Mathematics and logic have been central topics of concern since the dawn of philosophy. Since logic is the study of correct reasoning, it is a fundamental branch of epistemology and a priority in any philosophical system. Philosophers have focused on mathematics as a case study for general philosophical issues and for its role in overall knowledge- gathering. Today, philosophy of mathematics and logic remain central disciplines in contemporary philosophy, as evidenced by the regular appearance of articles on these topics in the best mainstream philosophical journals; in fact, the last decade has seen an explosion of scholarly work in these areas. This volume covers these disciplines in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 contributed chapters are by established experts in the field, and their articles contain both exposition and criticism as well as substantial development of their own positions. The essays, which are substantially self-contained, serve both to introduce the reader to the subject and to engage in it at its frontiers. Certain major positions are represented by two chapters-one supportive and one critical. The Oxford Handbook of Philosophy of Math and Logic is a ground-breaking reference like no other in its field. It is a central resource to those wishing to learn about the philosophy of mathematics and the philosophy of logic, or some aspect thereof, and to those who actively engage in the discipline, from advanced undergraduates to professional philosophers, mathematicians, and historians.

The Oxford handbook of philosophy of mathematics and logic

Tractatus 5.62 famously says: ‘ … what the solipsist means is quite correct; only it cannot be said but makes itself manifest. The world is my world: this is manifest in the fact that the limits of language (of that language which alone I understand) mean the limits of my world.’ The later part of this repeats what was said in summary at 5.6: ‘the limits of my language mean the limits of my world’. And the key to the problem ‘how much truth there is in solipsism’ has been provided by the reflections of TLP 5.61.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/F53C4BD60D31847966DD9E8894AE4CBF/S0080443600000285a.pdf/div-class-title-wittgenstein-and-idealism-div.pdf

Wittgenstein and Idealism

Comme le remarque Richardson, ‘c’est la limitation imposee a l’activite philosophique qui est la caracteristique de la philosophie de Wittgenstein que les philosophes contemporains ont le plus de difficultes a accepter’ (Richardson [1], p. 42). D’un bout a l’autre de son itineraire philosophique, Wittgenstein est reste convaincu que les problemes philosophiques resultent d’une incomprehension de la ‘logique de notre langage’ et doivent etre resolus (ou, plus exactement, elimines) non pas par des explications scientifiques ou pseudo-scientifiques, mais par une description correcte de cette logique. De ce point de vue, parler d’une ‘philosophie du langage’ de Wittgenstein est certainement trompeur; car son objectif n’etait manifestement pas de construire une theorie ou une philosophie du langage, mais uniquement de resoudre des problemes philosophiques en general, y compris des problemes de philosophie du langage, par une analyse du fonctionnement de nos expressions linguistiques qui ne presuppose pas, mais au contraire considere a priori comme un obstacle a surmonter, ce qu’on appelle habituellement une ‘philosophie du langage’ ou, plus generalement, ce que nous sommes tentes de dire sur le langage ‘quand nous philosophons’.

Wittgenstein et la philosophie du langage

Publisher Summary This chapter reviews Kreisel's work on the philosophy of mathematics. The data of foundations consist of the mathematical experience of the working mathematician; the general problem of foundations is to analyze the experience as a whole. Crude formalism holds that mathematics consists of assertions of the form: a concretely given configuration has been constructed by means of a given mechanical rule. No general statements about such configurations belong to mathematics. The positivist doctrine considers informal derivations either as unreliable or as irrelevant to mathematics. Realism is the assumption that there are basic elements (sets and the membership relation) with the properties assumed in the cumulative hierarchy; that is, the existential assumptions of set theory are valid. Strong realism is committed to the existence of sets, some containing an infinite number of members. Rejection of formalism has led to the development of systems of logic that cannot be represented mechanically, as by a Turing machine.

Kreisel's Work on the Philosophy of Mathematics-I. Realism

Wittgenstein and Idealism: Bernard Williams

ion is higher than the minimum actually* needed. As a corollary we have this: whatever the significant problems concerning the relation between the abstract and the concrete may be, they do not arise in the cases here considered. (More precisely, they do not arise vividly; for, of course, a 1 My own in this direction have certainly been influenced by the view of Wittgenstein's work here described, 1 A variant to Hilbert'j theory of the foundations of arithmetic, this Journal, 1953, 4, 107-139. Thu is a more thorough reduction to the concrete than might be supposed from the use of formal systems with arbitrarily long formulae, e tc For, m the interpretation of theorems that actually occur in ordinary mathematical practice, rather simple primitive recursive functional* are used, and those defined by means of tramfimte induction with respect to well ordering! near «0 are not needed. Similarly, in the next footnote, only arithmetical operations low down m the hyperanthmenc hierarchy are used (if the theorem permits a predicative interpretation at all). • Kleene and Mostowski, ' La prfdicatrvlti', Bulletin de la Soctitl MatMmatupie ie France, i960, 88. * Here again: within die limited context. For, girm the abstract conception the semi-nominalist description does not render the full content of the statement containing abstract terms, nor conversely. An example of the latter is the harmless statement:terms, nor conversely. An example of the latter is the harmless statement: for all natural numbers a and b, a+ b =* b+ a. For, the meaning of a quantifier is different on the finitist and the extennonal interpretation, the former implying that we have a proof of the statement. However, m the particular case, we may have to agree that either interpretation u consistent with our intuitive meaning on some given occasion.

Review: wittgenstein's theory and practice of philosophy

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Review: wittgenstein's theory and practice of philosophy