William W. Tait

“Finitism” (Tait 1981) presents an argument that finitist number theory is primitive recursive arithmetic (PRA). The argument is based on taking seriously the “finite” in “finitism”. But the question remained: what did Hilbert (and Bernays) mean in the early 1920’s through the early 1930’s by “finitism” and in particular, did they restrict finitist number theory to PRA. In his dissertation (Zach 2003), Richard Zach pointed out that Hilbert endorsed results as finitist that require more than PRA for their proofs. Tait 2002 and tait2005 argue that it is not clear that Hilbertwas aware that these results go beyond PRA. But that view is challenged in more recent times in Sieg/Ravaglia 2005 and by the editors of (the invaluable!) David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933 (Hilbert 2013). I will survey the old ground and then discuss the new challenge, which claims that, from the early 1920’s on, Hilbert accepted as finitist an enumeration function of the primitive recursive functions (which of course is not primitive recursive). The grounds for this are a reading of a passage in §7 of Grundlagen der Mathematik I and an argument for the consistency of PRA which goes back to 1922-1923 and is elaborated again in §7 of Grundlagen der Mathematik I. I will argue that their reading of the passage in question is a misreading and that the argument for the consistency of PRA uses, not an enumeration function for the primitive recursive functions, but rather mathematical induction on a Π02 predicate (i.e. of the form ∀x∃yϕ(x, y)), which was explicitly rejected by Hilbert as finitist - e.g. notably in Hilbert 1926.

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.

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.

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..

This paper contains a defense against anti-realism in mathematics in the light both of incompleteness and of the fact that mathematics is a ‘cultural artifact.’. Anti-realism (here) is the view that theorems, say, of aritltmetic cannot be taken at face value to express true propositions about the system of numbers but must be reconstrued to be about somctliiiig else or about nothing at all. A ‘bite-the-bullet’ aspect of the defease is that, adopting new axioms, liitherto independent, is not. a matter of recognizing trutlis wliich had previoasly been unrecognized, but of extending the domain of what is true.

There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper..

AT PHAEDO 96A-C Plato portrays Socrates as describing his past study of "the kind of wisdom known as περὶ φυσέως ἱστορία." At 96c-97b, Socrates says that this study led him to realize that he had an inadequate understanding of certain basic concepts which it involved. In consequence, he says at 97b, he abandoned this method and turned to a method of his own. But at this point in the dialogue, instead of proceeding immediately to describe his method, Plato has him interjecting a complaint concerning Anaxagoras and his view that everything should be explained in terms of Mind. His complaint is that Mind would order things in the best possible way and that, therefore, an account of things in terms of Mind would amount to showing that they are ordered in the best possible way. But Anaxagoras did not show this and, instead, offered other kinds of explanations of the various phenomena. Socrates is not just criticizing Anaxagoras here for not doing what he set out to do; he makes it clear, for example at 99b-c, that he believes that the best kind of explanation of the phenomena would be to show that they are ordered in the best possible way. But he was unable to discover an explanation of this kind, either for himself or from others, and so turned to a method which he calls "second best". The actual description of this method is contained in two passages, 99e5-100a7 and 101d5-e1. Between these passages, Socrates invokes the doctrine of Forms and, in particular, the formula.

An observation and a thesis: The observation is that, whatever the connection between Kant’s philosophy and Hilbert’s conception of finitism, Kant’s account of geometric reasoning shares an essential idea with the account of finitist number theory in “Finitism”, namely the idea of constructions f from ‘arbitrary’ or ‘generic’ objects of various types. The thesis is that, contrary to a substantial part of contemporary literature on the subject, when Kant referred to number and arithmetic, he was not referring to the natural or whole numbers and their arithmetic, but rather to the real numbers and their arithmetic.