This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove (...) the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof. (shrink)

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how individuals have valued generality and how they have worked with specific types of "general" entities, (...) procedures, and arguments. (shrink)

Les documents chinois, depuis le Iersiècle, indiens, depuis le Vesiècle, et arabes, depuis le IXesiècle, contiennent des procédures tabulaires similaires pour l'extraction de racines carrées et cubiques avec des systèmes de numération positionnels. Par ailleurs tant Jia Xian, astronome chinois du XIesiècle, qu'al-Samaw'al, mathématicien arabe du XIIesiècle, ont extrait des racines de degré plus élevé par la procédure dite de Ruffini-Horner. L'article tente de définir une méthode textuelle pour organiser ce corpus, en y distinguant des axes pertinents qui permettent de (...) dégager similarités et différences, d'un point de vue tant historique que conceptuel. Une première partie analyse trois états différents des descriptions d'algorithmes entre le Iersiècle et le XIesiècle en Chine, qui présentent chacun une stabilité historique certaine. La réécriture qui fait passer d'un état au suivant laisse émerger progressivement une uniformité dans les composantes de l'algorithme, laquelle culmine avec des procédures du type Ruffini-Horner. Des critères textuels font apparaître une affinité plus grande de certains algorithmes, tels ceux décrits par Kūshyār ibn Labbān (ca1000), avec les textes chinois qu'avec les textes indiens, plus proches, eux, des algorithmes décrits par al-Khwārizmī. Des critères de même nature lient, d'une part, les algorithmes de Jia Xian et d'al-Samaw'al, d'autre part les algorithmes de Kūshyār et d'al-Samaw'al. (shrink)

Cultural accounts of scientific ideas and practices have increasingly come to be welcomed as a corrective to previous—and still widely held—theories of scientific knowledge and practices as universal. The editors caution, however, against the temptation to overgeneralize the work of culture, and to lapse into a kind of essentialism that flattens the range and variety of scientific work. The book refers to this tendency as culturalism. The contributors to the volume model a new path where historicized and cultural accounts of (...) scientific practice retain their specificity and complexity without falling into the traps of culturalism. They examine, among other issues, the potential of using notions of culture to study behavior in financial markets; the ideology, organization, and practice of earthquake monitoring and prediction during China's Cultural Revolution; the history of quadratic equations in China; and how studying the "glass ceiling" and employment discrimination became accepted in the social sciences. Demonstrating the need to understand the work of culture as a fluid and dynamic process that directly both shapes and is shaped by scientific practice, Cultures without Culturalism makes an important intervention in science studies. Contributors. Bruno Belhoste, Karine Chemla, Caroline Ehrhardt, Fa-ti Fan,Kenji Ito, Evelyn Fox Keller, Guillaume Lachenal, Donald MacKenzie, Mary S. Morgan, Nancy J. Nersessian, David Rabouin, Hans-Jörg Rheinberger, Claude Rosental, Koen Vermeir. (shrink)

The book presents the outcomes of an innovative research programme in the history of science and implements a Text Act Theory which extends Speech Act Theory, in order to illustrate a new approach to texts and textual communicative acts. It examines assertives (absolute or conditional statements, forecasts, insurance, etc.), directives, declarations and enumerations, as well as different types of textual units allowing authors to perform these acts: algorithms, recipes, prescriptions, lexical templates for terminological studies and enumerative structures. The book relies (...) on the study of a broad range of documents of the past dealing with various domains: mathematics, zoology, medicine, lexicography. The documents examined come from scholarly sources from different parts of the world, such as China, Europe, India, Mesopotamia and are written in a variety of European languages as well as Chinese, Cuneiform and Sanskrit. This approach proves fruitful in both history of science and Text Act Theory. (shrink)

The ArgumentTo highlight speculative trends specific to the mathematical tradition that developed in China, the paper analyzes an excerpt of a third-century commentary on a mathematical classic, which arguably contains a proof. The paper shows that the following three tasks cannot be dissociated one from the other: (1) to discuss how the ancient text should be read; (2) to describe the practice of mathematical proof to which this text bears witness; (3) to bring to light connections between philosophy and mathematics (...) that it demonstrates were established in China. To this end the paper defines its use of the word “proof” and outlines a program for an international history of mathematical proof. It describes the sense in which the text conveys a proof and shows how it simultaneously fulfills algorithmic ends, bringing to light a formal pattern that appears to be fundamental both for mathematics and for other domains of reality. The interest in transformations that mathematical writings demonstrate in China at that time seems to have been influenced by philosophical developments based onThe Book of Changes (Yi-jing), which the excerpt quotes. This quotation within a mathematical context makes it possible to suggest an interpretation for a rather difficult philosophical statement. (shrink)

