Recent zbMATH articles in MSC 01https://zbmath.org/atom/cc/012023-09-22T14:21:46.120933ZUnknown authorWerkzeugThe meaning of proofs. Translated from the Italian by Bonnie McClellan-Broussard. With a foreword by Matilde Marcollihttps://zbmath.org/1517.000022023-09-22T14:21:46.120933Z"Lolli, Gabriele"https://zbmath.org/authors/?q=ai:lolli.gabriele``The Meaning of Proofs: Mathematics as Storytelling'' is a thought-provoking and innovative book that challenges traditional views of mathematical proofs by presenting them as narratives. Written by Gabriele Lolli, a researcher in logic, the book delves into the philosophical aspect of why mathematical proofs are expressed in the way they are and aims to convince readers that understanding proofs can be achieved through a narrative lens.
Lolli's book takes a refreshing approach to mathematics by exploring the idea that mathematical proofs are not mere technical and abstract constructs, but rather compelling narratives that unfold and captivate the reader. While it does not require biographical details about mathematicians, it focuses on the content and subject matter of mathematical pieces, emphasizing the significance of narratives in shaping mathematical pedagogy.
One of the key strengths of this book lies in its ability to make complex mathematical concepts accessible to a wide audience. By weaving narratives into mathematical proofs, Lolli demonstrates how concepts like limits and one-to-one correspondence can be comprehended through engaging stories such as Zeno's paradox and Hilbert's grand hotel. Through these narratives, readers are able to grasp the fundamental ideas behind these mathematical concepts in a more intuitive and relatable manner.
Moreover, ``The Meaning of Proofs'' highlights the historical and contemporary importance of metaphorical language and visual representations in mathematical proofs. Lolli emphasizes that these elements not only help us understand how narrative tools were employed in ancient times but also play a vital role in modern research on automated theorem-proving. By showcasing the relevance of metaphors and diagrams, the book underscores the dynamic and evolving nature of mathematical storytelling.
The author also explores the impact of narratives at both the grand and individual levels. Grand narratives, as exemplified by Klein's Erlangen program, are shown to have far-reaching effects on the field of mathematics and even spill over into other disciplines like theoretical physics. On the other hand, individual narratives focus on specific papers that employ unique styles and presentations to convey mathematical proofs. The book draws interesting connections between these individual narratives and ancient Greek composition forms, providing a rich historical context for mathematical storytelling.
``The Meaning of Proofs'' not only sheds light on the nature of mathematical proofs but also touches upon the broader significance of storytelling. It underscores the universal nature of mathematical ideas, emphasizing that these concepts are common to all humans. This notion is exemplified by Michael Frame's ``Geometry of Grief,'' where the author employs geometry to explore mental states and offer solace in times of loss.
Overall, Gabriele Lolli's book presents a compelling argument for embracing mathematical proofs as narratives. By bridging the gap between mathematics and literature, the book challenges conventional notions and reveals the beauty and accessibility of mathematical storytelling. With its engaging insights, historical perspectives, and interdisciplinary connections, ``The Meaning of Proofs'' is an invaluable contribution that will appeal to both math enthusiasts and intellectually curious readers alike.
Reviewer: Firdous Ahmad Mala (Srinagar)The book of wonders. The many lives of Euclid's \textit{Elements}https://zbmath.org/1517.000052023-09-22T14:21:46.120933Z"Wardhaugh, Benjamin"https://zbmath.org/authors/?q=ai:wardhaugh.benjaminPublisher's description: Euclid's Elements of Geometry was a book that changed the world. In a sweeping history, Benjamin Wardhaugh traces how an ancient Greek text on mathematics -- often hailed as the world's first textbook -- shaped two thousand years of art, philosophy and literature, as well as science and maths.
Thirteen volumes of mathematical definitions, propositions and proofs. Writing in 300 BC, Euclid could not have known his logic would go unsurpassed until the nineteenth century, or that his writings were laying down the very foundations of human knowledge.
Wardhaugh blasts the dust from Euclid's legacy to offer not only a vibrant history of mathematics, told through people and invention, but also a broader story of culture. Telling stories from every continent, ranging between Ptolemy and Isaac Newton, Hobbes and Lewis Carrol, this is a history that dives from Ancient Greece to medieval Byzantium, early modern China, Renaissance Italy, the age of European empires, and our world today.
How has geometry sat at the beating heart of sculpture, literature, music and thought? How can one unknowable figure of antiquity live through two millennia?Categorization and methodologyhttps://zbmath.org/1517.000062023-09-22T14:21:46.120933Z"Jedrzejewski, Franck"https://zbmath.org/authors/?q=ai:jedrzejewski.franckSummary: Au cours des trente dernières années, les mathématiques ont connu des bouleversements majeurs dans la façon de poser et de résoudre les problèmes. L'un de ces bouleversements auquel on assiste aujourd'hui est la catégorification. Comme son processus inverse la décatégorification, le mot signifie que l'on transpose un problème posé dans le vocabulaire classique ensembliste en un problème d'un plus haut degré d'abstraction posé dans le monde des catégories. La catégorication a suscité de nouveaux néologismes comme la combinatorialisation (Doron Zeilberger), l'homotopification ou la groupoïdification (John Baez). Si l'on parle de catégorification et non de catégorisation, c'est que l'on veut insister sur le processus de transposition. Il ne s'agit pas seulement de catégoriser, ce qui est une façon de définir des catégories en vue de ranger des objets, mais de catégorifier, ce qui suppose que l'on va, à travers une approche catégorique, donner un plus haut sens à un objet ou à un problème mathématique. Aujourd'hui, seule la théorie des catégories permet de faire ce travail et de donner une plus grande compréhension de certains aspects ou objets mathématiques. En réalité, le mot catégorification a plusieurs sens.
For the entire collection see [Zbl 1506.00095].Distributivity-like results in the medieval traditions of Euclid's \textit{Elements}. Between geometry and arithmetichttps://zbmath.org/1517.010012023-09-22T14:21:46.120933Z"Corry, Leo"https://zbmath.org/authors/?q=ai:corry.leoMultiplication distributes over addition: \(a(b+c+d)=ab+ac+ad\), or ``the products of several parts is equal to the product of the whole,'' as Viète put it (p. 2). This is true in geometry, in unit-based number theory, and in the arithmetic of the real-number continuum. So how mathematicians dealt with distributivity-related results says something about how they viewed the relations between these different domains of mathematics.
Euclid largely kept these different domains separate and hence developed distributivity-related results in three separate theoretical contexts. At least, that seems to have been his plan. In the details, his treatment leaves something to be desired.
Book II of the \textit{Elements} treats distributivity results in the case where addition means concatenation of line segments and multiplication means area formation. Euclid's approach hardly illuminates the essential algebraic core of these results, for one thing since he gives separate proofs for numerous propositions that could have been more efficiently derived from earlier ones (p. 7). Heron later gave the latter kind of proofs for these propositions (p. 28).
Also somewhat contrary to an algebraic flavour is Euclid's habit of stating and proving separately results that are the same except for a sign change when interpreted algebraically (p. 14) and his phraseology of describing \(a=nb\) as \(a\) being equal to or a multiple of \(b\) (p. 10). (Reviewer's remark: Modern textbooks still state formulas for \(\sin(a+b)\) and \(\sin(a-b)\) separately, and differentiation rules for \(e^x\) and \(e^{kx}\) separately, even though doing so is algebraically redundant since \(\sin(a+b)\) already includes negative values of \(b\) and \(e^{kx}\) includes the case \(k=1\).)
