On Three Appearances of Difference in Mathematics. This article proposes to examine three types of relations between man and equality, as they are embodied in the relation to the minimal condition of the mathematical: the sentence of identity: I=I. Starting our examination from the current common conception of science and mechanism, we aim to reveal that behind the dominating logic of identity there are two other systems of logic, which have arisen during the history of humanity and that of mathematics. (...) Following Serres, the first one might be revealed during the passage from the Babylonian mathematics to the interaction between Thales and the Greek gnomon. The Babylonian epoch marks an era, in which the logic of sameness operates; logic which offers a place for the mathematician. This era is followed by a science, working under the second logic: the logic of identity, which consecrates an axiomatic definition‐based approach, at the expense of the human being. The third logic looms after the discovery of Gödel's incompleteness theorems: that an undecidable statement can be derived within every arithmetical system. Under this logic, the roles of man and the allegedly mathematical automatic machine become even more intertwined: the newly found logic promises instability not only for man, but also for the machinic‐mathematics. (shrink)

A theme like “town and science” invites to comparative analysis, and suggests questions like these: Is the urban context a particularly fertile soil for the development of scientific thinking? Or rather the contrary? Is it fertile or barren under specific circumstances? Or does it favour a particular kind of scientific activity?General answers to such questions can hardly be found; still, they may provide case studies with a guiding perspective. Case studies, on the other hand, may lead to better understanding of (...) the implications of the general questions if not to answers. In the following I therefore try to read the emergence of Mesopotamian mathematics into the context of the “urban revolution” and the rise of states (c. 3300 to c. 2300 B.C.).Mesopotamian city states appeared in the late fourth millennium B.C., and were eventually absorbed in larger territorial states a thousand years later. Contemporary with the incipient state formation was the creation of a script and of genuine, coherent mathematics. Both were based on earlier accounting techniques, both were used by Temple officials for administrative purposes, and both were marked decisively by this application. At the same time, both presuppose modes of thought which are not produced directly by their application but only as mediated by a school system.In the mid‐third millennium, an autonomous scribal profession appears for the first time. At the same time emerge the first literary texts and the first “pure” mathematical problems (i.e., problems not directed at or following the pattern of genuine application). It seems as if the scribes, as soon as they were emancipated as a craft, tried out the range of their professional tools.Neither the use of writing for literary purposes nor the possibility to use numbers for “higher” computation were forgotten when the city states were absorbed in larger social systems. Both, however, were subsumed by the interest of the new territorial state, the former as a means to propagate royal ideology, the latter in the royal administration.The final section of the article compares this early Mesopotamian development with other instances of the “urban revolution” and with the development of scientific thought in the Ancient Greek city state and in the context of Medieval city communes. (shrink)

Differenti sono i modi di conoscere che sono emersi nel corso della storia umana. Ian Hacking ha proposto una nozione, quella di "stile di pensiero" ("style of reasoning"), che fornisce un modello per caratterizzarli ed esaminare la loro genesi e il loro sviluppo. L'articolo mette in luce alcune implicazioni di questa nozione concernenti l'evoluzione del nostro sapere.

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.

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)