First published in 1974. Despite the tendency of contemporary analytic philosophy to put logic and mathematics at a central position, the author argues it failed to appreciate or account for their rich content. Through discussions of such mathematical concepts as number, the continuum, set, proof and mechanical procedure, the author provides an introduction to the philosophy of mathematics and an internal criticism of the then current academic philosophy. The material presented is also an illustration of a new, more general method (...) of approach called substantial factualism which the author asserts allows for the development of a more comprehensive philosophical position by not trivialising or distorting substantial facts of human knowledge. (shrink)

4,12f. über Erkenntnismöglichkeiten - Bleistijtzusatz 11 4,15 über eigene Erkenntnismöglichkeit - Bleistijtzusatz 11 4,18ff..... müssen wir zunächst zweifellose Fälle haben von Erkenntnissen oder Erkennt­ nismöglichkeiten, die Erkenntnis wirklich treffen, und daher nicht unbesehen Erkenntnis als Erkenntnis hinnehmen; - der Satz in seiner ursprünglichen Form 114,22f. von sonst hätten wir.... bis volles Ziel Bleistijtzusatz 115,5 und Geisteswissenschaften - Bleistijtzusatz nach 1922 11 5,20f. Dieser Satz ist eine Bleistijtergänzung 11 5,28 voll und ganz adäquat - Bleistiftergänzung 11 5,33 adäquat - Bleistiftzusatz 116,2-16 (...) Der Text von: Was ich will bis sie nicht anzuknüpfen steht in eckiger Bleistiftklammer ; am Rande von Husserl vermerkt: Das ist unklar oder nicht gut passend. Beilage. 11 6,5 der Möglichkeit - Bleistiftzusatz 11 6,23 ihrer Leistung Hinzufügung nach 1922 11 6,32 Zusatz nach 1922: als Prämisse, selbst als Hypothese 11 7,2 ihrer Leistung - Bleistift­ verbesserung für: irgend einer Idee wie es ursprünglich hieß 11 7,3ff. Der letzte Teil dieses Satze8, von es heißt an, bis herleiten steht in eckiger Bleistiftklammer 117,15 Hinzugefügt: der Mensch 117,20 Hinzu­ gefügt: dieser Mensch 11 7,30 Bewußtsein des Menschen - Verbesserung von: Ichbewußtsein wie es ursprünglich hieß 11 7,32ff. Spätere Rand­ bemerkung zum Text: Das Immanente bezweifelt man nicht, aber die Erkenntnis des Immanenten ist genau so problematisch und auch ein schwieriges Problem. 11 9,18f. (shrink)

In explaining his concept of set Cantor intimates a connection with the metaphysical scheme put forward in Plato’s Philebus to determine the place of pleasure. We argue that these determinations capture key ideas of Cantorian set theory and, moreover, extend to intuitions which continue to play a central role in the modern mathematics of infinity.

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians (...) might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish. (shrink)

Kurt Gödel is almost as famous—one might say “notorious”—for his extreme platonist views as he is famous for his mathematical theorems. Moreover his platonism is not a myth; it is well-documented in his writings. Here are two platonist declarations about set theory, the first from his paper about Bertrand Russell and the second from the revised version of his paper on the Continuum Hypotheses.Classes and concepts may, however, also be conceived as real objects, namely classes as “pluralities of things” or (...) as structures consisting of a plurality of things and concepts as the properties and relations of things existing independently of our definitions and constructions.It seems to me that the assumption of such objects is quite as legitimate as the assumption of physical bodies and there is quite as much reason to believe in their existence.But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.The first statement is a platonist declaration of a fairly standard sort concerning set theory. What is unusual in it is the inclusion of concepts among the objects of mathematics. This I will explain below. The second statement expresses what looks like a rather wild thesis. (shrink)

Convinced that the classically undecidable problems of mathematics possess determinate truth values, Gödel issued a programmatic call to search for new axioms for their solution. The platonism underlying his belief in the determinateness of those questions in combination with his conception of intuition as a kind of perception have struck many of his readers as highly problematic. Following Gödel's own suggestion, this article explores ideas from phenomenology to specify a meaning for his mathematical realism that allows for a defensible epistemology.

