetic: the case of Edward Hattonhttps://zbmath.org/1521.010092023-11-13T18:48:18.785376Z"Melville, Duncan J."https://zbmath.org/authors/?q=ai:melville.duncan-jAuthor's abstract: ``For a period of some 40 years from 1695 onwards, Edward Hatton was a dominant author in the burgeoning world of commercial arithmetic publishing in London. Hatton was a prolific as well as successful author, selling some 40,000 copies across a dozen titles, some of which went through many editions. In this paper, we offer a survey of his works with emphasis on those that were key to his commercial success.''
Reviewer's remarks: In the introduction, the motivation of this historical overview, the role of Edward Hatton in commercial arithmetic publishing in London, as well as increasing trade and a growing mercantile class in England at the dawn of the eighteenth century, are briefly described.
This paper contains the following items:
\begin{itemize}
\item[--] ``Edward Hatton and his publishers''. Here are also noted peculiarities in the activity of the first insurance companies in London.
\item[--] ``A textbook: the \textit{Merchant's Magazine}''. The main attention is given to the first publication of Hatton, as well as to his positioning and work in the marketplace.
\item[--] ``Ready reckoners: \textit{Comes commercii}''. It is noted that this publication was probably most successful Hatton's work (in terms of sales).
\item[--] ``Ready reckoners: \textit{Index to interest}''.
\item[--] ``Posters''.
\item[--] ``Other works''.
\item[--] ``Appendix: A summary Hatton bibliography''.
\end{itemize}
The main overview is focused on the mathematical content in Hatton's works such that ``Hatton had become a brand in commercial mathematical publishing and his name went on being used for the rest of the century''.
For the entire collection see [Zbl 1495.01006].
Reviewer: Symon Serbenyuk (Kyïv)Mathematics in astronomy at Harvard College before 1839 as a case study for teaching historical writing in mathematics courseshttps://zbmath.org/1521.010102023-11-13T18:48:18.785376Z"Ackerberg-Hastings, Amy"https://zbmath.org/authors/?q=ai:ackerberg-hastings.amyThe central problem of the present historical overview is the following:
``How mathematical was the teaching of astronomy by the five men who served Harvard College as the Hollis Professor of Mathematics and Natural Philosophy in the years before the college established an observatory?''
Some attention and brief comments are also given to some related problems in investigations in history of mathematics. In addition, the primary source materials for this paper are briefly described.
The author notes the following:
``Writing a research paper is an important but challenging milestone for secondary and undergraduate students, one that does not need to be limited to disciplines that have been traditionally writing-oriented, such as history or English. Yet, instructors in any discipline may find that teaching the research and writing process -- especially in a way that produces, for instance, both sound history and valid mathematics -- can be challenging.
This paper brings together principles useful for teaching historical research and writing with the specific historical problem of how mathematics was employed in the teaching of astronomy at Harvard before the college established an observatory in 1839. It thus provides students and instructors with an opportunity to experience the application of historical thinking skills as a warm-up for undertaking original research in the history of mathematics.''
This historical overview contains the following items:
\begin{itemize}
\item[--] ``Methodological moment: establishing topic parameters''.
\item[--] ``Methodological moment: philosophical foundations for historical research questions''.
\item[--] ``John Winthrop''. This section includes ``Methodological moment: identifying relevant primary sources'' and ``Methodological moment: approaches to historical interpretation''.
\item[--] ``Samuel Williams''. Such subsections as ``Methodological moment: making sense of primary sources'' and ``Methodological moment: visual sources'' are given.
\item[--] ``John Farrar''. This section contains the subsection ``Methodological moment: verify generalizations and appreciate complexity''.
\end{itemize}
For the entire collection see [Zbl 1495.01006].
