This article aims to present how paper-folding activities were integrated into recreational mathematics during the 17th and the 18th centuries. Recreational mathematics was conceived during these centuries as a way not only to pique one’s curiosity, but also to communicate mathematical knowledge to the literate classes of the population. Starting with Leurechon’s 1624 Récréation mathématique, which did not contain any exercise concerning paper folding, we show how two other traditions—Dürer’s folded nets on the one hand and (...) napkin folding on the other hand—prompted and influenced the integration of folding within subsequent books and manuscripts, especially those of Georg Philipp Harsdörffer and Daniel Schwenter. In Germany, but also to a lesser extent in France, folding was henceforth re-conceptualised within recreational mathematics as a way to transmit geometrical knowledge. Following Harsdörffer, the paper will claim that practising folding activities enabled the acquiring of a geometrical knowledge, which was haptic rather than symbolical or merely visual. This tactility reflects the Baconian conception of science and scientific experiment; and the paper will try to illuminate how folding, by advancing practice and tactility via experiments, was representing these traditions and conceptions. (shrink)

This Is A New Release Of The Original 1898 Edition.

A review of Geoffrey Marnell, Mathematical Doodlings: Curiosities, conjectures, and challenges.

This book is based on two premises: one cannot understand philosophy of mathematics without understanding mathematics and one cannot understand mathematics without doing mathematics. It draws readers into philosophy of mathematics by having them do mathematics. It offers 298 exercises, covering philosophically important material, presented in a philosophically informed way. The exercises give readers opportunities to recreate some mathematics that will illuminate important readings in philosophy of mathematics. Topics include primitive recursive arithmetic, (...) Peano arithmetic, Gödel's theorems, interpretability, the hierarchy of sets, Frege arithmetic, and intuitionist sentential logic. The book is intended for readers who understand basic properties of the natural and real numbers and have some background in formal logic. (shrink)

This paper critically assesses the claim by Gavin Menzies that Regiomontanus knew about the Chinese Remainder Theorem (CRT) through the Shù shū Jiǔ zhāng (SSJZ) written in 1247. Menzies uses this among many others arguments for his controversial theory that a large fleet of Chinese vessels visited Italy in the first half of the 15th century. We first refute that Regiomontanus used the method from the SSJZ. CRT problems appear in earlier European arithmetic and can be solved by the method (...) of the Sun Zi, as did Fibonacci. Secondly, we pro-vide evidence that remainder problems were treated within the European abbaco tradition independently of the CRT method. Finally, we discuss the role of recreational mathematics for the oral dissemination of sub-scientific knowledge. (shrink)

The paper examines the structure of the mathematical instrument making trade in London from the mid-sixteenth century to the opening of the Hanoverian era. This analysis of the trade is primarily based on evidence drawn from contemporary advertising. A distinction between informal editorial recommendations and advertising per se is made. It is concluded that up to the mid-seventeenth century mathematical instrument makers worked in either wood or metal. After that date a growing number of workshops advertised that they manufactured in (...) all media. Advertising was aimed at informing professional users from whom particular instruments could be purchased, but not on informing customers in specific terms of the range of instruments manufactured. It is concluded that until the early eighteenth century most mathematical instruments were commissioned. Only towards the end of the period is there evidence of over-the-counter sales, and advertising aimed at encouraging the growing consumer market to buy mathematical instruments for the practice of science as a social or recreational activity. (shrink)

The fundamental idea of a Neoaristotelian inherence ontology of mathematical entities parallels that of an Aristotelian approach to the ontology of universals. It is proposed that mathematical objects are nominalizations especially of dimensional and related structural properties that inhere as formal species and hence as secondary substances of Aristotelian primary substances in the actual world of existent physical spatiotemporal entities. The approach makes it straightforward to understand the distinction between pure and applied mathematics, and the otherwise enigmatic success of (...) applied mathematics in the natural sciences. It also raises an interesting set of challenges for conventional mathematics, and in particular for the ontic status of infinity, infinite sets and series, infinitesimals, and transfinite cardinalities. The final arbiter of all such questions on an Aristotelian inherentist account of the nature of mathematical entities are the requirements of practicing scientists for infinitary versus strictly finite mathematics in describing, explaining, predicting and retrodicting physical spatiotemporal phenomena. Following Quine, we classify all mathematics that falls outside of this sphere of applied scientific need as belonging to pure, and, with no prejudice or downplaying of its importance, ‘recreational’, mathematics. We consider a number of important problems in the philosophy of mathematics, and indicate how a Neoaristotelian inherence metaphysics of mathematical entities provides a plausible answer to Benacerraf’s metaphilosophical dilemma, pitting the semantics of mathematical truth conditions against the epistemic possibilities for justifying an abstract realist ontology of mathematical entities and truth conditions. (shrink)

