To assert that the ancient planetary theory proposed by Ptolemy was irrefutable – at least until the telescope discovery – is a bit of a cliché. The aim of this paper is to analyze in what sense it could be said that the epicycle and deferent model proposed by Ptolemy to explain the planetary movement is irrefutable and in what sense it is not. To do this, we will use the conceptual framework developed by the Structuralist Conception, and in particular, (...) the Moulines’ analysis of the “guiding principles”. DOI:10.5007/1808-1711.2010v14n2p211. (shrink)

Tarski’s conceptual analysis of the notion of logical consequence is one of the pinnacles of the process of defining the metamathematical foundations of mathematics in the tradition of his predecessors Euclid, Frege, Russell and Hilbert, and his contemporaries Carnap, Gödel, Gentzen and Turing. However, he also notes that in defining the concept of consequence “efforts were made to adhere to the common usage of the language of every day life.” This paper addresses the issue of what relationship Tarski’s analysis, and (...) Béziau’s further generalization of it in universal logic, have to reasoning in the everyday lives of ordinary people from the cognitive processes of children through to those of specialists in the empirical and deductive sciences. It surveys a selection of relevant research in a range of disciplines providing theoretical and empirical studies of human reasoning, discusses the value of adopting a universal logic perspective, answers the questions posed in the call for this special issue, and suggests some specific research challenges. (shrink)

ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as (...) a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. (shrink)

It has been argued in relation to Old Babylonian mathematical procedure texts that their validity or correctness is self-evident. One “sees” that the procedure is correct without it having, or being accompanied by, any explicit arguments for the correctness of the procedure. Even when agreeing with this view, one might still ask about how is the correctness of a procedure articulated? In this work, we present an articulation of the correctness of ancient Egyptian and Old Babylonian mathematical procedure texts – (...) mathematical texts presenting the solution of problems. We endeavor to make explicit and explain how and why the procedures are reliable over and above the fact that their correctness is intuitive. (shrink)

La recherche historique dans le cours du dernier demi-siècle a amélioré notre connaissance des mathématiques de I 'Antiquité. Les textes en provenance d'Égypte et de Mésopotamie ont été mieux compris et leur interprétation a dépassé l'alternative sommaire entre empirisme et rationalisme. Le panorama offert par la science grecque s'est enrichi et diversifié: il n'est plus possible de le réduire à la seule théorie géométrique. Les principaux problèmes que posait son histoire ont été l'objet de discussions approfondies. À partir de là (...) des questions plus générales d'ordre épistémologique et philosophique sont soulevées. Y a-t-il un sens à chercher dans un lointain passé une « origine » unique des mathématiques? Quel rapport y at- il entre la forme des mathématiques dans une civilisation et la structure de la société? L'anthropologie culturelle peut-elle aider à interpréter la diversité et l'unité des mathématiques des divers peuples? À partir de quand et sous quelles conditions une histoire unique des mathématiques peut-elle commencer? (shrink)

In a somewhat polemical article on ‘Solstices, Equinoxes, and the Presocratics’ D. R. Dicks has recently challenged the usual view that the Presocratics in general, and the Milesians in particular, made significant contributions to the development of scientific astronomy in Greece. According to Dicks, mathematical astronomy begins with the work of Meton and Euctemon about 430 B.C. What passes for astronomy in the earlier period ‘was still in the pre-scientific stage’ of ‘rough-and-ready observations, unsystematically recorded and imperfectly understood, of practical (...) men’ whose chief concern was to fix the seasons for ploughing, seed-time, sailing voyages and religious festivals. Ionian speculation, says Dicks, took very little note of such observation: ‘some of its wilder flights of fancy might have been avoided, if it had taken more’. In this account of the rise of Greek astronomy, the natural philosophers have no part to play. Their theories represent a speculative enterprise without a scientific future, a philosophic sideline with no impact on the development of observational science from Hesiod to Meton or the development of mathematical astronomy from Meton to Ptolemy.I believe that such a dichotomy between early philosophy and early science in Greece is misguided in principle, and that it seriously distorts our picture of the initial phases of each discipline. It also imposes a considerable strain upon our credulity. Take the case of Anaxagoras who, according to Plato and Theophrastus, had given a causal explanation of eclipses of the moon a generation before Meton. (shrink)

This study considers the short list of Nature of Science features frequently published and widely known in the science education discourse. It is argued that these features were oversimplified and a refinement of the claims may enrich or sometimes reverse them. The analysis shows the need to address the range of variation in each particular aspect of NOS and to illustrate these variations with actual events from the history of science in order to adequately present the subject. Another implication of (...) the proposal is the highlighting of the central role of science educators who, facing various strong claims of researchers in education and philosophy of science, often have difficulty in making a choice of what to teach about NOS. It is suggested that a representative variation with regard to the traditional NOS claims may be appropriate for a genuine understanding of the subject. In that, using the discipline-culture structure of the fundamental theories of physics and addressing the plurality of scientific methods may be helpful in the actual teaching and learning of NOS. (shrink)

