This biography of Hypatia, the female philosopher and mathematician in Christian Egypt, provides background on her work and her life as an elite woman at this time. There are many myths about Hypatia, including her research, inventions and the impact of her murder, all based on a handful of contemporary resources. Through presenting the different theories and myths alongside the available evidence, this book will enable the reader to make their own interpretations about her life. Whilst the evidence does leave (...) many questions unanswered, this book provides the evidence as it stands, separating the myth from reality. There is very little published on Hypatia and she forms quite a niche market in the history of ancient Egypt. However, she is an interesting example of how multicultural Alexandria functioned at such an unstable political time, and provides anecdotal evidence of the atrocities that occurred. This book will appeal to scholars, lay people and political and religious researchers, and will show that the history of Egypt does not end at Cleopatra. (shrink)

This paper reconsiders a well-known Diophantine problem-“the coconut puzzle”-with two philosophically humorous motives. The first motive is methodological; the second, ontological. First, different methodological approaches to solving this problem, namely those of the stereotypical engineer, physicist, and mathematician, are informed by different conceptualizations of the mathematical puzzle itself. This is illustrated by comparing stereotypical solutions by an engineer, a physicist, and a mathematician. Second, ontological implications of Dirac’s unorthodox solution precipitate heuristic questions concerning correspondence between mathematically admissible answers and physically (...) possible states. An anecdotal history of this problem is also cited, in order to prepare the ground for a concluding satirical solution by an erstwhile Carlylean economist. (shrink)