The “unknown heritage” is the name usually given to a problem type in whose archetype a father leaves to his first son 1 monetary unit and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} (n usually being 7 or 10) of what remains, to the second 2 units and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{n}}$$\end{document} of what remains, and so on. In the end, all sons get the same, and nothing remains. The earliest known (...) occurrence is in Fibonacci’s Liber abbaci, which also contains a number of much more sophisticated versions, together with a partial algebraic solution for one of these and rules for all which do not follow from his algebraic calculation. The next time the problem turns up is in Planudes’s late thirteenth century Calculus according to the Indians, Called the Great. After that the simple problem type turns up regularly in Provençal, Italian and Byzantine sources. It seems never to appear in Arabic or Indian writings, although two Arabic texts (one from c. 1190) contain more regular problems where the number of shares is given; they are clearly derived from the type known from European and Byzantine works, not its source. The sophisticated versions turn up again in Barthélemy de Romans’ Compendy de la praticque des nombres (c. 1467) and, apparently inspired from there, in the appendix to Nicolas Chuquet’s Triparty (1484). Apart from a single trace in Cardano’s Practica arithmetice et mensurandi singularis, the sophisticated versions never surface again, but the simple version spreads for a while to German practical arithmetic and, more persistently, to French polite recreational mathematics. Close examination of the texts shows that Barthélemy cannot have drawn his familiarity with the sophisticated rules from Fibonacci. It also suggests that the simple version is originally either a classical, strictly Greek or Hellenistic, or a medieval Byzantine invention; and that the sophisticated versions must have been developed before Fibonacci within an environment (located in Byzantium, Provence, or possibly in Sicily?) of which all direct traces has been lost, but whose mathematical level must have been quite advanced. (shrink)

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.

Akin to the mathematical recreations, John Wilkins' Mathematicall Magick (1648) elaborates the pleasant, useful and wondrous part of practical mathematics, dealing in particular with its material culture of machines and instruments. We contextualize the Mathematicall Magick by studying its institutional setting and its place within changing conceptions of art, nature, religion and mathematics. We devote special attention to the way Wilkins inscribes mechanical innovations within a discourse of wonder. Instead of treating ‘wonder’ as a monolithic category, we present (...) a typology, showing that wonders were not only recreative, but were meant to inspire Wilkins' readers to new mathematical inventions. (shrink)

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some (...) mathematicians urgedThe Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

L’objectif de cet article est de rendre compte de la création et de la transmission d’un savoir mathématique lié aux combinaisons par le biais d’un support matériel particulier : les pavés mi-partis. Ces derniers sont des carrés du plan divisés par une diagonale en deux parties de couleur différente. Tour à tour objet d’ornement, objet de réflexion mathématique à partir du xviiie siècle grâce au Mémoire sur les combinaisons du père Truchet à l’Académie royale des sciences, objet pédagogique dans les (...) Récréations mathématiques d’Édouard Lucas au xixe siècle ou objet ludique, le pavé mi-parti présente une patrimonialisation originale d’un savoir mathématique, porté par un objet souvent considéré comme un simple jouet. (shrink)

For the purposes of this monograph, "by a degree" is meant a degree of recursive unsolvability. A degree [script bold]m is said to be minimal if 0 is the unique degree less than [script bold]m. Each of the six chapters of this self-contained monograph is devoted to the proof of an existence theorem for minimal degrees.

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and updated (...) by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, (...) there was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

The article tries to answer the following questions: Why did Lewis Carroll\'s ideas, expressed in the form of fairy tales, fascinate numerous analytical philosophers? What does Carroll\'s contribution to the contemporary logic and philosophy consist in? The basic thesis of the article is that Lewis Carroll - remaining in the Anglo-Saxon tradition of David Hume\'s and George Berkeley\'s philosophy - supplied material illustrating the problems connected with the use of language. He showed how improper use of language leads to formation (...) of philosophical problems. The article presents Carroll\'s output. He was one of the pioneers of symbolic logic that he developed in Boole\'s and De Morgan\'s tradition - in the field of the so-called recreation mathematics that he popularized in the form of riddles, puzzles, doublets and puns, compared by some logicians to a formal system. The article presents the essence of the theory of language developed by Carroll, in which language may be something hermetic with only one person having access to it (the case of Humpty Dumpty), but also something common, something social (Alice\'s conversation with the White King). Attention is paid to the fact that Carroll differentiated between what is nonsensical and what is absurd (the criterion being its relation to logic). It is pointed that Carroll\'s aim was first of all discovering the nonsense that is hidden behind the formulation of a metaphysical problem. In the article also the connections are studied between Carroll and Ludwig Wittgenstein, the relation between them being seen in the view that absurd introduced by means of humor is a kind of vaccine that is supposed to protect us from forming absurdity in philosophy. In conclusion it is stated that Carroll\'s ideas that are the most significant for analytic philosophers are concerned with the nature of language that is not a transparent medium for him, but something that offers resistance when we communicate with others, as well as something that may be flexible and adjusted to our will. By manipulating language Carroll shows in what way philosophy balances between sense and nonsense and how often philosophical questions arise from erroneous use of language and erroneous posing of problems. (shrink)

This is a book about infinity, specifically the infinity of numbers and sequences. Amazing properties arise, for instance, some kinds of infinity are argued to be greater than others. Along the way the author will demonstrate how infinity can be made to create beautiful 'art', guided by the development of underlying mathematics. This book will provide a fascinating read for anyone interested in number theory, infinity, math art, and/or generative art, and could be used a valuable supplement to any (...) course on these topics. (shrink)

Collects a variety of mathematics and logic puzzles, some based on the theorems of the mathematician Kurt Godel.