Reviewer: Symon Serbenyuk (Kyïv)Helmholtz and the geometry of color space: gestation and development of Helmholtz's line elementhttps://zbmath.org/1521.010112023-11-13T18:48:18.785376Z"Peruzzi, Giulio"https://zbmath.org/authors/?q=ai:peruzzi.giulio"Roberti, Valentina"https://zbmath.org/authors/?q=ai:roberti.valentinaThe authors describe Helmholtz's way from experimental findings to his mathematical derivation of the line element directing the geometry of color space. His assistants' experiments performed in Berlin had shown him that equal distances in the known color models did not correspond to equal perceptual differences in his color diagram. This made him turn toward non-Euclidean geometry to define a line element in color space. Beside Helmholtz, Riemann had the same conjecture, as their famous essays on the foundation of geometry demonstrate [\textit{B. Riemann}, Gött. Abh. 13, 133--152 (1868; JFM 01.0022.02); \textit{H. von Helmholtz}, Gött. Nachr. 1868, 193--221 (1868; JFM 01.0022.03)]. For Riemann this idea was of greatest importance because such the color space became a candidate for an experimental test of non-Euclidean geometry. The authors show that, as far as the theory of color is concerned, contrary to Riemann, Helmholtz went further proposing the first metrically significant model of color space basing on the Weber-Fechner law. They state: ``Beside the deep knowledge of the latest achievements in the field of geometry, Helmholtz could rely on a solid background in experimental psychology and, in particular, the new born field of psychophysics, from which he borrowed fundamental ideas for the development of his line element in color space.'' In a detailed analysis, they show how, by extending the one-dimensional Weber-Fechner law to two- and three-dimensional spaces, Helmholtz has founded his line element.
Reviewer: Horst-Heino von Borzeszkowski (Berlin)Alice without quaternions: another look at the mad tea-partyhttps://zbmath.org/1521.010122023-11-13T18:48:18.785376Z"van Weerden, Anne"https://zbmath.org/authors/?q=ai:van-weerden.anneAuthor's abstract: ``Ever since the publication of Lewis Carroll's \textit{Alice's adventures in Wonderland} (1866), interpretations of its apparent nonsense have been given. In [``Alice's adventures in algebra: Wonderland solved'', online feature article, \url{https://www.newscientist.com/article/mg20427391-600-alices-adventures-in-algebra-wonderland-solved}], \textit{M. Bayley} added new interpretations, for instance, that the chapter about the mad tea-party mocked the quaternions of Sir William Rowan Hamilton. In 2017, Francine Abeles argued against Bayley's quaternion interpretation of the tea-party, and these arguments will be supported and extended by showing that Bayley's interpretation is based on erroneous assumptions about quaternions. It can be concluded that it is indeed very unlikely that Dodgson had the quaternions in mind when writing the tea-party chapter.''
Reviewer's comments: The author gives a nice overview about the personalities occurring in Chapter 7 (``The mad tea party'') from Lewis Carroll's book \textit{Alice's adventures in Wonderland}, i.e. as far as eventual appearences of quaternions are involved à la Hamilton. The contents of the paper are convincing and well-done. Not only mathematicians but also laymen will get a good impression of it all.
The book [\textit{L. Carroll}, The annotated Alice. New York, NY: Bramhall House (1960)] edited by Martin Gardner (known from his years-long monthly columns in Scientific American) does also reflect on the mad tea party, but nowhere in his book he takes the opportunity to compare its nonsense with notions in mathematical vocabulary, let alone quaternions. He considers connections to politicians and philosophers.
In summary, the reader will find the author's observations, as presented in the paper, easy to grasp, not only in a serious but also in a casual way.
Reviewer: Robert W. van der Waall (Huizen)Charles Peirce and Bertrand Russell on Euclidhttps://zbmath.org/1521.010132023-11-13T18:48:18.785376Z"Anellis, Irving H."https://zbmath.org/authors/?q=ai:anellis.irving-hThe paper begins with a brief history of non-Euclidean geometry. It then discusses and contrasts two post-non-Euclidean-geometry criticisms of Euclid's original reasoning, by Charles Sanders Peirce and Bertrand Russell. The author argues that, while Peirce and Russell agreed that Euclid's reasoning was intuitive and empirical rather than logical and mathematical, they disagreed upon the role of diagrams: while Russell viewed them as the crutch that allowed Euclid to construct his shoddy proofs, Peirce had no such objections to the use of visual elements in mathematical proof. It should be noted that this is the posthumous publication of a manuscript and appears to be incomplete and unedited in places.
