Summary. The notion of mechanical process has played a crucial role in mathematical logic since the early thirties; it has become central in computer science, artificial intelligence, and cognitive psychology. But the discussion of Church's Thesis, which identifies the informal concept with a mathematically precise one, has hardly progressed beyond the pioneering work of Church, Gödel, Post, and Turing. Turing addressed directly the question: What are the possible mechanical processes a human computor can carry out in calculating values of a number-theoretic Junction? He claimed that all such processes can be simulated by machines, in modern terms, by deterministic Turing machines. Turing's considerations for this claim involved, first, a formulation of boundedness and locality conditions (for linear symbolic configurations and mechanical operations); second, a proof that computational processes (satisfying these conditions) can be carried out by Turing machines; third, the central thesis that all mechanical processes carried out by human computors must satisfy the conditions. In Turing's presentation these three aspects are intertwined and important steps in the proof are only hinted at. We introduce Kgraph machines and use them to give a detailed mathematical explication of the first two aspects of Turing's considerations for general configurations, i.e. K-graphs. This generalization of machines and theorems provides, in our view, a significant strengthening of Turing's argument for his central thesis.

Introduction

Turing's analysis of effective calculability is a paradigm of a foundational study that (i) led from an informally understood concept to a mathematically precise notion, (ii) offered a detailed investigation of the new mathematical notion, and (iii) settled an important open question, namely the Entscheidungsproblem. The special character of Turing's analysis was recognized immediately by Church in his review of Turing's 1936 paper. The review was published in the first issue of the 1937 volume of the Journal of Symbolic Logic, and Church contrasted in it Turing's mathematical notion for effective calculability (via idealized machines) with his own (via definability) and Gödel's general recursiveness and asserted: “Of these, the first has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately….”

Abstract. Alonzo Church's mathematical work on computability and undecidability is well known indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was “Church's Thesis” put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of Gödel's general recursiveness, not his own λ-definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of effective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the character of my answers is reflected by an alternative title for this paper, Why Church needed Gödel's Recursiveness for his Thesis.

Abstract. Two young logicians, whose work had a dramatic impact on the direction of logic, exchanged two letters in early 1931. Jacques Herbrand initiated the correspondence on 7 April and Kurt Gödel responded on 25 July, just two days before Herbrand died in a mountaineering accident at La Bérarde (Isère). Herbrand's letter played a significant role in the development of computability theory. Gödel asserted in his 1934 Princeton Lectures and on later occasions that it suggested to him a crucial part of the definition of a general recursive function. Understanding this role in detail is of great interest as the notion is absolutely central. The full text of the letter had not been available until recently, and its content (as reported by Gödel) was not in accord with Herbrand's contemporaneous published work. Together, the letters reflect broader intellectual currents of the time: they are intimately linked to the discussion of the incompleteness theorems and their potential impact on Hilbert's Program.

Introduction. Two important papers in mathematical logic were published in 1931, one by Jacques Herbrand in the Journal für reine und angewandte Mathematik and the other by Kurt Gödel in the Monatshefte für Mathematik und Physik. At age 25, Gödel was Herbrand's elder by just two years. Their work dramatically impacted investigations in mathematical logic, but also became central for theoretical computer science as that subject evolved in the fifties and sixties.

Reflections, a symposium on the foundations of mathematics, was held at Stanford University on December 11–13, 1998. The symposium was organized to honor Solomon Feferman who has played an enormous role in shaping the field over the last 40 years. It was timed so that its last day would coincide with Feferman's 70-th birthday; this provided a very special occasion to celebrate him and his career-long dedication to foundational research.

Jon Barwise and Wilfried Sieg, both doctoral students of Feferman, took the initiative in early 1996 of planning what became playfully called the Feferfest; Carolyn Talcott and Rick Sommer soon joined as the local Stanford organizers. Jon was instrumental in our subsequent venture to shape a program; he opened the symposium and gave a lecture on his latest work; he helped with the initial steps towards this volume, even after he had been diagnosed with cancer. Wemiss him.

The symposium was structured around proof-theoretically inspired themes. True to their origin in the work of David Hilbert and Paul Bernays, prooftheoretic investigations have sustained a special emphasis on or, at least a genuine connection to, broad philosophical issues. Stanford University has had an important role in fostering such work through actively engaged faculty, doctoral students, and visitors. Feferman has been at the very center of these activities.