In this entertaining and challenging collection of logic puzzles, Raymond Smullyan-author of Forever Undecided-continues to delight and astonish us with his gift for making available, in the thoroughly pleasurable form of puzzles, some of the most important mathematical thinking of our time.

John Barrow is increasingly recognized as one of our most elegant and accomplished science writers, a brilliant commentator on cosmology, mathematics, and modern physics. Barrow now tackles the heady topic of impossibility, in perhaps his strongest book yet. Writing with grace and insight, Barrow argues convincingly that there are limits to human discovery, that there are things that are ultimately unknowable, undoable, or unreachable. He first examines the limits on scientific inquiry imposed by the deficiencies of the human mind: (...) our brain evolved to meet the demands of our immediate environment, Barrow notes, and much that lies outside this small circle may also lie outside our understanding. Barrow investigates practical impossibilities, such as those imposed by complexity, uncomputability, or the finiteness of time, space, and resources. Is the universe finite or infinite? Can information be transmitted faster than the speed of light? The book also examines the deeper theoretical restrictions on our ability to know, including Godel's theorem--which proved that there were things that could not be proved--and Arrow's Impossibility theorem about democratic voting systems. Finally, having explored the limits imposed on us from without, Barrow considers whether there are limits we should impose upon ourselves. For instance, if the secrets of the atom are to be found only by recreating extreme environments at great financial cost, just how much should we devote to that quest? Weaving together this intriguing tapestry, he illuminates some of the most profound questions of science, from the possibility of time travel to the very structure of the universe. (shrink)

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my (...) cue from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way. (shrink)

Activities, projects, and games introduce the concepts of logic, probability, graphic analysis, and problem solving. Suggested level: primary.

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School—a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical studies and (...) teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines. The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School, accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name—Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period—the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of continuators are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

The Introduction outlines, in a concise way, the history of the Lvov-Warsaw School – a most unique Polish school of worldwide renown, which pioneered trends combining philosophy, logic, mathematics and language. The author accepts that the beginnings of the School fall on the year 1895, when its founder Kazimierz Twardowski, a disciple of Franz Brentano, came to Lvov on his mission to organize a scientific circle. Soon, among the characteristic features of the School was its serious approach towards philosophical (...) studies and teaching of philosophy, dealing with philosophy and propagation of it as an intellectual and moral mission, passion for clarity and precision, as well as exchange of thoughts, and cooperation with representatives of other disciplines.The genesis is followed by a chronological presentation of the development of the School in the successive years. The author mentions all the key representatives of the School (among others, Ajdukiewicz, Lesniewski, Łukasiewicz,Tarski), accompanying the names with short descriptions of their achievements. The development of the School after Poland’s regaining independence in 1918 meant part of the members moving from Lvov to Warsaw, thus providing the other segment to the name – Warsaw School of Logic. The author dwells longer on the activity of the School during the Interwar period – the time of its greatest prosperity, which ended along with the outbreak of World War 2. Attempts made after the War to recreate the spirit of the School are also outlined and the names of followers are listed accordingly. The presentation ends with some concluding remarks on the contribution of the School to contemporary developments in the fields of philosophy, mathematical logic or computer science in Poland. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Math Anxiety:Making Room to BreatheValerie Allen (bio) and Todd Stambaugh (bio)"Don't do that to me, Professor," the student said, and everybody laughed, for by this late in the semester, the atmosphere was relaxed. The instructor in question had just reached the point in a worked problem when they could move from reasoning about specific numbers to stating a general principle: x≤y≤z, meaning that y—the value we sought—was always going (...) to be greater than or equal to one value (x) and less than or equal to the other (z). The student who had spoken had earned everyone's affection over the weeks, open about their difficulties with math, ready to risk being wrong, and to ask obvious questions that others may have been reluctant to owning. They had hung in there, despite presenting a classic case of math anxiety, a syndrome long recognized in the scholarship of teaching and learning.Among math anxiety's many symptoms, breathing disruption ranks high: hyperventilation (rapid breathing) and hypoventilation (barely or not breathing). When the energy-conserving parasympathetic nervous system is overtaken by the energy-expending sympathetic nervous system, the long exhalations that decelerate the heart rate are replaced by short, panting puffs from the upper lungs. The timed test provides the perfect environment for math anxiety to flourish. Stress diverts cognitive resources needed for procedural execution, so working memory consequently shrinks.Self-rated metrics for math anxiety have been in place for decades (Richardson and Suinn). Relatively trivial interventions can yield nontrivial results: deep breathing exercises and positive affirmations have a beneficial impact, as does writing about one's nerves immediately prior to an exam (Brunyé et al.). More systemic changes require a reorientation in teaching whereby conceptual mistakes are positively rewarded as opportunities for learning; students compose questions instead of answers; students solve problems together rather than perform solo (Boaler).Return now to that opening scenario to watch it become stereotype incarnate: the student was mature, Black, and female; the teacher, younger, white, and male; her comment was triggered by his shift from a specific [End Page 217] set of numbers into symbolic notation. There is robust evidence to demonstrate that certain demographics are consistently underrepresented in the U.S. math community: women and "African-Americans, Hispanics, American Indians, Alaska Natives, and Pacific Islanders," according to the Mathematical Association of America (MAA) ("Resources for Minority"). Routinely feeling that they do not belong, such groups are particularly vulnerable to math anxiety.Several remedies for such disparity suggest themselves: appointing more women and people of color in the profession; one-on-one tutorial assistance; research projects about the students' personal environment; privileging pre-test, low-stakes work over timed tests; assigning group rather than individual work. When all is said and done, however, such strategies "have little to do with mathematics content" (Martin 304). Mathematicians still need to be able to conceptualize, describe, and manipulate abstract objects, and there seems to be a suspicion that, at heart, abstraction is gendered masculine and colored white.It is possible that the rhetorical immediacy and context-dependence of Black English has misguidedly reinforced such a perception. For poet June Jordan, Black English "devolves from a culture that abhors all abstraction" (Jordan 367). Psychiatrist Alvin Poussaint asks, "Is there a way to say metamorphosis in Ebonics?" (O'Malley). Yet, despite Black English's avoidance of nominalization and similar verbosities that afflict academic prose, abstract logical reasoning flourishes in vernacular idiom, as linguist William Labov demonstrated half a century ago. Our aim is not to explain the asymptote in Black English but rather to consider a feature characteristic of those very minority cultures historically underrepresented in math—storytelling—and to ask how it may be foregrounded in mathematics, where it is decisively sidelined. Storytelling has been claimed by every culture and ethnic group at some point as the repository of wisdom, practical skills, religious beliefs, and shared identity and values; but when it gets corralled into the category of "fiction," and when it gets opposed to "fact," a narrowed concept of truth dispatches storytelling to kindergarten and recreational math. Word problems may supply scenarios but offer none of a plot's satisfactions... (shrink)