Euclid later seems to tacitly assume that the results from Book II carry over to arithmetic, hence ``deviating from his self-imposed, strict separation of domains'' (p. 21). Later commentators al-Nayrizi (p. 31) and Campanus (p. 62) evidently disapproved of this and included the missing arithmetic versions of the results of Book II. ``Readers of the Campanus version \dots\ would now have good grounds -- and by all means better grounds than those of a reader of any previous treatise -- for seeing [key distributivity results] as inherently arising within the purely arithmetic realm, without any need for additional support coming from geometric considerations.'' (pp. 62--63)
But regardless of such theoretical subtleties, switching between numerical and geometrical paradigms was commonly taken for granted as unproblematic and straightforward in medieval works. ``For example, in his formulation of Euclid's II.4, Bar Hiyya speaks, as in Euclid's original, about a line that is divided at an arbitrary point and about the squares that are built on the resulting segments. But to this purely geometric formulation he added: `and I give you an example with numbers'.'' (p. 66)
Euclid also tacitly assumes that for any magnitude \(a\) one can form the magnitude \(a/n\) (in other words, one can divide any magnitude into as many equal parts as one wishes). In the figures associated with these proofs, the magnitudes in question are schematically represented by line segments, although the theorems are supposed to be equally valid for any kind of magnitudes, including areas, volumes, and angles. In the case of angles in particular, it is well known that \(a/3\) should not be assumed. (p. 12)
Al-Nayrizi's strategy for dealing with the issue of proving something for ``magnitudes'' was to think of such theorems ``as embodying several different, but specifically geometric situations, each of which required its own kind of justification. In other words, he was not thinking of magnitudes as a completely general concept on which we can argue in abstract terms, without a specific justification for each case. Properties of equimultiplicity, so it seems, were for him differently rooted in basic properties specific to each kind of magnitude that can be considered.'' (p. 32)
Outside the strictures of Euclid's \textit{Elements}, in the Arabic algebraic tradition, some algebraic rules were proved geometrically, others not at all (pp. 34--35). According to al-Khayyam, ``Algebra is something geometrical which is demonstrated in Book II of the \textit{Elements}'' (p. 36): ``In as much as these proofs are understood other than in this way (i.e., geometrically; while initially they were conceived from a purely arithmetic point of view), the art of algebra does not become truly scientific, even when this method requires that we address some difficulties.'' (p. 41) ``Difficulties'' included how to justify geometrically cancellations such as reducing \(x^3 = cx + bx^2\) to \(x^2 = c + bx\), which al-Khayyam treats somewhat awkwardly (pp. 38--41). Distributivity considerations are here essential, although ``al-Khayyam did not formulate any kind of explicit distributivity rules, nor are such rules found in other arithmetic or algebraic contexts of Islamicate mathematics.'' (p. 41)
``An underlying tension manifests itself, arising from the attempt to understand the basic properties of numbers without thereby giving away the traditional, Euclidean centrality of geometry as the field that is better understood and axiomatically founded.'' (pp. 48--49)
A 12th-century arithmetic treatise explicitly states its intention ``to add to what Euclid stated in the second book, in order to explain with respect to numbers what he himself explained with respect to lines'' (p. 44). But to prove such results the author avails himself of propositions from all over the \textit{Elements} with insufficient regard for their proper domain of validity. Thus, VII.18 is used to prove general results for numbers, even though ``Euclid's proof of VII.18 specifically depends on counting units'' (p. 46) whereas the ``numbers'' the author is trying to apply the result to are supposed to include not only integers but ``fractions or even roots'' (p. 45).
Fibonacci dealt with distributivity results, ``considering them simultaneously as related to both geometry and arithmetic, and he could move quite freely from one realm to the next when necessary. Some of the geometric results he relied upon he took for granted, some he just illustrated with numerical examples. He introduced new proofs for some of the Euclidean propositions but also used such canonical propositions in new contexts. And within this account, distributive-like results and rules appear in different ways in these two treatises: sometimes he relies on them explicitly (referring to the first propositions of Book II), sometimes he uses them implicitly and perhaps inadvertently, and sometimes he developed roundabout arguments aimed at establishing results of that kind.'' (p. 56)
The basic operations of place-value arithmetic with Hindu-Arabic numerals is ``permeated an implicit use of distributive-like arguments at the core of arithmetic'' (p. 57). ``The constant presence and use of distributive-like ideas sometimes led to explicitly formulating them, and sometimes to just use them in ingenious, yet tacit and perhaps even inadvertent ways.'' (p. 57)
Many tricks for mental arithmetic are ultimately based on distributivity rules, such as using \(n^2=(n-1)^2+n+(n-1)\) to square \(n\) when \(n-1\) is easier to square (p. 75). ``The source typically mentioned in Islamicate texts for methods of mental multiplication is Euclid's Book II'' (p. 77).
In conclusion, the author has collected many snippets form medieval works that can retrospectively be seen as having to do with distributivity. Since this was not an actor's category, and since it involves jumping abruptly between different contexts, it does not make for a very coherent narrative.
Some serious typos make the text harder to read: the displayed equations on page 40 are wrong in several ways; ``put to use in additional propositions'' (p. 18) should be ``put to use additional propositions''; ``Heron went like Euclid'' (p. 31) should be ``Al-Nayrizi went like Euclid''.
Reviewer: Viktor Blåsjö (Utrecht)Axiomatics. Mathematical thought and high modernismhttps://zbmath.org/1517.010022023-09-22T14:21:46.120933Z"Steingart, Alma"https://zbmath.org/authors/?q=ai:steingart.almaThe title of the book is \textit{Axiomatics}, but an equally good title could have been \textit{Abstraction}. Indeed, the word ``abstraction'' appears in the titles of all six chapters, and the book's main theme is the rise of abstraction in mathematics in the middle of the 20th century, with a particular emphasis on the situation in the United States. The author explains various reasons behind the shift of mathematics toward abstraction. One of them is the natural idea that abstract theories based on the axiomatic method are capable of unifying various mathematical disciplines, and therefore should have a broader applicability. Other reasons are less obvious: For example, during World War II, there was no doubt about the importance of applied mathematics for military purposes, but mathematicians began to be concerned with the role of mathematics in the post-war era. One possible attitude was to recognize mathematics not only as a science, but also as an art, with modern abstract mathematics being especially close to modern art in their separation from the real world. Other parts of the book deal with the rise of axiomatic and abstract thinking in economics and social science, exemplified by von Neumann's and Morgenstern's game theory, as well as Arrow's theory of social choice. But the mathematicians' enthusiasm for abstraction had sometimes gone too far and, as described in the final parts of the book, many mathematicians of the later 20th century felt the need to return to the historical roots of the subject, intuition, and visualizable mathematics.
The book offers many thought-provoking questions about the role of mathematics, the relation between mathematics and art, and about the border between pure and applied mathematics. It can be recommended to all historians of modern science, as well as to working mathematicians who enjoy thinking about the nature of their subject.
Reviewer: Antonín Slavík (Praha)Publishing Sacrobosco's \textit{De sphaera} in early modern Europe. Modes of material and scientific exchangehttps://zbmath.org/1517.010032023-09-22T14:21:46.120933ZPublisher's description: This open access volume focuses on the cultural background of the pivotal transformations of scientific knowledge in the early modern period. It investigates the rich edition history of Johannes de Sacrobosco's Tractatus de sphaera, by far the most widely disseminated textbook on geocentric cosmology, from the unique standpoint of the many printers, publishers, and booksellers who steered this text from manuscript to print culture, and in doing so transformed it into an established platform of scientific learning. The corpus, constituted of 359 different editions featuring Sacrobosco's treatise on cosmology and astronomy printed between 1472 and 1650, represents the scientific European shared knowledge concerned with the cosmological worldview of the early modern period until far after the publication of Copernicus' \textit{De revolutionibus orbium coelestium} in 1543. The contributions to this volume show how the academic book trade influenced the process of homogenization of scientific knowledge. They also describe the material infrastructure through which such knowledge was disseminated, and thus define the premises for the foundation of modern scientific communities.
The articles of this volume will be reviewed individually.How the estimate of \(\sqrt{2}\) on YBC 7289 may have been calculatedhttps://zbmath.org/1517.010042023-09-22T14:21:46.120933Z"Buckle, David"https://zbmath.org/authors/?q=ai:buckle.david-jThis article proposes a way to arrive at the very good approximation of $\sqrt{2}$ found on the Babylonian clay tablet YBC 7289, viz. 1:24:51:10 (sexagesimal notation). Only numbers are written on the tablet, inscribed within a drawing of a square. No procedure is given how the numbers were found. Since its first publication, several explanations have been provided by modern mathematicians.
In some Babylonian mathematical exercises, a method for estimating square roots is used that can be formulated in modern algebra as follows:
\[
1/2(a + d/a),\text{ where $a$ is the estimated square root of }d.
\]
Applied to finding $\sqrt{2}$, this becomes $a/2 + 1/a$, $a$ being the initial estimate.
This method can be used in iteration to find ever more accurate estimates of the square root. However, there is a limitation inherent in Babylonian arithmetic. Division is performed as multiplication by the reciprocal of the divisor. If the divisor is not a terminating sexagesimal number (i.e. if it contains other factors than 2, 3, or 5), an accurate division is not possible. Numbers containing only the factors 2, 3, or 5 are called ``regular'' in the literature on Babylonian mathematics.
Therefore, the initial estimate to be chosen has to be a regular number. Starting with 1:30 (i.e., 1 1/2), one finds 1:25. Its square is greater than 2. To improve this estimate, one can choose a smaller initial value; but this must be a regular number. Using 1:20, the resulting estimate is the same 1:25. 1:25 however is not regular, so the procedure cannot be repeated with 1:25. Another regular number slightly smaller than 1:25 has to be found. Among the possible numbers, 1:24:22:30 occurs in several mathematical tablets and was certainly known among scribes. It also happens to have a reciprocal which is only two digits long. The author then applies the above-mentioned formula to 1:24:22:30, finding in two steps the result 1:24:51:15 for an improved estimate of $\sqrt{2}$. To test the quality of this estimate, he squares it, using a method devised by himself (to be published later). The result is greater than 2, therefore a better value must be slightly smaller, i.e. between 1:24:51 and 1:24:51:15. This he estimates to be 1:24:51:10. He then squares this value which shows that it is closer to $\sqrt{2}$ than previous estimates. He has thus found a procedure that would deliver the number found on the tablet YBC 7289.
In the remainder of the article, the author rejects other proposals for explaining the number on the tablet. He also briefly comments on the other numbers on the tablet which obviously are the lengths of the side of the square drawn and its diagonal.