This was an opportune moment to reflect broadly on such investigations, but also to connect them systematically with topics in Feferman's work. His primary contributions have been to proof theory, recursion theory and, in more recent years, to an analysis of the development of mathematical logic in the twentieth century. Indeed, all of thesematters are of intense interest in the current discussion concerning modern mathematical thought.

The symposium had six sessions. The details of the program - with the names of contributors and chairs - can be found at the very back of the book. The papers in this volume were submitted by symposium participants, as well as by some of Feferman's students and former collaborators, as a tribute to Feferman. They are grouped, somewhat differently from the symposium program, into four parts: Proof-theoretic analysis, Logic and computation,Applicative and self-applicative theories, and Philosophy of modern mathematical and logical thought.

This symposium (a.k.a. the Feferfest) is centered around proof theoretically inspired foundational investigations. Solomon Feferman has been a contributor to these investigations during the last forty years in a most systematic and significant way, and the main themes of the Symposium are themes in his work. The Symposium is a tribute to him on the occasion of his 70-th birthday—a tribute both to his specific contributions and to his influence on the direction of current research.

Program

December 11; 100 Cordura Hall

Opening remarks. J. Barwise

Session I: Proof theoretic ordinals

(Chair: G. Mints).

W. Pohlers; Proof-theoretic ordinals for theories in the language of (second order) arithmetic and set theory

J. Avigad; Ordinal analysis without proofs

R. Sommer; Iterating reflection

Session II: Foundational reductions

(Chair: W. Sieg).

P. Martin-Lof; Modelling versus Tarski semantics.

J. Barwise; Symbolic and presymbolic logic

J. van Bentham; Logical constants: the variable fortunes of an elusive notion December 12; Gates Building, B03

Session III: Formalizations in restricted systems

(Chair: S. Buss).

G. Takeuti; Godel sentences of bounded arithmetic

R. Constable; Admiring proof reflections and working with them

U. Kohlenbach; Classical analysis in weak systems of finite type

S. Simpson; Predicativity: the outer limits

Session IV: Applicative and self-applicative theories

(Chair: M. Beeson).

D. Scott; Project update: logics of types and computations

M. Rathjen; Monotone inductive definitions in explicit mathematics

A. Cantini; On extensionality, uniformity and comprehension in explicit mathematics

I. Mason/ C. Talcott; Feferman-Landin logic

December 13; Gates Building, B03

Session V: Philosophy and history of modern mathematical thought

(Chair: D. Follesdal).

C. Parsons; Reflections on predicativity

J. Dawson; The unity of mathematics—a foundational touchstone?

W. Sieg; Calculations by man and machine

W. Tait; Some remarks about finitism

Session VI: Generalized computation and reflective closure (Chair: J. Etchemendy).

Analysis & history To investigate calculations is to analyze symbolic processes carried out by calculators; that is a lesson we owe to Turing. Taking the lesson seriously, I will formulate restrictive conditions and well-motivated axioms for two types of calculators, namely, for human (computing) agents and mechanical (computing) devices. My objective is to resolve central foundational problems in logic and cognitive science that require a deeper understanding of the nature of calculations. Without such an understanding, neither the scope of undecidability and incompleteness results in logic nor the significance of computational models in cognitive science can be explored in their proper generality. The claim for logic is almost trivial and implies the claim for cognitive science; after all, the relevant logical notions have been used when striving to create artificial intelligence or to model mental processes in humans.

The foundational problems come to the fore in arguments for Church's or Turing's Thesis, asserting that an informal notion of effective calculability is captured fully by a particular precise mathematical concept. Church's Thesis, for example, claims in its original form that the effectively calculable number theoretic functions are exactly those functions whose values are computable in Gödel's equational calculus. My strategy, when arguing for the adequacy of a notion, is to bypass theses altogether and avoid the fruitless discussion of their (un-)provability. This can be achieved by conceptual analysis, i.e., by sharpening the informal notion, formulating its general features axiomatically, and investigating the axiomatic framework. Such an analysis will be provided for the two types of calculators I mentioned, examining closely and recasting thoroughly work of Turing and Gandy. My paper builds on systematic and historical work I have pursued for more than a decade, much of it in collaboration with John Byrnes, Daniele Mundici, and Guglielmo Tamburrini. The considerations presented here reshape and extend the earlier systematic work in a novel and, for me, unexpected way; their aim is nevertheless extremely classical, namely, to provide what Hilbert called eine Tieferlegung der Fundamente. It will also become evident that they are embedded in an illuminating historical context.

There is general agreement that Turing, in 1936, gave the most convincing analysis of effective calculability in his paper “On computable numbers - with an application to the Entscheidungsproblem”.