This Is A New Release Of The Original 1897 Edition.

What is a paradox? -- Knowing and believing -- Vagueness and identity -- Logic and truth -- Mathematical paradoxes -- Probability paradoxes -- Space and time -- Impossibilities -- Deciding and acting -- Index of philosophers.

“Arisbe,” the Peirce home near Milford, Pennsylvania, belongs to the National Park Service, and the Delaware Water Gap National Recreation Area is responsible for its care. In 1979 a geodetic triangulation station was installed in the front yard and named the “C. S. Peirce Station.” This was intended, at least in part, as a recognition of the fact that Peirce's scientific career was in the service of the Coast and Geodetic Survey, and that the first of his more than thirty (...) years in its service was spent with triangulating parties along the coasts of Maine and of the Gulf states. It offers a suggestive metaphor for the present occasion. If the questions, methods, answers and reasons of another thinker, or of a whole movement of thought, whether earlier or later than Peirce, are illuminated by locating them, directly or indirectly, positively or negatively, in relation to those of the C. S. Peirce station, we may count that as part of the range of Peirce's relevance. In the case of an earlier thinker or movement, the relevance does not depend on Peirce's awareness or acknowledgment; in the case of a later thinker or movement, it does not depend on awareness or acknowledgment by that thinker or by one or more representatives of that movement. If the thinker is oneself, the triangulating will of course require a certain detachment. At least within limits, and perhaps even without limit, degrees of nearness or remoteness, likeness or difference, do not as such constitute degrees of relevance. And even if our interest is primarily in philosophical relevance, mathematical or scientific relevance may entail philosophical, and so be counted. (shrink)

This accessible book gives a new, detailed and elementary explanation of the Gödel incompleteness theorems and presents the Chaitin results and their relation to the da Costa-Doria results, which are given in full, but with no ...

Professor Rudolph A. Marcus, recipient of the 1992 Nobel Prize in Chemistry, is a distinguished theoretical chemist. Two important theories happen to bear his name: the Rice Ramsperger Kassel Marcus theory of unimolecular reactions and the Marcus theory of electron transfer reactions. When considering Marcus’ work, one finds characteristics of it that bear striking similarity to those that can be found in the work of some famous scientists. Such characteristics appear then as common recurring patterns in the work of theoreticians. (...) Among them we find the following: the creation of new theories, driven by experiments that preceding theories were unable to explain; the use of thought experiments; the generalization and extensions from one realm of science to another; the making of predictions and the finding of new properties from correlations among series of data; the use of variational principles; the providing of different demonstrations of an equation; the introduction of new types of coordinates; the innovative capacity to use, by somehow recreating it under the pressure of the physical problem, a mathematical method previously unknown to the author; and finally, the presentation of formulations of the same theory in different mathematical and physical languages. Before discussing in some detail the above points, a brief description of Marcus’ work is first presented for the convenience of the reader. (shrink)