Reviewer: Alexander Blum (Berlin)On the history of variational methods of non-linear equations investigations and the contribution of Soviet scientists (1920s--1950s).https://zbmath.org/1521.010142023-11-13T18:48:18.785376Z"Bogatov, Egor Mikhailovich"https://zbmath.org/authors/?q=ai:bogatov.egor-mikhailovichVariational calculus has a long history beginning with the work of Euler. The author gives a historical account of the development of the related variational methods for nonlinear equations in the period 1920's--1950's and argues that it owes much to the contributions of Soviet mathematicians that include L. A. Lyusternik, L. G. Shnirelman, S. L. Sobolev and others. He is especially focused on results by the Kyiv mathematicians M. A. Krasnoselskii and M. M. Vainberg.
This interesting article begins with the prehistory of Dirichlet's principle, its use by Riemann, critique by Weierstrass and justification by Hilbert. After a short discussion of semicontinuity in variational-type problems, he discusses the Ritz method but does not mention its massive use by Soviet mathematicians such as B. G. Galerkin. Variational methods served as a catalyst for the emergence of Sobolev spaces in functional analysis. The author puts the work of the Soviet mathematicians in the context of the development of the subject and compares their achievements with those of their foreign colleagues in the period 1920's--1950's: L. Tonelli, W. Ritz, L. Lichtenstein, G. D. Birkhoff, M. Morse, A. Hammerstein, M. Golomb and others.
The use of topological methods started by Poincaré and continued by Birkhoff and Morse, has been used by Lyusternik and Shnirelman, who among other things, extended the concept of category to the case of functionals to estimate the number of solutions of variational problems. As another application of the theory of categories, Lyusternik in 1939 estimated the number of critical points of functionals in a Hilbert space.
The author describes in more detail and with an example of buckling of rods the Krasnoselskii principle of linearization in the problem of bifurcation points [\textit{M. A. Krasnosel'skiĭ}, Топологические методы в теорий нелинейных интегральных уравнений (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0070.33001)]. Then he finishes his article with a list of solved problems by Vainberg.
Reviewer: Martin Lukarevski (Skopje)Emmy Noether steps onstage: her place in mathematical communities, past and presenthttps://zbmath.org/1521.010152023-11-13T18:48:18.785376Z"Rowe, David E."https://zbmath.org/authors/?q=ai:rowe.david-eThe paper tells the story of \textit{Diving into math with Emmy Noether}, a theatre performance by the Portraittheater Vienna directed by Sandra Schüddekopf, with Anita Zieher in the role of Emmy Noether. The original German version \textit{Mathematische Spaziergänge mit Emmy Noether} had a premiere in Berlin on June 4, 2019, 100 years after Emmy Noether's habilitation at the University of Göttingen. In 2022, the English version of the play had a successful tour in the United States, including a stop at the Bryn Mawr College, where Noether found refuge after begin expelled from Göttingen in 1933.
Reviewer: Antonín Slavík (Praha)The first and most elementary construction of real numbers -- by Karl Weierstraßhttps://zbmath.org/1521.010162023-11-13T18:48:18.785376Z"Spalt, Detlef D."https://zbmath.org/authors/?q=ai:spalt.detlef-dThis survey is devoted to an unknown concept of real numbers. This concept was formulated by Karl Weierstraß. One can note the following:
``In the Mathematics Library of Goethe University Frankfurt, a hitherto unknown manuscript of 171 pages was discovered by the librarian Boram Schröter in summer 2016. It contains the beginning of Weierstraß' lecture from winter 1880/81. These notes (from Emil Strauss (1859--1892)) give a painstaking record of Weierstraß' discourse, enabling the historian of mathematics, for the first time, to disclose Weierstraß' true concept of real numbers.
Weierstraß created an all-embracing concept of number. He started with the natural numbers \(1, 2, 3, \dots\), and \(0\), enlarged them to the fractions and the rational numbers and then added the irrational numbers. Lastly, he completed his monoid (only zero needs to be added) of fractions and irrational numbers to a group, which of course is a ring with 1; and moreover he gave procedures of division for fractions as well as for the irrational numbers and the general real numbers (the last two, however, are false). In the end he obtained a huge world of numbers, wherein each number (except the fractional numbers and the general real numbers) is unique.''