The main arguments of the author for his proposal are its simplicity and the fact that it could easily have been performed by a Babylonian mathematician.
Reviewer: Hermann Hunger (Wien)On Tusi's classification of cubic equations and its connections to Cardano's formula and Khayyam's geometric solutionhttps://zbmath.org/1517.010052023-09-22T14:21:46.120933Z"Kalantari, Bahman"https://zbmath.org/authors/?q=ai:kalantari.bahman"Zaare-Nahandi, Rahim"https://zbmath.org/authors/?q=ai:zaare-nahandi.rahimThis article explores the problem of determining the real roots of a cubic equation through an examination of the contributions made by Sharaf al-Din Tusi, a Persian mathematician from the 12th century. The authors establish a canonical representation for cubic equations, approximate the real roots, and reveal noteworthy connections between Tusi's work and Cardano's formula, which was discovered several centuries later. Additionally, the article highlights the association with Omar Khayyam's geometric solution. The primary objective of the article is to unite three influential historical works on cubic equations: Khayyam's geometric solution, Tusi's classification, and Cardano's formula. According to the translation of Tusi's work by the noted mathematical historian Rashed, Tusi analyzed the problem for five different types of equations one of which is the canonical form the authors refer here as the \textit{Tusi form}, \(x^2-x^3 = c\). Tusi argued the maximum of \(x^2-x^3\) on (0; 1) occurs at \(\frac{2}{3}\) and concluded when \(c = \frac{4}{27}\delta\), \( \delta \in (0; 1)\), there are roots in \((0; \frac{2}{3})\) and \((\frac{2}{3}; 1)\), while ignoring the root in \((-\frac{1}{3};0)\) as negative numbers were unaccepted at the time. On the one hand they show that a cubic equation in the \textit{reduced form} \(x^3+px+q = 0\) with \(p < 0\) is reducible to a \textit{Tusi form} with \(\delta = \frac{1}{2} +\frac{3\sqrt{3}q}{4\sqrt{-p^3}}\). It follows that there are three real roots if and only if the \(\textit{discriminant}\) \(\Delta = -(\frac{q^2}{4} + \frac{p^3}{27})\) is positive. This gives an explicit connection between \(\delta\) in Tusi form and \(\Delta\) in Cardano's formula. In particular, when \(\delta \in (0; 1)\), rather than using Cardano's formula in complex numbers which in turn needs to be approximated, one can use the intervals in Tusi form to directly approximate the roots iteratively. On the other hand, for \(p > 0\) the authors give a novel proof of Cardano's formula for the unique real root. While Rashed attributes Tusi's computation of the maximum to the use of derivatives, according to Hogendijk, ``Tusi probably found his results by means of manipulation of squares and rectangles on the basis of Book II of Euclid's \textit{Elements}.'' Here, the authors show the maximizer in Tusi form is directly computable via elementary algebraic manipulations. Indeed for an analogous \textit{quadratic Tusi form}, \(x - x^2 = \frac{\delta}{4}\), Tusi's approach results in a simple derivation of the quadratic formula, comparable with the pedagogical approach of Po-Shen Loh. Moreover, the authors derive analogous results for the general Tusi form, \(x^{n-1} - x^n = \frac{\delta(n-1)^{(n-1)}}{n^n}\). Finally, gaining insights from Tusi form, they present a concise derivation of Khayyam's geometric solution for all cubic equations. The results here complement previous findings on Tusi's work and reveal further facts on history, mathematics and pedagogy in solving cubic equations.
Reviewer: Jebrel M. Habeb (Irbid)Mathematics and philosophy: from the Greeks to Descarteshttps://zbmath.org/1517.010062023-09-22T14:21:46.120933Z"Bicudo, Irineu"https://zbmath.org/authors/?q=ai:bicudo.irineu"Vaz, Duelci Aparecido de Freitas"https://zbmath.org/authors/?q=ai:vaz.duelci-aparecido-de-freitasThis article offers a mathematically informed account of Descartes' geometry and its relationship to ancient Greek mathematics.
For the Greeks, ``analysis'' was the creative part of the mathematical method for proving theorems and solving problems. In the case of theorem-proving, the analysis assumes the conclusion to be true and searches for its consequences. The reverse operation, the ``synthesis'', was accomplished by deducing the truth of the theorem starting from the outcomes of the analysis and using the data provided by the latter.
While, for the Greeks, the method was limited to mathematical procedures, Descartes believed that only analysis, by virtue of its creative nature, could become a veritable method of discovery that was susceptible to wider application in mathematics and other sciences.
It is certainly impossible to aim at a comprehensive treatment of the history of mathematics from the Greek antiquity to Descartes on a few pages only; however, surprisingly, the article does not mention certain historical episodes, such as Arabic mathematics, which are conceptually and historically relevant to understand the genesis of Descartes' mathematics. Even more surprising for a scientific publication are a few historically misguided claims and extreme oversimplifications (for instance, the authors state that after the Greeks, no new philosophy or mathematics appeared until the 16th century). Despite these shortcomings, and even without offering original contributions, this paper can be read as a useful presentation of Descartes' mathematics for a Portuguese-speaking public, secondary school teachers and students.
Reviewer: Davide Crippa (Praha)How Leibniz tried to tell the world he had squared the circlehttps://zbmath.org/1517.010072023-09-22T14:21:46.120933Z"Strickland, Lloyd"https://zbmath.org/authors/?q=ai:strickland.lloydIn this paper, the author analyses four strategies through which Leibniz tried to spread his discovery on the approximation of \(\pi\) through an infinite rational series, specifically his identity \(\frac{\pi}{4}=1-\frac13+\frac15-\frac17+\frac19-\frac{1}{11}+\frac{1}{13}\), etc. The author identifies four strategies he calls ``conventional'' (pp. 21--23), ``the grand design'' (pp. 23--28), and ``impostrous'' (pp. 28--32), ``extravagant'' (pp. 33--35), respectively.
The ``conventional strategy'' consisted in divulgating the formula through letters or through the publication in a scientific journal. From an autograph note by Leibniz, we know that his discovery dates to the first half of 1674. In July 1674, he sent a letter to Oldenburg announcing his discovery, and in October a letter on the same subject was addressed to Mariotte (p. 21). In the same month, Leibniz sent Huygens the treatise \textit{Arithmetical quadrature} where he offered not only his formula, but its demonstration too. Huygens greatly appreciated this work of Leibniz. Besides these communications, there is no doubt that Leibniz let know his formula to his friends in France (Ozaman, Malebranche, Gallois, Mariotte himself, p. 22). In the last months of 1675, he expressed to Gallois his intention to publish his formula in a scientific journal, the \textit{Journal des Sçavants}. Leibniz conceived a letter for de La Roche, who was the editor of the \textit{Journal des Sçavants} at that time, but never sent it: it was lengthy and full of explanations, whereas the \textit{Journal} usually published short reports of 1--3 pages. Probably, Leibniz regarded his letter not suitable to be published in the \textit{Journal} (pp. 22--23).
The ``grand design'' consisted in inserting the formula to find \(\pi\) within the treatise \textit{De quadratura arithmetica circuli ellipseos et hyperbolae} [\dots], a text which Leibniz began to write in June 1676 (p. 23). In particular, he found the development in series of arctan (Prop. 31), from which the series which offers the development of \(\frac{\pi}{4}\) can be immediately obtained (p. 24). Starting from 1677, Leibniz tried to publish his treatise. As the author explains, Mariotti and Soudry were involved in this affair, but at the end of 1678 the book had not yet been published and in December Soudry died (p. 25). At that time, the story of the treatise on arithmetic quadrature was connected with Leibniz's attempt to become a member of the Royal Academy. Within the general strategy he developed, Leibniz sent his treatise to Huygens, who urged the former to publish such a work (pp. 26--27). In 1680, Leibniz hoped to publish his \textit{De quadratura} with the Dutch publisher Daniel Elzevier. The initial approach was unsuccessful and, furthermore, on 13th October 1680 Elzevier died, so that the text was not published (pp. 27--28).
Next, the ``impostrous attempt'': Leibniz lost interest in publishing his whole treatise, but he intended to disseminate at least the formula to calculate \(\pi\), which happened in 1682 (\textit{De vera proportione circuli} [\dots], p. 28): in 1681, Leibniz wrote a pseudonymous essay (there are two versions, one in Latin and one in French), referring to him himself in third person. He claimed that Mr. Leibniz gave the arithmetical quadrature of the circle (p. 28). This unpublished paper has several similarities with \textit{De vera proportione circuli} (p. 29). There are other examples in which Leibniz used pseudonyms. On one occasion, he wrote a letter to the \textit{Journal des Sçavants} concerning his own results on infinite series connected with the arithmetical quadratures, pretending the letter's author to be Cassini (pp. 29--30). As the author writes: ``Leibniz may have thought that there was benefit to be had in having his work sponsored [\dots] by luminaries in the Republic of Letters, especially by members of the Academy, the institution he still wished to join'' (p. 30). We ignore whether Cassini was aware of Leibniz's letter and if, in the affirmative case, he had accepted this stratagem. Afterwards, the author analyses the hypothesis that the pseudonymous essay was written for the Royal Academy of Sciences and discusses its plausibility. He concludes that, although Leibniz never sent the essay to the Academy, a priori this hypothesis is not absurd. In this regard, no certain conclusion can be drawn (pp. 31--32).