IntroductionIn May 2006, the small group of doctoral students working on ecophilosophy at the Higher Institute of Philosophy at K.U.Leuven invited the Dutch environmental philosopher Martin Drenthen to a workshop to discuss his writings on the concept of wilderness, its metaphysical and moral meaning, and the challenge social constructivism poses for ecophilosophy and environmental protection. Drenthen’s publications on these topics had already been the subject of intense discussions in the months preceding the workshop. His presentation on the workshop and the (...) three critical responses by Wim Bollen, Glenn Deliège, and Richard Kover are published here in a thoroughly revised form on the basis of further and ongoing discussions about the issues involved. Three further contributions to the discussion are added: one by Nathan Edward Kowalski, a young Canadian philosopher who studied in Leuven and wrote his as yet unpublished dissertation on “Evil in Nature,” a second one by Kingsley Goodwin who is currently writing a dissertation on deep ecology at University College in Dublin and, finally, my own unsystematic thoughts inspired by Martin Drenthen’s provocative and stimulating hermeneutics of our postmodern understanding of nature and wilderness, as well as the diverse critical responses to Drenthen’s views by the other contributors. I owe gratitude to all of them for enriching my own ecophilosophical thought and enlarging its horizon although as my fragmentary reflections show at the cost of yet greater uncertainty and more questions about the possibilities of arguing for a radical environmentalist position. Such a position would somehow connect or ground our moral obligation to preserve genuine non-human otherness, wild and free nature, in the non-human meaning of such nature and in our essential need to be in direct and concrete contact with that field of wild non-human meaning because we ourselves belong to nature, because we are not only and purely civilized but are also and most fundamentally wild and free. Three fundamental questions need to be distinguished: the question about our moral obligations regarding nature and its preservation, the metaphysical question about the nature of nature or the meaning of nature, and the question of the metaphysical grounding of our morals in general, and of our moral obligation regarding nature in particular, in the nature or meaning of nature. The complex intertwinement between these three questions is at the heart of Martin Drenthen’s reflections.The authors contributing to the present discussion approach the various issues from markedly different philosophical backgrounds and concerns: critical theory, anthropology, analytic philosophy, phenomenology, hermeneutics, and deep ecology are all to some degree or other brought to bear on the question of what wilderness means for us, why we are fascinated by it and what it actually and in the first place is.Another interesting difference between the five respondents to Martin Drenthen’s wilderness-philosophy relates to the place from which they write: while the Canadians Richard Kover and Nathan Edward Kowalski have wilderness of some sort still close at hand, the Belgians Wim Bollen and Glenn Deliège write from a place which has been thoroughly domesticated, cultivated, and industrialised for a long time. Martin Drenthen himself, of course, writes from the same kind of place as these last two, which explains why he only refers to the wilderness that he has read about in novels, seen in movies, or experienced as recreated in the reserves of the Netherlands. The Irish Kingsley Goodwin writes perhaps from a third, intermediate place that is not wild anymore but still much more natural than Belgium or the Netherlands are.The differences in philosophical approach and concern combined with the difference of place generate a rich tapestry of perspectives and voices wrestling with the fundamental questions of our perceived homelessness and alienation from nature, our cultural discontent, our longing for and need for radical alterity, of immanence and transcendence, of who we are and where we belong, and more concretely of whether there is more to ground our efforts to preserve nature than enlightened self-interest and subjective preference. Martin Drenthen deserves praise for inaugurating this fruitful discussion with his intriguing and highly paradoxical effort to hybridize a postmodern deconstruction of the idea and ideal of wilderness as unspoilt and true nature on the one hand with a new metaphysics of wilderness on the other. Arne Naess, who is generally regarded as the founding father of deep ecology, pointed out the positive and stimulating role that vagueness in the expression of ideas can play in political and philosophical debate. Such general and vague but highly suggestive expressions as ‘all life is one’ are not empty at all, as is often claimed; they are rather full of possibilities of ‘precisation’ and differentiation. Drenthen’s wilderness-philosophy is certainly not vague to the degree of ‘all life is one,’ but his expositions are similarly pregnant with a wealth of further questions, precisations, distinctions, and conceptual clarifications. Good examples of different kinds are to be found in the contributions of Nathan Edward Kowalski and Richard Kover. Kowalski points out that Drenthen’s notions of culture and wilderness need to be made more precise by distinguishing them from the notions of civilization and wildness such that wild cultures become a conceptual possibility. Richard Kover takes the lead from Drenthen’s question about the reason for our postmodern fascination for wilderness to confront it with our primordial fascination in terms of the dependence in the evolution of our subjectivity and our consciousness on the focused perceptual attention on wild forms.Glenn Deliège situates Martin Drenthen’s wilderness-philosophy in its original context of the Dutch debate on the preservation of nature, in particular on the new wilderness paradigm. According to this paradigm, the nature to be preserved should be true and pure, truly natural and wild nature, that is nature outside of and free from the cultivating and domesticating influence of humans. In the Netherlands, however, such nature no longer exists; it can only be preserved after it has first been artificially recreated. Deliège gives an informative summary account of the highly charged three-cornered debate between the adepts of this wilderness-paradigm, the proponents of the more traditional form of conservation of remnants of the typical European landscapes created by pre-industrial farming practices and the modern technocratic functionalists. This debate makes obvious that there are quite different views of the nature of nature and its value. Enter social constructivism: the contested nature of nature is nothing more than a social construct, an idea in a collective mind. The contestation itself is a struggle for