The heart of mathematics is its elegance; the way it all fits together. Unfortunately, its beauty often eludes the vast majority of people who are intimidated by fear of the difficulty of numbers. Mathematical Elegance remedies this. Using hundreds of examples, the author presents a view of the mathematical landscape that is both accessible and fascinating. At a time of concern that American youth are bored by math, there is renewed interest in improving math skills. Mathematical Elegance stimulates students, along (...) with those already experienced in the discipline, to explore some of the unexpected pleasures of quantitative thinking. Invoking mathematical proofs famous for their simplicity and brainteasers that are fun and illuminating, the author leaves readers feeling exuberant--as well as convinced that their IQs have been raised by ten points. A host of anecdotes about well-known mathematicians humanize and provide new insights into their lofty subjects. Recalling such classic works as Lewis Carroll's Introduction to Logic and A Mathematician Reads the Newspaper by John Allen Paulos, Mathematical Elegance will energize and delight a wide audience, ranging from intellectually curious students to the enthusiastic general reader. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Histories of Science in Early Modern Europe:IntroductionRobert GouldingIn 1713, Pierre Rémond de Montmort wrote to the mathematician Nicolas Bernoulli:It would be desirable if someone wanted to take the trouble to instruct how and in what order the discoveries in mathematics have come about.... The histories of painting, of music, of medicine have been written. A good history of mathematics, especially of geometry, would be a much more interesting and (...) useful work.... Such a work, if done well, could be regarded to some extent as a history of the human mind, since it is in this science, more than in anything else, that man makes known that gift of intelligence that God has given him to rise above all other creatures.1Such a history of mathematics was attempted by Jean-Etienne Montucla in his Histoire des mathématiques (first printed in 1758, and reissued in a [End Page 33] greatly expanded form in 1799).2 Montucla's great work is generally acknowledged as the first genuine history of mathematics. According to some modern historians, previous attempts at such a history had amounted to little more than collections of anecdotes, biographies or exhaustive bibliographies: "jumbles of names, dates and titles," as one writer in the Dictionary of Scientific Biography characterized them.3 Montucla, on the other hand, was thoroughly animated by the Enlightenment project expressed in Rémond's letter, and his Histoire was a philosophical history of the "development of the human mind," as he himself described it.4 It was precisely Montucla's vision of what mathematics meant and his conviction that mathematics itself must change in order to reflect the historical elevation of the human intellect, that allowed him to transform the scattered dates and anecdotes of his predecessors into a genuine history. All subsequent histories of mathematics—until the most recent social histories5 —have been in a sense "footnotes to Montucla."The papers here argue that there were indeed histories of mathematics before Montucla which are worthy of scholarly attention, and that the dismissal of Renaissance histories of science as nothing more than a "cloud of fine adjectives and metaphors" is unfair (although it contains a germ of truth). Anthony Grafton has argued in an article on the historical writings of Cardano, Rheticus, and Kepler, that the purpose of such texts was not so much to trace what actually happened, as to justify the study of a subject often derided—by both humanist and scholastic writers—as obscure, useless and undignified.6 The following four papers go further in arguing that writings in this genre (or, at least, some writings in this genre) were not limited to this characteristically humanist objective. Authors did not just use their histories to persuade others. They themselves relied on historical narratives in order to think about their discipline, define its parts, distinguish among its acceptable and unacceptable forms and prescribe its content [End Page 34] and method of teaching. By placing their discipline into a historical context shared by other, more mainstream humanistic arts, moreover, they could draw upon the large, narrative structures which Renaissance humanists had adapted to understand human intellectual and cultural development, origins, progress and decline.James Byrne's article highlights a disparity of vision between those humanists who treated the origin of the arts as just another subject for humanistic display; and those (like Regiomontanus) who were invested in the arts and wrote their histories from an insider's perspective. Alongside Regiomontanus, one could include several other authors treated in this collection, such as Thomas Vaughan, Peter Ramus, and Henry Savile. In each case, history provided a means for working through problems of the legitimacy and nature of their disciplines. While their writings are not "histories of the human mind," their interests went beyond mere rehearsal of anecdotes. Ramus's histories, for instance, written 200 years before Montucla's Histoire, had the kind of unifying vision which de Montmort demanded, and shared Montucla's concern to find a meaning in the development of mathematics. Ramus, and the supporters and detractors he spawned, were convinced that constructing the history of this science was crucial to understanding how and why human beings have... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:In Memoriam: Meyer Schapiro (1904–1996)David RosandMeyer Schapiro, University Professor Emeritus at Columbia University, died at home in New York City on Sunday 3 March, at the age of 91. With his death, the worlds of art and learning have lost a legendary figure.“The humanity of art lies in the artist and not simply in what he represents,” Schapiro had declared in his lectures on abstract art. “It is the (...) painter’s constructive activity, his power of impressing a work with feeling and the qualities of thought that gives humanity to art.” Informing his own life and work, these were the values that enabled him to give such expressive voice to so much of the art of the past and of the present, to articulate the passions of eighth-century Northumbrian manuscript illuminators and Romanesque stonecarvers as well as of the most creative painters of our own century. Schapiro’s range as an art historian was universal, for he believed in the universality of art.Meyer Schapiro was born 23 September 1904, in Siauliai, Lithuania, and emigrated to the United States with his family when he was three. He grew up in the Brownsville section of Brooklyn, where he was first exposed to art in evening classes taught by John Sloan at the Hebrew Settlement House. He entered Columbia College in 1920 at the age of 16, having won both Pulitzer and New York State Regents Scholarships, and received his AB in 1924. Five years later he submitted his dissertation for the Ph.D., the first in fine arts and archaeology awarded by Columbia. “The Romanesque Sculpture of Moissac,” parts of which were published in The Art Bulletin of 1931, opened an entirely new critical perspective on Romanesque art; recognizing its creative and expressive imperatives, Schapiro’s study articulated the aesthetic quality of that art as well as situating it historically and culturally with new precision. Supported by a grant from the Carnegie Corporation, he travelled widely in Europe and the Near East in 1926–27, leaving behind a trail of anecdotal legends that continued to be recounted long after by those who met the brilliant young graduate student—including Bernard Berenson, who compared the eloquence of the “very handsome youth” to Solomon himself.Schapiro began his teaching career at Columbia in 1928 as a lecturer and rose through the professorial ranks, becoming full professor in 1952; he was named University Professor in 1965 and became University Professor Emeritus in 1973. Throughout his career, he moved between the university campus on Morningside Heights and his home neighborhood in Greenwich Village. Lecturing as well at the New School for Social Research from 1936 to 1952, he reached the larger community of the New York art world, especially the artists, in the years when New York was becoming the most dynamic center of contemporary art. And he brought the artists uptown; auditing and participating in his classes at Columbia, their presence in his classroom seemed to validate his wisdom. At the other end of Schapiro’s range were the mathematicians, [End Page 547] scientists, and philosophers, in whose worlds he moved with equal confidence and, importantly, enthusiasm.During the ‘30s Schapiro’s was an important voice of moral balance in the cultural debates of the political left. Although he demonstrated with new insight and undogmatic focus the social bases of art, he insisted on recognizing “its own conditions which distinguish it from other activities.” Even as he avidly pursued knowledge—of history, culture, languages—Schapiro remained an artist. He talked about pictures as an artist. His criticism was an act of re-creation, as he lovingly reconstructed decisions made by the painter on the canvas; every stroke demanded attention. Choice was important, that freedom of the individual responding to the world around him and to the challenge of his own creation. In the anonymous art of the early Middle Ages Schapiro discovered the artist, the human maker; he intuited the feeling individual responsible for the invention of such expressive form. Distant art took on life and became accessible through his recognition of and insistence on its humanity.Schapiro’s learning was legendary, and yet, however intimidating it may have... (shrink)

This book explores the research of Professor Hilary Putnam, a Harvard professor as well as a leading philosopher, mathematician and computer scientist. It features the work of distinguished scholars in the field as well as a selection of young academics who have studied topics closely connected to Putnam’s work. It includes 12 papers that analyze, develop, and constructively criticize this notable professor's research in mathematical logic, the philosophy of logic and the philosophy of mathematics. In addition, it features a short (...) essay presenting reminiscences and anecdotes about Putnam from his friends and colleagues, and also includes an extensive bibliography of his work in mathematics and logic. The book offers readers a comprehensive review of outstanding contributions in logic and mathematics as well as an engaging dialogue between prominent scholars and researchers. It provides those interested in mathematical logic, the philosophy of logic, and the philosophy of mathematics unique insights into the work of Hilary Putnam. (shrink)

