In den letzten Jahrzehntel! hat sich das Interesse an der Grund­ legung der Mathematik immer gesteigert. Fanden frtiher die wenigen Forscher, die sich emsthaft mit dieser 'Frage beschaftigten, wenig Be­ achtung, heute ist die Teilnahme sowohl von mathematischer wie von philosophischer Seite fast allgemein. Zu diesem Umschwung hat sieher die CANToRSche Mengenlehre, die gleich nach ihrem Entstehen lebhafte Erorterungen tiber ihre Berechtigung hervorrief, den AnstoB gegeben, und besonders die bei riicksichtsloser Durchfiihrung ihrer Grundgedanken auftretenden Widerspriiche zogen die allgemeine Aufmerksamkeit auf (...) sich. Doch ist die bisweUen noch geauBerte Behauptung, der Zweck -der Grundlagenforschung liege in der Beseitigung der Widersprtiche, verfehlt. In philosophischer und in mathematischer Richtung geht diese weit tiber eine solche Zielsetzung hinaus. Philosophisch untersucht man -das Wesen der mathematischen Erkenntnis, ihre Voraussetzungen und Endziele, ihr Verhaltnis zu anderen Wissensgebieten, insbesondere der Physik, und ihre Abgrenzung gegen diese dem lnhalt und der Methode nacho An diese philosophischen Erorterungen schlieBen sich umfangreiche mathematische Untersuchungen tiber den Aufbau der Mathematik aus den philosophisch gegebenen Voraussetzungen und tiber die Struktur der mathematischen Beweisftihrungen. Einzelne Teilgebiete dieser Unter­ suchungen entwickeln sich schon zu selbstandigen Disziplinen, die sich in ihren Methoden und Problemstellungen von der eigentlichen Grund­ lagenforschung unabhangig machen; ein Beispiel eines solchen neuen Zweiges der Mathematik, der sein Entstehen der Grundlagenforschung verdankt, ist die mathematischp. Logik. Allmahlich haben sich drei J. auptrichtungen gebildet, die je einer eigenen Auffassung tiber das Wesen der Mathematik entsprechen undo je zu verschieden gearteten mathematischen Untersuchungen geftihrt haben. (shrink)

The role of critique in the Anglophone analytical tradition of philosophy of education is outlined and some of its shortcomings are noted, particularly its apparent claim to methodological objectivity in arriving at what are clearly contestable positions about the normative basis of education. Many of these issues can be seen to have a long history within European, and especially German, philosophy of education. In the light of this the discussion moves on to a consideration of similarities and contrasts between the (...) Anglophone and German-inspired deployment of the concept of critical rationality in philosophy of education. The claims to objectivity of the Anglophone tradition are contrasted with a more self-conscious concern for social justice and improvement in other European traditions, which has been followed more recently by a greater scepticism concerning the potential of critique for delivering social justice and improvement in education. This has parallels with the growing Anglophone disillusion with ‘classical’ analytic philosophy of education. This in turn has resulted in a greater awareness of the limitations of critique: its ideological character, its rootedness in specific contexts, its own potential dogmatism and its ambiguities. The various contributions to this volume are briefly described and related to each other. (shrink)

ZusammenfassungDie Arbeit enthält Bemerkungen über den Intuitionismus and über seine Beziehungen zu anderen Gebieten der Grundlagenforschung. Innerhalb der intuitionistischen Mathematik werden, im Anschluss an die Kritik von Griss gegen den Gebrauch der Negation, Evidenzstufen unterschieden, abhängend von der Art, in der bedingte Konstruktionen zugelassen werden. Auch werden gewisse Schwierigkeiten in der Theorie der endlichen Spezies diskutiert. Was die Grundlagenforschung im Aligemeinen betrifft, wird bemerkt, dass sie die klassische Mathematik weitgehend in ihre intuitiven, formalen and platonischen Bestandteile zerlegt hat. Es wird (...) näher eingegangen auf die These von Church in der Theorie der rekursiven Funktionen.RésuméL'article contient des remarques sur l'intuitionisme et sur ses rapports avec les autres directions dans la recherche des fondements. Dans la mathématique intuitioniste, et en connexion avec la critique de Griss contre l'usage de la négation, on distingue des degrés d'évidence, dépendant de l'espèce de constructions conditionnées qui est admise. En outre, on discute certaines difficultés dans la théorie des espèces finies. Concernant la recherche des fondements en général, on remarque que les mathématiques classiques ont été scindées en leurs éléments intuitifs, formels et platonistes. Quelques remarques sur la thèse de Church dans la théorie des fonctions récursives terminent l'article.The paper contains remarks on intuitionism and its relations with other domains of foundational research. Inside the intuitionistic mathematics, in connection with Griss' criticism against the use of negation, different degrees of evidence are distinguished, depending upon the way in which conditioned constructions are admitted. Some difficulties in the theory of finite species are discussed. Concerning the foundational research in general it is observed that it has separated intuitive, formal and platonistic constituents in classical mathematics. Some remarks are made on Church's thesis in the theory of recursive functions. (shrink)