authority and power between different such social constructs, any claim to scientific objectivity and objective truth for one’s image of nature is an arrogation of a truth which doesn’t exist and is potentially oppressive by silencing other views. Nature then is up for the grabs of politics, political negotiation, and compromise between various interest groups who all want to get some benefit from nature, some use-value.The problem with Drenthen’s effort to articulate a normative concept of nature which cannot be deconstructed in this way is the extreme formality and emptiness of his metaphysical concept of wilderness as that which is always ‘beyond’, always ‘more’ and which can never be exhausted by any image and construction of nature. Any more concrete determination is mere appearance; the thing in itself is a border concept of that which in fact cannot be grasped conceptually. Glenn Deliège is critical of this formality and emptiness from which it will be impossible to derive any concrete guidance for our conservation practices. He then goes on to defend the Arcadian tradition of our appreciation of nature as giving us a historically well-grounded orientation for those practices. Of course that Arcadian tradition is obviously a cultural construction but that doesn’t make it completely arbitrary and contingent since it is itself ultimately grounded in and open to modification by our concrete sensuous experiences of nature.Deliège here raises the very important issue of the relation between our concrete perceptual and bodily experiences in and with nature and our concepts and metaphysical images of nature. Drenthen himself approaches this issue from the hermeneutical perspective of our need to make sense of our experiences and to articulate their meaning. His confrontation with social constructivism and his effort to salvage something from its relativizing acid lead him, however, into the direction of a highly abstract and empty concept of wilderness.Richard Kover shows that perception is, indeed, the key to a radical critique of social constructivism. He does this with the help of the ecological anthropology of Paul Shepard, the structuralist anthropology of Claude Levi-Strauss and the phenomenology of Maurice Merleau-Ponty. It needs to be said here that Kover takes his leave from an unpublished paper by Martin Drenthen with the title “How to appropriate wilderness appropriately?” – the paper is accessible though on Drenthen’s web-site - in which Drenthen comments and reflects upon three contemporary wilderness-tales, one of them being Werner Herzog’s much-discussed documentary on Timothy Treadwell, an American environmentalist drop-out who lived thirteen summers among wild grizzlies in the Alaskan wilderness before being killed together with his girlfriend by a grizzly bear.Kover tries to show that we are not imprisoned in the mirror-cabinet of our mind or of our human sociality. To make his case, he starts from Paul Shepard’s thoughtful observation that the evolutionary emergence of the mind did not remove us from the natural world but actually connected us deeper with it. Through our mind we evolved more fully into it because our mind is primordially in our senses and in the ability to articulate a perceptual world. The natural world in the form of the system of natural species and their relations is the primary model for perceptual discriminations, for the fundamental discovery of order with its twin-aspects of distinction and discontinuity on the one side and relations and continuity on the other side. Our cognition is essentially perceptually based, our subjectivity is primordially informed by our inherence in the sensible world. Perceptual cognition is inherently intentional and relational, it is not an autonomous mental construction, it is rather in the world and informed by that which is perceived. The other is always intertwined with the subject. We are always already hooked up with the world, with the others, before we start reflecting, thinking, and conceptualizing. This is one of the major themes in Shepard’s work: what this insight means with regard to our relations to non-human otherness, in particular to our perceptual encounter with wild animals. In our encounter with non-human otherness, we are confronted with an essentially ambiguous sense of both affinity and disjuncture. Wild animals demonstrate to us that we are not homeless or alien to this world, that we fundamentally belong to it and are part of it. Their difference, however, shows us that there are different perspectives and ways of being in the world. “Thus the encounter with non-human others inform humans that we are both of the world and simultaneously not the world and there are horizons of significance beyond our own perspective.”Nathan Edward Kowalski comes to a similar conclusion when he situates our morality in the larger moral order of wild nature. According to Kowalski it is indeed a moral order though not our own, an Other to our morality but not simply a negative one. It is, as Drenthen says, a highly paradoxical amoral morality. But what can it mean to speak of morality here? For Kowalski it means that the issue is not merely the theoretical recognition and acknowledgment of axiological transcendence, of non-human values out there, but whether we will appropriately conform to and comport ourselves towards that axiological transcendence so that we can find a human homeland in wild nature instead of turning it into a wasteland. The question then is, how to fit our own human morality into this larger order and greater reality, to subordinate it to that greater reality as a part of it and encompassed but not denied by it so that we can be moral and natural at the same time, tender carnivores hunting the sacred game. Indigenous people follow a moral protocol in their relationship with non-human others which acknowledge and respects their autonomy, their alterity, and the sacred mystery of their different ways of being. They are not befriended but some of them are hunted and eaten – as an exchange in the gift-economy of nature.The problem of the homelessness and alienation from nature is discussed by Wim Bollen from the radically different vantage point of critical theory. In Adorno’s and Horkheimer’s Dialectic of Enlightenment the human story unfolds as a tragic effort to liberate ourselves from nature and to become fully human through rational control when in the end this effort turns against ourselves and instead of liberation and human flourishing we get social domination and repression of our natural needs. The frustrations and mutilations imposed upon us by a repressive civilization give rise to the regressive desire for a return to a natural existence. Referring to the myth of Odysseus and Circe as interpreted by Adorno and Horkheimer, Wim Bollen equates such a natural existence with animal life. In the light of contemporary knowledge about primitive existence, however, this equation is highly