Darstellung des Noema in 12 Thesen.\nverwendete Textstellen: Ideen 1: S. 203, 22-23; S. 204, 20-21; S. 357, 19-20: Handlungen sind zielgerichtet. Dabei bedarf eines keines physischen Objekts. Husserl setzt and diese Stelle das Noema. Somit wird auch zielgerichtetes Handeln aufgrund einer Halluzination m{ö}glich, Zielgerichtet zu sein bedeutet ein Noema zu haben.\n1. Follesdal´sche These: Noema ist eine intensionale Entit{ä}t, eine Generalisierung des Begriffs Sinn/Bedeutung.\n2. These: Das Noema hat zwei Bestandteile, a) der noematische Sinn, der allen thetischen Handlungen (erinnern, sich vorstellen usw.) (...) gem{ä}{ß} desselben Objekts mit denselben Eigenschaften in derselben Gerichtetheit gemeinsam ist und b) die Gegebenheitsweise (hat auch Sinncharakter), die sich unterscheidet in Handlungen mit verschiedenem thetischen Charakter.\n3. These: Der noematische Sinn ergibt sich aus der Art und Weise, wie Bewu{ß}tsein zum Objekt in Relation steht.\n!! 4. These: Das Noema einer Handlung ist nicht das Objekt dieser Handlung\n5. These: Das Objekt einer Handlung ist eine Funktion des noematischen Sinns, d.h. mit einem bestimmten Noema kann nur ein Objekt korrespondieren.\n6. These: Jedoch k{ö}nnen mit einem bestimmten Objekt verschiedene Noemata korrespondieren. (z.B. kann ein Objekt das Ziel verschiedener thetischer Handlunge sein, etwa erinnern, wahrnehmen, daher:)\n6. These a): Mit einem bestimmten Objekt korrespondieren verschiedene noematische Sinne.\n7. These: Jede Handlung besteht aus nur einem Noema. Sie wird durch dieses Noema charakterisiert. (Umgekehrt trifft dies nicht zu.)\n8. These: Noemata sind abstrakte Entit{ä}ten. (Sinne sind nicht real. Sie sind bezogen auf Handlungen in tempor{ä}ren Intervallen.)\n9. These: Noemata werden nicht durch unsere Sinnesorgane wahrgenommen.\n10. These: Noemata gelangen durch ph{ä}nomenologische Reflektion ins Bewu{ß}tsein.\n11. These: Eine ph{ä}nomenologische Reflektion kann iterativ geschehen (ann{ä}herungsweise; in Schritte gegliedert). Dies ist so, weil schon die Korrespondenz zwischen Sinn und Objekt wieder ein Objekt ergibt. Es folgt ein unendlicher Regress.\nPhysische Objekte sind transzendent in dem Sinne, dass sie unendlich viele Noemata haben. Sinn kann sie nie ersch{ö}pfend erfassen.\nNoemata k{ö}nnen andererseits ein physisches Objekt repr{ä}sentieren in der Form, dass mehr intendiert wird als gesehen wird. So denkt man sich etwa bei einem Stuhl, von dem man nur zwei Beine sieht, die beiden {ü}brigen hinzu.\n12. These: Diese Muster der Vorherbestimmung bilden zusammen mit den Gegebenheitsweisen das Noema (s. These 2). (shrink)

This paper explores the metaphysical roots of Cantor’s conception of absolute infinity in order to shed some light on two basic issues that also affect the mathematical theory of sets: the viability of Cantor’s distinction between sets and inconsistent multiplicities, and the intrinsic justification of strong axioms of infinity that are studied in contemporary set theory.

