HILBERT BETWEEN THE FORMAL AND THE INFORMAL SIDE OF MATHEMATICS

Abstract: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

The demonism of creation in Goethe's philosophy

Goethe's philosophy of creativity revolves around what he called das Dämonische. This essay is not meant as a definition or an explanation of demonic creation, but instead presents a demonic work par excellence, as the term "demonic" is defined by Goethe in the Elegy from Marienbad. The process of the creation of this work, as it is described by Goethe, also represents a strange exorcism, as the entire daemonic creative force of the author is transposed in this lyrical masterpiece of German and universal literature. After writing the Elegy Goethe, it is no longer demonic.

Nietzsche and Neo-Kantian historiography: points of contact

In the German academies of Nietzsche's period of writing, the Kantian tradition was largely displaced in favor of two independent schools that have since been labeled "Neo-Kantianism." This paper presents four key theses about philosophy of history from four Neo-Kantian thinkers, how they follow from their adaptation of the Kantian tradition, and how Nietzsche critically engaged the very same issues in the formation of his own historical theory. Although there is little direct influence between orthodox Neo-Kantianism and Nietzsche, their comparison on these points will illuminate their unique adaptations of the Kantian tradition.

The notion of process in nonstandard theory and in Whiteheadian metaphysics

In this article I intend to show that certain aspects of A.N. Whitehead's philosophy of organism and especially his epochal theory of time, as mainly exposed in his well-known work Process and Reality, can serve in clarify the underlying assumptions that shape nonstandard mathematical theories as such and also as metatheories of quantum mechanics. Concerning the latter issue, I point to an already significant research on nonstandard versions of quantum mechanics; two of these approaches are chosen to be critically presented in relation to the scope of this work. The main point of the paper is that, insofar as we can refer a nonstandard mathematical entity to a kind of axiomatical formalization essentially 'codifying' an underlying mental process indescribable as such by analytic means, we can possibly apply certain principles of Whitehead's metaphysical scheme focused on the key notion of process which is generally conceived as the becoming of actual entities. This is done in the sense of a unifying approach to provide an interpretation of nonstandard mathematical theories as such and also, in their metatheoretical status, as a formalization of the empirical-experimental context of quantum mechanics.

The role of selection in functional explanations

In this essay I will argue that natural selection is more important to functional explanations than what has been thought in some of the literature in philosophy of biology. I start by giving a brief overview of the two paradigms cases of functional explanations: etiological functions and causal-role functions. i then consider one particular attempt to conciliate both perspectives given by David Buller (1998). Buller's trial to conciliate both etiological functions and causal-role functions results in what he calls a weak etiological theory. I argue that Buller has not succeeded in his construal of the weak etiological theory: he underestimates the role that selective processes have in functional explanations and so his theory may not be classified as an etiologial theory. As an alternative, I consider the account of etiological functions given by Ruth Millikan (1984) and I argue that Millikan's theory is more comprehensive to assess contentious case in biology like exaptations. Finally, I conclude by analyzing where the adoption of Millikan's theory leave us. I argue, contrary to Millikan and others, that once we assume the importance of natural selection in functional explanations, there is no strong reason to resist a linguistic reform of the word function and hence that the attempts to conciliate both etiological functions and causal-role functions are misplaced.

TOWARDS A PHILOSOPHICAL UNDERSTANDING OF THE LOGICS OF FORMAL INCONSISTENCY

Abstract In this paper we present a philosophical motivation for the logics of formal inconsistency , a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency within the object language in such a way that consistency may be logically independent of non-contradiction. We defend the view according to which logics of formal inconsistency may be interpreted as theories of logical consequence of an epistemological character. We also argue that in order to philosophically justify paraconsistency there is no need to endorse dialetheism, the thesis that there are true contradictions. Furthermore, we show that mbC , a logic of formal inconsistency based on classical logic, may be enhanced in order to express the basic ideas of an intuitive interpretation of contradictions as conflicting evidence.

Classificações de signos de C.S.Peirce: de ‘On the Logic of Science’ ao ‘Syllabus of Certain Topics of Logic’

As classificações dos signos de C.S.Peirce começam a ser desenvolvidas em 1865 e se estendem a até, pelo menos, 1909. Vou apresentar o período que tem início em 1865, e possui dois momentos de intensa produção - "On a New List of Categories" e "On the Algebra of Logic: a contribution to the philosophy of notation". Em seguida apresento as dez classes de signos, uma morfologia que aparece no "Syllabus of Certain Topics of Logic", e é desenvolvida a partir de 1903. Meu propósito aqui é familiarizar o leitor com as intrincadas classificações sígnicas de Peirce.