The paper contains the main and additional descriptions. The main consideration consists of explanations of Weierstraß' ideas and are related to such items:
\begin{itemize}
\item[--] ``Natural numbers and quantities -- the philosophical foundations''.
\item[--] ``Fractions -- the start of arithmetics''.
\item[--] ``Irrational numbers -- the start of analysis''. Here such Weierstraß' notions as finite and infinite irrational numbers are noted.
\item[--] ``General real numbers -- completing the basis of calculus''.
\end{itemize}
An additional part of the consideration contains a historical overview, certain views of some mathematicians, and the section ``Retrospect: where Weierstraß succeeded and where he did not succeed \dots.''.
Reviewer: Symon Serbenyuk (Kyïv)On the 85th birthday of Anatolii Mykhailovych Samoilenko (02.01.1938 -- 04.12.2020)https://zbmath.org/1521.010172023-11-13T18:48:18.785376Z(no abstract)Anatoly Pavlovich Markeev. On the occasion of his 80th birthdayhttps://zbmath.org/1521.010182023-11-13T18:48:18.785376Z(no abstract)Faina Mikhaĭlovna Kirillova (on the occasion of her 90th birthday)https://zbmath.org/1521.010192023-11-13T18:48:18.785376Z(no abstract)Igor' Rostislavovich Shafarevich (03.06.1923--19.02.2017)https://zbmath.org/1521.010202023-11-13T18:48:18.785376Z(no abstract)Biography of Robert Pertsch Gilberthttps://zbmath.org/1521.010212023-11-13T18:48:18.785376Z(no abstract)PhD students of Robert Pertsch Gilberthttps://zbmath.org/1521.010222023-11-13T18:48:18.785376Z(no abstract)Publications of Robert Pertsch Gilberthttps://zbmath.org/1521.010232023-11-13T18:48:18.785376Z(no abstract)Igor Moiseevich Krichever (obituary)https://zbmath.org/1521.010242023-11-13T18:48:18.785376Z"Buchstaber, V. M."https://zbmath.org/authors/?q=ai:bukhshtaber.viktor-matveevich"Novikov, S. P."https://zbmath.org/authors/?q=ai:novikov.sergei-petrovich"Taimanov, I. A."https://zbmath.org/authors/?q=ai:taimanov.iskander-a(no abstract)On the 90th birthday of Professor Oleg Vladimirovich Besovhttps://zbmath.org/1521.010252023-11-13T18:48:18.785376Z"Burenkov, V. I."https://zbmath.org/authors/?q=ai:burenkov.viktor-i"Tararykova, T. V."https://zbmath.org/authors/?q=ai:tararykova.tamara-vassilevna|tararykova.tamara-vasilevnaFrom the text: This issue of the Eurasian Mathematical Journal is dedicated to the 90th birthday of Oleg Vladimirovich Besov.In memoriam: Marco Avellaneda (1955--2022)https://zbmath.org/1521.010262023-11-13T18:48:18.785376Z"Cont, Rama"https://zbmath.org/authors/?q=ai:cont.rama(no abstract)Yueh-Gin Gung and Dr. Charles Y. Hu Award for 2023 to Victor J. Katz for distinguished service to mathematicshttps://zbmath.org/1521.010272023-11-13T18:48:18.785376Z"Coufal, Vesta"https://zbmath.org/authors/?q=ai:coufal.vesta"Dorée, Suzanne"https://zbmath.org/authors/?q=ai:doree.suzanne"Dray, Tevian"https://zbmath.org/authors/?q=ai:dray.tevian"Farris, Frank"https://zbmath.org/authors/?q=ai:farris.frank-a"Shell-Gellasch, Amy"https://zbmath.org/authors/?q=ai:shell-gellasch.amy-e"Smith, Derek"https://zbmath.org/authors/?q=ai:smith.derek-j|smith.derek-h|smith.derek-a|smith.derek-l(no abstract)Abel interview 2023: Luis Ángel Caffarellihttps://zbmath.org/1521.010282023-11-13T18:48:18.785376Z"Dundas, Bjørn Ian"https://zbmath.org/authors/?q=ai:dundas.bjorn-ian"Skau, Christian F."https://zbmath.org/authors/?q=ai:skau.christian-fr(no abstract)Obituary: Richard Kenneth Guy (1916--2020)https://zbmath.org/1521.010292023-11-13T18:48:18.785376Z"Falconer, Kenneth J."https://zbmath.org/authors/?q=ai:falconer.kenneth-j(no abstract)Obituary. Marco Avellaneda: mathematician and traderhttps://zbmath.org/1521.010302023-11-13T18:48:18.785376Z"Gatheral, Jim"https://zbmath.org/authors/?q=ai:gatheral.jim(no abstract)My memories of Vaughan Joneshttps://zbmath.org/1521.010312023-11-13T18:48:18.785376Z"Hoste, Jim"https://zbmath.org/authors/?q=ai:hoste.jim(no