The article explores the relationship between the philosopher and historian of mathematics Jacob Klein’s account of the transformation of the concept of number coincident with the invention of algebra and Husserl’s early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein’s research is shown to present the historical context for Husserl’s twin failures in the Philosophy of Arithmetic, to provide a psychological foundation for the proper concept of number (...) and to show how this concept of number functions as the mathematical foundation of universal arithmetic. The argument is advanced that one significant result of bringing together Klein’s and Husserl’s thought on these issues is the need to fine-tune Husserl’s Crisis project of desedimenting the mathematization of nature. Specifically, Klein’s research shows that “a ‘sedimented’ understanding of numbers” “is superposed upon the first stratum of ‘sedimented’ geometrical ‘evidences’” uncovered by Husserl’s fragmentary analyses of geometry in the Crisis. In addition, then, to the task of “the intentional-historical reactivation of the origin of geometry” recognized by Husserl as intrinsic to the reactivation of the origin of mathematical physics, Klein discloses a second task, that of “the reactivation” of the “complicated network of sedimented significances” that “underlies the ‘arithmetical’ understanding of geometry.”. (shrink)

This volume goes beyond presently available phenomenological analyses based on the structures and constitution of the lifeworld. It shows how the science of history is the mediator between the human and the natural sciences. It demonstrates that the distinction between interpretation and explanation does not imply a strict separation of the natural and the human sciences. Finally, it shows that the natural sciences and technology are inseparable, but that technology is one-sidedly founded in pre-scientific encounters with reality in the lifeworld. (...) In positivism the natural sciences are sciences because they offer causal explanations testable in experiments and the humanities are human sciences only if they use methods of the natural sciences. For epistemologists following Dilthey, the human sciences presuppose interpretation and the human and natural sciences must be separated. There is phenomenology interested in psychology and the social sciences that distinguish the natural and the human sciences, but little can be found about the historical human sciences. This volume fills the gap by presenting analyses of the material foundations of the "understanding" of expressions of other persons, and of primordial recollections and expectations founding explicit expectations and predictions in the lifeworld. Next, it shows, on the basis of history as applying philological methods in interpretations of sources, the role of a universal spatio-temporal framework for reconstructions and causal explanations of "what has really happened". (shrink)

Important study focuses on the revival and assimilation of ancient Greek mathematics in the 13th–16th centuries, via Arabic science, and the 16th-century development of symbolic algebra. This brought about the crucial change in the concept of number that made possible modern science — in which the symbolic "form" of a mathematical statement is completely inseparable from its "content" of physical meaning. Includes a translation of Vieta's Introduction to the Analytical Art. 1968 edition. Bibliography.

Dieses Buch ist in erster Linie als ein Beitrag zur phänomenologi­ schen Aufklärung der mathematischen Erkenntnis gedacht. Phä­ nomenologie als Methode kann nur im handanlegenden Bearbei­ ten von Bewußtseinsleistungen ihre Angemessenheit und ihre Lei­ stung erweisen. Weiterhin ist eine phänomenologische Klärung der Möglichkeit der Erkenntnis in Mathematik und Logik nahelie­ gend, weil es deren Erkenntnisprobleme waren, die Husserl zur Philosophie und schließlich zur Phänomenologie geführt haben. Will man sich bei diesem Vorhaben an der phänomenologischen Methode orientieren, so kann eine Sammlung (...) und Darstellung der verstreuten Husserlschen Stellungnahmen zur Philosophie der Mathematik nicht ausreichen. Jede dieser Stellungnahmen muß darüberhinaus an der phänomenologischen Methode gemessen werden, die Husserl mit viel größerer Mühe ausgearbeitet hat als seine Ansätze zu Einzelfragen, wie z. B. zur Klärung der Mathema­ tik, die im Rahmen seines philosophischen Gesamtvorhabens schließlich nur noch ein Problem neben anderen darstellten. In dem Vorhaben, Phänomenologie als Philosophie der Mathematik zu entfalten, mußten die wichtigsten, von Husserl vorgegebenen Lösungsansätze seinem eigenen Willen gemäß kritisch geprüft werden. Wo es sich als unumgänglich erwies, mußte im Namen der Methode auch eine andere Lösung bevorzugt werden. Die vorlie­ gende Schrift unterscheidet sich im wesentlichen von den bisheri­ gen Darstellungen von "Husserls Philosophie der Mathematik" durch die weiterführende Anwendung der phänomenologischen Methode auf Fragen der mathematischen Erkenntnisgewinnung. Husserls Äußerungen zu philosophischen Fragen der Mathema­ tik finden sich in allen seinen veröffentlichten phänomenologi- 2 EINLEITUNG sehen Schriften verstreut. Die Ausführungen der 4. und 6. (shrink)

Burt C. Hopkins presents the first in-depth study of the work of Edmund Husserl and Jacob Klein on the philosophical foundations of the logic of modern symbolic mathematics. Accounts of the philosophical origins of formalized concepts—especially mathematical concepts and the process of mathematical abstraction that generates them—have been paramount to the development of phenomenology. Both Husserl and Klein independently concluded that it is impossible to separate the historical origin of the thought that generates the basic concepts of mathematics from their (...) philosophical meanings. Hopkins explores how Husserl and Klein arrived at their conclusion and its philosophical implications for the modern project of formalizing all knowledge. (shrink)

The article explores the relationship between the philosopher and historian of mathematics Jacob Klein’s account of the transformation of the concept of number coincident with the invention of algebra, together with Husserl’s early investigations of the origin of the concept of number and his late account of the Galilean impulse to mathematize nature. Klein’s research is shown to present the historical context for Husserl’s twin failures in the Philosophy of Arithmetic: to provide a psychological foundation for the proper concept of (...) number, and to show how this concept of number functions as the mathematical foundation of universal arithmetic. This context establishes that Husserl’s failures are ultimately rooted in the historical transformation of number documented in Klein’s research, from its premodern meaning as the unity of a multitude of determinate objects to its modern meaning as a symbolic representation with no immediate relation to a concrete multiplicity. The argum... (shrink)