Translator's introduction -- Fundamental questions of geometry -- The decidability requirement -- The origin of the concept of number -- Implicit definition and the proper grounding of mathematics -- Rigid bodies in geometry -- Prelude to geometry : the essential ideas -- Physical and mathematical geometry -- Natural geometry -- The concept of the differential -- Reflections on the proper grounding of mathematics I -- Concepts and proofs in mathematics -- Dimension and space in mathematics -- Reflections on the proper (...) grounding of mathematics II -- The axiomatic method in modern mathematics. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Name der Zeitschrift: Zeitschrift für Religionswissenschaft Jahrgang: 23 Heft: 2 Seiten: 285-319.

Moritz Pasch (1843ber neuere Geometrie (1882), in which he also clearly formulated the view that deductions must be independent from the meanings of the nonlogical terms involved. Pasch also presented in these lectures the main tenets of his philosophy of mathematics, which he continued to elaborate on throughout the rest of his life. This philosophy is quite unique in combining a deductivist methodology with a radically empiricist epistemology for mathematics. By taking into consideration publications from the entire span (...) of Paschs highly original, but virtually unknown, philosophy of mathematics is presented. (shrink)

RÉSUMÉ: Pasch est généralement considéré comme le premier à avoir proposé une axiomatisation de la géométrie. Mais ses Vorlesungen über neure Geometrie (1882) contiennent plusieurs éléments étrangers au paradigme hilbertien. Pasch soutient ainsi que la « géométrie élémentaire », dont il propose une axiomatisation complète, est une théorie empiriquement vraie. Les commentateurs considèrent généralement les différences entre la méthode de Pasch et celle qui deviendra standard après Hilbert comme autant de défauts affectant une pensée encore inaboutie. Notre (...) but consiste au contraire à reconstruire la cohérence et l’originalité de l’approche de Pasch. Nous donnerons d’abord un aperçu du contexte historique dans lequel les Vorlesungen s’inscrivent, pour tenter ensuite de réconcilier l’effort d’axiomatisation et l’empirisme de Pasch. Nous insisterons notamment sur les remarquables procédures logiques que Pasch met sur pied afin d’adapter sa pratique mathématique aux contraintes dictées par sa réflexion philosophique.ABSTRACT: Pasch is usually credited with having presented the first axiomatization of a geometrical theory, but the Vorlesungen über neuere Geometrie (1882) contains many features which do not fit the Hilbertian paradigm. Thus Pasch, while axiomatizing his “elementary geometry,” claims that it is an empirically true theory. Scholars usually regard the discrepancies between Pasch and the post-Hilbertian standard method as mere inconsistencies of Pasch’s theory. On the contrary, this article aims at reconstructing the coherence and originality of the Vorlesungen. We will first display the historical background of this work and then try to reconcile Pasch’s logical axiomatic claim with his empiricist stance. More importantly, we will insist on the remarkable logical procedures worked out by Pasch in order to adapt his mathematical development to the strictures of his broad philosophical position. (shrink)

The extant correspondence, consisting of ten letters from the period from 1882 to 1902, from Moritz Pasch to Felix Klein is presented together with an English translation and a short introduction. These letters provide insights into the views of Pasch and Klein regarding the role of intuition and axioms in mathematics, and also into the hiring practices of mathematics professors in the 1880s.

We prove that, in the framework of ordered geometry, the inner form of the Pasch axiom does not imply its outer form . We also show that OP can be properly split into IP and the weak Pasch axiom.

On a tente de dresser le tableau des caracteristiques multiples, changeantes et souvent contradictoires de la matiere, afin de re-ouvrir le proces qu'on lui a toujours intente : ainsi, ou bien elle s'offre a toutes les influences qu'elle subit, d'ou sa passivite et son inconsistance, sa non substantialite, ou bien elle resiste, mais on lui reprochera son insensibilite et sa tenacite obtuse. Correlativement, on a voulu reprendre les analyses argumentees des philosophes a son sujet : celle des anciens (Platon, Aristote, (...) Epicure et Lucrece), celle de l'age classique (principalement les Cartesiens), celles du monde du XVIIIe (les Newtoniens), celle des modernes (le materialisme dialectique meme), sans oublier celles de ceux qui ont cru l'atteindre a travers des experiences-limites du moi, dans la resistance a l'effort, au coeur de l'oubli ou de l'habitude. Troisieme moment de l'analyse, on a souhaite mettre en lumiere comment la technologie des actuels materiaux, de meme que l'art le plus contemporain, renouvelaient entierement la conception du substrat. Une revolution s'accomplit sous nos yeux : l'histoire au present des sciences experimentales et la metamorphose de la plasticite permettent d'en prendre la mesure. Du coup, on ne peut plus reprendre les vieux enonces du requisitoire. Deux consequences s'en degagent : outre l'urgence de ce que nous appelons la rematerialisation, la guerre sans merci qu 'on declare ouvertement ici a l'un des piliers de la metaphysique, le dualisme qui eloigne l'une de l'autre, pour de nombreuses raisons, l'intellectualite et la materialite. On apprend au contraire a les lier inseparablement. Comment envisager cette possible reconciliation? (shrink)

How mathematicians conceive of the nature of mathematics is reflected in the metaphors they use to talk about it. In this paper I investigate a change in the use of metaphors in the late nineteenth and early twentieth centuries. In particular, I argue that the metaphor of mathematics as a tree was used systematically by Pasch and some of his contemporaries, while that of mathematics as a building was deliberately chosen by Hilbert to reflect a different view of mathematics. (...) By taking these metaphors seriously we attain a new vantage point for understanding historical changes in conceptions of mathematics. (shrink)

Hans Vaihinger tried to explain how mathematical theories can be useful without being true or even coherent, arguing that mathematicians employ a special kind of fictional or “as if” reasoning that reliably extracts truths from absurdities. Moritz Pasch insisted that Vaihinger was wrong about the incoherence of core mathematical theories, but right about the utility of fictional discourse in mathematics. This essay explores this area of agreement between Pasch and Vaihinger. Pasch’s position raises questions about structuralist interpretations (...) of mathematics. (shrink)