In this paper, we discuss the prevailing view amongst philosophers and many mathematicians concerning mathematical proof. Following Cellucci, we call the prevailing view the “axiomatic conception” of proof. The conception includes the ideas that: a proof is finite, it proceeds from axioms and it is the final word on the matter of the conclusion. This received view can be traced back to Frege, Hilbert and Gentzen, amongst others, and is prevalent in both mathematical text books and logic text books.

This paper consists of three main sections. In the first section, we consider how early attempts at understanding the relationship between mathematics and philosophy in Leibniz’s thought were often made within the framework of grand reconstructions guided by intellectual trends such as the search for “the ideal of system”. In the second section, we proceed to recount Leibniz’s first encounter with contemporary mathematics during his four years of study in Paris presenting some of the earliest mathematical successes which he made (...) there. In particular, we argue that recently published letters and papers reveal how his youthful mathematical reflexions were deeply intertwined with important philosophical insights that, in turn, acted as guiding ideas for his mathematical research. Finally, in the third section, we situate the central themes of the essays of the present volume within the new understanding of the interrelations between philosophy and mathematics in Leibniz’s thought briefly indicated in the opening section. (shrink)

Johann Wolfgang von Goethe is most famous for his work on color theory, but he was also an accomplished all-round scientist, studying and writing on anatomy, geology, botany, zoology and meteorology. This book draws together, in Goethe's own words, his key ideas on nature, science and scientific method. Goethe believed that we should study our world and nature as people at home in it, rather than removedly, as if we were aliens from another planet. He adopted a (...) qualitative approach to science at odds with Newton's quantitative methods that were so popular in his day. His is a sensitive science which does not ignore our relationship to nature. The extracts in this book are fascinating and essential reading for anyone who feels that we've lost our spiritual connection to nature. (shrink)

Die Arbeit untersucht die Bedeutung der Analogie an der Schnittstelle von Wissenschaft und Ästhetik um 1800. Hierfür analysiert sie ihre mathematischen und rhetorischen Ursprüngen in der Antike und zeichnet davon ausgehend ihre Rückkehr in der M.

This paper revisits some of Chateaubriand’s critical considerations with regard to representing our reasoning practices in logic and mathematics by means of “idealized syntax”. I focus on the persistently critical side of these considerations which aim to prepare the ground for “an interesting epistemology of logic and mathematics” that ought to make room for understanding the pragmatic dimensions of proofs as explanatory rational displays. First, I discuss the 20th century “syntactic conception” of the logical and the underlying set of values (...) it upholds. Secondly, I revisit the syntactic constraints on systematizing our formal forms of reasoning and ask about the relationship between “idealized” proofs construed as “syntactic objects” and the variety of formal forms of reasoning with its uses of the logical by the research mathematician. Finally, I consider the reasons why Chateaubriand thinks the syntactic requirements of “logical rigor” cannot be fulfilled, and why they ought not to be on the agenda. I conclude my paper by pointing to a deeper assumption which needs to be critically revisited as it stands in the way to what the author envisages as an “interesting epistemology of logic and mathematics”.O presente artigo reconsidera algumas das considerações críticas de Chateaubriand com relação a representar nossa prática de raciocínios em lógica e matemática por meio de uma “sintaxe idealizada”. Concentro-me no aspecto invariavelmente crítico dessas considerações, que têm por objetivo preparar o terreno para “uma epistemologia interessante da lógica e da matemática”, a qual deve abrir caminho para a compreensão da dimensão pragmática de provas como exibição racional explicativa. Em primeiro lugar, discuto a “concepção sintática” da noção de lógica do século XX e o conjunto de valores que ela sustenta. Em segundo lugar, eu reconsidero as restrições sintáticas impostas à sistematização de nossos raciocínios formais e pergunto sobre a relação entre provas “idealizadas” construídas como “objetos sintáticos” e a variedade de modos formais de raciocínios com os seus usos do lógico pelos pesquisadores em matemática. Por fim, eu considero as razões pelas quais Chateaubriand pensa que os requisitos sintáticos do “rigor lógico” não podem ser satisfeitos, e por que eles não deveriam ser parte da agenda. Concluo meu artigo apontando uma assunção mais profunda que precisa ser reconsiderada criticamente uma vez que ela representa um obstáculo àquilo que o autor vislumbra como uma “epistemologia interessante da lógica e da matemática”. (shrink)

Quem quer que de algum modo tenha se ocupado da teoria da poesia, e particularmente da tragédia, recordar-se-á de uma passagem em Aristóteles que causou muita dificuldade aos intérpretes, sem que pudessem concordar completamente sobre o seu significado. Numa caracterização mais precisa da tragédia, o grande homem parece esperar dela que, por meio da encenação de ações e acontecimentos que suscitam compaixão e medo, purifique (reinigen) o ânimo do espectador das paixões mencionadas. Acredito poder comunicar de melhor maneira meus pensamentos (...) e minha convicção sobre a passagem mencionada por meio de uma tradução dela. (shrink)

