“Dialectic” has been a matter of growing interest in contemporary philosophy. The present article analyzes dialectical methods and positions them by reference to two paradigmatic texts of German idealism and analytic philosophy, i.e. J.G. Fichte’s Science of Knowing (1804) and J. McDowell’s Mind and World. Both dialectical approaches will be interpreted with regard to their contribution in the debate on reductionism and anti-reductionism: both Fichte and McDowell claim that philosophical positions and logical terms stand in a dualistic relationship to one (...) another, on the one hand, but are separated by a gulf, on the other. I will argue that for McDowell dialectic seems to be an alternative to one-sided reductionisms as well as to normal anti-reductionistic holism. Furthermore, for Fichte dialectic is an adequate method for describing the relationship of reductionism and anti-reductionism itself. Both see in dialectic a technique for bridging the gulf between binary opposite terms of logic as well as mutually exclusive positions, such as mind and world, subject and object, or idealism and realism. (shrink)

In this paper, I offer a close reading of Wittgenstein's remarks on inconsistency, mostly as they appear in the Lectures on the Foundations of Mathematics. I focus especially on an objection to Wittgenstein's view given by Alan Turing, who attended the lectures, the so-called ‘falling bridges’-objection. Wittgenstein's position is that if contradictions arise in some practice of language, they are not necessarily fatal to that practice nor necessitate a revision of that practice. If we then assume that we have adopted (...) a paraconsistent logic, Wittgenstein's answer to Turing is that if we run into trouble building our bridge, it is either because we have made a calculation mistake or our calculus does not actually describe the phenomenon it is intended to model. The possibility of either kind of error is not particular to contradictions nor inconsistency, and thus contradictions do not have any special status as a thing to be avoided. (shrink)

Drawing mainly from the Tractatus Logico-Philosophicus and his middle period writings, strategic issues and problems arising from Wittgenstein’s philosophy of mathematics are discussed. Topics have been so chosen as to assist mediation between the perspective of philosophers and that of mathematicians on their developing discipline. There is consideration of rules within arithmetic and geometry and Wittgenstein’s distinctive approach to number systems whether elementary or transfinite. Examples are presented to illuminate the relation between the meaning of an arithmetical generalisation or theorem (...) and its proof. An attempt is made to meet directly some of Wittgenstein’s critical comments on the mathematical treatment of infinity and irrational numbers. (shrink)

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the (...) ground for a desirable integration of their strenghts. (shrink)