The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop's constructive mathematics (BISH). It gives a sketch of both Myhill's axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof's theory of types as a formal system for BISH. 1 What is Constructive Mathematics? The story of modern constructive mathematics begins with the publication, in 1907, of L.E.J. Brouwer's doctoral dissertation Over de Grondslagen der Wiskunde [18], in which he gave the first exposition of his philosophy of intuitionism (a general philosophy, not merely one for mathematics). According to Brouwer, mathematics is a creation of the human mind, and precedes logic: the logic we use in mathematics grows from mathematical practice, and is not some a priori given before mathematical activity c...

The type free -calculus is powerful enough to contain all the polymorphic and higher order nature of functional programming and furthermore types could be constructed inside it. However, mixing the type free -calculus with logic is not very straightforward (see [Aczel 80] and [Scott 75]). In this paper, a system that combines polymorphism and higher order functions with logic is presented. The system is suitable for both the functional and the logical paradigms of programming as from the functional paradigms point of view, the system enables one to have all the polymorphism and higher order that exist in functional languages and much more. In fact even the fixed point operator Y which is defined as f:(x:f(xx))(x:f(xx)) can be type checked to ((ff ! ff) ! ff)) where ff is a variable type. (x:xx)(x:xx) can be type checked too, something not allowed in functional languages. From the point of view of theorem proving, the system is expressive enough to allow self referential sentences and ...

Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a far-away planet. Would their mathematics be set-based? What are the alternatives to the set-theoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.

Logic, due to the paradoxes, is absent from the type free -calculus. This makes such a calculus an unsuitable device for Natural Language Semantics. Moreover, the problems that arise from mixing the type free -calculus with logic lead to type theory and hence formalisations of Natural Language were carried out in a strictly typed framework. It was shown however, that strict type theory cannot capture the self-referential nature of language ([Parsons 79], [Chierchia, Turner 88] and [Kamareddine, Klein 93]) and hence other approaches were needed. For example, the approach carried out by Parsons is based on creating a notion of floating types which can be instantiated to particular instances of types whereas the approaches of Chierchia, Turner and Kamareddine, Klein are based on a type free framework. In this paper, we will embed the typing system of [Parsons 79] into a version of the one proposed in [Kamareddine, Klein 93] giving an interpretation of Parsons' system in a type free theory...

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 ...

In this paper, a theory TH based on combining type freeness with logic is introduced and is then used to build a theory of properties which is applied to determiners and quantifiers. keywords: type freeness, logic, property theory, determiners, quantifiers. 1 THE THEORY TH It is well known that mixing type freeness and logic leads to contradictions. For example, by taking the following syntax of terms: t := xjx:tjt 1 t 2 j:t and applying the term x::xx to itself one gets a contradiction (known as Russell's paradox). Church was aware of the problem when he started the -calculus which he intended to be a theory of functions and logic. But his first theory of the -calculus was type free and so was inconsistent. The paradox could be described as follows: take a to be x:(xx ! ?). Then from Modus Ponens (MP), the Deduction Theorem (DT), and fi-conversion, we could derive Curry's paradox: 1: aa = aa ! ? by fi -conversion 2: aa ` aa 3: aa ` ? by MP + 1+2 4: ` aa ! ? by DT +3 5: aa from 1 ...