abstract)Obituary: Peter Michael Neumann (1940--2020)https://zbmath.org/1521.010322023-11-13T18:48:18.785376Z"Liebeck, Martin W."https://zbmath.org/authors/?q=ai:liebeck.martin-w"Praeger, Cheryl E."https://zbmath.org/authors/?q=ai:praeger.cheryl-e(no abstract)Pierced by a sun ray. An obituary of Yuri Ivanovich Manin (1937--2023)https://zbmath.org/1521.010332023-11-13T18:48:18.785376Z"Marcolli, Matilde"https://zbmath.org/authors/?q=ai:marcolli.matilde(no abstract)Orbituary: Gordon Douglas James, 1945--2020https://zbmath.org/1521.010342023-11-13T18:48:18.785376Z"Mathas, Andrew"https://zbmath.org/authors/?q=ai:mathas.andrew(no abstract)Boris Teodorovich Polyak (04.05.1935 -- 03.02.2023)https://zbmath.org/1521.010352023-11-13T18:48:18.785376Z"Novikov, D. A."https://zbmath.org/authors/?q=ai:novikov.dmitri-a|novikov.d-a"Khlebnikov, M. V."https://zbmath.org/authors/?q=ai:khlebnikov.mikhail-vladimirovich(no abstract)Edmond Malinvaud: a tribute to his contributions in econometricshttps://zbmath.org/1521.010362023-11-13T18:48:18.785376Z"Phillips, Peter C. B."https://zbmath.org/authors/?q=ai:phillips.peter-c-b(no abstract)Obituary: Peter Vámos (1940--2020)https://zbmath.org/1521.010372023-11-13T18:48:18.785376Z"Sharp, Rodney Y."https://zbmath.org/authors/?q=ai:sharp.rodney-y(no abstract)ICERM: 10 years laterhttps://zbmath.org/1521.010382023-11-13T18:48:18.785376Z"Pausader, Benoît"https://zbmath.org/authors/?q=ai:pausader.benoit(no abstract)How do mathematicians publish? -- Some trendshttps://zbmath.org/1521.010392023-11-13T18:48:18.785376Z"Hulek, Klaus"https://zbmath.org/authors/?q=ai:hulek.klaus"Teschke, Olaf"https://zbmath.org/authors/?q=ai:teschke.olafSummary: We have already discussed bibliometric measures for the mathematics corpus in this column before. This included the unusual longevity of mathematics citations, effects of delayed publication due to often long and complex refereeing processes, and the specifics of different mathematical areas. It has become clear that purely numerical criteria are often unsuitable to measure mathematical quality or the scientific impact of publications. At the same time, the bibliometric results often depend on mathematical subfields, thus reflecting the structure and different behaviour of mathematical communities. In this column we concentrate on an author-oriented viewpoint. We will derive some quantities which illustrate how the landscape of mathematical publications has changed over the past decades.Robert Meyer's publications on relevant arithmetichttps://zbmath.org/1521.030052023-11-13T18:48:18.785376Z"Ferguson, Thomas Macaulay"https://zbmath.org/authors/?q=ai:ferguson.thomas-macaulay"Priest, Graham"https://zbmath.org/authors/?q=ai:priest.grahamSummary: The bibliography appearing below collects the publications in which Meyer's investigations into relevant arithmetic saw print. Each bibliographic item is accompanied by a short description of the text or other remarks. We include papers on relevant arithmetic coauthored by Meyer, but omit both Meyer's work on relevant \textit{logic} and the work published independently by his collaborators.Where do axioms come from?https://zbmath.org/1521.030062023-11-13T18:48:18.785376Z"Smoryński, Craig"https://zbmath.org/authors/?q=ai:smorynski.craigThe structure of this brief and dense chapter may be defined by means of the concept introduced by Moritz Pasch in order to extend the Euclidean geometry: betweenness. I am playing with the meaning of a word. I hope that the reader will understand, at the end of this review, the quite naive game that I am suggesting. Let's see why. At the beginning of this work, a question and a problem are raised with lucidity: ``The Greek view, held by Frege, is that one chooses evident truths about some subject as axioms and generates additional truths through logical reasoning. Hilbert more-or-less