The fourth attempt by Leibniz is defined the ``extravagant method'' (p. 33): he intended to divulge his result by means of a medal. Leibniz had attempted to immortalise another of his discoveries in a medal: that of binary arithmetic. There are several designs on this (p. 33). The author refers to sketches by Leibniz where drawings appear that show medals depicting results related to binary arithmetic and the squaring of the circle. For, at least until 1682, Leibniz thought of a connection between these two parts of his mathematics, an idea which he abandoned later (p. 34).
This paper is a valuable source to understand how mathematics was spread in the 17th century and, particularly, how Leibniz attempted to divulge his discovery concerning the squaring of the circle. As a matter of fact, the picture described in the article goes far beyond the specific problem, because the author enters many details on the relations between Leibniz and the French scientific community. For, if you read this work, you will get a good idea of what the practices were at the time for disseminating scientific knowledge and what the relations were between the scholars who gravitated around the Royal Academy of Sciences. Therefore, this paper is surely recommended to achieve a good idea as to the history concerning the spread of science and mathematics in the 17th century.
Reviewer: Paolo Bussotti (Udine)Transfer principles, Klein's Erlangen program, and methodological structuralismhttps://zbmath.org/1517.010082023-09-22T14:21:46.120933Z"Schiemer, Georg"https://zbmath.org/authors/?q=ai:schiemer.georg\textit{Summary}. The author wants to discuss three questions: (i) whether Klein contributed to the development of the structural turn in mathematics; (ii) whether his group-theoretic approach to geometry is structuralist in character; and (iii) whether Klein's account anticipated debates in contemporary (philosophical) structuralism (p.~107). He argues, among others, for the following claims; in regards to (i): ``Klein developed his own account of geometry in direct continuation with the [\dots] `structuralist' methods of Plücker and Hesse'' (p.~138); in regards to (ii): structuralism is ``implicit'' (p.~135) in Klein's ``use of structure-preserving mappings in reciprocity and transfer principles'' as well as in his ``focus on invariant form in the analytic presentation of geometrical figures and their properties'' (p.~138); in regards to (iii) there seem to be two different claims: first, that Klein's account is a form of ``methodological structuralism'', that is, a kind of ``mathematical methodology'' that ``usually impl[ies] some form of abstraction'' (p.~128), and, second, that ``Klein's approach presents a particular version of [\dots], \textit{in re} structuralism [\dots\ i.e.,] the view that mathematical theories describe abstract structures as their subject matter but that these structures do not exist independently of concrete representations instantiating them'' (p.~132).
\textit{Contents}. After a brief introduction (pp.~106--108), Section 2 ``Duality and transfer principles'' (pp.~108--118) reviews the work by Plücker and Hesse in particular on duality in projective geometry and its extension to more general transfer principles; Section 3 (pp.~118--124) offers a summary of Klein's Erlangen program under the two headers of ``A group-theoretic approach'' and ``Transfer by mapping''; the final Section 4 ``Structuralist themes'' (pp.~128--137), divided into the subsections ``Invariance and structural indiscernibility'' and ``Transfer principles and structural equivalence'', is followed by a brief conclusion (pp.~137f.). The goal of Sections 2 and 3 is not to present new historical research but, building on familiar primary sources and secondary studies, to develop a new narrative that makes explicit some structuralist undertows implicit in 19th-century (projective) geometry. Similar to what \textit{J.-P. Marquis} did for Klein's program and category theory in the first chapter of [From a geometrical point of view. A study of the history and philosophy of category theory. Berlin: Springer (2009; Zbl 1165.18002)] the author tries to do for Klein's program and modern theories of structuralism.
The article is well-organized and clearly written but may on occasion tax the patience of those familiar with the subject matter. In Section 2.2, Plücker is often quoted as ``19xx'' instead of ``18xx,'' while under the references the entry for (Hesse 1866b) has the wrong name attached to it. But the only potentially troublesome oversight happens on p.~117: the given page number, 32, for the two quotations from Hesse 1866b is correct for the off-print, published separately as a small booklet not mentioned in the bibliography, while the correct page number for the original journal publication, which is listed in the bibliography, is 400.
The reviewer sees two reasons that make it difficult to say whether the author achieves his goals or not. First, precise definitions of the various kinds of structuralism are hard to come by; in their absence it is difficult, if not impossible -- for anyone, to be clear -- to say whether Klein's program is a good fit or not. Second, much of the historical evidence depends on reading the sources in a certain way. For example, Hesse, referring to pairs of dual theorems we obtain from the same analytic formula by interchanging their point and line coordinates, writes: ``the law of reciprocity stays the same from one pair of theorems to the next, while the proving formulas change.'' Does this justify its paraphrase as ``the existence of a structure-preserving mapping''; especially when Hesse himself frames it in terms of economy of proof: get two (theorems) for the price of one (proof)? Similar reservations may apply to some of the structuralist readings of Klein, namely, as anachronistic. But in light of the first reason mentioned (i.e., the lack of precise criteria), everyone's mileage may vary here.
For the entire collection see [Zbl 1440.03007].
Reviewer: Bernd Buldt (Fort Wayne)On Gauss's area formulahttps://zbmath.org/1517.010092023-09-22T14:21:46.120933Z"Wittmann, Axel"https://zbmath.org/authors/?q=ai:wittmann.axel-dThe paper is devoted to a formula developed by Gauss around 1810. This formula allows one to calculate the area of a plane region bounded by a closed polygon from the knowledge of the Cartesian coordinates of the corner points only. The Gaussian formula is described in the paper and it is shown how it can be applied -- the formula is used to calculate the area of the site of the Göttingen Observatory.
Reviewer: Roman Murawski (Poznań)Gerhard Hermann Waldemar Kowalewski and his two Prague periodshttps://zbmath.org/1517.010102023-09-22T14:21:46.120933Z"Bečvářová, Martina"https://zbmath.org/authors/?q=ai:becvarova.martinaThis is a collecting rather than analytical paper on the German mathematician Gerhard Kowalewski (1876--1950), who is most known for his many textbooks on analysis, determinants, integral equations etc. He had a rather turbulent life, both personally and professionally. His two stays in Prague, one of them under Nazi occupation, are the focus of this article. The author quotes long passages from Kowalewski's books, mostly in the German original, without comments and partly with typos which even leave German readers clueless (e.g. p. 147). The English of the article could have been easily improved by using translating tools such as DeepL, which covers Czech as well. Kowalewski's interesting although in parts unreliable German autobiography from 1950 [Bestand und Wandel. Meine Lebenserinnerungen, zugleich ein Beitrag zur neueren Geschichte der Mathematik. München: Oldenbourg (1950; Zbl 0037.00203)] is not critically evaluated. About his historical book, also in German, [Große Mathematiker. Eine Wanderung durch die Geschichte der Mathematik vom Altertum bis zur Neuzeit. München, Berlin: J. F. Lehmann (1938; Zbl 0017.38501; JFM 64.0001.03)] is erroneously said: ``No national, political or ideological ideas had a place in his book'' (p. 128). This is in contrast to the following passage in the book (p. 17): ``Einstein was not afraid to radically change the concept of time and created his much-disputed theory of relativity.'' (``Einstein scheute sich nicht, den Zeitbegriff radikal zu ändern und schuf seine viel umstrittene Relativitätstheorie.''). Kowalewski compares this to the recognition of irrationality in Greek mathematics, what amounts to support for Einstein in an ideologically oppressive environment of anti-Semitic Nazi Germany.
The value of the paper consists in alerting the reader to printed and unpublished material which is awaiting further and deeper analysis.
Reviewer: Reinhard Siegmund-Schultze (Kristiansand)Gino Fano in Switzerland (1939--1945)https://zbmath.org/1517.010112023-09-22T14:21:46.120933Z"Luciano, Erika"https://zbmath.org/authors/?q=ai:luciano.erikaAuthor's summary: ``Dispossessed of his chair, banned from scientific and academic arenas and unable to tolerate ``the reduction to a caste of pariahs'', Gino Fano left Turin for Switzerland in 1938, settling in Lausanne. He would remain there until 1945, engaging in three areas: solidarity, teaching in the courses organized for Italian university students interned in the Lausanne and Huttwil camps; and dissemination, through a series of conferences on Italian algebraic geometry that he held at the Cercle Mathématique. In this paper we will focus on the Swiss period of Fano's personal and professional trajectory, seven years generally dismissed as the unpleasant and unjust epilogue of a highly successful scientific life, and that by contrast are not, in the measure that they return a broader narrative: that of Jewish scholars, trained in the Belle Époque of scientific internationalism, who saw the principles of the rule of law denied by race theories, and who witnessed the perversion of collective consciences under totalitarian regimes.''
Reviewer's comments: This study shows the difficult circumstances in the years 1938--1945, in which Italian mathematicians like Gino Fano had to deal with their families and still had to do some mathematics.