This book offers an engaging and comprehensive introduction to scientific theories, and the evolution of science and mathematics through the centuries. It discusses the history of scientific thought and ideas and the intricate dynamic between new scientific discoveries, scientists, culture, and societies. Through stories and historical accounts, the volume illustrates the human engagement and preoccupation with science and the interpretation of natural phenomena. It highlights key scientific breakthroughs from the ancient to later ages, giving us accounts of the work of (...) ancient Greek and Indian mathematicians and astronomers as well as of the work of modern scientists like Descartes, Newton, Planck, Mendel and many more. The author also discusses the vast advancements which have been made in the exploration of space, matter, and genetics and their relevance in the advancement of the scientific tradition. He provides great insights into the process of scientific experimentation and the relationship between science and mathematics. He also shares amusing anecdotes of scientists and their interactions with the world around them. Detailed and accessible, this book will be of great interest to students and researchers of science, mathematics, the philosophy of science, science and technology studies, and history. It will also be useful for general readers who are interested in the history of scientific discoveries and ideas. (shrink)

The preeminent explicator of mathematical logic to non-mathematicians, John Allen Paulos is familiar to general readers not only from his bestselling books but also from his media appearances, including The David Letterman Show and National Public Radio's "Talk of the Nation" and "Science Friday," as well as articles in _Newsweek, Nature, Discover, Business Week, the New York Times Book Review, The Nation, New York Review of Books,_ and _The London Review of Books._ Paulos originally wrote this charming little book (...) on analytic logic, its mathematics, and its puzzles in 1985. And as in his later books, he uses jokes, stories, parables, and anecdotes to elucidate difficult concepts, in this case, some of the fundamental problems in modern philosophy. (shrink)

The preeminent explicator of mathematical logic to non-mathematicians, John Allen Paulos is familiar to general readers not only from his bestselling books but also from his media appearances, including The David Letterman Show and National Public Radio's "Talk of the Nation" and "Science Friday," as well as articles in _Newsweek, Nature, Discover, Business Week, the New York Times Book Review, The Nation, New York Review of Books,_ and _The London Review of Books._ Paulos originally wrote this charming little book (...) on analytic logic, its mathematics, and its puzzles in 1985. And as in his later books, he uses jokes, stories, parables, and anecdotes to elucidate difficult concepts, in this case, some of the fundamental problems in modern philosophy. (shrink)

The present article explores the productive afterlife of the Vitruvian anecdote concerning Aristippus’s shipwreck on the shore of Rhodes. Known to several medieval scholars, the anecdote came into vogue during the Renaissance, when it was transformed into a potent metaphor mobilised by moralists, educators and religious authors. Not until the sixteenth century, however, did mathematicians come to recognise the value in appropriating the metaphor as a means to elevate the dignity of their discipline. Two centuries later, having accomplished their (...) mission, mathematicians elided much of the anecdote, preserving only the geometrical part. (shrink)

