"With exceptional clarity, but with no evasion of essential ideas, the author outlines the fundamental structure of mathematics."--Carl B. Boyer, Brooklyn College. This enlightening survey of mathematical concept formation holds a natural appeal to philosophically minded readers, and no formal training in mathematics is necessary to appreciate its clear exposition. Contents include examinations of arithmetic and geometry; the rigorous construction of the theory of integers; the rational numbers and their foundation in arithmetic; and the rigorous construction of elementary arithmetic. Advanced (...) topics encompass the principle of complete induction; the limit and point of accumulation; operating with sequences and differential quotient; remarkable curves; real numbers and ultrareal numbers; and complex and hypercomplex numbers. 1959 ed. 27 Figures. Index. (shrink)

In this study Friedrich Waismann gives a systematic presentation of insights into philosophical problems which can be achieved by clarifying the language in which the problems are posed. Much of the material and the method itself derive from Wittgenstein's work in the early 30s. The book was originally envisaged as a lucid and well organized account of Wittgenstein's distinctive form of linguistic philosophy to enable the Vienna Circle to incorporate these valuable methods into their own programme of analysis. The (...) project evolved over many years into a wide-ranging survey of the dissolution of many philosophical problems and the construction of a systematic philosophical grammar. Waismann shows in detail how puzzlement can be removed by careful description of the uses of the terms employed in framing problems. At the same time, he sketches a general framework for analysis of language, including chapters on names, general terms, logical operators, propositions, questions, etc. This book is an authoritative presentation of Wittgenstein's influential philosophy of language in a format much easier to follow than his own intricate and elusive texts. Waismann exhibits clearly the merits of this method of philosophizing. This book should serve as an important and useful text for those who wish to understand the method and power of linguistic philosophy. It has no equal as an introduction to Wittgenstein's philosophy. (shrink)

The Voices of Wittgenstein brings for the first time, in both the original German and in English translation, over one hundred short essays in philosophical logic and the philosophy of mind. This text is of key historical importance to understanding Wittgenstein's philosophical thought and development in the 1930's. Transcribed from the papers of Friedrich Waismann and dating from 1932 to 1935, the majority are highly important dictations by Wittgenstein to Waismann. It also includes texts of redrafted material by (...) Waismann, closely based on these dictations. (shrink)

Friedrich Waismann (1896–1959) was one of the most gifted students and collaborators of Moritz Schlick. Accepted as a discussion partner by Wittgenstein from 1927 on, he functioned as spokesman for the latter’s ideas in the Schlick Circle, until Wittgenstein’s contact with this most faithful interpreter was broken off in 1935 and not renewed when exile took Waismann to Cambridge. Nonetheless, at Oxford, where he went in 1939, and eventually became Reader in Philosophy of Mathematics (changing later to Philosophy (...) of Science), Waismann made important and independent contributions to analytic philosophy and philosophy of science (for example in relation to probability, causality and linguistic analysis). The full extent of these only became evident later when the larger (unpublished) part of his writings could be studied. His first posthumous work The Principles of Linguistic Philosophy (1965, 2nd edn.1997; German 1976) and his earlier Einführung in das mathematische Denken (1936) have recently proved of fresh interest to the scientific community. This late flowering and new understanding of Waismann’s position is connected with the fact that he somewhat unfairly fell under the shadow of Wittgenstein, his mentor and predecessor. Central to this book about a life and work familiar to few is unpublished and unknown works on causality and probability. These are commented on in this volume, which will also include a publication of new or previously scattered material and an overview of Waismann’s life. (shrink)

This work brings in both the original German and English translation of over one hundred short essays in philosophical logic and the philosophy of mind of historical importance to understanding Wittgenstein's philosophical thought and development in the 1930's. Transcribed from the papers of Friedrich Waismann and dating from 1932-35, the majority are highly important dictations by Wittgenstein to Waismann, but also includes texts of redrafted material by Waismann closely based on the dictations. Many of these texts become (...) the ultimate sources of material on philosophy of language that constitutes the bulk of Waismann's book Logic, Sprache, Philosophy. The transcriptions throw light on the development of Wittgenstein's thought in this period and some of the topics are not covered in any of the existing publications of his writings, in particular his detailed attention to causal theories of meaning. They also contain texts covering psychological concepts such as expecting, intending and experience; these have no counterparts in Waismann's publications. (shrink)

The author attempts to sketch a new picture of language: language is stratified into layers, each layer having a logic of its own and being separated from the others by gaps over which one may jump but which cannot be bridged by logical processes. Philosophers try to bridge the gaps and become entangled in pseudo-problems. Law statements exemplify one stratum, thing statements another, sense-datum statements another, ethical statements another, and so on. The different subject-matters are to be characterized by reference (...) to the different strata, rather than conversely; a sense impression is something that is describable in a language of such-and-such structure; a material object is something which can be described in such-and-such language; and so on. A consequence of the theory is said to be that logic loses its universal validity: logic can only be applied to statements that are homogeneous. However, relations between the layers do claim the attention of the logician. (shrink)