William Tait is one of the most distinguished philosophers of mathematics of the last fifty years. This volume collects his most important published philosophical papers from the 1980's to the present. The articles cover a wide range of issues in the foundations and philosophy of mathematics, including some on historical figures ranging from Plato to Gdel.
There can be no doubt about the value of Frege's contributions to the philosophy of mathematics. First, he invented quantification theory and this was the first step toward making precise the notion of a purely logical deduction. Secondly, he was the first to publish a logical analysis of the ancestral R* of a relation R, which yields a definition of R* in second-order logic.1 Only a narrow and arid conception of philosophy would exclude these two achievements. Thirdly and very importantly, (...) the discussion in §§58-60 of the G r u n d l a g e n defends a conception of mathematical existence, to be found in Cantor (1883) and later in the writings of Dedekind and Hilbert, by basing it upon considerations about meaning which have general application, outside mathematics.2.. (shrink)
The background of these remarks is that in 1967, in ‘’Constructive reasoning” [27], I sketched an argument that finitist arithmetic coincides with primitive recursive arithmetic, P RA; and in 1981, in “Finitism” [28], I expanded on the argument. But some recent discussions and some of the more recent literature on the subject lead me to think that a few further remarks would be useful.
Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will (1) criticize this foundation, (2) propose a quite different one, and (3) note that essentially the latter foundation also underlies the Curry-Howard type theory, and hence Heyting's intuitionistic conception of logic. Thus the Dialectica interpretation (in (...) so far as its aim was to give constructive content to intuitionism) is superfluous. (shrink)
Foundations of a General Theory of Manifolds [Cantor, 1883], which I will refer to as the Grundlagen, is Cantor’s first work on the general theory of sets. It was a separate printing, with a preface and some footnotes added, of the fifth in a series of six papers under the title of “On infinite linear point manifolds”. I want to briefly describe some of the achievements of this great work. But at the same time, I want to discuss its connection (...) with the so-called paradoxes in set theory. There seems to be some agreement now that Cantor’s own understanding of the theory of transfinite numbers in that monograph did not contain an implicit contradiction; but there is less agreement about exactly why this is so and about the content of the theory itself. For various reasons, both historical and internal, the Grundlagen seems not to have been widely read compared to later works of Cantor, and to have been even less well understood. But even some of the more recent discussions of the work, while recognizing to some degree its unique character, misunderstand it on crucial points and fail to convey its true worth. (shrink)
The reduction of the lambda calculus to the theory of combinators in [Sch¨ onfinkel, 1924] applies to positive implicational logic, i.e. to the typed lambda calculus, where the types are built up from atomic types by means of the operation A −→ B, to show that the lambda operator can be eliminated in favor of combinators K and S of each type A −→ (B −→ A) and (A −→ (B −→ C)) −→ ((A −→ B) −→ (A −→ C)), (...) respectively.1 I will extend that result to the case in which the types are built up by means of the general function type ∀x : A.B(x) as well as the disjoint union type ∃x : A.B(x)– essentially to the theory of [Howard, 1980]. To extend the treatment of −→ to ∀ we shall need a generalized form of the combinators K and S, and to deal with ∃ we will need to introduce a new form of the combinator S.. (shrink)
These essays present new analyses of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.
We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is definable (...) in the Curry-Howard theory. (shrink)
These essays present new analyses of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.
The classical theory of types in question is essentially the theory of Martin-Löf [1] but with the law of double negation elimination. I am ultimately interested in the theory of types as a framework for the foundations of mathematics and, for this purpose, we need to consider extensions of the theory obtained by adding ‘well-ordered types,’ for example the type N of the finite ordinals; but the unextended theory will suffice to illustrate the treatment of extensional equality.
1. A Road to Philosophy of Mathematics l became interested in philosophy and mathematics at more or less the same time, rather late in high school; and my interest in the former certainly influenced my attitude towards the latter, leading me to ask what mathematics is really about at a fairly early stage. I don ’t really remember how it was that I got interested in either subject. A very good math teacher came to my school when I was in (...) 9th grade and I got caught up in his course on solid geometry; but he soon left and math then lost its luster again in the hands of teachers who neither liked nor understood it. Calculus wasn’t taught in high school in those days, or at least not in mine: besides geometry we learned some algebra and trigonometry. I doubt that even the word “ philosophy ” passed the lips of any of my teachers. My mother, who worked for a publishing house, brought home for me copies of, among other works, the Jowett translations of Plato’s Dialogues, Will Durant’s Story of Philosophy and Courant and Robbins’ What Is Mathematics?; but I can’t remember why she did that: She wasn’t at all intellectual and, as far as I recall, my interests at the time were mostly confined to sports and girls—in some order. Maybe she just thought it was time for me to develop new interests. After high school, I went in 1948 to Lehigh University, then at least primarily an engineering school, on an athletic scholarship. There I had the good fortune in my first year to have an introduction to philosophy course with Lewis White Beck. He had just moved there from the University of Delaware and shortly thereafter moved on to the University of Rochester, where he became one of the leading lights of American Kant studies. My good luck was compounded when, in my second year, Adolph Gr¨ unbaum arrived at Lehigh, fresh from graduate school at Yale, and stayed at least long enough for me to graduate, before moving to the University of Pittsburgh as Andrew Mellon Professor of Philosophy of Science.. (shrink)