This book explores the research of Professor Hilary Putnam, a Harvard professor as well as a leading philosopher, mathematician and computer scientist. It features the work of distinguished scholars in the field as well as a selection of young academics who have studied topics closely connected to Putnam's work. It includes 12 papers that analyze, develop, and constructively criticize this notable professor's research in mathematical logic, the philosophy of logic and the philosophy of mathematics. In addition, it features a short (...) essay presenting reminiscences and anecdotes about Putnam from his friends and colleagues, and also includes an extensive bibliography of his work in mathematics and logic. The book offers readers a comprehensive review of outstanding contributions in logic and mathematics as well as an engaging dialogue between prominent scholars and researchers. It provides those interested in mathematical logic, the philosophy of logic, and the philosophy of mathematics unique insights into the work of Hilary Putnam. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 66.4 (2005) 557-576 [Access article in PDF] The Writing in the Wittenberg Sky: Astrology in Sixteenth-Century Germany Claudia Brosseder University of Munich It probably was a delightful summer day when the celebrated humanist Willibald Pirckheimer, best known as a friend of Albrecht Dürer and Erasmus of Rotterdam, strolled, with an unknown friend, through the streets of Nuremberg. When they saw a girl standing (...) at the streetside, it occurred to the friend to predict the girl's future by reading her palm. So he did, foretelling a forthcoming marriage and a blissful life. After the girl had passed on, Pirckheimer's unknown friend admitted that, unfortunately, he had not seen the girl's blithe life but her impending death. In order not to scare her, however, he had not revealed this message to the girl; only his learned friend Pirckheimer could know the truth. Some years later, the Nuremberg philologist Joachim Camerarius, who narrated this rather trivial event with great excitement in 1576, does not display any sign of bewilderment that in this instance a chiromantic had told untruths to somebody.1 On the contrary, he praises this art and admires Italy's scholars, who practiced, besides chiromancy, other wonderful divinatory arts based on astrology. Camerarius was perceptibly impressed by Italian astrology and her divinatory stepsisters. Presumably he was thinking about De subtilitate (1554) by Girolamo Cardano who, at least in this work, counted astrology and chiromancy among the arts of natural divination.2 And so, as if inspired by Cardano's acute judgment years afterward, Camerarius [End Page 557] read in Cardano, in the same breath, the idea that there had been only twelve significant scholars in the history of civilization, who were capable of divination: such were, among others, Aristotle, Ptolemy, Alchindi, and Vitruvius. However, no contemporary of Cardano was among them.This anecdote provides but a glimpse of the sixteenth-century enthusiasm for the divinatory art of chiromancy and, relatedly, astrology. My contribution will show that academic Protestant Germany was actually in the grips of an astrology fever. Throughout the sixteenth century, many German scholars were under the spell of astrology. In this, Germany did not essentially differ from Italy and France, but in one specific place astrology presented itself in a peculiar shape.In the sixteenth century astrology flourished particularly in one German site of erudition: in Wittenberg, 51°52' latitude, 12°38' longitude. Particularly here at the University of Wittenberg, attracted by Philipp Melanchthon's pedagogic enterprise, a series of scholars came together who shaped the image of astrology in sixteenth-century Germany. They were humanists, physicians, astronomers, mathematicians, municipal surgeons, and scholars of Greek. There were at least forty-six of them. In one way or another, each of them was interested in astrology. Many of them composed astrological analyses for wealthy clients. "Pene innumberabiles" mathematicians came from the University of Wittenberg, as the Tübingen professor of rhetorics Nikodemus Frischlin grudgingly asserted. By this he meant astrologers, and he did not try to conceal his aversion to them.3 He was unequivocal: astrologers are swindlers.With this opinion, Frischlin joined the ranks of Giovanni Pico della Mirandola,4 Martin Luther,5 Erasmus of Rotterdam,6 and others. This is not a novelty, and yet Frischlin's caustic remark can still teach us something today: regarding the history of German astrology, one should resist the temptation to see Melanchthon as the one and only spiritus rector of German sixteenth-century astrology. Instead, it is reasonable to concede much more weight to the sway of Wittenberg and its numerous astronomers, mathematicians, and physicians, who did not only gravitate toward the personality of Melanchthon. The large number of scholars and the variety of ideas preempts any attempt [End Page 558] to reduce the number of astrologers to a so-called Melanchthon circle.7 This circle did not exist, not least because his pupils did not monistically adhere to Melanchthon's thoughts on astrology. In particular, a personality as industrious in publishing as Melanchthon's son-in... (shrink)

The author claims to have developed interactional expertise in gravitational wave physics without engaging with the mathematical or quantitative aspects of the subject. Is this possible? In other words, is it possible to understand the physical world at a high enough level to argue and make judgments about it without the corresponding mathematics? This question is empirically approached in three ways: anecdotes about non-mathematical physicists are presented; the author undertakes a reflective reading of a passage of physics, first without (...) going through the maths and then after engaging with it and discusses the difference between the experiences; the aforementioned exercise gives rise to a table of Levels of Understanding of mathematics, and physicists are asked about the level mathematical understanding they applied when they last read a paper. Each phase of empirical research suggests that mathematics is not as central to gaining an understanding of physics as it is often said to be. This does not mean that mathematics is not central to physics, merely that it is not essential for every physicist to be an accomplished mathematician, and that a division of labour model is adequate. This, in turn, suggests that a stream of undergraduate physics education with fewer mathematical hurdles should be developed, making it easier to train wider groups of people in physical science comprehension.Keywords: Physics; Mathematics; Interactional expertise; Physics education; Mathematical literacy; Scientific literacy. (shrink)

Birch’s criterion for the precautionary principle imposes a high evidential standard that many cases will fail to meet. Reliable, relevant anecdotal evidence suggestive of animal sentience should also fall within the scope of the precautionary principle. This would minimize potential suffering (as happened in the case of cephalopods) while further evidence is gathered.

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the (...) existence of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life. The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought. (shrink)

In this article, we report a study in which 109 research-active mathematicians were asked to judge the validity of a purported proof in undergraduate calculus. Significant results from our study were as follows: (a) there was substantial disagreement among mathematicians regarding whether the argument was a valid proof, (b) applied mathematicians were more likely than pure mathematicians to judge the argument valid, (c) participants who judged the argument invalid were more confident in their judgments than those (...) who judged it valid, and (d) participants who judged the argument valid usually did not change their judgment when presented with a reason raised by other mathematicians for why the proof should be judged invalid. These findings suggest that, contrary to some claims in the literature, there is not a single standard of validity among contemporary mathematicians. (shrink)

This paper introduces the first English-language book-length study to focus on the rhetorical function of anecdotal narratives across several literary genres of early China. In this volume we seek to clarify the nature and function of early Chinese anecdotes by raising the following questions: What are their characteristic features? What are their generic boundaries, that is to say, how do they relate to other types of narrative? What degree of historical authenticity do they display? How malleable were the stories? (...) What different framing techniques did authors use to fit stock anecdotes into larger narrative contexts? What was the rhetorical power of anecdotes when used in argumentation? How does the early Chinese preference for using anecdotes in argumentation differ from modes of argumentation preferred in other eras and cultures? In addressing these and other questions, this book will advance the idea that anecdotes were an essential rhetorical tool that early Chinese writers used effectively to persuade their audience of one or another point of view. (shrink)

This is the first book to evaluate the significance and usefulness of the practices of anthropomorphism and anecdotalism for understanding animals.