Volume 7 of Joseph Needham’s Science and Civilisation in China (SCC) gives the issue of Chinese language in scientific practice pride of place. Its arguments are mainly devised to oppose views, put forward by Marcel Granet in 1920 and then by Derk Bodde in the 1970s, to the effect that the Chinese script (for Granet) or “literary Chinese” (for Bodde) impeded the development of science. The essay outlines the way in which Christoph Harbsmeier, in Part 1 of Volume 7, and (...) then Needham and Kenneth Robinson, in Part 2, expose the fallacies of these views. It suggests that Needham and Robinson’s rebuttals to Bodde’s position, innovative though they might be, share essential features with the latter’s theses. In particular, they all view the Chinese language primarily from the viewpoint of its function of communicating knowledge. The essay argues that Chinese-writing communities of practitioners of mathematics constantly shaped technical languages that drew on but also added to the resources of the Chinese language available to them to pursue their goals. The technical languages thereby shaped were not simply a means of communication with fellow-practitioners; they were also a tool of work. In conclusion, the essay argues that a language and the work done with this language are co-constructed. (shrink)

This is the first book-length analysis of the techniques and procedures of ancient mathematical commentaries. It focuses on examples in Chinese, Sanskrit, Akkadian and Sumerian, and Ancient Greek, presenting the general issues by constant detailed reference to these commentaries, of which substantial extracts are included in the original languages and in translation, sometimes for the first time. This makes the issues accessible to readers without specialized training in mathematics or in the languages involved. The result is a much richer understanding (...) than was hitherto possible of the crucial role of commentaries in the history of mathematics in four different linguistic areas, of the nature of mathematical commentaries in general, of the contribution that the study of mathematical commentaries can make to the history of science and to the study of commentaries in general, and of the ways in which mathematical commentaries are like and unlike other kinds of commentaries. (shrink)

Alors qu'ils présentent tous deux, sous l'impulsion de la tentative comtienne, des histoires globales des sciences, Paul Tannery et Joseph Needham mettent en place des cadres historiographiques de fait distincts. Tannery fonde son histoire générale sur la notion de civilisation, dont la science est un élément intégrant. L'outil méthodologique qu'il utilise pour dégager les traits saillants qui caractérisent une civilisation donnée à une époque donnée sous l'aspect de la science est la synthèse. Needham part de la science moderne, et se (...) donne pour objectif de mettre en évidence qu'elle fut constituée par un processus de synthèse des apports de différentes traditions. Pour lui, la formation de ce savoir moderne a procédé d'un effort continu qui a tour à tour été mené par les divers groupes composant l'humanité. Tannery, en revanche, est très préoccupé par les ruptures et les crises, qui peuvent être à l'origine de la disparition de pans entiers du savoir, mais aussi d'intenses périodes de création. Ce qu'il est convenu d'appeler« révolution scientifique » prend alors un sens très différent pour les deux historiens. (shrink)

The earliest extant mathematical books from China contain a lot of problems and data about grains. They also betray a close relationship with imperial bureaucracy in this respect. Indeed, these texts quote administrative regulations about grains. For instance, the Book on mathematical procedures 筭數書, found in a tomb sealed ca. 186 BCE, has a section in common with the “regulations on granaries” from the Qin statutes in eighteen domains, known thanks to slips excavated at Shuihudi. Mathematical writings also deal with (...) official vessels used to measure grains. They cast light on statements from, and practices evidenced by, official histories and administrative documents. This article addresses the following issues. Which information about the concrete management of grains can we derive from mathematical writings in relation to administrative documents? Which data can we find in these writings about continuities and changes in the management of grains in the time span between the Qin and Han dynasties? In particular, how can we account for the fact that in a later mathematical text, namely, The Nine Chapters, probably completed in the first century CE, there was a change in the form in which the data about grain equivalences were given, by comparison with the “regulations on granaries”? Finally, what do our conclusions imply with respect to the nature of the earliest extant mathematical writings. In this article, we gather the various types of statement that mathematical writings contain about grains and offer several elements of interpretation for the “regulations on granaries” and the related text in The Nine Chapters. From this perspective, we offer several hypotheses about the management of grain in the Qin and Han dynasties. (shrink)

In the first part of chapter 2 of book II of the Physics Aristotle addresses the issue of the difference between mathematics and physics. In the course of his discussion he says some things about astronomy and the ‘ ‘ more physical branches of mathematics”. In this paper I discuss historical issues concerning the text, translation, and interpretation of the passage, focusing on two cruxes, the first reference to astronomy at 193b25–26 and the reference to the more physical branches at 194a7–8. In (...) section I, I criticize Ross’s interpretation of the passage and point out that his alteration of has no warrant in the Greek manuscripts. In the next three sections I treat three other interpretations, all of which depart from Ross's: in section II that of Simplicius, which I commend; in section III that of Thomas Aquinas, which is importantly influenced by a mistranslation of, and in section IV that of Ibn Rushd, which is based on an Arabic text corresponding to that printed by Ross. In the concluding section of the paper I describe the modern history of the Greek text of our passage and translations of it from the early twelfth century until the appearance of Ross's text in 1936. (shrink)