states that one chooses axioms arbitrarily and, should they be consistent, they are true about something (and so too are the theorems derived from them [\dots] but not all axiom systems are equal and some theories are deemed more significant and worthy of our attention than others (p.185). It is not enough for mathematics to choose axioms because of their self-evident truth. In fact, truth and evidence are elusive and must be handled with delicacy. Hence ``axioms may be chosen for a number of reasons'' (p.191). The core of the opus consists in describing these reasons: Truth (p.186); Necessity (pp.186--1879; Proof-Generation (pp.187--188); Convenience (p.188); System-Refinement (pp.188--189); Analogy (p.190); Pure Formalism (p.190). At the end the conclusion is subtle and may be divided into two parts: (1) ``I am now inclined to believe that one genuinely does have a complete freedom in choosing a set of axioms to study'' and (2) ``but I also believe that one should be aware that the significance of the set chosen is guaranteed by what it does, most obviously measured by its connections with the rest of mathematics'' (p.191). Between evident axioms and arbitrary axioms lies the realm of mathematical thought. And the marvel of mathematics appears in the fact that its validity consists in establishing new relations within its own domain.
For the entire collection see [Zbl 1497.03004].
Reviewer: Godofredo Tomas Iommi Amunátegui (Valparaíso)Some interrelations between geometry and modal logichttps://zbmath.org/1521.030372023-11-13T18:48:18.785376Z"Pledger, Ken E."https://zbmath.org/authors/?q=ai:pledger.ken-eSummary: This is a reprinting of \textit{K. E. Pledger}'s PhD thesis [Some interrelations between geometry and modal logic. Warsaw: University of Warsaw (PhD Thesis) (1980)], submitted to the University of Warsaw in 1980 with the degree awarded in 1981. It develops a one-sorted approach to the theory of plane geometry, based on the idea that the usually two-sorted theory ``can be made one-sorted by keeping careful account of whether the incidence relation is iterated an even or odd number of times''.
The one-sorted structures can also serve as Kripke frames for modal logics, and the thesis defines and studies two such logics that are validated by projective planes and elliptic planes respectively. It raises questions of logical completeness for these systems that are addressed in the first article of this journal issue.Relevant arithmetichttps://zbmath.org/1521.030432023-11-13T18:48:18.785376Z"Meyer, Robert K."https://zbmath.org/authors/?q=ai:meyer.robert-k|meyer.robert-k.1Summary: This is a republication of \textit{R. K. Meyer}'s ``Relevant arithmetic'', which originally appeared in [Bull. Sect. Logic, Pol. Acad. Sci. 5, 133--137 (1976)]. It sets out the problems that Meyer was to work on for the next decade concerning his system, \(R^\sharp\).Laver and set theoryhttps://zbmath.org/1521.031792023-11-13T18:48:18.785376Z"Kanamori, Akihiro"https://zbmath.org/authors/?q=ai:kanamori.akihiroSummary: In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Inconsistent models for relevant arithmeticshttps://zbmath.org/1521.032282023-11-13T18:48:18.785376Z"Meyer, Robert K."https://zbmath.org/authors/?q=ai:meyer.robert-k.1|meyer.robert-k"Mortensen, Chris"https://zbmath.org/authors/?q=ai:mortensen.chrisOriginally published in [J. Symb. Log. 49, 917--929 (1984; Zbl 0599.03015)].Algebraic curves and surfaces. A history of shapeshttps://zbmath.org/1521.140012023-11-13T18:48:18.785376Z"Busé, Laurent"https://zbmath.org/authors/?q=ai:buse.laurent"Catanese, Fabrizio"https://zbmath.org/authors/?q=ai:catanese.fabrizio"Postinghel, Elisa"https://zbmath.org/authors/?q=ai:postinghel.elisaThe book contains the notes of three courses given by the authors during the summer school for young researchers TiME2019, held in Levico Terme (Italy).