Although the term “anecdote” entered the modern European languages fairly recently and remains to this day ill-defined, the short, freestanding accounts of particular events, true or invented, that are usually referred to as anecdotes have been around from time immemorial. They have also always stood in a close relation to the longer, more elaborate narratives of history, sometimes in a supportive role, as examples and illustrations, sometimes in a challenging role, as the repressed of history—“la petite histoire.” Historians’ relation (...) to them, in turn, varied from appreciative to dismissive in accordance with their own objectives in writing history. It appears that highly structured anecdotes of the kind that are remembered and find their way into anecdote collections depend on and tend to confirm established views of history, the world, and human nature. In contrast, loosely structured anecdotes akin to the modern fait divers have usually worked to undermine established views and stimulate new ones, either by presenting material known to few and excluded from officially authorized histories, or by reporting “odd” occurrences for which the established views of history, the world, and human nature do not easily account. (shrink)

Anecdotal pluralism (AP) is the claim that, when two individuals disagree on the truth of a religious belief, the right move to make is to engage in a communal epistemic process of evidence sharing and evaluation, motivated by the willingness to learn from each other, understand the adversary's views and how these challenge their own, and re-evaluate their own epistemic position in regards to external criticisms. What I will do in my paper is to provide a presentation of AP and (...) give a few reasons in support. I will begin with showing how pluralism can be promoted by religious experiences inhering in any (historical) tradition. To this regard, my purpose is to analyse such experiences as conducive to the assumption of the two main principles defining any pluralist view. Subsequently, I will construe AP by seven claims, and I will focus my efforts on justifying its superiority both to exclusivism/inclusivism and other varieties of pluralism. My next and final move is to list a few reasons which support my view. (shrink)

Mathematicians often conduct aesthetic judgements to evaluate mathematical objects such as equations or proofs. But is there a consensus about which mathematical objects are beautiful? We used a comparative judgement technique to measure aesthetic intuitions among British mathematicians, Chinese mathematicians, and British mathematics undergraduates, with the aim of assessing whether judgements of mathematical beauty are influenced by cultural differences or levels of expertise. We found aesthetic agreement both within and across these demographic groups. We conclude that judgements (...) of mathematical beauty are not strongly influenced by cultural difference, levels of expertise, and types of mathematical objects. Our findings contrast with recent studies that found mathematicians often disagree with each other about mathematical beauty. (shrink)

Claims about the occurrence of future events play an important role in pragmatic argumentation. Such claims can be supported by inductive arguments employing anecdotal, statistical, or causal evidence. In an experiment, the actual and perceived persuasiveness of these three types of evidence were assessed. A total of 324 participants read a newspaper article in which it was claimed that the building of a cultural centre would be profitable. This claim was supported by either anecdotal, statistical or causal evidence. The statistical (...) evidence proved to be more convincing than the anecdotal and causal evidence. Although the latter two evidence types were equally unconvincing, the anecdotal evidence was perceived as less persuasive than the causal evidence. Therefore, the actual and perceived persuasiveness of the evidence did not correspond. These results partly replicate the results obtained in previous experiments. They also underscore the need to distinguish between the perceived and the actual persuasiveness of an argument. (shrink)

Recent years have seen a growing acknowledgement within the mathematical community that mathematics is cognitively/socially constructed. Yet to anyone doing mathematics, it seems totally objective. The sensation in pursuing mathematical research is of discovering prior (eternal) truths about an external (abstract) world. Although the community can and does decide which topics to pursue and which axioms to adopt, neither an individual mathematician nor the entire community can choose whether a particular mathematical statement is true or false, based on the given (...) axioms. Moreover, all the evidence suggests that all practitioners work with the same ontology. (My number 7 is exactly the same as yours.) How can we reconcile the notion that people construct mathematics, with this apparent choice-free, predetermined objectivity? I believe the answer is to be found by examining what mathematical thinking is (as a mental activity) and the way the human brain acquired the capacity for mathematical thinking. (shrink)

This brief notice records anecdotes about Wittgenstein gathered from Br. Herbert Kaden OSB.

