In this paper I show that proofs by contradiction were a serious problem in seventeenth century mathematics and philosophy. Their status was put into question and positive mathematical developments emerged from such reflections. I analyse how mathematics, logic, and epistemology are intertwined in the issue at hand. The mathematical part describes Cavalieri's and Guldin's mathematical programmes of providing a development of parts of geometry free of proofs by contradiction. The logical part shows how the traditional Aristotelean doctrine that perfect demonstrations (...) are causal demonstrations influenced the reflection on proofs by contradiction. The main protagonist of this part is Wallis. Finally, I analyse some epistemological developments arising from the Cartesian tradition. In particular, I look at Arnauld's programme of providing an epistemologically motivated reformulation of Geometry free of proofs by contradiction. The conclusion explains in which sense these epistemological reflections can be compared with those informing contemporary intuitionism. (shrink)

Leibniz's overall view of the relationship between reasoning and computation is discussed on the basis of two broad claims that one finds in his writings, concerning respectively the nature of human reasoning and the possibility of replacing human thinking by a mechanical procedure. A joint examination of these claims enables one to appreciate the wide scope of Leibniz's interests for mechanical procedures, concerning a variety of philosophical themes further developed both in later logical investigations and in methodological contributions to cognitive (...) psychology. (shrink)

Leibniz’s universal characteristic is a fundamental aspect of his theory of cognition. Without symbols or characters it would be difficult for the human mind to define several concepts and to achieve many demonstrations. In most disciplines, and particularly in mathematics, the mind must then focus on symbols and their combinatorial rules rather than on mental contents. For Leibniz, mental perception is most of the time too confused for attaining distinct notions and valid deductions. In this paper, I argue that the (...) functions of symbolization differ depending upon the kind of concepts that are replaced with characters. In my view, most commentators did not sufficiently underline the distinction between two main functions of formal substitution in Leibniz’s characteristic: either increasing our knowledge or simply structuring it. In the first case, we complete our knowledge because formal substitution makes sensible and imaginary concepts more distinct. In the second case, symbolization helps to organize contents that are already conceived of by reason. Thus the process of substitution is not always identically applicable, for symbols replace different types of concepts. (shrink)

O verdadeiro infinito, afirma Leibniz em seus Novos ensaios, não é um modo da quantidade, é anterior a qualquer composição e não é formado pela adição de partes. O infinito, para Leibniz, é atual e é propriedade de todas as coisas. Como criaturas finitas conhecem o infinito? Neste artigo, investigamos que tipo de relação pode ter o infinito matemático, quantitativo, para o conhecimento da infinitude divida e do infinito atual que existe no mundo. A ordem ideal da matemática instrui sobre (...) a ordem real do mundo? E quais os limites da ordem ideal na explicação do infinito atual que caracteriza Deus e o mundo? (shrink)

Two books on Whitehead, a major study by the noted philosopher of science, Isabelle Stengers, and a shorter one by Didier Debaise are reviewed, along with two earlier mathematical and scientific works by Whitehead himself, which have been re-issued. This provides the basis for a wide-ranging discussion of the relationships between Whitehead's love of poetry and Heidegger's approach to it, Whitehead's background in mathematics and theoretical physics and his attitude to empirical science and more general problems of the philosophy of (...) the event, in particular how radical change can come about. (shrink)