We investigate both iteration hypotheses and extender models at the level of one supercompact cardinal. The HOD Conjecture is introduced and shown to be a key conjecture both for the Inner Model Program and for understanding the limits of the large cardinal hierarchy. We show that if the HOD Conjecture is true then this provides strong evidence for the existence of an ultimate version of Gödel's constructible universe L. Whether or not this "ultimate" L exists is now arguably the central (...) issue for the Inner Model Program. (shrink)

Godel began to seriously study Husserl's phenomenology in 1959, and the Godel Nachlass is known to contain many notes on Husserl. In this paper I describe what is presently known about Godel's interest in phenomenology. Among other things, it appears that the 1963 supplement to "What is Cantor's Continuum Hypothesis?", which contains Godel's famous views on mathematical intuition, may have been influenced by Husserl. I then show how Godel's views on mathematical intuition and objectivity can be readily interpreted in a (...) phenomenological theory of intuition and mathematical knowledge. (shrink)

Husserl (1859-1938) hatte sich in seinem Werk "Formale und transzendentale Logik" das Ziel gesetzt, den inneren Sinn, die Gliederung und Zusammengehörigkeit all dessen nachzuweisen, was bislang an logischen Problemen behandelt worden war, und die Notwendigkeit einer phänomenologischen Durchleuchtung der gesamten logischen Problematik darzutun. Ein Hauptstück der analytisch-deskriptiven Untersuchungen, die einer solchen phänomenologischen Begründung der Logik dienen, ist "Erfahrung und Urteil". Das Buch entstand in Zusammenarbeit mit Schülern und Mitarbeitern.

Die Cartesianischen Meditationen sind aus Vorträgen hervorgegangen, die Edmund Husserl Mitte Februar 1929 an der Sorbonne gehalten hat. Bei der Grundfragestellung Descartes’ einsetzend, entfaltet Husserl die transzendentale Phänomenologie als »Umbildung und Neu­bildung« des Cartesischen Programms der prima philosophia im Sinne einer Reform der Philosophie zu einer absoluten Wissenschaft aus absoluter Begründung. Eine französische Ausgabe, in der Übersetzung von Emmanuel Levinas und Gabrielle Pfeiffer, erschien 1931 bei A. Colin in Paris. Husserls Arbeiten an dem Manuskript für die deutsche Ausgabe, die gegenüber (...) der französischen erheblich erweitert werden sollte, blieben, nicht zuletzt wegen der 1933 veränderten politischen Situation, unvollendet. So erschien die erste deutschsprachige Ausgabe der Cartesianischen Meditationen erst postum in dem von Stephan Strasser herausgegebenen Band 1 der Husserliana , Den Haag 1950.Die Studienausgabe basiert auf dem Text der zweiten Auflage des Bandes in den Husserliana von 1963, hg. von Rudolf Boehm. (shrink)

Hao Wang was one of the few confidants of the great mathematician and logician Kurt Gödel. _A Logical Journey_ is a continuation of Wang's _Reflections on Gödel_ and also elaborates on discussions contained in _From Mathematics to Philosophy_. A decade in preparation, it contains important and unfamiliar insights into Gödel's views on a wide range of issues, from Platonism and the nature of logic, to minds and machines, the existence of God, and positivism and phenomenology. The impact of Gödel's theorem (...) on twentieth-century thought is on par with that of Einstein's theory of relativity, Heisenberg's uncertainty principle, or Keynesian economics. These previously unpublished intimate and informal conversations, however, bring to light and amplify Gödel's other major contributions to logic and philosophy. They reveal that there is much more in Gödel's philosophy of mathematics than is commonly believed, and more in his philosophy than his philosophy of mathematics. Wang writes that "it is even possible that his quite informal and loosely structured conversations with me, which I am freely using in this book, will turn out to be the fullest existing expression of the diverse components of his inadequately articulated general philosophy." The first two chapters are devoted to Gödel's life and mental development. In the chapters that follow, Wang illustrates the quest for overarching solutions and grand unifications of knowledge and action in Gödel's written speculations on God and an afterlife. He gives the background and a chronological summary of the conversations, considers Gödel's comments on philosophies and philosophers, and his attempt to demonstrate the superiority of the mind's power over brains and machines. Three chapters are tied together by what Wang perceives to be Gödel's governing ideal of philosophy: an exact theory in which mathematics and Newtonian physics serve as a model for philosophy or metaphysics. Finally, in an epilog Wang sketches his own approach to philosophy in contrast to his interpretation of Gödel's outlook. (shrink)

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, (...) … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

