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)

"--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.

In 1807 the first life insurance society was established in The Netherlands. In the second half of the century, life insurance societies underwent considerable expansion. During the intervening period, the lines had to be laid along which this new phenomenon was to develop in the future: between 1827 and 1830, the government started discussing the nature of its responsibility in this field and the kind of policy to be developed, and in 1830, a book on the organization of life insurance (...) societies, the calculation of life annuities and widows' fund premiums was published, written by the mathematician Rehuel Lobatto. This book played an important role in the government's discussion. Royal Decrees which prescribed government approval for the establishment of life assurance societies were promulgated in 1830, 1833 and 1840. In 1832, Lobatto became the government's scientific adviser on the assessment of the calculations performed by these societies, and in the same year, he was also appointed adviser to the first life insurance society. From 1832 until his death in 1866 he advised the company on the use of life tables for life as well as for reversionary annuities, and he calculated the premiums based on these life tables. Another decree was promulgated in 1864 prescribing exactly which life tables were to be used. Because Lobatto probably played a part in this decree, he was responsible for a very ‘conservative’ government policy, which was no longer adequate in the second half of the century. (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 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)

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)

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)

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)

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)

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)

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)

Piero della Francesca is best known as a painter but he was also a mathematician. His treatise De prospectiva pingendi is a superb example of a union between the fne arts and mathemati‑ cal sciences of arithmetic and geometry. In this paper, I explain some reasons why his paint‑ ing is considered as a part of perspective and, therefore, can be identifed with a branch of geometry.

Mathematical practitioners in seventeenth-century London formed a cohesive knowledge community that intersected closely with instrument-makers, printers and booksellers. Many wrote books for an increasingly numerate metropolitan market on topics covering a wide range of mathematical disciplines, ranging from algebra to arithmetic, from merchants’ accounts to the art of surveying. They were also teachers of mathematics like John Kersey or Euclid Speidell who would use their own rooms or the premises of instrument-makers for instruction. There was a high degree of interdependency (...) even beyond their immediate milieu. Authors would cite not only each other, but also practitioners of other professions, especially those artisans with whom they collaborated closely. Practical mathematical books effectively served as an advertising medium for the increasingly self-conscious members of a new emerging professional class. Contemporaries would talk explicitly of ‘the London mathematicians’ in distinction to their academic counterparts at Oxford or Cambridge. The article takes a closer look at this metropolitan knowledge culture during the second half of the century, considering its locations, its meeting places and the mathematical clubs which helped forge the identity of its practitioners. It discusses their backgrounds, teaching practices and relations to the London book trade, which supplied inexpensive practical mathematical books to a seemingly insatiable public. (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)

Intended for logicians and mathematicians, this text is based on Dr. Hamilton's lectures to third and fourth year undergraduates in mathematics at the ...

This fascinating exploration of the great discoveries of history's most important mathematicians seeks an answer to the eternal question: Does mathematics hold the key to understanding the mysteries of the physical world?

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)

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)

In mid-Victorian Britain, reconciling elite mathematical expertise with practical mechanical experience presented both engineering and social challenges. Nowhere was this more apparent than in the construction of the Westminster Clock at Britain’s Houses of Parliament. Realizing this scheme engendered the collaboration between Cambridge mathematicians George Biddell Airy and Edmund Beckett Denison, and the clockmaker Edward John Dent. Transforming theoretical mathematical drawings into physical apparatus challenged existing relations between conveyors of privileged scientific knowledge and those with practical experience of what (...) was, and what was not, mechanically possible. My article demonstrates how, within this project, physical models and devices provided material solutions to ambiguities over authority and social disorder in Victorian Britain. (shrink)

Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of (...) mathematical claims. In this paper, I argue that none of the epistemic objectives of mathematicians that are currently on the table provide a satisfactory explanation of this rejection of probabilistic proofs. (shrink)

Invoking a form of quantifier variance promises to let us explain mathematicians’ freedom to introduce new kinds of mathematical objects in a way that avoids some problems for standard platonist and nominalist views. In this paper I’ll note that, despite traditional associations between quantifier variance and Carnapian rejection of metaphysics, Siderian realists about metaphysics can naturally be quantifier variantists. Unfortunately a variant on the Quinean indispensability argument concerning grounding seems to pose a problem for philosophers who accept this hybrid. (...) However I will charitably reconstruct this problem and then argue for optimism about solving it. (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)

New York University mathematical physicist Alan Sokal published in the postmodern humanities journal Social Text a parody entitled Transgressing the Boundaries: Toward a Transformative Hermeneutics of Quantum Gravity [1]. His point in doing so was to test whether the field of ``cultural studies of science'' was seriously lacking in ``intellectual standards.'' His article is nonsense from start to finish, but was still published. He revealed the hoax in another article in Lingua Franca [2]. The incident, and reactions to it, now (...) being called the Sokal Affair, have received wide coverage in The New York Times [3]. (shrink)

info:eu-repo/semantics/publishedVersion.