The literature on mathematical explanation contains numerous examples of explanatory, and not so explanatory proofs. In this paper we report results of an empirical study aimed at investigating mathematicians’ notion of explanatoriness, and its relationship to accounts of mathematical explanation. Using a Comparative Judgement approach, we asked 38 mathematicians to assess the explanatory value of several proofs of the same proposition. We found an extremely high level of agreement among mathematicians, and some inconsistencies between their assessments and (...) claims in the literature regarding the explanatoriness of certain types of proofs. (shrink)

We suggest that anecdotes have evidentiary value in interpreting nonhuman primate behavior. We also believe that any outcome from the experiments proposed by Heyes can be interpreted as a product of previous experience with trainers or as associative learning using the experimental cues. No potential outcome is clearcut evidence for or against the theory of mind proposition.

This paper examines Timothy Gowers’ attempt to counter a mythology of genius in mathematics: that to be a mathematician one has to be a mathematical genius. Someone might take such attacks on the myth of genius as expressions of envy, but I propose that there is another reason for cautioning against placing a high value on genius, by turning to research in the humanities.

My main claim is that, contrary to the assumptions of mainstream literature, epistemic religious diversity is not a matter of an abstract comparison among the belief systems of different religions or denominations; rather, it is a relation arising from the epistemic encounter among individuals who adhere to different doxastic groups. Particularly, while epistemic symmetry inclines to treat our doxastic opponents as peers, epistemic peerhood is not the starting point of doctrinal comparisons, but the potential outcome of the epistemic process of (...) the construal of shared evidence. A key point in my approach is that such a process is anecdotally constituted. My working plan is the following. In the first section I will introduce the challenge of religious diversity. In the next section, I will distinguish between two characterisations of the relation among seminal claims of doxastic groups, namely, beliefs which stand in a competing relation and beliefs which are merely alternative. I will provide a degree view of this distinction. The subsequent section consists in a overview of the outcomes from the epistemology of disagreement. In the fourth section, I will provide two basic motivations in support of TE. Finally, I will conclude by sketching how TE motivates the assumption of a schema for handling religious disagreements which I name anecdotal pluralism. (shrink)

The way of thinking of mathematicians and chemists in their respective disciplines seems to have very different levels of abstractions. While the firsts are involved in the most abstract of all sciences, the seconds are engaged in a practical, mainly experimental discipline. Therefore, it is surprising that many luminaries of the mathematics universe have studied chemistry as their main subject. Others have started studying chemistry before swapping to mathematics or have declared some admiration and even love for this discipline. (...) Here I reveal some of these mathematicians who were involved in chemistry from a biographical perspective. Then, I analyze what these remarkable mathematicians and statisticians could have learned while studying chemical subjects. I found analogies between code-breaking and molecular structure elucidation, inspiration for statistics in quantitative analytical chemistry, and on the role of topology in the study of some organic molecules. I also analyze some parallelisms between the way of thinking of organic chemists and mathematicians in terms of the use of backward analysis, search for patterns, and use of pictures in their respective researches. (shrink)

We present a case study of how mathematicians write for mathematicians. We have conducted interviews with two research mathematicians, the talented PhD student Adam and his experienced supervisor Thomas, about a research paper they wrote together. Over the course of 2 years, Adam and Thomas revised Adam’s very detailed first draft. At the beginning of this collaboration, Adam was very knowledgeable about the subject of the paper and had good presentational skills but, as a new PhD student, (...) did not yet have experience writing research papers for mathematicians. Thus, one main purpose of revising the paper was to make it take into account the intended audience. For this reason, the changes made to the initial draft and the authors’ purpose in making them provide a window for viewing how mathematicians write for mathematicians. We examined how their paper attracts the interest of the reader and prepares their proofs for validation by the reader. Among other findings, we found that their paper prepares the proofs for two types of validation that the reader can easily switch between. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Clinical Anecdotes: A Painful Lack of WoundsChristopher Bailey (bio)Keywordsdepression, post-traumatic stress disorder (PTSD), evolution, fight-or-flight, veteran (treatment of)Colin came to me complaining of depression, which started after he got back from Iraq in 2005. Although he had served in the National Guard, he volunteered absolutely nothing about his time in Iraq as we spoke, instead focusing on other factors, like problems at his job and a family history (...) of depression.Colin's depression seemed serious but not severe. He had been functioning fairly well at work and was not suicidal, although his home life had begun to suffer. As the interview proceeded, I myself fell into Colin's blind spot, all but forgetting his tour of duty as I forged ahead with my list of questions. My avoidance was conscious at first, a deliberate reluctance to come across as one more morbidly curious onlooker recklessly inflaming the wounds of war.When Colin complained of irritability, however, and being kept awake by sinister, shadowy “flashes” at night, I knew I had to ask about Iraq. He was probably suffering from post-traumatic stress disorder, I decided, and was thus trying to avoid the very memories that were haunting and hunting him. And for fear of retraumatizing him (if not suffering second-hand trauma myself), I had colluded in his denial.So when I finally got around to asking about Iraq, I fully expected to hear horror stories, half-articulated perhaps, but all the more grisly for the graininess of their images. Instead, Colin surprised me. He had not experienced anything he considered traumatic. No bombs, bloodshed, or shattered bodies. And in fact, he felt like a “wimp” for having been assigned to a noncombat detail where he missed what might have been his last, best chance “to be a hero.”Most of us would consider this a good thing, but in Colin's case, it had left him feeling somehow uninitiated, like he had not done his part or proven his manhood. Contrary to my assumption, it was the lack of trauma that seemed to trigger his depression upon returning from Iraq. In a relatively safe homeland, rife with middle-class comforts and lacking all but metaphorical rites of passage, perhaps we all feel this way at some level.I pointed out to Colin that people have been dying everywhere in Iraq and under all different circumstances, not just combat. From my stateside perspective, just being there is brave. Although I meant to honor him, he seemed to take my offering as more evidence of his failure, reassuring me that nothing he did could ever compare with the unspeakable horrors endured by veterans of Viet Nam and World War II.Now, as a civilian, he questions whether or not he would respond with sufficient ferocity if, say, he had to defend his family against an attacker. Although powerfully built, Colin (to his credit) has never been a brawler or a bully. Nonetheless, [End Page 223] I have no doubt he could do whatever it took to protect his family. What concerns me more is the opposite: That in trying to prove himself, he might unwittingly put himself and his family in harm's way, if not hurt someone who was never really a threat to begin with.With a new family member, I too feel the burden of hypervigilance. But although terrorists, serial killers, and gangs grab headlines, activating our visceral alarm systems, data from the Centers for Disease Control and Prevention suggest that a sedentary lifestyle still exacts a higher death toll among white American men, like Colin and myself. Although the effect of inertia on the body has been discussed extensively, its impact on mental health has received relatively little attention. Maybe for a creature who evolved on the African savannah, competing over territory and trying not to get eaten, the absence of a visceral challenge can be as debilitating as too much of one.Colin seemed receptive to my concerns, which I took to be a good sign. Although insecure men are quick to fear emasculation, Colin did not seem to be the least bit threatened when I suggested a course... (shrink)