Part of the ambiguity lies in the various points of view from which this question might be considered. The crudest di erence lies between the point of view of the working mathematician and that of the logician concerned with the foundations of mathematics. Now some of my fellow mathematical logicians might protest this distinction, since they consider themselves to be just more of those \working mathematicians". Certainly, modern logic has established itself as a very respectable branch of mathematics, and there (...) are quite a few highly technical journals in logic, such as The Journal of Sym-. (shrink)

We introduce insertion domains that support the placement of new, higher, vertices into finite trees. We prove that every nonincreasing insertion domain has an element with simple structural properties in the style of classical Ramsey theory. This result is proved using standard large cardinal axioms that go well beyond the usual axioms for mathematics. We also establish that this result cannot be proved without these large cardinal axioms. We also introduce insertion rules that specify the placement of new, higher, vertices (...) into finite trees. We prove that every insertion rule greedily generates a tree with these same structural properties; and every decreasing insertion rule generates (or admits) a tree with these same structural properties. It is also necessary and sufficient to use the same large cardinals (in the precise sense of Corollary D.25). The results suggest new areas of research in discrete mathematics called "Ramsey tree theory" and "greedy Ramsey theory" which demonstrably require more than the usual axioms for mathematics. (shrink)

One of the standard ways of postulating large cardinal axioms is to consider elementary embeddings,j, from the universe,V, into some transitive submodel,M. See Reinhardt–Solovay [7] for more details. Ifjis not the identity, andκis the first ordinal moved byj, thenκis a measurable cardinal. Conversely, Scott [8] showed that wheneverκis measurable, there is suchjandM. If we had assumed, in addition, that, thenκwould be theκth measurable cardinal; in general, the wider we assumeMto be, the largerκmust be.

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.

5 sehr merkwürdiger Tatsachen zutage gefördert, die vordem verborgen waren, und wirklich psychologische Tatsachen, wenn auch die Physiologen manche große Gruppen von ihnen ihrer eigenen Wissenschaft mit zurechnen. Mag die Einstimmigkeit 5 in der theoretischen Interpretation dieser Tatsachen auch sehr weit zurückstehen hinter derjenigen der exakten naturwissen­ schaftlichen Disziplinen, so ist sie in gewisser Hinsicht doch wieder eine vollkommene, nämlich was den methodischen Stil der gesuchten Theorien anlangt. Jedenfalls ist man in den inter- 10 nationalen Forscherkreisen der neuen Psychologie der (...) festen Überzeugung, einer bis vor kurzem ungebrochenen Überzeugung, daß nun endlich die allein wahre und echte Psychologie in den Gang gebracht sei, als eine strenge Wissenschaft, auf deren Wegen die Gesamtheit aller psychologischen Probleme, aller 15 zur individuellen und Kulturgeistigkeit gehörigen, liegen müssen. Es bedürfe nur, wie in jeder auf elementaren Aufbau und auf die Erklärung aus elementaren Gesetzen bedachten Erfahrungs­ wissenschaft, geduldiger Zurückhaltung und eines ganz vor­ sichtigen Emporschreitens ; man dürfe nur nicht voreilig nach 20 Problemen greifen, die noch nicht zu wissenschaftlicher Be­ arbeitung reif, für die noch nicht die Tatsachenunterlage bereit­ gestellt und die nötigen Erfahrungsbegriffe geschaffen sind. Einen nicht geringen Zuwachs an innerer Sicherheit hat die neue Psychologie durch die gelingende Schöpfung einer Psycho- 25 technik erhalten. Nun schien diese Psychologie wirklich der exakten Physik gleichzustehen. Sie war nun sogar so weit, um ihre psychologische Erkenntnis, ganz so wie physikalische und chemische, technisch nutzbar zu machen. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method 'ideation'. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. (shrink)

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections (...) in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Uber Den Psychologischen Ursprung Der Raumvorstellung Carl Stumpf S. Hirzel, 1873 Space perception. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method ‘ideation’. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. What would be the study that would draw the soul away from the world of becoming to the world of being?… Geometry and arithmetic would be among the studies we are seeking … a philosopher must learn them because he must arise out of the region of generation and lay hold on essence or he can never become a true reckoner … they facilitate the conversion of the soul itself from the world of generation to essence and truth… they are knowledge of that which always is and not of something which at some time comes into being and passes away. (shrink)