Alexander George

Dieses Buch blickt in eine bedeutende Epoche der Philosophie der Mathematik zurück, deren Strömungen die heutige Gestalt der Mathematik prägten. In der Wende vom 19. zum 20. Jahrhundert befand sich die Mathematik in einem fundamentalen Umbruch, der die Mathematiker dieser Zeit herausforderte. Sie mussten Stellung beziehen. Die Grundsätze und Wege der philosophischen Richtungen, die dieses Buch verständlich, kritisch und anerkennend beschreibt, wurden von Mathematikern formuliert. Eine Zeit gravierender Disharmonien begann, die bis in Streit und Feindschaften mündeten und zugleich faszinierende und fruchtbare Ergebnisse hervorbrachten, mathematisch wie philosophisch. Es war ein aufregendes, intellektuelles Abenteuer zu Beginn des 20. Jahrhunderts auf einem außergewöhnlich scharfsinnigen und kreativen Niveau. Die Debatte über die unversöhnlichen Ansichten versiegte allmählich und inzwischen ist wieder relative Ruhe in die Gemeinde der Mathematiker eingekehrt. Zentrale philosophische Fragen aber, die damals die Protagonisten spalteten, sind nach wie vor unbeantwortet. Die Suche nach dem Wesen der Mathematik geht weiter und greift auf die Ideen dieser Kontroversen zurück.

Ludwig Wittgenstein has a recognizable approach that he regularly pursues in his philosophical investigations. There is a problem that he often presses, a form of criticism that he often develops, against traditional pursuits of philosophy. It is surprisingly difficult to say clearly what this problem is. But it is worthwhile to try, for this criticism is not only a hallmark of his thought but is also closely connected to other central features of it, for instance, to his conceptions of language and of the nature of philosophical investigation. These features can be properly understood only in concert with a correct view of his terms of criticism of traditional philosophy. In this essay, Alexander George articulates a problem Wittgenstein sees with philosophy, shows how it illuminates otherwise peculiar features of Wittgenstein’s investigations, and finally considers an interesting situation in which Wittgenstein’s goals might be thwarted.

Intuitionism is occasionally advanced on the grounds that a classical understanding of mathematical discourse could not be acquired, given limitations of the experience available to the language learner. In this note, focusing on the acquisition of the universal quantifier, I argue that this route of attack against a classical construal results, at best, in a Pyrrhic victory. The conditions under which it is successful are such as to redound upon the tenability of intuitionism itself. Adjudication will not follow merely from attending to the learner''s experience. The nature of the agent''s ability to engage in conceptual extrapolation from that experience must be considered as well. (And divergent views regarding this are likely to recapitulate the original disagreement.).

The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his position on the philosophical significance of the theorem as a result of a shift in his background beliefs. The nature of this shift and possible explanations for it are investigated. Ironically, Skolem's own case provides a historical example of the philosophical flexibility of his theorem. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something of this sort might be called ?a puffed-up proof?. Ludwig Wittgenstein, Remarks on the foundations of mathematics (revised edition), vol. 2, 21

Few thinkers in the past three decades have exerted more influence on the philosophy of language than Quine, Dummett, and Chomsky. No investigation into the current state of philosophy of language can omit consideration of their views. Yet I believe that their work has often been seriously misinterpreted. I begin by trying to clear up some unfortunate and prevalent misunderstandings. In particular, I examine in detail the relationship between Quine's and Chomsky's thought and argue that rumors of their incommensurability have been greatly exaggerated. I lay out the many affinities between their approaches and isolate the crucial junction at which they part company . I also reconstruct Dummett's arguments against truth-based theories of meaning and in favor of verificationist ones, and argue that some recent criticisms of these have rested on misunderstanding. During my exposition of Dummett's thought, various puzzling impasses are noted. I show that these are symptomatic of a deep tension within his view. Essentially, this consists in his desire to follow both Quine and Chomsky, to adopt simultaneously a Chomskyan position and a Quinean one on the very issue that I argued fundamentally divided these two thinkers. The resultant schizophrenia about the nature of linguistic knowledge is studied closely and suggestions for its cure are tendered. Its significance for the future of the philosophy of language is assessed

The foundations of linguistics continue to generate philosophical debate. Jerrold Katz claims that the subject matter of linguistics consists of abstract objects and that, as a consequence, the discipline cannot be viewed as part of psychology. I respond by arguing (1) that Katz misinterprets work in the philosophy of mathematics which he believes sheds light on foundational questions in linguistics; (2) that he misunderstands aspects of Noam Chomsky's position, against whose conception of linguistics many of his claims are directed; (3) that Katz fails to dispose of a much more plausible analysis, according to which linguistics remains an empirical inquiry in spite of its abstract subject matter; and, finally, (4) that his arguments against what he calls‘generativism’, appealing to the existence of an infinitely long grammatical sentence of English, are flawed.