Excerpt from Substanzbegriff und Funktionsbegriff: Untersuchungen Uber die Grundfragen der Erkenntniskritik Die erste Anregung zu den Untersuchungen, die dieser Band enthalt, ist mir aus Studien zur Philosophie der Mathe matik erwachsen. Indem ich versuchte, von Seiten der Logik aus einen Zugang zu den Grundbegriffen der Mathematik zu gewinnen, erwies es sich vor allem als notwendig, die B e g r i f f s f u n k t i 0 n e'lhsimaher zu zergliedern und auf ihre Voraussetzungen zuruckzufuhren. Hier (...) aber machte sich alsbald eine eigentumliche Schwierigkeit geltend: die her kommliche logische Lehre vom Begriff zeigte sich in ihren bekannten Hauptzugen als unzureichend, die Probleme, zu denen die Prinzipienlehre der Mathematik hinfuhrt, auch nur vollstandig z u b e z e i c h n e n. Die exakte Wissenschaft war hier, wie sich mir immer deutlicher zu ergeben schien, zu Fragen gelangt, fur welche die Formensprache der tradi tionellen Logik kein genaues Correlat besitzt. Der sachliche Gehalt der mathematischen Erkenntnisse wies auf eine Grund form des Begriffs zuruck, die in der Logik selbst nicht zu klarer Bezeichnung und Anerkennung gekommen war. Ins besondere waren es Untersuchungen jiber den Reihenbegriff und den Grenzbegriff (deren spezielles Ergebnis in die allgemeineren Erorterungen dieses Buches nich? 3iffge nommen werden konnte), die diese Uberzeugung in mir be festigten und damit zu einer erneuten Analyse der Prinzipien der Begriffsbildung selbsfhindrangten. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Gary Hatfield examines theories of spatial perception from the seventeenth to the nineteenth century and provides a detailed analysis of the works of Kant and Helmholtz, who adopted opposing stances on whether central questions about spatial perception were fully amenable to natural-scientific treatment. At stake were the proper understanding of the relationships among sensation, perception, and experience, and the proper methodological framework for investigating the mental activities of judgment, understanding, and reason issues which remain at the core of philosophical psychology (...) and cognitive science. (shrink)

Universally recognized as bringing about a revolutionary transformation of the notions of space, time, and motion in physics, Einstein's theory of gravitation, known as "general relativity," was also a defining event for 20th century philosophy of science. During the decisive first ten years of the theory's existence, two main tendencies dominated its philosophical reception. This book is an extended argument that the path actually taken, which became logical empiricist philosophy of science, greatly contributed to the current impasse over realism, whereas (...) new possibilities are opened in revisiting and reviving the spirit of the more sophisticated tendency, a cluster of viewpoints broadly termed transcendental idealism, and furthering its articulation. It also emerges that Einstein, while paying lip service to the emerging philosophy of logical empiricism, ended up siding de facto with the latter tendency. Ryckman's work speaks to several groups, among them philosophers of science and historians of relativity. Equations are displayed as necessary, but Ryckman gives the non-mathematical reader enough background to understand their occurrence in the context of his wider philosophical project. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics, on the possibility of quantitative psychology, and on the meaning of temperature measurement. Late nineteenth-century scrutinisers of the foundations of mathematics made little of Helmholtz’s essay. Yet it inspired two mathematicians with an eye on physics, and a few philosopher-physicists. The aim of the present paper is to situate Helmholtz’s contribution (...) in this complex array of nineteenth-century philosophies of number, quantity, and measurement.Author Keywords: Helmholtz; Measurement; Arithmetic; P. Du Bois-Reymond; H. and R. Grassmann; J. von Kries. (shrink)

The object in this article is to discuss the philosophical bearing of recent inquiries concerning geometrical axioms and the possibility of working out analytically other systems of geometry with other axioms than Euclid's. Digital edition compiled by Gabriele Dörflinger, Heidelberg University Library.

