Viewed this way, the texts yield striking examples of language and notation that are irreducibly ambiguous and productive because they are ambiguous.

The transformation of mathematics from ancient Greece to the medieval Arab-speaking world is here approached by focusing on a single problem proposed by Archimedes and the many solutions offered. In this trajectory Reviel Netz follows the change in the task from solving a geometrical problem to its expression as an equation, still formulated geometrically, and then on to an algebraic problem, now handled by procedures that are more like rules of manipulation. From a practice of mathematics based on the localized (...) solution we see a transition to a practice of mathematics based on the systematic approach. With three chapters ranging chronologically from Hellenistic mathematics, through late Antiquity, to the medieval world, Reviel Netz offers an alternate interpretation of the historical journey of pre-modern mathematics. (shrink)

Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide (...) a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth. (shrink)

Lists the practical problems found in the manuscript of Book III of the Algebra which were not included in the printed text. The author believes that their omission reflects the influence of Bombelli's discovery of Diophantus.

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 is a companion paper to my preceding one on Harriot's experimentations in the field of the sign-rule of multiplication in algebra. Cardano had earlier attacked the conventional rule in a chapter of his De Aliza regula liber, published in 1570 as an appendix to the second edition of his Ars magna. He returned to the subject in a brief tract, published nearly a century later in his collected works as Sermo de plus et minus. Only Cardano's valid contention that (...) the standard ‘proof’ of the accustomed sign-rule was faulty made impact, in the Euclidean background to which the ‘proof’ belonged. His cause was to preserve the purity of the Euclidean tradition. It overshadows the true aim in the De Aliza chapter, but the Sermo reveals it plainly as far from retrograde: he wished to retain all the new techniques of algebra that brought in not only ‘minus’ quantities but also their square roots, and to escape the ‘impossible’ status of these last. This the rebel sign-rule did for him by making every root of a ‘minus’ just another ‘minus’. (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)