A multiplicity of circumstances - including geographic and political isolation, and differences of social organization and customs - has led different groups of people to develop mathematical knowledge independently of each other. Yet history has shown us again and again that by some necessity these separate groups often encounter the same problems. The solutions they propose, however, are often different. This suggests a series of questions. First of all: what is the relationship between the solutions? Is one solution an alternative (...) to the other, and if so, how to account for their differences? Or, on the contrary, are the solutions complementary? Both of these configurations can be found in the history of mathematics: in the following article we will investigate two elementary examples of these phenomena and trace the historical circumstances under which different solutions have come into contact. (shrink)

Suszko’s problem is the problem of finding the minimal number of truth values needed to semantically characterize a syntactic consequence relation. Suszko proved that every Tarskian consequence relation can be characterized using only two truth values. Malinowski showed that this number can equal three if some of Tarski’s structural constraints are relaxed. By so doing, Malinowski introduced a case of so-called mixed consequence, allowing the notion of a designated value to vary between the premises and the conclusions of an argument. (...) In this article we give a more systematic perspective on Suszko’s problem and on mixed consequence. First, we prove general representation theorems relating structural properties of a consequence relation to their semantic interpretation, uncovering the semantic counterpart of substitution-invariance, and establishing that mixed consequence is fundamentally the semantic counterpart of the structural property of monotonicity. We use those theorems to derive maximum-rank results proved recently in a different setting by French and Ripley, as well as by Blasio, Marcos, and Wansing, for logics with various structural properties. We strengthen these results into exact rank results for nonpermeable logics. We discuss the underlying notion of rank, and the associated reduction proposed independently by Scott and Suszko. As emphasized by Suszko, that reduction fails to preserve compositionality in general, meaning that the resulting semantics is no longer truth-functional. We propose a modification of that notion of reduction, allowing us to prove that over compact logics with what we call regular connectives, rank results are maintained even if we request the preservation of truth-functionality and additional semantic properties. (shrink)

In this article we refine the design of context shifting experiments, which play a central role in contextualist debates, and we subject a large number of scenarios involving different types of expressions of interest to contextualists, including ‘know’ and color adjectives like ‘green’, to experimental investigation. Our experiment (i) reveals an effect of changing contexts on the evaluation of uses of the sentences that we examine, thereby overturning the absence of results reported in previous experimental studies (so‐called null results), (ii) (...) uncovers evidence for a ‘truth bias' in favor of positive over negative sentences, and (iii) reveals previously unnoticed distinctions between the strength of the contextual effects displayed by scenarios involving knowledge ascriptions and scenarios concerning color and other miscellaneous scenarios. (shrink)

The presupposition triggered by an expression E is generally satisfied by information that comes before rather than after E in the sentence or discourse. In Heim’s classic theory (1983), this left-right asymmetry is encoded in the lexical semantics of dynamic connectives and operators. But several recent analyses offer a more nuanced approach, in which presupposition satisfaction has two separate components: a general principle (which varies from theory to theory) specifies under what conditions a presupposition triggered by an expression E is (...) satisfied; and an ‘incremental’ component specifies that the principle must be checked on the basis of information that comes before E. Several researchers take this incremental component to be a processing bias, which can be overcome at some cost. If so, it should be possible, though costly, to satisfy presuppositions ‘symmetrically’, i.e. by taking into account linguistic material that comes both before and after the presupposition trigger. We test this claim with experimental means. Using inferential (and to some extent acceptability) tasks involving the anaphoric trigger aussi (‘too’) in French, we argue that symmetric readings are indeed possible (albeit degraded) in environments involving the connectives if, or, and unless. (shrink)

Presupposition, vagueness, and oddness can lead to some sentences failing to have a clear truth value. The homogeneity property of plural predication with definite descriptions may also create truth-value gaps: The books are written in Dutch is true if all relevant books are in Dutch, false if none of them are, and neither true nor false if, say, half of the books are written in Dutch. We study the projection property of homogeneity by deploying methods of general interest to identify (...) truth-value gaps. Method A consists in collecting both truth judgments and, independently, falsity judgments. The second method, employed in experiment series B and C, is based on one-shot ternary judgments: completely true vs. completely false vs. neither. After a calibration of these methods, we use them to demonstrate that homogeneity projects out of negation, the scope of universal sentences and the scope of non-monotonic quantifiers such as exactly two, to some extent. We assess our results in light of different theoretical approaches to homogeneity—approaches based on presuppositions, scalar implicatures, and something like supervaluations, respectively. We identify free parameters in these theories and assess various variants of them based on our results. Our experimental paradigms may be of broader significance insofar as they can be applied to other phenomena which result in the failure of a sentence to have a definite truth value. (shrink)

How does science create knowledge? Epistemic cultures, shaped by affinity, necessity, and historical coincidence, determine how we know what we know. In this book, Karin Knorr Cetina compares two of the most important and intriguing epistemic cultures of our day, those in high energy physics and molecular biology. The first ethnographic study to systematically compare two different scientific laboratory cultures, this book sharpens our focus on epistemic cultures as the basis of the knowledge society.