The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.

One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)

This paper studies Paul Cohen’s philosophy of mathematics and mathematical practice as expressed in his writing on set-theoretic consistency proofs using his method of forcing. Since Cohen did not consider himself a philosopher and was somewhat reluctant about philosophy, the analysis uses semiotic and literary textual methodologies rather than mainstream philosophical ones. Specifically, I follow some ideas of Lévi-Strauss’s structural semiotics and some literary narratological methodologies. I show how Cohen’s reflections and rhetoric attempt to bridge what he experiences as an (...) uncomfortable tension between reality and the formal by means of his notion of intuition. (shrink)

"Roy Wagner is a one-of-a-kind anthropologist whose books provide intense intellectual stimulation. His way of connecting the world of New Guinea to the world of anthropology is unique and, well, mind-blowing.... He writes books that you actually want to and will read more than once."--Steven Feld, author of Sound and Sentiment "Wagner asks, daringly, what it would be like to imagine one of the most significant of human activities, the activity of description or representation, as a self-scaling phenomenon.... (...) One begins to glimpse a genuine 'alternative anthropology.'"--Marilyn Strathern, author of The Gender of the Gift. (shrink)

This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post-structuralist train of thought going from Mauss and L vi-Strauss to Baudrillard and Derrida. We import these authors' semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of mathematical (...) signs to reinterpretation, and bridge some barriers between philosophy of mathematics and critical approaches to knowledge. (shrink)

This article interprets Józef Maria Hoëné Wronski’s (1776–1853) use of actual infinities in his mathematical work. The interpretation places this usage, which undermined Wronski’s acceptance as a mathematician, in his contemporary mathematical and philosophical context and in the context of his own sociopolitical-philosophical project.

This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida and de Man.

This paper considers two models of embodied mathematical cognition, and analyses the image of mathematics that they support.

This article studies Ibn al-Haytham’s treatment of the common notions from Euclid’s Elements (usually referred to today as the axioms). We argue that Ibn al-Haytham initiated a new approach with regard to these foundational statements, rejecting their qualification as innate, self-evident, or primary. We suggest that Ibn al-Haytham’s engagement with experimental science, especially optics, led him to revise the framing of Euclidean common notions in a way that would fit his experimental approach.

This note opens with brief evaluations of classical foundationalist endeavors – those of Frege, Russell, Brouwer and Hilbert. From there we proceed to some pluralist approaches to foundations, focusing on Putnam and Wittgenstein, making a note of what enables their pluralism. Then, I bring up approaches that find foundations potentially harmful, as expressed by Rav and Lakatos. I conclude with a brief discussion of a late medieval Indian case study in order to show what an “unfounded” mathematics could look like. (...) The general purpose is to re-evaluate the desiderata of foundational programs in mathematics. (shrink)

We present the notion of finite high-order Gowers games, and prove a statement parallel to Gowers's Combinatorial Lemma for these games. We derive ‘quantitative’ versions of the original Gowers Combinatorial Lemma and of Gowers's Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces.

ArgumentThis paper argues for the viability of a different philosophical point of view concerning classical Greek geometry. It reviews Reviel Netz's interpretation of classical Greek geometry and offers a Deleuzian, post-structural alternative. Deleuze's notion of haptic vision is imported from its art history context to propose an analysis of Greek geometric practices that serves as counterpoint to their linear modular cognitive narration by Netz. Our interpretation highlights the relation between embodied practices, noisy material constraints, and operational codes. Furthermore, it sheds (...) some new light on the distinctness and clarity of Greek mathematical conceptual divisions. (shrink)

This paper will apply post-structural semiotic theories to study the texts of Gödel's first incompleteness theorem. I will study the texts’ own articulations of concepts of ‘meaning’, analyze the mechanisms they use to sustain their senses of validity, and point out how the texts depend (without losing their mathematical rigor) on sustaining some shifts of meaning. I will demonstrate that the texts manifest semiotic effects, which we usually associate with poetry and everyday speech. I will conclude with an analysis of (...) how the picture I paint relates to an ethics of mathematical production. (shrink)

