The thirty year long friendship between Imre Lakatos and the classic scholar and historian of mathematics Árpád Szabó had a considerable influence on the ideas, scholarly career and personal life of both scholars. After recalling some relevant facts from their lives, this paper will investigate Szabó's works about the history of pre-Euclidean mathematics and its philosophy. We can find many similarities with Lakatos' philosophy of mathematics and science, both in the self-interpretation of early axiomatic Greek mathematics as Szabó reconstructs it, (...) and in the general overview Szabó provides us about the turn from the intuitive methods of Greek mathematicians to the strict axiomatic method of Euclid's Elements. As a conclusion, I will argue that the correct explanation of these similarities is that in their main works they developed ideas they had in common from the period of intimate intellectual contact in Hungarian academic life in the mid-twentieth century. In closing, I will recall some relevant features of this background that deserve further research. (shrink)

The purpose of this paper is to reassess some mathematical examples in Plato’s dialogues which at a first glance may appear to be nothing more than trivial puzzles. In order to provide the necessary background for this analysis, I shall begin by sketching a brief overview of Plato’s mathematical passages and discuss the criteria for aptly selecting them. Second, I shall explain what I mean by ‘mathematical examples,’ and reflect on their function in light of the discussion on παραδείγματα outlined (...) in theSophistand theStatesman. Against this framework, I shall move to a close examination of specific examples drawn from theTheaetetus(154c–55d), theRepublic(523c–24d), and thePhaedo(96d–97b, 100e–101d). By placing these examples in the broader context of pre-Euclidean mathematics, I shall show that their mathematical content is often less elementary than might appear at first sight. Moreover, by placing emphasis on the specific philosophical concerns that motivated their introduction, I shall argue that the examples are not merely nonsensical jokes. Even if their illustrative purpose is not always immediately clear, and even if they can sound playful and bizarre, they in fact fulfil an important function: by virtue of their power to generate wonder or confusion, they serve as exercises and act as a trigger with respect to the difficult philosophical issues they are intended to clarify. (shrink)

Ein Beispiel für die Rezeption und Fortführung der schopenhauerschen Logik findet man in den Mathematiklehrbüchern Friedrich Adolph Wilhelm Diesterwegs (1790–1866), In diesem Aufsatz werden die historische und systematische Dimension dieser Anwendung von Logikdiagramme auf die Mathematik skizziert. In Kapitel 2 wird zunächst die frühe Rezeption der schopenhauerschen Logik und Philosophie der Mathematik vorgestellt. Dabei werden einige oftmals tradierte Vorurteile, die das Werk Schopenhauers betreffen, in Frage gestellt oder sogar ausgeräumt. In Kapitel 3 wird dann die Philosophie der Mathematik und der (...) Logik Schopenhauers vorgestellt. Es wird gezeigt, dass Schopenhauers Logikdiagramme auf Ideen der Mathematiker Leonhard Euler (1707–1783), Bernhard Friedrich Thibaut (1775–1832) und Franz Ferdinand Schweins (1780–1856) zurückgehen und dass Schopenhauer diese Logikdiagramme auch bereits zur Analyse mathematischer Begriffe verwendet. Kapitel 4 beschäftigt sich mit der Schopenhauer-Rezeption Diesterwegs. Es wird die Frage behandelt, inwieweit Diesterweg in der Mathematik von Schopenhauer beeinflusst war und welche Rolle dieser Einfluss auf sein Verständnis von Logik und Mathematik hatte. In Kapitel 5 wird schließlich gezeigt, dass Diesterweg die schopenhauersche Begriffsanalyse der Mathematik mit Hilfe von Logikdiagrammen aufgreift und weiterführt. Da Diesterweg einer der einflussreichsten Pädagogen und Mathematikdidaktiker des 20. Jahrhunderts war, ist anzunehmen, dass sich Spuren dieser Methode noch heute finden lassen. Inwiefern dies wahrscheinlich ist, wird in der Zusammenfassung (Kapitel 6) thematisiert. (shrink)

ArgumentThe Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various anachronisms. (...) It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. (shrink)

We discuss the concept of a cardinal number and its history, focussing on Cantor's work and its reception. J'ay fait icy peu pres comme Euclide, qui ne pouvant pas bien >faire< entendre absolument ce que c'est que raison prise dans le sens des Geometres, definit bien ce que c'est que memes raisons. (Leibniz) 1.

Qu’est-ce que l’un qui fait l’objet de la deuxième partie du Parménide? C’est-à-dire le long « exercice » que l’Éléate entreprend pour « muscler » le jeune Socrate? Cet article vise à montrer qu’une réponse attentive et articulée à cette question épineuse mais absolument centrale est en mesure de définir les traits principaux d’une lecture unitaire du Parménide de Platon, capable non seulement de raccorder de manière efficace les deux parties de l’œuvre mais aussi de rendre raison de la grandiose (...) complexité de la deuxième partie, à commencer par ce qu’on appelle traditionnellement la « deuxième hypothèse ». J’entends à cette fin montrer que l’un peut être assimilé à l’idée, comme de plus en plus de chercheurs semblent l’affirmer, mais tout en soulignant qu’il est aussi et à la fois le premier nombre, la monade, et l’être monadique que le Parménide historique décrit dans son poème. Ce qui en ressort est une conception de l’un qui, en plus de permettre l’unification de trois domaines d’étude apparemment différents, la mathématique, l’ontologie et la dialectique, est en mesure d’éclairer d’un jour nouveau notre compréhension de la notion d’idée, de sa genèse et des nombreuses difficultés qu’il fallait résoudre pour arriver à la perfectionner. (shrink)

With reference to an already existing and relatively widespread use of the expression in question, mathematical “thought experiments” (“TEs”) involve mathematical reasoning in which visualisation plays a relatively more important role. But to ensure an unambiguous and consistent use of the term, certain conditions have to be met: (1) Contrary to what has happened so far in the literature, the distinction between logical-formal thinking and experimental-operational thinking must not be ignored; (2) The separation between the context of discovery and the (...) context of justification is to be rejected, at least in one of the main senses in which it was defended by the logical empiricists and Popper (this excludes any position which, ascribing to mathematical TEs only a heuristic role, regards them an intermediate step to attain more traditional forms of rigour); (3) The distinction between mathematical TEs and formal proofs must be regarded as one of degree, and not as a qualitative one, although this distinction may be used in a de facto way for particular or local purposes. (shrink)