Between History and Philosophy is the first book-length study in English to focus on the rhetorical functions and forms of anecdotal narratives in early China. Edited by Paul van Els and Sarah A. Queen, this volume advances the thesis that anecdotes—brief, freestanding accounts of single events involving historical figures, and occasionally also unnamed persons, animals, objects, or abstractions—served as an essential tool of persuasion and meaning-making within larger texts. Contributors to the volume analyze the use of anecdotes from (...) the Warring States Period to the Han Dynasty, including their relations to other types of narrative, their circulation and reception, and their central position as a mode of argumentation in a variety of historical and philosophical literary genres. (shrink)

To open a newspaper or turn on the television it would appear that science and religion are polar opposites - mutually exclusive bedfellows competing for hearts and minds. There is little indication of the rich interaction between religion and science throughout history, much of which continues today. From ancient to modern times, mathematicians have played a key role in this interaction. This is a book on the relationship between mathematics and religious beliefs. It aims to show that, throughout scientific (...) history, mathematics has been used to make sense of the 'big' questions of life, and that religious beliefs sometimes drove mathematicians to mathematics to help them make sense of the world. Containing contributions from a wide array of scholars in the fields of philosophy, history of science and history of mathematics, this book shows that the intersection between mathematics and theism is rich in both culture and character. Chapters cover a fascinating range of topics including the Sect of the Pythagoreans, Newton's views on the apocalypse, Charles Dodgson's Anglican faith and Gödel's proof of the existence of God. (shrink)

In this paper we try to convert the mathematician who calls himself, or herself, “a formalist” to a position we call “meth-odological pluralism”. We show how the actual practice of mathe-matics fits methodological pluralism better than formalism while preserving the attractive aspects of formalism of freedom and crea-tivity. Methodological pluralism is part of a larger, more general, pluralism, which is currently being developed as a position in the philosophy of mathematics in its own right.1 Having said that, henceforth, in this (...) paper, we abbreviate “methodological pluralism” with “pluralism”. (shrink)

In seeking the truth, philosophers have long used fiction and counterfactual scenarios to raise and answer questions, to foster dialogue or give a commentary on some facet of life. In this paper I will present a few well-known thought experiments from contemporary Western philosophers and highlight some characteristic traits of such thought experiments. I will then discuss some of the fictitious anecdotes that appear in the Zhuangzi. In comparing the features of Western thought experiments to fables from Zhuangzi, we (...) will see that although Zhuangzi’s stories would likely fail to be considered good philosophical thought experiments (according to the Western tradition), the very thing that makes them fail is arguably what allows the writings of Zhuangzi to continue to inspire philosophical reflection and dialogue. (shrink)

The coming of mathematicians to the United States fleeing the spread of Nazism presented a serious problem to the American mathematical community. The persistence of the Depression had endangered the promising growth of mathematics in the United States. Leading mathematicians were concerned about the career prospects of their students. They feared that placing large numbers of refugees would exacerbate already present nationalistic and anti-Semitic sentiments. The paper surveys a sequence of events in which the leading mathematicians reacted (...) to the foreign-born and to the spread of Nazism, culminating in the decisions by the American Mathematical Society to found the journal Mathematical reviews and to form a War Preparedness Committee in September 1939. The most obvious consequence of the migration was an enlarged role for applied mathematics. (shrink)