S(zp,zp) performs an innovative analysis of one of modern logic's most celebrated cornerstones: the proof of Gödel's first incompleteness theorem. The book applies the semiotic theories of French post- structuralists such as Julia Kristeva, Jacques Derrida and Gilles Deleuze to shed new light on a fundamental question: how do mathematical signs produce meaning and make sense? S(zp,zp) analyses the text of the proof of Gödel's result, and shows that mathematical language, like other forms of language, enjoys the full complexity of (...) language as a process, with its embodied genesis, constitutive paradoxical forces and unbounded shifts of meaning. These effects do not infringe on the logico-mathematical validity of Gödel's proof. Rather, they belong to a mathematical unconscious that enables the successful function of mathematical texts for a variety of different readers. S(zp,zp) breaks new ground by synthesising mathematical logic and post-structural semiotics into a new form of philosophical fabric, and offers an original way of bridging the gap between the "two cultures". (shrink)

This essay asks: Is “culture” the subject of a communication among anthropologists, or are anthropologists subjects to a communication among cultures? Put more simply, is there only one culture, comprised of multiplex variations recovered from various parts of an ever-changing world, or are there a great many, all of them variations on a single theoretical insight, which anthropologists have made up in secret and carefully keep as a secret from themselves? (Why not? the author asks, adding that such is exactly (...) how the modern state operates.) Is it possible, he further asks, for a memory to have an independent existence, untroubled by the people who constantly keep occurring to it? It is of course absurd to ascribe agency to what amount to mere figures of speech, metaphors, or enigmatic perceptual cues, “but if a metaphor could not think, as an agency in and of itself, then neither could we.” (Memories would have to be metaphors; otherwise how could we forget them?) Finally, the essays asks: What is more natural than the agency of the one you see in the mirror, that steals your act of looking, but only to view itself? (shrink)

A few years ago, a manuscript by Jost Bürgi (1552–1632) was brought to scholarly attention, which included an ingenious sine calculation method. The purpose of this paper is to discuss two aspects of this manuscript. First, we wish to improve the current understanding of Bürgi’s method of sine calculation, especially with respect to the calculation of sines at a resolution of 1 min. Second, we wish to suggest a possible transfer of knowledge between India’s Kerala School of mathematical astronomy and (...) Bürgi. The evidence for the latter seems to be stronger than the evidence for other available case studies, but still revolves mainly around analogies, and can therefore not be considered as conclusive proof of transmission. We also append a translation of the relevant chapter of Bürgi’s treatise. (shrink)

This introduction to the Common Knowledge symposium titled “Comparative Relativism” outlines a variety of intellectual contexts where placing the unlikely companion terms comparison and relativism in conjunction offers analytical purchase. If comparison, in the most general sense, involves the investigation of discrete contexts in order to elucidate their similarities and differences, then relativism, as a tendency, stance, or working method, usually involves the assumption that contexts exhibit, or may exhibit, radically different, incomparable, or incommensurable traits. Comparative studies are required to (...) treat their objects as alike, at least in some crucial respects; relativism indicates the limits of this practice. Jensen argues that this seeming paradox is productive, as he moves across contexts, from Lévi-Strauss's analysis of comparison as an anthropological method to Peter Galison's history of physics, and on to the anthropological, philosophical, and historical examples offered in symposium contributions by Barbara Herrnstein Smith, Eduardo Viveiros de Castro, Marilyn Strathern, and Isabelle Stengers. Comparative relativism is understood by some to imply that relativism comes in various kinds and that these have multiple uses, functions, and effects, varying widely in different personal, historical, and institutional contexts that can be compared and contrasted. Comparative relativism is taken by others to encourage a “comparison of comparisons,” in order to relativize what different peoples—say, Western academics and Amerindian shamans—compare things “for.” Jensen concludes that what is compared and relativized in this symposium are the methods of comparison and relativization themselves. He ventures that the contributors all hope that treating these terms in juxtaposition may allow for new configurations of inquiry. (shrink)

Regular musical activity as a complex multimodal lifestyle activity is proposed to be protective against age-related cognitive decline and Alzheimer’s disease. This cross-sectional study investigated the association and interplay between musical instrument playing during life, multi-domain cognitive abilities and brain morphology in older adults from the DZNE-Longitudinal Cognitive Impairment and Dementia Study study. Participants reporting having played a musical instrument across three life periods were compared to controls without a history of musical instrument playing, well-matched for reserve proxies of education, (...) intelligence, socioeconomic status and physical activity. Participants with musical activity outperformed controls in global cognition, working memory, executive functions, language, and visuospatial abilities, with no effects seen for learning and memory. The musically active group had greater gray matter volume in the somatosensory area, but did not differ from controls in higher-order frontal, temporal, or hippocampal volumes. However, the association between gray matter volume in distributed frontal-to-temporal regions and cognitive abilities was enhanced in participants with musical activity compared to controls. We show that playing a musical instrument during life relates to better late-life cognitive abilities and greater brain capacities in OA. Musical activity may serve as a multimodal enrichment strategy that could help preserve cognitive and brain health in late life. Longitudinal and interventional studies are needed to support this notion. (shrink)