In the references, the reader will find very good sources, also by and due to Fano himself; of course also due to others. The paper under review collects these in a good way. Highly recommended! In addition, see also Wikipedia and the MacTutor Archive regarding Gino Fano.
The contents of the paper deserve a translation into English.
Reviewer: Robert W. van der Waall (Huizen)Obituary: Professor Leonid I. Manevitchhttps://zbmath.org/1517.010122023-09-22T14:21:46.120933Z"Andrianov, Igor V."https://zbmath.org/authors/?q=ai:andrianov.igor-v"Gendelman, Oleg V."https://zbmath.org/authors/?q=ai:gendelman.oleg-v"Kovaleva, Margarita A."https://zbmath.org/authors/?q=ai:kovaleva.margarita-alekseevna"Mikhlin, Yuri V."https://zbmath.org/authors/?q=ai:mikhlin.yuri-vladimirovich"Pilipchuk, Valery N."https://zbmath.org/authors/?q=ai:pilipchuk.valery-n(no abstract)Václav Láska in Polandhttps://zbmath.org/1517.010132023-09-22T14:21:46.120933Z"Bečvářová, Martina"https://zbmath.org/authors/?q=ai:becvarova.martinaFor the entire collection see [Zbl 1258.01002].Roman Żuliński and his \textit{Elements of the differential and integral calculus}https://zbmath.org/1517.010142023-09-22T14:21:46.120933Z"Dawidowicz, Antoni Leon"https://zbmath.org/authors/?q=ai:dawidowicz.antoni-leonFor the entire collection see [Zbl 1258.01002].Set theory in Józef Puzyna's \textit{Theory of analytic functions}https://zbmath.org/1517.010152023-09-22T14:21:46.120933Z"Domoradzki, Stanisław"https://zbmath.org/authors/?q=ai:domoradzki.stanislawFor the entire collection see [Zbl 1258.01002].Bolesław Maleszewski (1844--1912): a biographical sketchhttps://zbmath.org/1517.010162023-09-22T14:21:46.120933Z"Ermolaeva, Natalia"https://zbmath.org/authors/?q=ai:ermolaeva.nataliaFor the entire collection see [Zbl 1258.01002].Mysteries of sequences in ``Observations cyclometricae'' by Adam Adamandy Kochańskihttps://zbmath.org/1517.010172023-09-22T14:21:46.120933Z"Fukś, Henryk"https://zbmath.org/authors/?q=ai:fuks.henrykFor the entire collection see [Zbl 1258.01002].Obituary: Prof D. T. Mookhttps://zbmath.org/1517.010182023-09-22T14:21:46.120933Z"Hajj, Muhammad R."https://zbmath.org/authors/?q=ai:hajj.muhammad-r"Preidikman, Sergio"https://zbmath.org/authors/?q=ai:preidikman.sergio"Balachandran, Balakumar"https://zbmath.org/authors/?q=ai:balachandran.balakumar"Lacarbonara, Walter"https://zbmath.org/authors/?q=ai:lacarbonara.walter(no abstract)Obituary: Alan Baker, FRS, 1939--2018https://zbmath.org/1517.010192023-09-22T14:21:46.120933Z"Masser, David"https://zbmath.org/authors/?q=ai:masser.david-williamSummary: Alan Baker, Fields Medallist, died on 4 February 2018 in Cambridge, England, after a severe stroke a few days earlier. In 1970 he was awarded the Fields Medal at the International Congress in Nice on the basis of his outstanding work on linear forms in logarithms and its consequences. Since then he received many honours, including the prestigious Adams Prize of Cambridge University, the election to the Royal Society (1973) and the Academia Europeae; and he was made an honorary fellow of University College London, a foreign fellow of the Indian Academy of Science, a foreign fellow of the National Academy of Sciences, India, an honorary member of the Hungarian Academy of Sciences, and a fellow of the American Mathematical Society.Vasily Ivanovich Bernik (to the 75th anniversary)https://zbmath.org/1517.010202023-09-22T14:21:46.120933Z"Nesterenko, Yu. V."https://zbmath.org/authors/?q=ai:nesterenko.yuri-v"Bykovskiĭ, V. A."https://zbmath.org/authors/?q=ai:bykovskii.v-a"Bukhshtaber, V. M."https://zbmath.org/authors/?q=ai:bukhshtaber.viktor-matveevich"Chirskiĭ, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevich"Chubarikov, V. N."https://zbmath.org/authors/?q=ai:chubarikov.vladimir-nikolaevich"Laurinčikas, A. P."https://zbmath.org/authors/?q=ai:laurincikas.antanas"Dobrovol'skiĭ, N. M."https://zbmath.org/authors/?q=ai:dobrovolskii.n-m"Dobrovol'skiĭ, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, I. Yu."https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Budarina, N. V."https://zbmath.org/authors/?q=ai:budarina.natalia-v"Beresnevich, V. V."https://zbmath.org/authors/?q=ai:beresnevich.victor-v"Vasil'ev, D. V."https://zbmath.org/authors/?q=ai:vasilev.denis-vladimirovich"Kalosha, N. I."https://zbmath.org/authors/?q=ai:kalosha.nikolai-ivanovichSummary: This paper commemorates the seventy-fifth anniversary of Doctor of Physical and Mathematical Sciences, Professor Vasily Ivanovich Bernik. His curriculum vitae is presented, together with a brief analysis of his work in scientific research, education and management. 13 major scientific papers of V. I. Bernik are referenced.E. E. Slutsky/W. Bortkiewicz: correspondence 1923--1928https://zbmath.org/1517.010212023-09-22T14:21:46.120933Z"Rauscher, Guido"https://zbmath.org/authors/?q=ai:rauscher.guido"Sheynin, Oscar"https://zbmath.org/authors/?q=ai:sheynin.oscar-b"Wittich, Claus"https://zbmath.org/authors/?q=ai:wittich.clausFor the entire collection see [Zbl 1258.01002].In memory of Vadim Fedorovich Kirichenkohttps://zbmath.org/1517.010222023-09-22T14:21:46.120933Z"Rustanov, A. R."https://zbmath.org/authors/?q=ai:rustanov.aligadzhi-rabadanovich"Shelekhov, A. M."https://zbmath.org/authors/?q=ai:shelekhov.alexander-m"Arsen'eva, O. E."https://zbmath.org/authors/?q=ai:arseneva.olga-evgenevna"Kirichenko, V. F."https://zbmath.org/authors/?q=ai:kirichenko.vadim-fedorovich"Burlakov, M. P."https://zbmath.org/authors/?q=ai:burlakov.mikhail-p"Banaru, M. B."https://zbmath.org/authors/?q=ai:banaru.mihail-b"Guseva, N. I."https://zbmath.org/authors/?q=ai:guseva.nadezhda-ivanovna"Kharitonova, S. V."https://zbmath.org/authors/?q=ai:kharitonova.svetlana-vladimirovna"Ivanov, A. O."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-olegovich"Chubarikov, V. N."https://zbmath.org/authors/?q=ai:chubarikov.vladimir-nikolaevich"Dobrovol'skiĭ, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, I. Yu."https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Dobrovol'skiĭ, N. M."https://zbmath.org/authors/?q=ai:dobrovolskii.n-m(no abstract)Obituary: John Corcoran (1937--2021)https://zbmath.org/1517.010232023-09-22T14:21:46.120933Z"Sagüillo, José M."https://zbmath.org/authors/?q=ai:saguillo.jose-miguel"Scanlan, Michael"https://zbmath.org/authors/?q=ai:scanlan.michael-j"Shapiro, Stewart"https://zbmath.org/authors/?q=ai:shapiro.stewartSummary: We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications.Ubiratan D'Ambrosio (1932--2021): in memoriamhttps://zbmath.org/1517.010242023-09-22T14:21:46.120933Z"Saraiva, Luís"https://zbmath.org/authors/?q=ai:saraiva.luis-manuel-ribeiro(no abstract)Number theory in the works of Franciszek Mertenshttps://zbmath.org/1517.010252023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].Wacław Sierpiński's contribution to number theoryhttps://zbmath.org/1517.010262023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].Number theory and algebra in the works of Salomon Lubelskihttps://zbmath.org/1517.010272023-09-22T14:21:46.120933Z"Schinzel, Andrzej"https://zbmath.org/authors/?q=ai:schinzel.andrzejFor the entire collection see [Zbl 1258.01002].Yuri Valentinovich Nesterenko (to the 75th