Abstract: Critics of Havi Carel's 2008 book, Illness: The Cry of the Flesh, have contended that Carel's deployment of phenomenological philosophy adds little to commonsense views about illness and that Carel relies too heavily on emotion-laden autobiographical anecdotes. Against these contentions this article argues: first, that a perfectly respectable task of philosophy is to find reasons to support pre-existing beliefs; and secondly, that Carel's use of anecdotes, while certainly appealing to readers' emotions, constitutes part of a legitimate argumentative (...) strategy. The article links these proposals with broader debates concerning the proper task of philosophy and the role of emotionality and imagination in philosophical inquiry. (shrink)

After World War II, quite a few mathematicians, including Mark Kac, John von Neumann, and Nobert Wiener, worked on the physical problem of phase transitions, i.e. changes in the state of matter caused by gradual changes of physical parameters such as the condensation of a gas to a liquid and the loss of magnetization of a ferromagnet above a certain temperature. The significance of these mathematicians was not so much that they brought mathematical rigor to the theoretical description (...) of the phenomena,1 but that they applied their mathematical skills and tools to solve concrete mathematical problems encountered in the microscopic modeling of phase transitions. The present paper deals... (shrink)

Nevil Maskelyne, the Cambridge-trained mathematician and later Astronomer Royal, was appointed by the Royal Society to observe the 1761 transit of Venus from the Atlantic island of St Helena, assisted by the mathematical practitioner Robert Waddington. Both had experience of measurement and computation within astronomy and they decided to put their outward and return voyages to a further use by trying out the method of finding longitude at sea by lunar distances. The manuscript and printed records they generated in this (...) activity are complemented by the traditional logs and journals kept by the ships’ officers. Together these records show how the mathematicians came to engage with the navigational practices that were already part of shipboard routine and how their experience affected the development of the methods that Maskelyne and Waddington would separately promote on their return. The expedition to St Helena, in particular the part played by Maskelyne, has long been regarded as pivotal to the introduction of the lunar method to British seamen and to the establishment of theNautical Almanac. This study enriches our understanding of the episode by pointing to the significant role played by the established navigational competence among officers of the East India Company. (shrink)

"--David Ruelle, author of "Chance and Chaos" "This is an important book, one that should cause an epoch-making change in the way we think about mathematics.

Hans Blumenberg’s work is characterized by a seemingly insatiable predilection for anecdotes — about Thales and Pyrrhus, Goethe and Fontane, Husserl and Wittgenstein, Polgar and Jünger. This essay explores the theoretical status of anecdotes by juxtaposing Carl Linnaeus’s Nemesis Divina with Blumenberg’s Care Crosses the River, both read alongside Aristotle’s notion of exemplarity and Joel Fineman’s delineation of the anecdote as the literary-historical form for expressing contingency. As a mode of thought at the nexus of literature and experience, (...) anecdotes immediately pose the question of evidence. Even if true, an anecdote opens the issue of whether it is the exception or the norm, whether it only speaks for itself or assumes exemplary force, and what such exemplarity might mean. By juxtaposing two theorists qua anecdotists — Carl Linnaeus and Hans Blumenberg — this essay aims to show the inextricable relation between anecdote and example, and thus between anecdote and rhetoric, a counterpart and supplement to the poetics of metaphor within what one can call with Blumenberg a theory of nonconceptuality. (shrink)

ArgumentUp until the French Revolution, European mathematics was an “aristocratic” activity, the intellectual pastime of a small circle of men who were convinced they were collaborating on a universal undertaking free of all space-time constraints, as they believed they were ideally in dialogue with the Greek founders and with mathematicians of all languages and eras. The nineteenth century saw its transformation into a “democratic” but also “patriotic” activity: the dominant tendency, as shown by recent research to analyze this transformation, (...) seems to be the national one, albeit accompanied by numerous analogies from the point of view of the processes of national evolution, possibly staggered in time. Nevertheless, the very homogeneity of the individual national processes leads us to view mathematics in the context of the national-universal tension that the spread of liberal democracy was subjected to over the past two centuries. In order to analyze national-universal tension in mathematics, viewed as an intellectual undertaking and a profession of the new bourgeois society, it is necessary to investigate whether the network of international communication survived the political, social, and cultural upheavals of the French Revolution and the European wars waged in the early nineteenth century, whether national passions have transformed this network, and if so, in what way. Luigi Cremona's international correspondence indicates that relationships among individuals have been restructured by the force of national membership, but that the universal nature of mathematics has actually been boosted by a vision shared by mathematicians from all countries concerning the role of their discipline in democratic and liberal society as the basis of scientific culture and technological innovation, as well as a basic component of public education. (shrink)

Growing up in Ionia -- Travels far and wide -- Settling in Croton -- Pythagorean beliefs -- A lasting legacy.