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.

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.

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)

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)

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)

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)

"Die Gedanken, die ich in diesem Buche mitteile, sollen die Grundstimmung festhalten, die ich in der Weltanschauung Goethes beobachtet habe. Im Lauf vieler Jahre habe ich immer wieder und wieder das Bild dieser Weltanschauung betrachtet. Besonderen Reiz hatte es fur mich, nach den Offenbarungen zu sehen, welche die Natur uber ihr Wesen und ihre Gesetze den feinen Sinnes- und Geistesorganen Goethes gemacht hat. Ich lernte begreifen, warum Goethe diese Offenbarungen als so hohes Gluck empfand, dass er sie zuweilen hoher (...) schatzte als seine Dichtungsgabe. Die Eindrucke, welche Goethe von den Erscheinungen der Natur empfangen hat, muss man kennen, wenn man den vollen Gehalt seiner Dichtungen verstehen will. Die Geheimnisse, die er dem Wesen und Werden der Schopfung abgelauscht hat, leben in seinen kunstlerischen Erzeugnissen und werden nur demjenigen offenbar, der hinhorcht auf die Mitteilungen, die der Dichter uber die Natur macht. Niemand kann in die Tiefen der Goetheschen Kunst hinuntertauchen, dem Goethes Naturbeobachtungen unbekannt sind." [...] Rudolf Steiner war ein osterreichischer Esoteriker und Philosoph. Auf Grundlage seiner Lehre entstand unter anderem die Waldorfpadagogik. Der Verlag der Wissenschaften verlegt historische Literatur bekannter und unbekannter wissenschaftlicher Autoren. Dem interessierten Leser werden so teilweise langst nicht mehr verlegte Werke wieder zugangig gemacht. Das vorliegende Buch ist ein unveranderter Nachdruck der historischen Originalausgabe aus dem Jahr 1897.". (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)

Did Goethe devise an empirically viable theory of classical ray optics? Or can we at least make use of his ideas to propose one? And if so, does this confront us with an intriguing case of theory underdetermination? In this paper, which is mainly a comment on the recent work of Olaf Müller, I shall address these three questions and argue for ‘no, yes, no’. This is in contrast to Müller, who has recently launched a vivid defense of (...) class='Hi'>Goethe-style ray optics :569–573, 2015b; Z Philos Forsch 69:588–598, 2015c; Br J Hist Philos 24:322–346, 2016). Müller aims to give an almost positive answer to all three questions: ‘perhaps, yes, yes’. My overall line of argument will be that the rather restricted regime of classical geometrical optics of spectral colors allows at best for a weak form of transient theory underdetermination that, in turn and more straightforwardly, also allows for a structuralist reading in terms of two structurally equivalent formulations of one and the same theory. However, extending any of the rivaling models of ray optics other than Newton’s beyond the mentioned regime and embedding them into physics in total—especially in view of thermodynamics—leads to a contradiction. Hence, Newton’s theory is confirmed as the only consistent theoretical interpretation of ray optics. (shrink)

Malgré sa défiance pour la théorie, la recherche scientifique de Goethe n’est pas allée sans un soin tout particulier porté à la méthode qu’elle devait mettre en œuvre. Précisément parce qu’il fallait rendre compte du phénomène dans sa diversité et dans sa totalité, il importait de réfléchir aux moyens qui assureraient la réussite d’une telle démarche. Pour Goethe, il s’agissait de mettre en œuvre un concours harmonieux des différentes facultés de l’esprit humain, seul capable de répondre à la (...) richesse du réel tel qu’il se manifeste à nous, et donc seul capable de fonder adéquatement une démarche expérimentale en science. (shrink)

Fink explores how Goethe's scientific activities contributed to the growing literature in the history and philosophy of science.

Als Goethe in seiner monumentalen Farbenlehre einen Angriff auf Newtons Theorie des Lichts und der Farben lancierte, setzte er eine Methode ein, die er als Vermannigfachung der Erfahrungen bezeichnete: Er variierte verschiedene Parameter der newtonischen Experimente, um neuen Spielraum für Alternativen zur Theorie Newtons zu gewinnen. Dabei erzielte er durchaus Erfolge. U.a. entdeckte er das Komplement zum newtonischen Spektrum (das aussieht wie dessen Farbnegativ und durch Vertauschung der Rollen von Licht und Finsternis entsteht). Kürzlich hat der Wiener Maler Ingo (...) Nussbaumer Goethes Methode kongenial fortgeführt. Dabei hat er sechs weitere Farbspektren entdeckt. Sie entstehen, wenn man anstelle des Hell/Dunkel-Kontrasts (in Newtons und Goethes Experimenten) mit Paaren bunter Komplementärfarben arbeitet. Die neuen Farbspektren sehen genauso differenziert aus wie Newtons und Goethes Spektrum; doch anders als diese enthalten sie die unbunten "Farben" Schwarz und Weiß. Die vielfältigen Ordnungsbeziehungen und Symmetrien, die Ingo Nussbaumer in der Farbenwelt der insgesamt acht Spektren ausgemacht hat, verhelfen uns vielleicht zu einem tieferen Verständnis der Prinzipien menschlicher Farbwahrnehmung. Und sie tauchen die überkommenen Prinzipien der additiven und der subtraktiven Farbmischung in neues Licht. (shrink)

Goethes Protest gegen Newtons Theorie des Lichts und der Farben ist besser, als man gemeinhin denkt. Man kann diesem Protest in den wichtigsten Elementen folgen, ohne Newton in der physikalischen Sache unrecht zu geben. Laut meiner Interpretation hat Goethe in Newtons wissenschaftsphilosophischer Selbsteinschätzung eine entscheidende Schwäche aufgedeckt: Newton glaubte, mithilfe prismatischer Experimente beweisen zu können, dass das Licht der Sonne aus Lichtstrahlen verschiedener Farben zusammengesetzt sei. Goethe zeigt, dass dieser Übergang vom Beobachtbaren zur Theorie problematischer ist, als Newton (...) wahrhaben wollte. Wenn Goethe darauf beharrt, dass uns der Übergang zur Theorie nicht von den Phänomenen aufgezwungen wird, dann kommt dadurch unser eigener, freier und kreativer Beitrag zur Theorie-Bildung ans Tageslicht. Und diese Einsicht Goethes gewinnt eine überraschende Schärfe, weil Goethe plausibel machen kann, dass sich alle entscheidenden prismatischen Experimente Newtons ebenso gut mit einer alternativen Theorie vereinbaren lassen. Wenn ich recht sehe, war Goethe der erste Wissenschaftsphilosoph, der mindestens eine empirisch äquivalente Alternative zu einer wohletablierten physikalischen Theorie gesehen hat: Damit war Goethe seiner Zeit um ein gutes Jahrhundert voraus. (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)

The relationship between Goethe and Salomon Maimon has only been touched on once in the literature, and further clarification of the link between them remains a desideratum. Below I propose that the way to grasp their seeing eye to eye is through Spinoza, and specifically Spinoza's notion of scientia intuitiva. Initially I provide some context in order to illustrate what makes Maimon's role here rather unique. Then I sketch the relationship Maimon and Goethe had both to Spinoza and (...) each other while emphasizing what is at stake. I conclude with various new findings. (shrink)

Written by major scholars and practitioners of Goethean science today, this book considers the philosophical foundations of Goethe's approach and applies the ...