I formulate the Zermelo-Russell paradox for naive set theory. A sketch is given of Zermelo’s solution to the paradox: the cumulative type structure. A careful analysis of the set formation process shows a missing component in this solution: the necessity of an assumed imaginary jump out of an infinite universe. Thus a set is formed by a suitable combination of concrete and imaginary operations all of which can be made or assumed by a Turing machine. Some consequences are drawn from (...) this improved analysis of the concept of set, for the theory of sets and for the philosophy and foundations of mathematics. (shrink)

This essay is a critical analysis of some themes in Wittgenstein’s later philosophy. It is not primarily Wittgenstein-exegesis. Much more modestly, my purpose is to express my own thoughts about some questions which Wittgenstein has treated in his writings. It is the second in a series of two articles. The first article, “Remarks on Wittgenstein’s Philosophy: Private Language and Meaning", was published in Volume 42, 2007, of the present YEARBOOK. Section 1, “Philosophical Method”. Wittgenstein’s conception of philosophy as language therapy (...) is criticised. Instead philosophy is construed as foundational research. Wittgenstein’s statement that mathematical logic cannot contribute to progress in philosophy is repudiated. Several examples of ideas and results in mathematical logic which have led to the solution of philosophical problems are given. Section 2, “Contradictions: The Wittgenstein- Turing Debate”. In lectures on the foundations of mathematics given in 1939, Wittgenstein claimed that contradictions in mathematical theories are harmless. A debate ensued on this question between him and Alan Turing. In support of Turing's standpoint, I use the theorem on Taylor series, Church's Theorem, and Gentzen’s Cut-Elimination Theorem to show that Wittgenstein’s standpoint is untenable. For orientation, I also include here the abstract for the first article in the series, “Remarks on Wittgenstein’s Philosophy: Private Language and Meaning”. Section 1, “The Private Language Argument”. An independent argument is given for Wittgenstein’s thesis that there is no private language. I show that psychological terms in ordinary language, in contrast to an implication of Wittgenstein’s own private language argument, in their meanings do contain references to inner states, processes, and events. Section 2, “Meaning”. Wittgenstein’s definition of meaning as use in the language is criticised. Meaning is instead identified with something in the content of a conscious mind. Applications are given to some suggestions in the philosophy of language by Chomsky, Harman and Fodor, Grice, and Kripke. (shrink)

This essay is a critical analysis of some themes in Wittgenstein’s later philosophy. It is not primarily Wittgenstein-exegesis. Much more modestly, my purpose is to express my own thoughts about some questions which Wittgenstein has treated in his writings. It is the first in a series of two articles. The second article, “Remarks on Wittgenstein’s Philosophy: Philosophical Method and Contradictions”, will occur in next year’s issue of the present YEARBOOK. Section 1, “The Private Language Argument”. An independent argument is given (...) for Wittgenstein’s thesis that there is no private language. I show that psychological terms in ordinary language, in contrast to an implication of Wittgenstein’s own private language argument, in their meanings do contain references to inner states, processes, or events. Section 2, “Meaning”. Wittgenstein’s definition of meaning as use in the language is criticised. Meaning is instead identified with something in the content of a conscious mind. Applications are given to some suggestions in philosophy of language by Chomsky, Grice, Harman and Fodor, and Kripke. For orientation, I also include here the abstract for the second article, “Remarks on Wittgenstein’s Philosophy: Philosophical Method and Contradictions”. Section 1, “Philosophical Method”. Wittgenstein’s conception of philosophy as language therapy is criticised. Instead philosophy is construed as foundational research. Wittgenstein’s statement that mathematical logic cannot contribute to progress in philosophy is repudiated. Several examples of ideas and results in mathematical logic which have led to the solution of philosophical problems are given. Section 2, “Contradictions: The Wittgenstein-Turing Debate”. In lectures on the foundations of mathematics given in 1939, Wittgenstein claimed that contradictions in mathematical theories are harmless. A debate ensued on this question between him and Alan Turing. In support of Turing’s standpoint, I use the theorem on Taylor series, Church’s Theorem, and Gentzen’s Cut-Elimination Theorem to show that Wittgenstein’s standpoint is untenable. (shrink)

Eight derivations of Bell’s inequality are given. First a simple and concrete derivation is given drawing on the full strength of the two hypotheses of locality and hidden variables. Two attempts at deriving Bell’s inequality without locality are made. They fail, but give valuable insight into the form which nonlocality must take in quantum physics. Arguments are given against this form of nonlocality. Ontological ideas which allow separation, realism, and locality in quantum mechanics are indicated. In the following five derivations, (...) a number of variants of the assumption of hidden variables are tried. Among the insights which these derivations give rise to are: Particles cannot be assumed to have spin or polarisation values for as much as one fixed direction. The classical picture of electromagnetic radiation is incompatible with QM. Heisenberg’s idea of potentiality in elementary particles is incompatible with QM. There is a weakest form of hidden variables, called realisability, which is sufficient to yield Bell’s inequality. Quantum states cannot be identified with physical states. The mathematical problem of the integration of quantum states into physical states is nontrivial. (shrink)