In the first section, the second author outlines the classification of projective surfaces \(S\), described by Castelnuovo and Enriques at the beginning of the XXth century. The classification is based on the Kodaira dimension \(\kappa(S)\), which corresponds to the dimension of the image of the map associated with \(mK_s\) for \(m\gg 0\). The case \(\kappa(S)=2\) corresponds to surfaces of general type, whose moduli spaces are still intensively studied. The case of rational and ruled surfaces is studied in the first two lectures. Then, the exposition focuses on the \(P_{12}\)-Theorem of Castelnuovo-Enriques, which shows how the birational structure of \(S\) depends on the dimension of the linear system \(12K_S\). In particular, when \(P_{12}>1\) and \(K_S^2=0\), the surface has a fibration \(S\to B\) over a curve, and the fibers are smooth elliptic curves. The classification of this case is based on the isotriviality of the fibration, i.e. the property that the elliptic fibers are isomorphic or, equivalently, their moduli are trivial. In the last lecture, several strategies for the proof of isotriviality are collected and illustrated.
In the second section of the book the third author illustrates the main techniques and some recent achievements in the study of polynomial interpolation with multiplicities. The problem is to determine the dimension of linear systems of polynomials vanishing at \(k\) general points with multiplicities greater or equal than preassigned values. Even in the case of homogeneous polynomials in three variables over the complex field (corresponding to curves in the complex projective plane) the situation is not totally understood. A general method for the construction of linear systems whose dimension is bigger than the expected value is known; it is based on the geometry of the blow up of the plane at several general points. Long ago it has been conjectured that the method exhausts all the cases in which the dimension is bigger than expected. The conjecture is still open. The book contains a discussion on the conjecture, with a description of the cases in which it is known to hold, and an illustration of the main methods used to attack the problem (as Cremona transformations, and the study of the nef cone of blow up's). Possible extensions of the conjecture to higher dimensional spaces are also presented.
The third section contains an outline of algebraic and geometric methods for the computation of implicit (polynomial) equations that describe algebraic varieties defined by parametric polynomials (hence defined as the image of suitable polynomial maps). The construction of implicit equations can provide a way to solve some problems, like the membership problem, relevant for applications. The straightforward method to obtain implicit equations is based on the classical elimination theory. Yet, when the number of variables and parameters increases, a direct use of elimination theory becomes unpractical. The first author describes a series of algebraic tools that can simplify the problem. These tools are based on a matricial representation of the associated ideals, and the author introduces and explains the role of the corresponding elimination matrices. The main application discussed in the section concerns implicit equations that define the intersection of parametric (rational) hypersurfaces, a problem that arises naturally in the reconstruction of images.
Reviewer: Luca Chiantini (Siena)The Jameson wayhttps://zbmath.org/1521.760032023-11-13T18:48:18.785376Z"Löhner, Rainald"https://zbmath.org/authors/?q=ai:lohner.rainald(no abstract)The age of innocence. Nuclear physics between the First and Second World Warshttps://zbmath.org/1521.810032023-11-13T18:48:18.785376Z"Stuewer, Roger H."https://zbmath.org/authors/?q=ai:stuewer.roger-hSee the review of the original hardback edition in [Zbl 1496.81021].