The aim of this text is to present the evolution of the relation between the concept of number and magnitude in ancient Greek mathematics. We will briefly revise the Pythagorean program and its crisis with the discovery of incommensurable magnitudes. Next, we move to the work of Eudoxus and present its advances. He improved the Pythagorean theory of proportions, so that it could also treat incommensurable magnitudes. We will see that, as the time passed by, the existence of incommensurable magnitudes (...) was no longer something strange. Already in the period of Plato and Aristotle, their existence was common place: up to the point of being considered absurd that all magnitudes were commensurable. Aristotle criticized the Pythagorean program and defended that, though belonging to the same category, number and magnitude are of distinct species: number is discrete and magnitude is continuous. We finish by presenting briefly how the concept of number was amplified throughout the centuries until it came also to include the notion of continuity. (shrink)

‘Recent studies of Babylonian sources have shown that we must revise former estimates of the extent to which the Greeks were indebted for the details of their astronomy to the Babylonians; the debt proves to have been much greater than had been imagined, and further researches may prove it to have been greater still.’ So wrote Sir Thomas Heath in 1932; in the previous year, Professor Filon had written, ‘It is gradually beginning to be realized that many of the achievements (...) of Greek culture in the fields of astronomy and mathematics did not spring, fully armed, from the Hellenic brain, but had their more remote origins in the civilizations of the ancient East.’There is available now sufficient evidence to show that a great deal of the astronomical knowledge which has come down to us from the Hellenistic period was not initially discovered during that period; and such new empiric discoveries as were made in that time were not all due to Greeks, for important contributions were still being made by Babylonians during the Seleucid Era.To a large extent it seems that the Greeks kept very closely, even in astronomy, to the mode of research advocated by Plato, who said in The Republic, ‘Which things truly are to be comprehended by reason and intellect, but not by sight’. The Greeks founded a ‘school’ of theoretical astronomy and, with their highly developed mathematics, were able to go far with it; but their source-material was in very many cases not Greek.The author of Epinomis states, ‘We may assume that whatever the Greeks take from the barbarians, they bring it to a finer perfection’. Adrastus wrote that the methods used by the Chaldeans and Egyptians in astronomy were imperfect because these people lacked physiologia; no doubt this was true, but it was people of these races who had done, and continued to do, most of the equivalent of modern observatory routine work. (shrink)

A Bayesian account of the virtue of unification is given. On this account, the ability of a theory to unify disparate phenomena consists in the ability of the theory to render such phenomena informationally relevant to each other. It is shown that such ability contributes to the evidential support of the theory, and hence that preference for theories that unify the phenomena need not, on a Bayesian account, be built into the prior probabilities of theories.

This article deals with the social production of knowledge in the exact sciences. After defining the term ?exact science?, it delineates the broad dynamic of its history. It, then, offers a socio-economic historical explanation of why the production of knowledge has become a major industry, if not the largest industry, in the last hundred years. The article concludes by drawing a detailed blueprint of the components, mechanisms, and specificities of the knowledge industry.

The paper lists all the predictions and observations of eclipses of Jupiter's satellites 1668–1678 and compares them with modern computations of the these eclipses by J. H. Lieske. We discuss Roemer's method that led to his discovery of the retardment of light and finally we shall interpret Roemer's calculations.

Ever since Proclus wrote his commentary on Plato's Republic, repeated attempts have been made to find a hidden number of cosmic significance in Rep. 8.546. For the Neo-Platonist it was natural to look for esoteric secrets in ancient works; among the men of the New Learning at the end of the Middle Ages there were enough astrologers and necromancers to ensure respect for the proposition; we are now again enamoured of irrationality. But the scholars who attempted such calculations around 1900 (...) must have considered Plato himself a mystery-monger. In this article I propose: to show why such attempts are mistaken, to discuss what early writers who mention the passage say about its meaning, to provide a mathematician's translation that fits the context, and to comment on it; for the currently accepted explanation is unsatisfactory. ‘There is fairly widespread agreement that the geometrical number is 12,960,000 = 3,6002 = 4,800 × 2,700, but on the method by which this number is reached the widest divergence exists’ or, from an earlier, different guess: ‘…one can, so to speak, state a priori that Plato's number is a multiple of 19 ten thousands’, i.e. the text is approached with a ready-made answer in mind. There are three further objections to the conventional view, It takes the text out of its context as if it had strayed from the Timaeus. (shrink)

Eine Reihe von mittelalterlichen Schriften zur Landmessung (vom 9. islamischen Jahrhundert bis zu Fibonacci und Pacioli) enthält eine besondere Art von „algebraischen” Aufgaben. Darin werden z.B. die Summe der Fläche und einer oder alle vier Seiten eines Quadrates beschrieben und nach der Seite gefragt. Es zeigt sich erstens, daß dieser Aufgabentyp mindestens seit dem frühesten 2. vorchristlichen Jahrtausend von geometrischen Praktikern tradiert wurde, und zweitens, daß er die Entwicklung einer „Algebra” in der altbabylonischen Schreiberschule inspirierte. Der Aufsatz untersucht, in welcher (...) Weise die Überführung der „sub-wissenschaftlichen” Praktikertradition in einen systematischen Schulunterricht den mathematischen Inhalt und den Denkstil künftiger Praktiker präge. Im letzten Kapitel wird diskutiert, inwieweit es sinnvoll ist, innerhalb dieser Schulalgebra von „Schulen” zu sprechen. (shrink)