problematic. Palaeolithic foragers were already fully human in all respects. A life without civilization need not be an animal-existence.According to Adorno/Horkheimer, our subjectivity and autonomy have to be wrested from animal-nature through an exertive struggle that cannot but proceed by a threefold strategy of domination: the domination of non-human otherness, of human others, and of our own inner nature. The siren song of nature consists in the temptation to give up that struggle for the promise of an unreflective, non-alienated natural existence without toil.When Odysseus’ men give in to Circe’s seduction they are, however, not transformed into wild animals but into swine. According to Bollen this signifies that we cannot re-wild ourselves, that we remain imprisoned in civilization, either as master-subjects or as domesticated animals. Bollen, by the way, like Drenthen, does not distinguish conceptually between civilization and culture. Since re-wilding for Bollen means the impossibility of returning to animal-existence, which is actually the impossibility of becoming another species since biologically we are animals already, to be imprisoned in civilisation means nothing else than to be imprisoned in our humanity. The fact that we cannot become wild animals again – according to Paul Shepard, however, we are still wild animals – is indeed, as Bollen stresses, a trivial form of alienation. A more meaningful concept of alienation which can be put to critical use must refer to a culturally and historically determined frustration and deficit inside the cultural order that is to be overcome not by an impossible regression but by progressive politics leading beyond the three intertwined forms of domination. Not back to nature but forward towards a reconciliation of culture with nature! Such a reconciliation would neither collapse culture into the immanence of nature nor submerge nature into the transcendence of culture. According with the three forms of domination it would be a three-dimensional form of reconciliation.Odysseus’ men did not got what they thought were promised them: the complete liberation from the burden of being autonomous subjects, of being civilized, but they got something worthwhile, namely a certain from of pleasure, a limited and provisional relief and escape. Bollen sees a critical potential in these piggish pleasures after all and consequently even in the siren song of nature since these pleasures were only experienced because Odysseus’ men gave in to it. But one has to give in to this siren song critically if it is to unfold its liberating potential. The problem with pleasure though in bourgeois society is that it is necessarily betrayed, instrumentalized and sold.Kingsley Goodwin’s paper starts by defending deep ecology against the charge of asserting a naïve and dogmatic metaphysics of nature. For convenience sake, but not quite unproblematically, he identifies deep ecology with the environmental thought of Arne Naess. A metaphysics of nature for Naess is always part of a personal worldview. Worldviews which are partly inspired by ecology Naess calls ecosophies. Naess always stresses the personal character of such a worldview and the irreducible plurality of them. It used to be a common misunderstanding of the platform of deep ecology which Naess conceived together with George Sessions that it expressed in a condensed form the metaphysical beliefs of deep ecology. Naess’ intention, however, was to try to articulate in a general fashion the basic viewpoints on which the supporters of the deep ecology movement agreed. It would be possible then to derive these viewpoints from quite different worldviews. In light of this, I think, that Goodwin’s characterisation of the platform as “pragmatic and activist” is not quite appropriate.Goodwin goes on to compare Drenthen’s conception of wilderness and wildness with the similar position developed by Neil Evernden in his book The Social Creation of Nature. He like Naess in his personal ecosophy stresses the crucial role of perceptual experience of wild otherness to find an antidote to the relativity and contingency of our interpretations of nature. Evernden refers to studies on child development to argue for a non-dualistic and relational conception of the self, similar to Naess’ relational ontology. According to Goodwin, Drenthen, in spite of his hermeneutical approach remains too much indebted to the Western subject/object-dualism. The object, in this case wild nature, is pure otherness.With regard to a compost heap, Goodwin shows that we are not separated from nature by an inseparable gulf, that nature is close at hand, and that we can experience and explore it, appreciate the complex play of identity and difference it consists of, the exchanges and transformations and that we can identify with the organisms involved in these exchanges.*The proceeding remarks only highlight some of the prominent themes and considerations in the various contributions to this topical issue of Ethical Perspectives. Hopefully the reader will feel invited and enticed to read the articles themselves, starting, preferably of course, with the essay by Martin Drenthen, who sparked off this lively discussion on the moral meaning of wild nature. I am honoured and pleased to be able to introduce these rich explorations by such talented young environmental philosophers who are eager to develop environmental philosophy further and lead it into new directions. They all have to wrestle with and find answers to the challenge postmodern deconstruction and social constructivism pose for environmental philosophy. In the face of that challenge, it is even more difficult than it already was to provide solid and convincing arguments for the defence of wild and free nature beyond merely pragmatic arguments in terms of its utilitarian value for us. The reason for that defence is not clear-cut and proves to be just as evasive as the moral meaning of nature itself. But neither that reason nor that meaning are mere phantoms and mirages that vanish when we approach them and try to grasp them. They are more substantial, even if they will never have the solidity of a mathematical proof or a scientific measurement. Their locale is the lifeworld which from the point of view of scientific and logical precision and clarity is a messy place where subject and object, reason and heart, fact and value, experience and thought and, yes, nature and culture cannot be clearly separated. This has important consequences for environmental philosophy and its methodology. Conceptual analysis and logical argumentation have to become sensitive to the richness of experience in the lifeworld and to the diverse efforts of its articulation: i.e., they have to open themselves up and maybe even to subordinate themselves to phenomenological and hermeneutical approaches. For me personally this is one of the most important conclusions I draw from the discussion presented here. (shrink)

