To 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 on The 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.

Critique involves reflection, specifically self-reflection, and as such it is inherently linked with philosophy. Critique calls for change, awareness, liberation from false conceptions, and reshaping of spheres of action and belief. Consequently it is closely linked with the moral and the political. Critique aspires to enhance truth, beauty, and justice and is thus an integral part of science, art, and social action. The present volume tackles issues of critique through a selection of papers originally presented at the workshop on “Models of Critique in the Sciences, Society, and the Arts,” held in Jerusalem and Tel Aviv during May 1996. As the full program indicates, this selection provides only a partial picture of the topics discussed at the workshop. A major consideration has been the publication policy of Science in Context, with its emphasis on science, which is reflected here. We wish to express our gratitude to all the participants for their presentations and contributions to a stimulating exchange of ideas.

This paper focuses on several Italian Jewish philosophers in the second half of the sixteenth century and the first third of the seventeenth century. It argues that their writings share a certain theology of nature. Because of it, the interest of Jews in the study of nature was not a proto-scientific but a hermeneutical activity based on the essential correspondence between God, Torah, and Israel. While the theology of nature analyzed in the paper did not prevent Jews from being informed about and selectively endorsing the first phase of the scientific revolution, it did render the Jews marginal to it. So long as Jewish thinkers adhered to this theology of nature, Jews could not adopt the scientific mentality that presupposed a qualitative distinction between the Book of Nature and the Book of Scripture.

It is well known that the Tractatus of Peter of Spain (later Pope John XXI) was one of the most popular logic textbooks in the Middle Ages and early Renaissance. Less known is the Tractatus's considerable reputation and diffusion among the Jews, as evidenced by five translations, two commentaries, and what appears to be an abbreviatio — if not of the Tractatus itself, then of a similar work. The present article attempts to understand the phenomenon of the Tractatus's popularity and offers an analysis of the three translations whose authors are known — those by Shemaryah ha-Ikriti (Greece, early to mid-fourteenth century), Abraham Abigdor (late fourteenth century), and Judah b. Samuel Shalom (either Italy or Spain, mid-fifteenth century) — and their subsequent fate. The more popular versions of Abraham Abidgor and Judah Shalom provided Jewish students, many of whom would likely become physicians, with a grounding in logic comparable to that of their Christian counterparts.

In this paper I argue that Kuhn's and Hanson's notion of incommensurable paradigms is rooted in the rhetoric of finality of the Copenhagen dogma — the orthodox philosophical interpretation of quantum physics. I also argue that arguments for holism of a paradigm, on which the notion of the impossibility of its gradual modification is based, misinterpret the Duhem-Quine thesis. The history of science (Copernican, Chemical, and Quantum Revolutions) demonstrates fruitful selective appropriation of ideas from seemingly “incommensurable” paradigms (rather than the impossibility of communication between their representatives). I argue in conclusion that such openness underlies scientific creativity, and that dialogue between different alternatives is indispensable for the growth of knowledge.

The belief in the existence of eternal mathematical truth has been part of this science throughout history. Bourbaki, however, introduced an interesting, and rather innovative twist to it, beginning in the mid-1930s. This group of mathematicians advanced the view that mathematics is a science dealing with structures, and that it attains its results through a systematic application of the modern axiomatic method. Like many other mathematicians, past and contemporary, Bourbaki understood the historical development of mathematics as a series of necessary stages inexorably leading to its current state — meaning by this, the specific perspective that Bourbaki had adopted and were promoting. But unlike anyone else, Bourbaki actively put forward the view that their conception of mathematics was not only illuminating and useful for dealing with the current concerns of mathematics, but that this was in fact the ultimate stage in the evolution of mathematics, bound to remain unchanged by any future development of this science. In this way, they were extending in an unprecedented way the domain of validity of the belief in the eternal character of mathematical truths, from the body to the images of mathematical knowledge.

Bourbaki were fond of presenting their insistence on the centrality of the modern axiomatic method as a way to ensure the eternal character of mathematical truth as an offshoot of Hilbert's mathematical heritage. A detailed examination of Hilbert's actual conception of the axiomatic method, however, brings to the fore interesting differences between it and Bourbaki's conception, thus underscoring the historically conditioned character of certain, fundamental mathematical beliefs.