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint (...) pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

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.

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)

197: on logon didonai as giving a proof. In answer to Plato's charge that mathematicians take as their starting point certain unproved assumptions, and call upon them to "give an account" of them in the sense of deriving them from some more basic principle or principles.

From ancient times to the present, the discovery and presentation of new proofs of previously established theorems has been a salient feature of mathematical practice. Why? What purposes are served by such endeavors? And how do mathematicians judge whether two proofs of the same theorem are essentially different? Consideration of such questions illuminates the roles that proofs play in the validation and communication of mathematical knowledge and raises issues that have yet to be resolved by mathematical logicians. The Appendix, (...) in which several proofs of the Fundamental Theorem of Arithmetic are compared, provides a miniature case study. (shrink)

I will present here a personal point of view on the commitment of mathematicians in medicine. Starting from my personal experience, I will suggest generalisations including favourable signs and caveats to show how mathematicians can be welcome and helpful in medicine, both in a theoretical and in a practical way.

I argue that recollection, in Plato's Meno , should not be taken as a method, and, if it is taken as a myth, it should not be taken as a mere myth. Neither should it be taken as a truth, a priori or metaphorical. In contrast to such views, I argue that recollection ought to be taken as an hypothesis for learning. Thus, the only methods demonstrated in the Meno are the elenchus and the hypothetical, or mathematical, method. What Plato's (...) Meno demonstrates, then, is that we cannot be philosophers if we fail to make use of the mathematician's hypothetical method. (shrink)

In their own words, many of the world's foremost mathematicians discuss the art and practice of their work in this book, which shines a light on some of the issues of mathematical creativity. It is neither a philosophical treatise nor the presentation of experimental results, but a compilation of reflections from top-calibre working mathematicians. This approach highlights the creative aspects of the field, illustrates the dramatic variation by individual, and hopes to express the vibrancy of creative minds at (...) work. Organised alphabetically, this book is meant for a general audience and is best read by browsing. It can be used as a supplementary text in mathematics history and mathematics education courses. (shrink)

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

Mathematicians only use deductive proofs to establish that mathematical claims are true. They never use inductive evidence, such as probabilistic proofs, for this task. Don Fallis (1997 and 2002) has argued that mathematicians do not have good epistemic grounds for this complete rejection of probabilistic proofs. But Kenny Easwaran (2009) points out that there is a gap in this argument. Fallis only considered how mathematical proofs serve the epistemic goals of individual mathematicians. Easwaran suggests that deductive proofs (...) might be epistemically superior to probabilistic proofs because they are transferable. That is, one mathematician can give such a proof to another mathematician who can then verify for herself that the mathematical claim in question is true without having to rely at all on the testimony of the first mathematician. In this paper, I argue that collective epistemic goals are critical to understanding the methodological choices of mathematicians. But I argue that the collective epistemic goals promoted by transferability do not explain the complete rejection of probabilistic proofs. (shrink)

Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. It (...) contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle. (shrink)

Socrates' dream puts in generalized form the difficulty that plato saw in the mathematician's procedure of hypothesis, I.E., Of positing undemonstrated first principles ("prota") or elements ("stoicheia") as starting-Points of demonstration. If the elements are unknown, How can what is constructed from them be known?--A difficulty to which plato had earlier called attention in the 'republic' (510cd, 533cd.) this interpretation accords with the mathematical setting and personages of the dialogue, And explains why the explicit refutation of theaetetus' third proposal, That (...) knowledge be defined as true belief accompanied by a logos, Is so perfunctory and unconvincing. Furthermore, The dilemma thus brought to light is reflected in 'posterior analytics' (book i, Chap. 3), Which was presumably written during aristotle's residence in the academy in association with plato and the other mathematicians gathered around him. (shrink)

Hermeneutic fictionalism about mathematics maintains that mathematics is not committed to the existence of abstract objects such as numbers. Mathematical sentences are true, but they should not be construed literally. Numbers are just fictions in terms of which we can conveniently describe things which exist. The paper defends Stephen Yablo’s hermeneutic fictionalism against an objection proposed by John Burgess and Gideon Rosen. The objection, directed against all forms of nominalism, goes as follows. Nominalism can take either a hermeneutic form and (...) claim that mathematics, when rightly understood, is not committed to the existence of abstract objects, or a revolutionary form and claim that mathematics is to be understood literally but is false. The hermeneutic version is said to be untenable because there is no philosophically unbiased linguistic argument to show that mathematics should not be understood literally. Against this I argue that it is wrong to demand that hermeneutic fictionalism should be established solely on the basis of linguistic evidence. In addition, there are reasons to think that hermeneutic fictionalism cannot even be defeated by linguistic arguments alone. (shrink)