In reaction to Doret De Ruyter's recent defence of the importance of ideals in education, I advocate cautiousness in three respects. First, I explain the importance of distinguishing ideals more sharply from goals by demonstrating the problems of considering ideals even approximately realisable. Second, I substantiate my doubts about their indispensability in human motivation, and question the desirability of encouraging the motivational use of ideals. Third, I question whether ideals could or should be ‘passed on’ in education, drawing attention to (...) their non-objective, personally created nature. (shrink)

Many philosophers of education emphasise the impossibility to really ‘solve’ philosophical—and with that, educational—problems these days. Philosophers have been trying to give philosophy a new, constructive turn in the face of this insolvability. This paper focuses on irony-based approaches that try to exploit the very uncertainty of philosophical issues to further philosophical understanding. We will first briefly discuss a few highlights of historical uses of irony as a philosophical tool. Then we concentrate on two different interpretations of irony, formulated by (...) Bransen and Rorty, that aim at gaining insight into how we make meaning of the world, while at the same time recognising that such an understanding would be impossible. After discussing some problematic aspects of these interpretations a third interpretation of irony is developed, based on a third view of the nature of meaning-making. Following these three interpretations, we will discuss their philosophical merits and the different kinds of insight they can produce for philosophy of education. (shrink)

Many philosophers of education emphasise the impossibility to really ‘solve’ philosophical—and with that, educational—problems these days. Philosophers have been trying to give philosophy a new, constructive turn in the face of this insolvability. This paper focuses on irony-based approaches that try to exploit the very uncertainty of philosophical issues to further philosophical understanding. We will first briefly discuss a few highlights of historical uses of irony as a philosophical tool. Then we concentrate on two different interpretations of irony, formulated by (...) Bransen and Rorty, that aim at gaining insight into how we make meaning of the world, while at the same time recognising that such an understanding would be impossible. After discussing some problematic aspects of these interpretations a third interpretation of irony is developed, based on a third view of the nature of meaning-making. Following these three interpretations, we will discuss their philosophical merits and the different kinds of insight they can produce for philosophy of education. (shrink)

This is the Spanish translation, by María Gabriela Fulugonio, of Arend Heyting’s classical text “Die logizistische Grundlegung der Mathematik”, which was originally presented at the Königsberg’s Symposium on Philosophy of Mathematics in 1930, and finally published in Erkenntnis in 1931.

This book gives a comprehensive account of methods in philosophy of education, it also examines their application in the 'real world' of education. It will therefore be of interest to philosophers and educators alike.

How can philosophy exert its critical function in society and in education if any appeal to independent and even relatively ‘certain’ criteria seems problematic? The epistemological doubts that foundationalist models of justification encounter unavoidably seem to raise this question. In particular, the relativist implications that seem to result from rejecting such models seem to paralyse the critical potential of philosophy of education. In order to explore the possibilities of a conception of educational critique that avoids the pitfalls of foundationalism, I (...) analyse the epistemological dimensions of this much-feared relativism, illustrating this with some characteristic examples. Solving the problems raised will require an interpretation of critique that leaves our daily sense of critique intact, without literally adopting its—foundationalist—basic assumptions. After systematically developing such an alternative interpretation of critical usage, a non-relativist but still non-foundationalist and powerful conception of philosophical critique seems possible. I illustrate results with some examples from philosophy of education. (shrink)

Differential academic language proficiency is an issue of major educational concern, bearing on problems varying from pupil performance, to social prospects, and citizenship. In this paper we develop a conception of the language‐acquiring subject, and we discuss the consequences for understanding differential language proficiency in schools. Starting from Wittgenstein's meaning‐as‐use theory we show that learning a language requires an activity that relates the subject both to the community of language users, and to the things language is about. In opposition to (...) Luntley, we contend that this does not mean that linguistic development involves linguistic adjustment to the world ‘as it is’. It is argued that, in as far as linguistic development involves a process of adjustment, this concerns conceptions about the world as it is presupposed to be—a ‘world’ that is subjected to doubt and revision time and again. With respect to dealing with differential academic language proficiency, this approach to linguistic development suggests bringing pupils into situations which require active participation in processes of ‘negotiating meaning’, including negotiating the prevailing presuppositions about what the world is like. This also puts novices in a different position—less assimilatory—recognising their co‐constructive potencies at a more fundamental level. (shrink)