Conventional wisdom has it that Ashkenazic rabbinic culture was far less receptive to non-Jewish learning and worldly disciplines than its Sephardic counterpart. Whereas great Sephardic rabbis such as Maimonides and many others were masters of philosophy, medicine, and science, Ashkenazic rabbis usually restricted their intellectual horizons to talmudic literature and, in the best of cases, “broadened” them to include the Bible and/or Kabbalah. Ashkenazic rabbinic culture was, according to this image, insular and unidimensional.

Two dominant models of criticism are identified and analyzed. One is selfconsciously normative. It conceives of criticism as subject to strict logical rules. The other views itself as essentially descriptive and accounts for the critical activity in terms of social factors. In spite of their different origins and purposes, it is argued that both models share a reductionistic thrust, which minimizes the role of the critic qua agent. It is further agreed that neither provides an adequate account of critical activity and its role in science.

The major part of the mathematical “classics” in Hebrew were translated from Arabic between the second third of the thirteenth century and the first third of the fourteenth century, within the northern littoral of the western Mediterranean. This movement occurred after the original works by Abraham bar Hiyya and Abraham ibn Ezra became available to a wide readership. The translations were intended for a restricted audience — the scholarly readership involved in and dealing with the theoretical sciences. In some cases the translators themselves were professional scientists (e.g., Jacob ben Makhir); in other cases they were, so to speak, professional translators, dealing as well with philosophy, medicine, and other works in Arabic.

In aketshing this portrait of the beginning of Herbrew scholarly mathematics, my aim has been to contribute to a better understanding of mathematical activity as such among Jewish communities during this period.

The group of mathematicians known as Bourbaki persuasively proclaimed the isolation of its field of research — pure mathematics — from society and science. It may therefore seem paradoxical that links with larger French cultural movements, especially structuralism and potential literature, are easy to establish. Rather than arguing that the latter were a consequence of the former, which they were not, I show that all of these cultural movements, including the Bourbakist endeavor, emerged together, each strengthening the public appeal of the others through constant, albeit often superficial, interaction. This codependency is partly responsible for their success and moreover accounts for their simultaneous fall from favor, which, however, can clearly be seen as also stemming from different internal problems. To understand this dynamics, I argue that Bourbaki's role can best be captured by using the notion of cultural connector, which I introduce here.

This paper makes use of the 1860 clash between T. H. Huxley and Richard Owen to examine the role of social context in scientific advance in the biological sciences. It shows how the social context of nineteenth-century England first favored the Coleridge-Owenite interpretation of the biological world and then, at mid-century and subsequently, allowed the Darwin-Huxley interpretation to win through. It emphasizes the complexity of the clash. Professional, personal, and generational agendas as well as scientific theory and fundamental philosophical intuition were at stake. The history, ultimately, provides a useful instance of Schutz ‘s and Scheler ‘s concept of societal factors controlling and selecting between flows of idealfactoren. Although Huxley is conventionally regarded as the victor I conclude by suggesting that the issue is not yet closed and that profound philosophical issues remain unresolved.

In 1903 Rabbi Philipp Bloch, of Posen (Poznan), published a unique Ashkenazic sixteenth-century polemical pamphlet which attested, so it seemed, to a heated controversy in yeshivah circles in the larger cities of the Ashkenazi cultural sphere in the late 1550s (Bloch 1903). Revolving around the place of philosophy in Judaism, the dispute reached one of its peaks in Prague some time before April 1559, probably in a public debate before a yeshivah audience, basically similar to the Disputationes then popular in European universities. The disputants were two young scholars, one of whom, the writer of the pamphlet, enthusiastically supported the teaching of classical philosophical texts in Ashkenazi yeshivot, while the other fiercely opposed what he called “philosophy.” He accused Maimonides of heresy and unbelief, as he did anything that he associated with either “philosophy” or Maimonides. The resolution adopted by the yeshivah audience — at least, so we read in the pamphlet, our only source for the event — was unequivocal: the opponent of Maimonides and philosophy was forbidden, from that time on, to broadcast his views.