anniversary)https://zbmath.org/1517.010282023-09-22T14:21:46.120933Z"Shafarevich, A. I."https://zbmath.org/authors/?q=ai:shafarevich.andrei-i"Fomenko, A. T."https://zbmath.org/authors/?q=ai:fomenko.anatolii-t"Chubarikov, V. N."https://zbmath.org/authors/?q=ai:chubarikov.vladimir-nikolaevich"Ivanov, A. O."https://zbmath.org/authors/?q=ai:ivanov.aleksandr-olegovich"Chirskiĭ, V. G."https://zbmath.org/authors/?q=ai:chirskii.vladimir-grigorevich"Bernik, V. I."https://zbmath.org/authors/?q=ai:bernik.vasili-i"Bykovskiĭ, V. A."https://zbmath.org/authors/?q=ai:bykovskii.v-a"Galochkin, A. I."https://zbmath.org/authors/?q=ai:galochkin.a-i"Demidov, S. S."https://zbmath.org/authors/?q=ai:demidov.sergei-s"Gashkov, S. B."https://zbmath.org/authors/?q=ai:gashkov.sergey-b"Nizhnikov, A. I."https://zbmath.org/authors/?q=ai:nizhnikov.aleksandr-ivanovich"Fomin, A. A."https://zbmath.org/authors/?q=ai:fomin.aleksandr-aleksandrovich|fomin.aleksandr-arkadevich"Deza, E. I."https://zbmath.org/authors/?q=ai:deza.elena-ivanovna"Kanel'-Belov, A. Ya."https://zbmath.org/authors/?q=ai:kanel-belov.alexei"Dobrovol'skiĭ, N. M."https://zbmath.org/authors/?q=ai:dobrovolskii.n-m"Dobrovol'skiĭ, N. N."https://zbmath.org/authors/?q=ai:dobrovolskii.n-n"Rebrova, I. Yu."https://zbmath.org/authors/?q=ai:rebrova.irina-yurevna"Salikhov, V. Kh."https://zbmath.org/authors/?q=ai:salikhov.v-kh|salikhov.vladislaus-khasanovich(no abstract)Professor Krishan L. Duggal: a biographical notehttps://zbmath.org/1517.010292023-09-22T14:21:46.120933Z"Sharma, Ramesh"https://zbmath.org/authors/?q=ai:sharma.ramesh"Şahin, Bayram"https://zbmath.org/authors/?q=ai:sahin.bayramSummary: In this note, we present a biographical sketch of the life and academic contributions of late Professor Krishan L. Duggal. His contributions span from Riemannian and Lorentzian geometries of manifolds with various structural groups of the tangent bundle, Lightlike curves and submanifolds, Cauchy-Riemann geometry, Symmetries of semi-Riemannian manifolds, to Killing horizons. In particular, his approach to the study of lightlike submanifolds is remarkable and drawn considerable interest of many geometers.Ladislaus von Bortkiewicz: a scientific biographyhttps://zbmath.org/1517.010302023-09-22T14:21:46.120933Z"Sheynin, Oscar"https://zbmath.org/authors/?q=ai:sheynin.oscar-bFor the entire collection see [Zbl 1258.01002].Georg Cantor: biographical material about his family and childhood in archives in St. Petersburghttps://zbmath.org/1517.010312023-09-22T14:21:46.120933Z"Sinkevich, Galina Ivanovna"https://zbmath.org/authors/?q=ai:sinkevich.galina-ivanovnaFor the entire collection see [Zbl 1258.01002].Doctorates in mathematics and logic from the Jan Kazimierz University in Lwów during the years 1920--1938https://zbmath.org/1517.010322023-09-22T14:21:46.120933Z"Prytuła, Jarosław"https://zbmath.org/authors/?q=ai:prytula.yaroslav-gFor the entire collection see [Zbl 1258.01002].The mathematical society in Lwówhttps://zbmath.org/1517.010332023-09-22T14:21:46.120933Z"Domoradzki, Stanisław"https://zbmath.org/authors/?q=ai:domoradzki.stanislawFor the entire collection see [Zbl 1258.01002].The AWM (Mathematics) Education Committeehttps://zbmath.org/1517.010342023-09-22T14:21:46.120933Z"Hsu, Pao-sheng"https://zbmath.org/authors/?q=ai:hsu.pao-sheng"Dewar, Jacqueline M."https://zbmath.org/authors/?q=ai:dewar.jacqueline-mFor the entire collection see [Zbl 1485.01005].Works in analytic functions in the Memoirs of the Society of Exact Sciences in Parishttps://zbmath.org/1517.010352023-09-22T14:21:46.120933Z"Jakóbczak, Piotr"https://zbmath.org/authors/?q=ai:jakobczak.piotrFor the entire collection see [Zbl 1258.01002].The education of a high school teacher of mathematics in Galicia, 1850--1918https://zbmath.org/1517.010362023-09-22T14:21:46.120933Z"Rakoczy-Pindor, Krystyna"https://zbmath.org/authors/?q=ai:rakoczy-pindor.krystynaFor the entire collection see [Zbl 1258.01002].The teaching of algebra and probability theory in the high schools of Galicia at the turn of the 19th and 20th centurieshttps://zbmath.org/1517.010372023-09-22T14:21:46.120933Z"Rakoczy-Pindor, Krystyna"https://zbmath.org/authors/?q=ai:rakoczy-pindor.krystynaFor the entire collection see [Zbl 1258.01002].The Society of Exact Sciences in Paris. I. Documentshttps://zbmath.org/1517.010382023-09-22T14:21:46.120933Z"Więsław, Witold"https://zbmath.org/authors/?q=ai:wieslaw.witoldFor the entire collection see [Zbl 1258.01002].Adam Ries \textit{Coß} 1. In 2 volumes. Volume 1. Text volume. Volume 2. Commentary volume. The commentary on \textit{Coß} 1. Transcribed and edited by Bernd Rüdiger and Rainer Gebhardthttps://zbmath.org/1517.010392023-09-22T14:21:46.120933Z"Folkerts, Menso"https://zbmath.org/authors/?q=ai:folkerts.menso"Hellmann, Martin"https://zbmath.org/authors/?q=ai:hellmann.martinIt was known from Adam Ries's (1492--1559) ``Third Rechenbuch'', published in 1550, that Ries had written a \textit{Coß}, an algebra in the German style of the century, and that he hoped to get it into print. He did not succeed before his death, and the manuscript was only located in 1855; a description with extracts of the text was published by Bruno Berlet in 1860. Berlet found out that it consisted of three main parts: a completed algebra, now known as \textit{Coß} 1, finished in 1524; a re-elaboration that was never brought to an end -- now \textit{Coß} 2; and Ries's paraphrase of Jordanus de Nemore's \textit{De numeris datis}, a re-interpetation of that work in algebraic key.
In 1992, Wolfgang Kaunzner and Hans Wußing published a facsimile of the manuscript, accompanied by a commentary and transcribed text excerpts. The script is careful and decorative, but the facsimile cannot be used by readers without palaeographic training and adequate familiarity with \textit{Frühneuhochdeutsch}.
Now, as part of a larger project, Bernd Rüdiger, Menso Folkerts and Rainer Gebhardt have published an edition of \textit{Coß} 1. The other parts of the project are a sorely needed restoration of the manuscript; and a digitization made publicly available at \url{https://sachsen.digital/sammlungen/adam-ries-museum-annaberg-buchholz} (accessed 9 June 2023).
Together with the digitized manuscript, this edition has everything that can be asked for. Volume 1 contains a careful transcription of the text prepared by Bernd Rüdiger and Rainer Gebhardt -- faithful to the division of the text in pages and lines, which makes comparison with the manuscript easy.
Volume 2 is dedicated to two commentaries. The first, due to Menso Folkerts with the assistance of Martin Hellmann, goes through Ries's preface and the dedication as well the initial arithmetic and the algebraic generalia (the text itself as well as the historical background). The well-known development of Latin Hindu-Arabic arithmetic from the 12th century onward is dealt with quite briefly by Folkerts and Hellmann. The early history of German algebra is less familiar and therefore gets more attention.
The second commentary, by Rainer Gebhardt, discusses the collection of problems (322 in total), offering for each single problem a close paraphrase in modern German but using Ries's algebraic notation, followed by a modern solution, mostly kept close to Ries's way.
The divisions of the commentaries correspond to those of Ries's text. After a pseudo-historical preface and an informative dedicatory letter it contains an arithmetic (manuscript pages 5--89); an introduction to algebra (ms. pp. 109--122 -- the pages from 90 to 108 are empty), and a collection of problems (ms. pp. 122--324).
The arithmetic contains algorisms for integers and fractions, generally quite traditional but provided with extensive well-explained examples and systematic proofs by reverse calculation and by casting out sevens and nines. Outside the habitual falls an algorithm for ``parts of parts'' -- \(\frac{2}{3}\) of \(\frac{1}{4}\) and similarly. Initially, it teaches the reduction of these composites expressions to simple fractions; calculations in the ensuing algorism build on these. The proofs reveal the real purpose of this section: they refer to the subunits of the monetary system and fractions of these. In the very end comes an algorism ``for signs, degrees, minutes, seconds, thirds, etc.'', restricted to addition and subtraction but explaining that the principle applies not only to the calculation of celestial distances but also to the calculation in weight metrology.
