Abstract not found

This article, based on an invited lecture at the Logic Colloquium '93 in Keele, is a sequel to van Lambalgen [1992]. Apart from presenting new results, it differs from its predecessor in the following respects: (i) the presentation of the axioms is simplified, following some suggestions of Wojciech Buszkowski, (ii) the axioms have been strengthened, and (iii) the philosophical discussion has (hopefully) been improved. The article has appeared in W. Hodges et al (eds.), Logic: from Foundations to Applications (European Logic Colloquium), Oxford University Press 1996

Contents 1 The continuity principle 1 2 A phenomenological consideration 8 2.1 An argument for G(raph)WC-N . . . . . . . . . . . . . . . . . 8 2.2 Two arguments against WC-N . . . . . . . . . . . . . . . . . . 13 3 Other arguments for continuity 15 3.1 Undecidability of equality of choice sequences . . . . . . . . . 15 3.2 Kripke's Schema and full PEM . . . . . . . . . . . . . . . . . 15 3.3 The KLST theorem . . . . . . . . . . . . . . . . . . . . . . . . 16 4 Conclusion 19 1 The continuity principle There are two principles that lend Brouwer's mathematics the extra power beyond arithmetic. Both are presented in Brouwer's writings with little or no argument. One, the principle of bar induction, will not concern us here. The other, the continuity principle for numbers, occurs for the rst time in print in [Brouwer 1918]. It is formulated and immediately applied to show that the set of numerical choice sequences is not enumerable. In fa

This paper appeared in T. Ferguson et al (eds.): Probability, statistics and game theory, papers in honor of David Blackwell, Institute for Mathematical Statistics 1996

The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...

notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 1887--1963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifier-free expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...

Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of them and has been misinterpreted. We show where these particular choice sequences appear in the work of Brouwer and we show how they should be handled.

It is known that in quantum mechanics, the set S of all possible states coincides with the set of all the complex-valued functions ψ(x) for which ∫ |ψ(x) | 2 dx = 1. From the mathematical viewpoint, this set is a unit sphere in the space L 2 of all the functions for which the value ∥ψ ∥ 2 def ∫ 2 = |ψ(x) | dx is finite. Because of this mathematical fact, usually the set S is considered with the topology induced by L 2, i.e., topology in which the basis of open neighborhood of a state ψ is formed by the open balls Bε(ψ) = {φ: ∥ψ − φ ∥ < ε}. This topology seem to work fine, but since this is a purely mathematical definition, a natural question appears: does this topology have a physical meaning? In this paper, we show that a natural physical definition of closeness indeed leads to the usual L 2-topology.