Logical pluralism is the view that there is more than one formal logic that correctly (or best, or legitimately) codifies the logical consequence relation in natural language. This essay provides a taxonomy of different variations on the logical pluralist theme based on a five-part structure, and then identifies an unoccupied position in this taxonomy: perspectival logical pluralism. Perspectival pluralism provides an attractive position from which to formulate a philosophy of logic from a feminist perspective (and from other, identity-based perspectives, such (...) as critical race theory). An example of how such an account might be developed is sketched. The essay concludes by defusing an obvious objection to the perspectival approach: the claim that the correct logic (or logics), in virtue of the formal nature of logic, should be independent of considerations regarding the identity of the reasoner. (shrink)

Can God create a stone too heavy for him to lift? Can time have a beginning? Which came first, the chicken or the egg? Riddles, paradoxes, conundrums--for millennia the human mind has found such knotty logical problems both perplexing and irresistible. Now Roy Sorensen offers the first narrative history of paradoxes, a fascinating and eye-opening account that extends from the ancient Greeks, through the Middle Ages, the Enlightenment, and into the twentieth century. When Augustine asked what God was doing before (...) He made the world, he was told: "Preparing hell for people who ask questions like that." A Brief History of the Paradox takes a close look at "questions like that" and the philosophers who have asked them, beginning with the folk riddles that inspired Anaximander to erect the first metaphysical system and ending with such thinkers as Lewis Carroll, Ludwig Wittgenstein, and W.V. Quine. Organized chronologically, the book is divided into twenty-four chapters, each of which pairs a philosopher with a major paradox, allowing for extended consideration and putting a human face on the strategies that have been taken toward these puzzles. Readers get to follow the minds of Zeno, Socrates, Aquinas, Ockham, Pascal, Kant, Hegel, and many other major philosophers deep inside the tangles of paradox, looking for, and sometimes finding, a way out. Filled with illuminating anecdotes and vividly written, A Brief History of the Paradox will appeal to anyone who finds trying to answer unanswerable questions a paradoxically pleasant endeavor. (shrink)

The paper addresses the nature of seemings in light of their role in inquiry. Seemings are mental states or events with propositional content that have a specific phenomenology often referred to as “felt truth”. In epistemology, seemings are mainly discussed as possible (non-inferential) justifications for belief. Yet, epistemology has recently taken a zetetic turn, that is, a turn toward the study of inquiry. I will argue that the role of seemings in epistemology should be re-assessed from the perspective of inquiry (...) and deliberation. Seemings may fall short of justifying beliefs, but there is an important role for seemings in the zetetic realm, which may also explain why seemings lack direct justificatory force for beliefs. Seemings, so I will suggest, can provide normative reasons for performing certain zetetic tasks, but they cannot provide normative reasons for making up one’s mind about the inquired question. In this sense, the zetetic role of seemings can be described as guiding rational inquiry. I will suggest that a modified version of the so-called Taking Evidence View about the nature of seemings is best fitted to accommodate their zetetic role. Yet, contra existing versions of the Taking Evidence View, my modified account acknowledges that seemings are genuine experiences (of one’s evidential standing during inquiry) that are accompanied by the characteristic phenomenology of what is called a “feeling of truth”. By doing so, my modified version of the Taking Evidence View incorporates the (as I take it) most important feature of the rival Experience View of seemings. (shrink)

This book examines the intricate dynamics of when individuals adhere to laws, taking into account the context in which laws attempt to shape human behavior. While existing literature touches upon various reasons why people comply with laws, the book focuses on a critical question which has been missing from the discussion: when do people obey laws? By treating law as a form of social communication, it develops an integrated framework to answer this question. It explores how social, psychological, and institutional (...) conditions shape compliance decisions of individuals. What does a law signify? When does the compulsion to obey arise? When do individuals comply out of a fear for legal sanctions or social repercussions? Why do some laws have high symbolic values and others fail despite harsh punishments? The book unveils the contextual intricacies that underlie obedience to law. It challenges conventional wisdom and offers a fresh perspective on the power and limitations of law in shaping human behavior. For scholars and academics seeking a deeper understanding of legal compliance and role of law in shaping behaviors, this book will be an indispensable resource. (shrink)