The algebraic generalities start (ms. p. 109) by presenting the abbreviations used for the powers of the unknown: \(\mathbf{\phi}\), ``\textit{dragma or numerus}'', the ``unit'' for pure numbers; a contracted \(re[s]\) explained to stand for ``\textit{radix or coß}''; a stylized \(z\), ``\textit{zensus or quadratus}'', the second power of the unknown; etc., until the ``\textit{cubus de cubo}'', the ninth power. Next, it points to the importance of observing their mutual distance when two or three powers occur in the same equation, which leads to a listing of eight equation types -- in modern translation (Greek letters stand for implicit coefficients),
(1) \(\alpha x^n = \beta x^{n+1}\),
(2) \(\alpha x^n = \beta x^{n+2}\),
(3) \(\alpha x^n = \beta x^{n+3}\),
(4) \(\alpha x^n = \beta x^{n+4}\),
(5) \(\alpha x^{n} = \beta x^{n+1}+\gamma x^{n+2}\),
(6) \(\alpha x^n+\beta x^{n+2} = \gamma x^{n+1}\),
(7) \(\alpha x^n+\beta x^{n+1} = \gamma x^{n+2}\),
(8) \(\alpha x^{2n}+\beta x^{n+2} = \gamma\).
For all cases, rules for solving them are given -- for all except (6), correct. For (6), the case with a double solution, the rule is botched up -- for the simple case \(\beta = 1, n = 0\), the rule given is
\[
x =\sqrt{\biggl({\frac{\gamma}{2}\biggr)^2} -\alpha \pm \frac{\gamma}{2}} \text{ instead of } x = \frac{\gamma}{2}\pm \sqrt{\biggl({\frac{\gamma}{2}\biggr)^2}-\alpha}.
\]
Another list of 24 equation types follows on ms. p. 115, in Ries's opinion developed from the eight that were just presented (the actual historical order is the opposite). It comprises the 22 equation types of up to the fourth degree that can be reduced to the second degree or solved by means of a root extraction; these had already been known as a group in Italian abbacus algebra. Beyond these, the list includes equations that can be translated \(\alpha x^2 = \sqrt{\beta x}\) and \(\alpha x^2 = \sqrt{\beta x^2}\). Then follow the 322 problems. All but three are immediately of the first degree; \#104, \#135 and \#322 seem to be of the second degree, but once the equation of formulated the zensus disappears. So, in spite of the initial presentation of many equation types of higher degree, Ries restricts himself to the domain where algebra can really serve to solve normal business problems -- which does not mean, however, that all of Ries's \textit{are} business problems or could be useful in normal business.
Many are pure-number problems. Others are genuine business problems, many more deal with situations involving trade or money but in a way that would never present itself in commercial real life, illustrating that commercial questions that can produce complicated mathematics tend to become recreational riddles; other recreational types are also present, often with unusual variations.
First-degree equations with a single unknown are never intricate. The complexity of Ries's problems, and what he trains extensively, is to be located in the formulation of problems and in their transformation into an equation, and that often asks for a clear mind.
As also illustrated by the immense success of his \textit{Rechenbücher}, Ries was a born pedagogue. Other algebraic writings since the beginning of abbacus algebra (indeed since al-Khwārizmī) had excelled in the treatment of higher-degree problems, their main purpose being to display virtuosity. Ries, as he explains in the dedication, aims at selecting from what he has learned from predecessors that which can be useful for the common man, and to present that. As a great pedagogue he also combines thorough training with so much variation that boredom can be avoided. For this purpose, a collection of 322 first-degree problems is justified.
Reviewer: Jens Høyrup (Roskilde)The story of the education column in the \textit{AWM Newsletter}https://zbmath.org/1517.010402023-09-22T14:21:46.120933Z"Dewar, Jacqueline M."https://zbmath.org/authors/?q=ai:dewar.jacqueline-mFor the entire collection see [Zbl 1485.01005].Twenty years of the AWM mentor networkhttps://zbmath.org/1517.010412023-09-22T14:21:46.120933Z"Ghazaryan, Anna"https://zbmath.org/authors/?q=ai:ghazaryan.anna|ghazaryan.anna-r"Kuske, Rachel"https://zbmath.org/authors/?q=ai:kuske.rachel-a"Lawrence, Emille"https://zbmath.org/authors/?q=ai:lawrence.emille-davieFor the entire collection see [Zbl 1485.01005].Mentoring and empowering with (sometimes) distressing mathematicshttps://zbmath.org/1517.010422023-09-22T14:21:46.120933Z"Grundman, Helen G."https://zbmath.org/authors/?q=ai:grundman.helen-gFor the entire collection see [Zbl 1485.01005].Mentoring and providing community for young womenhttps://zbmath.org/1517.010432023-09-22T14:21:46.120933Z"Haunsperger, Deanna"https://zbmath.org/authors/?q=ai:haunsperger.deanna-bFor the entire collection see [Zbl 1485.01005].A few memories and insights from a 50-year careerhttps://zbmath.org/1517.010442023-09-22T14:21:46.120933Z"Jochnowitz, Naomi"https://zbmath.org/authors/?q=ai:jochnowitz.naomiFor the entire collection see [Zbl 1485.01005].The SummerMath and SEARCH programs 1982--2009, Mount Holyoke Collegehttps://zbmath.org/1517.010452023-09-22T14:21:46.120933Z"Morrow, Charlene"https://zbmath.org/authors/?q=ai:morrow.charlene"Morrow, James"https://zbmath.org/authors/?q=ai:morrow.james-r-jun|morrow.james-aFor the entire collection see [Zbl 1485.01005].Celebrating AWM's fiftieth anniversary: mentoring PhD studentshttps://zbmath.org/1517.010462023-09-22T14:21:46.120933Z"Shu, Chi-Wang"https://zbmath.org/authors/?q=ai:shu.chi-wangFor the entire collection see [Zbl 1485.01005].Advisor actions: reach out, listen, provide timely information -- advisee reactions: overwhelmed, informed, more confident and connectedhttps://zbmath.org/1517.010472023-09-22T14:21:46.120933Z"Vélez, William Yslas"https://zbmath.org/authors/?q=ai:velez.william-yslas"Christensen, Alex Julia Hinojosa"https://zbmath.org/authors/?q=ai:christensen.alex-julia-hinojosaFor the entire collection see [Zbl 1485.01005].Women count! A quarter century of Sonia Kovalevsky Dayshttps://zbmath.org/1517.010482023-09-22T14:21:46.120933Z"Yanik, Betsy"https://zbmath.org/authors/?q=ai:yanik.betsyFor the entire collection see [Zbl 1485.01005].Tractarian logicism: operations, numbers, inductionhttps://zbmath.org/1517.030012023-09-22T14:21:46.120933Z"Landini, Gregory"https://zbmath.org/authors/?q=ai:landini.gregorySummary: In his Tractatus, Wittgenstein maintained that arithmetic consists of equations arrived at by the practice of calculating outcomes of operations \(\Omega^n(\bar{\xi })\) defined with the help of numeral exponents. Since \(\operatorname{Num} (x)\) and quantification over numbers seem ill-formed, Ramsey wrote that the approach is faced with ``insuperable difficulties.'' This paper takes Wittgenstein to have assumed that his audience would have an understanding of the implicit general rules governing his operations. By employing the Tractarian logicist interpretation that the \(N\)-operator \(N(\bar{\xi })\) and recursively defined arithmetic operators \(\Omega^n(\bar{\xi })\) are not different in kind, we can address Ramsey's problem. Moreover, we can take important steps toward better understanding how Wittgenstein might have imagined emulating proof by mathematical induction.The development Of Gödel's ontological proofhttps://zbmath.org/1517.030172023-09-22T14:21:46.120933Z"Kanckos, Annika"https://zbmath.org/authors/?q=ai:kanckos.annika"Lethen, Tim"https://zbmath.org/authors/?q=ai:lethen.timSummary: Gödel's ontological proof is by now well known based on the 1970 version, written in Gödel's own hand, and Scott's version of the proof. In this article new manuscript sources found in Gödel's Nachlass are presented. Three versions of Gödel's ontological proof have been transcribed, and completed from context as true to Gödel's notes as possible. The discussion in this article is based on these new sources and reveals Gödel's early intentions of a liberal comprehension principle for the higher order modal logic, an explicit use of second-order Barcan schemas, as well as seemingly defining a rigidity condition for the system. None of these aspects occurs explicitly in the later 1970 version, and therefore they have long been in focus of the debate on Gödel's ontological proof.Graph theory in America. The first hundred yearshttps://zbmath.org/1517.050012023-09-22T14:21:46.120933Z"Wilson, Robin"https://zbmath.org/authors/?q=ai:wilson.robin-j"Watkins, John J."https://zbmath.org/authors/?q=ai:watkins.john-j"Parks, David J."https://zbmath.org/authors/?q=ai:parks.david-jLet us begin with the following description:
``\textit{Graph Theory in America} focuses on the development of graph theory in North America from 1876 to 1976. At the beginning of this period, James Joseph Sylvester, perhaps the finest mathematician in the English-speaking world, took up his appointment as the first professor of mathematics at the Johns Hopkins University, where his inaugural lecture outlined connections between graph theory, algebra, and chemistry -- shortly after, he introduced the word graph in our modern sense. A hundred years later, in 1976, graph theory witnessed the solution of the long-standing four color problem by Kenneth Appel and Wolfgang Haken of the University of Illinois.