This paper seeks to describe the attitudes to science of the higher classes of Liverpool in the early nineteenth century. It does so by examining the roles which science played in the town's major cultural institutions. Consideration of the membership and activities of these societies suggests that most of Liverpool's wealthier citizens saw science as merely one component of a general literary culture; a polite, recreational form of science was best suited for this role.A small group of middle-class men (...) held science in much higher esteem, valuing it particularly as an educational tool. This function was best fulfilled, they believed, by a mathematical and experimental form of science, but few of their fellow citizens shared their opinions and successive attempts to teach this form of science foundered. (shrink)

This book focuses on the fundamental principles behind scientific methods. The author uses concrete examples and illustrations to introduce and explain principles. He also uses analogies to connect different methods or problems to arrive at a general principle or common notion. The book explains how the principles of scientific methods are not only applicable to scientific research but also in our daily lives. It shows how the scientific method is used to understand how and why things happen, to make predictions, (...) to present mistakes, and to solve problems. (shrink)

This is a concise book that introduces students to the basics of logical thinking and important mathematical structures that are critical for a solid understanding of logical formalisms themselves as well as for building the necessary background to tackle other fields that are based on these logical principles. Despite its compact and small size, it includes many solved problems and quite a few end-of-section exercises that will help readers consolidate their understanding of the material. This textbook is essential reading for (...) anyone interested in the logical foundations of Informatics, Computer Science, Data Science, Artificial Intelligence, and other related areas. Written with undergraduate students in these disciplines in mind, this book can very well serve the needs of interested and curious readers who wish to get a grasp of the logical principles upon which these fields are built. This book does not require readers to possess math skills beyond those learned in high school. (shrink)

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs. Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their (...) own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples. (shrink)

Topics covered in Discrete Mathematics have become essential tools in many areas of studies in recent years. This is primarily due to the revolution in technology, communications, and cyber security. The book treats major themes in a typical introductory modern Discrete Mathematics course: Propositional and predicate logic, proof techniques, set theory (including Boolean algebra, functions and relations), introduction to number theory, combinatorics and graph theory. An accessible, precise, and comprehensive approach is adopted in the treatment of each topic. (...) The ability of abstract thinking and the art of writing valid arguments are emphasized through detailed proof of (almost) every result. Developing the ability to think abstractly and roguishly is key in any areas of science, information technology and engineering. Every result presented in the book is followed by examples and applications to consolidate its comprehension. The hope is that the reader ends up developing both the abstract reasoning as well as acquiring practical skills. All efforts are made to write the book at a level accessible to first-year students and to present each topic in a way that facilitates self-directed learning. Each chapter starts with basic concepts of the subject at hand and progresses gradually to cover more ground on the subject. Chapters are divided into sections and subsections to facilitate readings. Each section ends with its own carefully chosen set of practice exercises to reenforce comprehension and to challenge and stimulate readers. As an introduction to Discrete Mathematics, the book is written with the smallest set of prerequisites possible. Familiarity with basic mathematical concepts (usually acquired in high school) is sufficient for most chapters. However, some mathematical maturity comes in handy to grasp some harder concepts presented in the book. (shrink)

Does intersectionality divide marginalized groups (e.g., women) along identity lines (e.g., race, class, and sexuality)? In response to the criticism that intersectional approaches to feminist and critical race theories lead to fragmentation and division, this paper notes that it relies on an ontological (mis)understanding of identity as a fixed entity. I argue against this notion of identity by engaging in a detailed case study of how Asian American women experience their Asian identity. The case study demonstrates that identity is lived (...) not as a pre-given static object but rather as a fluid and flexible process: what Asian identity means varies according to how this identity is related to the dynamics of power. I propose a tripartite taxonomy of identity-power relationships exhibited in Asian American women’s lives, in which identities are 1) constructed by and 2) used to reproduce power as oppression, as well as 3) used to build power as resistance. In particular, I explore Black and Asian feminist solidarities to show that those at the margins recreate their identities and, in doing so, build positive forms of power such as resistance and solidarity. The criticism that intersectionality leads to fragmentation does not hold, as it is grounded in a flawed notion of identity that fails to consider how identities are actually lived in their changing relationships to power—especially how marginalized groups, such as women of color, build solidarity to resist oppression by recreating their identities, rather than being fragmented into smaller and mutually exclusive subgroups. (shrink)