It has become customary to locate the origins of modern measurement theory in the works of Helmholtz and Hölder. If by ‘modern measurement theory’ is meant the representational theory, then this may not be an accurate assessment. Both Helmholtz and Hölder present theories of measurement which are closely related to the classical conception of measurement. Indeed, Hölder can be interpreted as bringing this conception to fulfilment in a synthesis of Euclid, Newton, and Dedekind. The first explicitly representational theory appears to (...) have been Russell's. He rejected the traditional concept of quantity and its connection to number, making the use of number in quantitative science problematical. He solved the problem via representationalism. (shrink)

The object in this article is to discuss the philosophical bearing of recent inquiries concerning geometrical axioms and the possibility of working out analytically other systems of geometry with other axioms than Euclid's. Digital edition compiled by Gabriele Dörflinger, Heidelberg University Library.

This book offers a new interpretation of Hermann von Helmholtz's work on the epistemology of geometry. A detailed analysis of the philosophical arguments of Helmholtz's Erhaltung der Kraft shows that he took physical theories to be constrained by a regulative ideal. They must render nature "completely comprehensible", which implies that all physical magnitudes must be relations among empirically given phenomena. This conviction eventually forced Helmholtz to explain how geometry itself could be so construed. Hyder shows how Helmholtz answered this question (...) by drawing on the theory of magnitudes developed in his research on the colour-space. He argues against the dominant interpretation of Helmholtz's work by suggesting that for the latter, it is less the inductive character of geometry that makes it empirical, and rather the regulative requirement that the system of natural science be empirically closed. (shrink)

The remarkable modern speculations concerning non-Euclidean sorts of space, of which Prof. Helmholtz gave some account in No. III. of MIND, are likely to be hailed as one of the chief difficulties with which the Kantian theory of space will have to deal. Digital edition compiled by Gabriele Dörflinger. Heidelberg University Library.

Das Interesse allgemeinerer Kreise an der Mathematik hat in neuerer Zeit entschieden zugenommen. Es ist dies die Folge teils der Anwendungen, welche die Technik von dieser Wissenschaft macht, teils der Bemiihungen der Philosophie um das Problem, das durch das Vorhandensein einer solchen Wissenschaft gestellt ist. Darum ist es auch kein Wunder, daB die Mathematiker selbst mehr als friiher geneigt sind, weiteren Kreisen von ihrer Wissenschaft Mitteilungen zukommen zu lassen und sich an der Erorterung der mit der Mathe matik verkniipften erkenntnistheoretischen (...) Fragen zu beteiligen. So habe ich mich schon vor einer Reihe von J ahren entschlossen, die Dberlegungen der Offentlichkeit zu unterbreiten, die seit meiner Studentenzeit, urspriinglich durch die Kantsche Philosophie an geregt, meine Facharbeit begleitet haben. Schon vorher hatte meine, im Jahre IgOO veroffentlichte Leip ziger Antrittsvorlesung mir den AnlaB geboten, mich iiber das Teil gebiet der Geometrie in der einer Gelegenheitsschrift entsprechenden kurzen Weise zu auBern. In den Jahren Igo3-IgIO sind dann die bekannten groBeren Werke von Russel, Couturat und Natorp iiber die Prinzipien der Mathematik und der exakten Wissenschaften er schienen. Diese Schriften haben viel Dankenswertes gebracht, ohne jedoch auf die logischen und erkenntnistheoretischen Fragen die jenigen Antworten zu geben, die mir als die vermutlich richtigsten vorschweben. Hieraus ergab sich mir der auBere AnstoB zu der aus fiihrlichen Darlegung meines eigenen Standpunktes. (shrink)

I said that the book is brilliant. This is not so much because of the conclusions eventually reached about the inadequacy of a purely naturalistic approach to mind. These conclusions are already familiar in the work of Donald Davidson and others. Rather, it is because of the accumulation of historical detail and insight on the basis of which these conclusions are reached. It is often said, for instance, that Kant is a watershed figure, in some sense synthesizing and then moving (...) beyond both empiricism and naturalism. Hatfield uses these terms. But he reinvests them with meaning and energy and in the process shows how radical Kant was in refocusing, by way of a sharp distinction between empirical and transcendental inquiries, the central questions of philosophy. (shrink)

The way in which geometrical knowledge has been obtained has always attracted the attention of philosophers. The fact that there is a science that concerns things outside our thinking and that proceeds inferentially appeared striking, and gave rise to specific theories of experience and space. Nonetheless, the geometrical method has not yet been sufficiently investigated. Philosophers who investigate the theory of knowledge discuss the question of whether geometry is an empirical science, but...