Tracing graph theory's trajectory across its first century, this book looks at influential figures in the field, both familiar and less known. Whereas many of the featured mathematicians spent their entire careers working on problems in graph theory, a few such as Hassler Whitney started there and then moved to work in other areas. Others, such as C. S. Peirce, Oswald Veblen, and George Birkhoff, made excursions into graph theory while continuing their focus elsewhere. Between the main chapters, the book provides short contextual interludes, describing how the American university system developed and how graph theory was progressing in Europe. Brief summaries of specific publications that influenced the subject's development are also included.
\textit{Graph Theory in America} tells how a remarkable area of mathematics landed on American soil, took root, and flourished.''
In addition to the publisher's description, one can note the following.
In the first section, entitled as ``Setting the scene: early American mathematics'', the focus is on the first institutions of higher education which were established in the American colonies. Some additional descriptions about Harvard University, Yale University, and Princeton University, as well as the Massachusetts Institute of Technology (MIT) and Johns Hopkins University are given. Also, some peculiarities of the development of mathematics education in the early years in the USA are considered, as well as the contributions of Benjamin Pierce and Eliakim Hastings Moore to this development are briefly described.
Chapter 1 ``The 1800s'' is devoted to the history of mathematical research under the influence of James Joseph Sylvester and Johns Hopkins University, as well as to the early interest in graph theory in America and to some scientists. The American Journal of Mathematics is mentioned as the oldest mathematics journal in continuous publication in North America. Special attention is also given to the four-color theorem.
The titles of the following chapters are: ``The 1900s and 1910s'', ``The 1920s'', ``The 1930s'', ``The 1940s and 1950s'', ``The 1960s and 1970s''. In these chapters, the attention mainly is given to peculiarities of the development of investigations, to some mathematicians (brief biographical data, descriptions of results, publications, and some proofs, etc.), as well as to certain conjectures, algorithms, and to explanations of several notions. Several problems, including a progress in solving the four-color problem, are discussed. The development of mathematics science is also briefly considered in the periods of the Great Depression, World War I, and World War II. The Zentralblatt für Mathematik and the Mathematical Reviews are reported on.
Reviewer: Symon Serbenyuk (Kyjiw)Comments on Risch's \textit{On the integration of elementary functions which are built up using algebraic operations}https://zbmath.org/1517.120012023-09-22T14:21:46.120933Z"Raab, Clemens G."https://zbmath.org/authors/?q=ai:raab.clemens-gThis survey article is part of the collection \textit{Integration in Finite Terms} of the book series \textit{Texts \& Monographs in Symbolic Computation}. The article provides much more than what the modest title suggests: it actually gives a brief but rather complete overview of the history and development of symbolic integration. While it does not go into any mathematical details, it sketches the different directions of research, variations of the problem, and recent algorithmic advancements. Moreover, it provides an extensive collection of references. This survey is therefore a very valuable resource for entering the field of symbolic integration and for getting an overview of its many different aspects.
For the entire collection see [Zbl 1490.26001].
Reviewer: Christoph Koutschan (Linz)Epistemological study of mathematical inequalitieshttps://zbmath.org/1517.260012023-09-22T14:21:46.120933Z"Espinoza, Edgardo Locia"https://zbmath.org/authors/?q=ai:espinoza.edgardo-locia"Carballo, Armando Morales"https://zbmath.org/authors/?q=ai:carballo.armando-morales"Santiesteban, José Luis"https://zbmath.org/authors/?q=ai:santiesteban.jose-luis"Sigarreta, José María"https://zbmath.org/authors/?q=ai:sigarreta-almira.jose-mariaSummary: Inequalities have proven to be one of the basic tools to solve multiple theoretical and practical problems of science and technology. In this article, guided by the Theory of Dialectical Knowledge, an epistemological study of the conditions of evolution and development of mathematical inequalities is carried out, taking into account their origin, systematization and formalization.Wacław Sierpiński's work on real functionshttps://zbmath.org/1517.260022023-09-22T14:21:46.120933Z"Wilczyński, Władysław"https://zbmath.org/authors/?q=ai:wilczynski.wladyslawFor the entire collection see [Zbl 1258.01002].Notices of the international congress of Chinese mathematicians, Vol. 10, No. 2 (December 2022)https://zbmath.org/1517.530032023-09-22T14:21:46.120933ZPublisher's description: This is the twentieth issue (Vol. 10, No. 2, December 2022) of the Notices of the International Consortium of Chinese Mathematicians, the organization's official periodical.
Formerly entitled Notices of the International Congress of Chinese Mathematicians, this journal brings research, news, and the presentation of various perspectives relevant to Chinese mathematics development and education.
Readers of the Notices will find research papers on various topics by prominent experts from around the world, interesting and timely articles on current applications and trends, biographical and historical essays, profiles of important institutions of research and learning, and more.
The articles of this volume were reviewed individually within the journal [ICCM Not. 10, No. 2 (2022)].Jordan, Schoenflies and position theory (an outline of the problem up to 1960)https://zbmath.org/1517.570012023-09-22T14:21:46.120933Z"Duda, Roman"https://zbmath.org/authors/?q=ai:duda.romanFor the entire collection see [Zbl 1258.01002].Reception of probability theory and statistical methods in Polandhttps://zbmath.org/1517.600022023-09-22T14:21:46.120933Z"Duda, Roman"https://zbmath.org/authors/?q=ai:duda.romanSummary: Chance has always accompanied man. He was afraid of the elusive forces behind him, but he was also fascinated. Relatively late from this fascination arose the understanding that in some circumstances chance can be measured and, consequently, with the help of mathematics to reveal the regularities behind it. It began with gambling games already known in ancient times, such as the dice game popular among Roman legionnaires, but the first serious attempts at mathematical approach begin only in the Renaissance. The history of this development is described and the purpose of this article is not to recall it, but only the account of its sounds in Poland and the circumstances of settling this account in Polish mathematics.Lectures in probability theory in Polish territories at the turn of the 19th and 20th centurieshttps://zbmath.org/1517.600032023-09-22T14:21:46.120933Z"Dudek, Dorota"https://zbmath.org/authors/?q=ai:dudek.dorota"Zięba, Wiesław"https://zbmath.org/authors/?q=ai:zieba.wieslawFor the entire collection see [Zbl 1258.01002].A journey through the history of numerical linear algebrahttps://zbmath.org/1517.650022023-09-22T14:21:46.120933Z"Brezinski, Claude"https://zbmath.org/authors/?q=ai:brezinski.claude"Meurant, Gérard"https://zbmath.org/authors/?q=ai:meurant.gerard-a"Redivo-Zaglia, Michela"https://zbmath.org/authors/?q=ai:redivo-zaglia.michelaThis voluminous tome of almost 800 pages is just what it says on its title: a journey through the history of linear algebra. Concerning the importance of the topic in mathematics and nearly all of its applications for decades it seems unbelievable that this `journey' appears in print as late as 2023! In the introduction, the authors state that they are mainly interested in matrix computations and warn their readers that the choice of topics may be biased by their personal interests. Another warning concerns the mathematical content of the book because the authors do not shy away to explain mathematical ideas in terms of mathematics for which the author of these lines is very grateful. Hence, the book is refreshingly different from books written for a wider public.
But now to a discussion of the content. Part I is simply titled `History' and is concerned with the birth of matrices starting with the works of Gauss on quadratic forms. Norms, ill-conditioning, Schur complements and matrix mechanics are treated in detail, historically as well as mathematically. A second chapter is concerned with elimination methods for linear systems and has to start in antiquity, of course. Developments in ancient China, Greece, India, and Persia are treated, the middle ages are touched upon (think of Fibonacci) and then the journey goes on from the 16th to the 21st century. Starting with the 17th century, the history of determinants is discussed in the 3rd chapter while the 4th chapter concerns matrix factorizations and canonical forms. Chapters 5 and 6 are devoted to iterative methods for the solution of linear system and eigenvalues and eigenvectors, respectively. Each of Chapters 1 to 6 is concluded with a very useful section called `Lifetimes'. In these chapters, one can view the development of techniques and theories as timelines plotted in form of coloured bar charts.
Since the 1950s, the development of electronic computers helped enormously in the development of algorithms in linear algebra and to spread linear algebraic methods into the wide range of applications. Chapter 7 of Part I is devoted to computing machines starting with the invention of logarithms and ending in the rise of microprocessors and parallel computers. Since a computer, however advanced its architecture may be, is useless without reliable software and so the 8th chapter discusses not only the development of programming languages but also the famous linear algebra packages BLAS, EISPACK, LAPACK and others. Even MATLAB is mentioned on two pages. Part I concludes with `Miscellaneous topics' containing a discussion of fast solvers, matrices of special forms, and ill-posed problems.
The flavour of the book changes drastically in Part II which bears the title `Biographies'. It consists of a single Chapter 10 of 200 pages describing the lives and works of 78 persons ranging alphabetically from A as Aitken to Z as Zuse.
The book with its two parts is certainly a gem, requires some mathematical maturity, and is well readable. The bibliography consists of (incredible) 3344 (!) titles.
Reviewer: Thomas Sonar (Braunschweig)