Da Goethes Farbenlehre im allgemeinen allein unter dem Blickwinkel der darin enthaltenen methodologischen Thesen untersucht wird, tritt hier die Rekonstruktion ihres physikalischen Gehalts in den Vordergrund. Es zeigt sich auf diese Weise, daß die Farbenlehre als eine Verbindung physikalischer und sinnesphysiologischer Aspekte einige der zentralen Versuche der Newtonschen Optik angemessen interpretieren kann. Die Diskussion der Methodologie zeitigt anschließend ein zukunftweisendes Element: die Erkenntnis der Bedeutung der Meßapparatur für die Konstitution der Theorie. Schließlich wird Goethes Naturbild und seine Aktualisierung in den (...) Bestrebungen der kritischen Theorie erörtert. (shrink)

In this essay, I will discuss a variety of considerations that Goethe expressed in his writings. I will with few exceptions address these writings in chronological order. I include both literary and scientific-philosophical works. In this way I hope to show that a certain theme is at the heart of Goethe’s thinking, and that Goethe’s later works expresses a sophisticated and “deep” account of this theme. In addition, I will try to explain how one can ascribe this (...) Goethean theme to major philosophers of the twentieth century – Cassirer, Merleau-Ponty, and Wittgenstein. The theme in question concerns the individuality of a human life in a metaphysical sense, characterizing the individual as situated “in between” Nature and Culture. By being both a child of Nature and a child of Culture, the fate of individuals is the transformation of previously given human concerns and practices. There never is a natural child nor a cultural formation securing human individuality. In Goethe’s words: The history of an individual human being is the individual human being. “Die Geschichte der Wissenschaft ist die Wissenschaft selbst, die Geschichte des Individuums, das Individuum”. See Hamacher. Hamacher’s book has been a major source for me! (shrink)

This essay is a journey into the phenomenology of place and Goethe's science of nature by an Australian lecturer on the philosophies and practices of place-based education. It takes the form of a series of encounters with leading figures in the field— David Seamon, Henri Bortoft and Isis Brook, as well as an application of Goethean science to some granite outcroppings on the Cornish coast of England. The profundity of the phenomenological concepts of 'natural attitude' and 'lifeworld' is discussed (...) together with ideas behind Goethe's participative and intuitive practices. Goethean science and phenomenology have enormous potential to deepen the experience, understanding and expression of place relationships, but they put challenging demands upon students and lecturers within the structure of a university subject. (shrink)

Goethe carried out an enormous number of experiments before criticizing Newton's theory of light and colours in the Farbenlehre (1810). He managed to show that Newton's reasoning is based on a rather narrow choice of experiments, in which parameters such as the distance between the prism and the screen are fixed arbitrarily: Newton's famous spectrum (with its green centre) occurs only at a specific distance. Once you reduce the distance, the green centre disappears, and you see the two border (...) spectra instead. Newton can of course explain these border spectra; but things can be explained just as well the other way around: Then you start with a theory based on the two border spectra in order to derive the Newtonian spectrum. Both accounts fit many data (available in Newton and Goethe's days) equally well. Thus by insisting that the step to theory is not forced upon us by "reason and experiments" alone (as Newton would have it), Goethe revealed our own free, creative contribution to theory construction. And surprisingly, his theory of the prismatic colours seems no worse off than Newton's, even taking into account additional criteria of theory choice, such as simplicity or elegance. (shrink)

Dentro de la perspectiva de lo que el autor denomina lecciones universales,los grandes pensamientos que de las conversaciones con Goethe pudo recoger Eckermann constituyen enseñanzas sobre lo que puede llamarse sabiduría de la vida, y también orientaciones para el diseño de una humanidad más alta, espiritualmente. El artículo contiene, además, información sobre quién era Eckermann y una serie de lecciones goethianas que van desde una profunda intuición metafísica hasta asuntos prácticos y concretos.

Text: Reproduktion der 2. Ausgabe Berlin 1924. - Eine gänzlich andere, eine fundamental neue Form der Beziehung des Besonderen aufs Allgemeine waltet in Goethes Naturbetrachtung, die von dem Urphänomen des Lebens ihren Ausgang nimmt. Darin zwar stimmt diese Betrachtung mit der wissenschaftlichen Physik überein, dass auch in ihr der Wert und die Kraft des Allgemeinen rückhaltlos anerkannt wird. Das Vergängliche, das zeitlich Wandelbare und Verfließende wird zum Gleichnis. Die morphologische Erkenntnis der Gestalten und des Gestaltenwandels, wie Goethe sie versteht, (...) geht niemals im Sinnlich-Einzelnen oder in einem Analogon des Sinnlich-Einzelnen auf. [Auszug aus dem Buch]. (shrink)

The influence of Goethe on Wittgenstein is just beginning to be appreciated. Hacker and Baker, Westphal, Monk, and Haller have all drawn attention to significant affinities between the two men's work, and the number of explicit citations of Goethe in Wittgenstein's texts supports the idea that we are not dealing simply with a matter of deeplying similarities of aim and method, but of direct and major influence. These scholarly developments are encouraging because they help to place Wittgenstein's work (...) within an important tradition of German letters which goes far beyond his contemporaries and immediate forebears in Vienna; and they show that Wittgenstein's profound interest in literature and music is ceasing to be merely a matter of biographical anecdote, and is being used to illuminate some of the most central areas of his work. (shrink)