It was commonly accepted in the middle ages that void within or outside the world is impossible. The paper presents a quite unusual conception of void, which is described in Yeda'aya ha-Penini's commentary on Ibn Rushd's epitome on Aristotle's Physics. According to this conception there is a thin layer of void between the water and the inner surface of the container. Ha-Penini describes two versions of this conception. According to one version this void layer is three-dimensional but thin, according to the other it is two-dimensional. The first part of the paper shows how ha-Penini “corrects” the text of Ibn Rushd, putting into it ideas which were unknown to Ibn Rushd. It is argued that, though the two views are rejected by Ibn Rushd, ha-Penini himself partly accepts (his version of) these views. The second part of the paper argues that ha-Penini could not have found these views in the Arabic-Hebrew tradition, and it seems that he relied on Christian sources. If this is indeed so, the paper presents an example of acquaintance of Hebrew scholars in southern France with Scholastic science in the first half of the fourteenth century.

A large part of our critical acumen depends on a clear distinction between what is real and what is constructed, what is out there in the nature of things and what is in there in the representation we make of them. Something has been lost however for the sake of this clarity and a heavy price has been paid for this dichotomy between ontological questions on the one hand and the epistemological questions on the other: it has become impossible to understand the simplest features of action. What the critical gesture smashes into pieces is precisely the very possibility of hearing as synonyms the two sentences, “I have fabricated it well, thus it is autonomous.” The paper tries to find a way to avoid the critical gesture and link again together facts and fetishes, producing this rather strange hybrid: factishes.

This paper introduces a distinction between two understandings of the pragmatic tradition: domesticated and radicalized pragmatism. The main difference between these two views concerns the feasibility and moral legitimacy of a radical critique of an existing practice such as science, politics, and so on. It is argued that domesticated pragmatism, with its emphasis on local rather than global perspective, has led to trivialization and degeneration of self-reflective critique. Without rejecting pragmatism as such, this paper urges a reinterpretation of this tradition so as to make room for more thorough forms of critique of both science and social practice.

Between 1550 and 1650, the intellectual elite of Ashkenazic (German-and Yiddish-speaking) Jews, including rabbis such as Yom Tov Lipmann Heller (1578–1654), showed a marked interest in astronomy, and to a lesser degree in the natural sciences generally. This is one aspect of the assimilation of medieval Jewish rationalism by that group. Passages from Heller‘s writings show his familiarity with medieval and early modern Hebrew astronomical texts, and his belief that astronomy should be studied by all Jewish schoolboys. Heller‘s astronomical views were then influenced by the discoveries and debates of his period. Between 1614 and the 1630‘s, Heller moved from an Aristotelian to a Tychonic view of the nature of the celestial bodies. Inspired, furthermore, by the notion of a natural order subject to change, and basing himself on the exegesis of ancient rabbinic texts, Heller offered what we have termed” midrashic natural histories”: namely, a hypothesis concerning the development of a certain type of animal, and another concerning the dimming of the moon and its movement into a lower orbit.

Levi ben Gerson, also known as Gersonides or Leo de Balneolis, was one of the most original Jewish thinkers of the Middle Ages, and he wrote on logic, philosophy, biblical exegesis, mathematics, and astronomy. During the last years of his life he maintained relations with the papal court of Clement VI (1342–52) at Avignon, and collaborated in the translation into Latin of his Sefer Tekhuna (Book of Astronomy). The object of this paper is to establish the main stages of the redaction of the Hebrew and Latin extant versions of his astronomical work. Although Levi declares that the work was finished in 1328,1 argue that this text was the preliminary draft of the preserved one, most of which was composed after 1338. A thorough revision of the work was undertaken at an indeterminate date before 1344. It is also argued that the final form of the work was probably due to the request of solar and lunar tables made to Levi by “great and noble Christians” around 1332.

European ethology and North American comparative psychology have been the two most prominent approaches to the study of animal behavior through most of the twentieth century. In this paper I analyze sets of popular articles by ethologist Nikolaas Tinbergen and psychologist Frank Beach, in an effort to understand the contrasting rhetorical styles of the two. Among the numerous ways in which Tinbergen and Beach differed were with respect to expressing the joy of research, the kind of scientific approach adopted, their treatment of animals, their sense of history, and the types of illustrations they chose. In general, Tinbergen‘s articles placed him closer to nature and Beach‘s more in the tradition of dispassionate scientific inquiry. These differences in rhetoric may have been one factor in the greater success of ethology after World War II.