Barion, J. Was es heisst, ein Philosoph zu sein.--Flach, W. Die Objektivität der Erkenntnis.--Schmidt, G. Die Transzendentalität des Seingedankens.--Winterhager, E. Das Sich-Haben des Subjekts.--Hartmann, K. Analytische und kategoriale Transzendentalphilosophie.--Marx, W. Systemidee und die Problematik ihrer Begründung.--Röd, W. Transzendentalphilosophie und deskriptive Philosophie als wissenschaftliche Theorien.--Vuillemin, J. Caractères et fonctions des signes.--Ritzel, W. Zur Theorie praktischer Wissenschaft.--Hufnagel, E. Relationen--zum Streit um die pädagogische Anthropologie.--Derbolav, J. Politik und Moral.--Wolandt, G. Standpunkte der Kunstphilosophie.

Paradoxes are arguments that lead from apparently true premises, via apparently uncontroversial reasoning, to a false or even contradictory conclusion. Paradoxes threaten our basic understanding of central concepts such as space, time, motion, infinity, truth, knowledge, and belief. In this volume Roy T Cook provides a sophisticated, yet accessible and entertaining, introduction to the study of paradoxes, one that includes a detailed examination of a wide variety of paradoxes. The book is organized around four important types of paradox: the semantic (...) paradoxes involving truth, the set-theoretic paradoxes involving arbitrary collections of objects, the Soritical paradoxes involving vague concepts, and the epistemic paradoxes involving knowledge and belief. In each of these cases, Cook frames the discussion in terms of four different approaches one might take towards solving such paradoxes. Each chapter concludes with a number of exercises that illustrate the philosophical arguments and logical concepts involved in the paradoxes. _Paradoxes_ is the ideal introduction to the topic and will be a valuable resource for scholars and students in a wide variety of disciplines who wish to understand the important role that paradoxes have played, and continue to play, in contemporary philosophy. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

An entertaining history of the idea of nothing - including absences, omissions, and shadows - from the Ancient Greeks through the 20th century How can nothing cause something? The absence of something might seem to indicate a null or a void, an emptiness as ineffectual as a shadow. In fact, 'nothing' is one of the most powerful ideas the human mind has ever conceived. This short and entertaining book by Roy Sorensen is a lively tour of the history and philosophy (...) of nothing, explaining how various thinkers throughout history have conceived and grappled with the mysterious power of absence -- and how these ideas about shadows, gaps, and holes have in turned played a very positive role in the development of some of humankind's most important ideas. Filled with Sorensen's characteristically entertaining mix of anecdotes, puzzles, curiosities, and philosophical speculation, the book is ordered chronologically, starting with the Taoists, the Buddhists, and the ancient Greeks, moving forward to the middle ages and the early modern period, then up to the existentialists and present day philosophy. The result is a diverting tour through the history of human thought as seen from a novel and unusual perspective. (shrink)

This introduction reflects on the themes of viscosity, death and the Other in three essays, written by John Milbank, Julia Hell, and Martin Jay, which provide a response — respectively — to three of Professor Zygmunt Bauman’s key works: Legislators and Interpreters, Modernity and the Holocaust, and Liquid Modernity.

In this book, Roy Bhaskar sets out to revindicate ontology, critiquing the reduction of being in favor of knowledge, which he calls the "epistemic fallacy".

The No-No Paradox consists of a pair of statements, each of which ?says? the other is false. Roy Sorensen claims that the No-No Paradox provides an example of a true statement that has no truthmaker: Given the relevant instances of the T-schema, one of the two statements comprising the ?paradox? must be true (and the other false), but symmetry constraints prevent us from determining which, and thus prevent there being a truthmaker grounding the relevant assignment of truth values. Sorensen's view (...) is mistaken: situated within an appropriate background theory of truth, the statements comprising the No-No Paradox are genuinely paradoxical in the same sense as is the Liar (and thus, on Sorensen's view, must fail to have truth values). This result has consequences beyond Sorensen's semantic framework. In particular, the No-No Paradox, properly understood, is not only a new paradox, but also provides us with a new type of paradox, one which depends upon a general background theory of the truth predicate in a way that the Liar Paradox and similar constructions do not. (shrink)