ON COMPUTABILITY Wilfried Sieg 1 INTRODUCTION Computability is perhaps the most significant and distinctive notion modern logic has introduced; in the guise of decidability and effective calculability it has a venerable history within philosophy and mathematics. Now it is also the basic theoretical concept for computer science, artificial intelligence and cognitive sci- ence. This essay discusses, at its heart, methodological issues that are central to any mathematical theory that is to reflect parts of our physical or intellec- tual experience. The discussion is grounded in historical developments that are deeply intertwined with meta-mathematical work in the foundations of mathemat- ics. How is that possible, the reader might ask, when the essay is concerned solely with computability? This introduction begins to give an answer by first describ- ing the context of foundational investigations in logic and mathematics and then sketching the main lines of the systematic presentation. 1.1 Foundational contexts In the second half of the 19th century the issues of decidability and effective calcu- lability rose to the fore in discussions concerning the nature of mathematics. The divisive character of these discussions is reflected in the tensions between Dedekind and Kronecker, each holding broad methodological views that affected deeply their scientific practice. Dedekind contributed perhaps most to the radical transforma- tion that led to modern mathematics: he introduced abstract axiomatizations in parts of the subject (e.g., algebraic number theory) and in the foundations for arithmetic and analysis. Kronecker is well known for opposing that high level of structuralist abstraction and insisting, instead, on the decidability of notions and the effective construction of mathematical objects from the natural numbers. Kronecker’s concerns were of a traditional sort and were recognized as perfectly legitimate by Hilbert and others, as long as they were positively directed towards the effective solution of mathematical problems and not negatively used to restrict the free creations of the mathematical mind. At the turn of the 20th century, these structuralist tendencies found an impor- tant expression in Hilbert’s book Grundlagen der Geometrie and in his essay Uber¨ den Zahlbegriff. Hilbert was concerned, as Dedekind had been, with the consis- tency of the abstract notions and tried to address the issue also within a broad set Handbook of the Philosophy of Science. Philosophy of Mathematics Volume editor: Andrew Irvine. General editors: Dov M. Gabbay, Paul Thagard and John Woods. c 2008 Elsevier BV. All rights reserved. 526 Wilfried Sieg theoretic/logicist framework. The framework could have already been sharpened at that point by adopting the contemporaneous development of Frege’s Begriffs- schrift, but that was not done until the late 1910s, when Russell and Whitehead’s work had been absorbed in the Hilbert School. This rather circuitous development is apparent from Hilbert and Bernays’ lectures [1917/18] and the many founda- tional lectures Hilbert gave between 1900 and the summer semester of 1917. Apart from using a version of Principia Mathematica as the frame for formalizing math- ematics in a direct way, Hilbert and Bernays pursued a dramatically different approach with a sharp focus on meta-mathematical questions like the semantic completeness of logical calculi and the syntactic consistency of mathematical the- ories. In his Habilitationsschrift of 1918, Bernays established the semantic complete- ness for the sentential logic of Principia Mathematica and presented a system of provably independent axioms. The completeness result turned the truth-table test for validity (or logical truth) into an effective criterion for provability in the logical calculus. This latter problem has a long and distinguished history in philosophy and logic, and its pre-history reaches back at least to Leibniz. I am alluding of course to the decision problem (“Entscheidungsproblem”). Its classical formula- tion for first-order logic is found in Hilbert and Ackermann’s book Grundz¨uge der theoretischen Logik. This problem was viewed as the main problem of mathemat- ical logic and begged for a rigorous definition of mechanical procedure or finite decision procedure. How intricately the “Entscheidungsproblem” is connected with broad perspec- tives on the nature of mathematics is brought out by an amusingly illogical argu- ment in von Neumann’s essay Zur Hilbertschen Beweistheorie from 1927: . it appears that there is no way of finding the general criterion for deciding whether or not a well-formed formula a is provable. (We cannot at the moment establish this. Indeed, we have no clue as to how such a proof of undecidability would go.) . the undecidability is even a conditio sine qua non for the contemporary practice of mathematics, using as it does heuristic methods, to make any sense. The very day on which the undecidability does not obtain any more, mathematics as we now understand it would cease to exist; it would be replaced by an absolutely mechanical prescription (eine absolut mechanische Vorschrift) by means of which anyone could decide the provability or unprovability of any given sentence. Thus we have to take the position: it is generally undecidable, whether a given well-formed formula is provable or not. If the underlying conceptual problem had been attacked directly, then something like Post’s unpublished investigations from the 1920s would have been carried out in G¨ottingen. A different and indirect approach evolved instead, whose origins can be traced back to the use of calculable number theoretic functions in finitist con- sistency proofs for parts of arithmetic. Here we find the most concrete beginning On Computability 527 of the history of modern computability with close ties to earlier mathematical and later logical developments. There is a second sense in which “foundational context” can be taken, not as referring to work in the foundations of mathematics, but directly in modern logic and cognitive science. Without a deeper understanding of the nature of calculation and underlying processes, neither the scope of undecidability and incompleteness results 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. These foundational problems come strikingly to the fore in arguments for Church’s or Turing’s Thesis, asserting that an informal notion of effective calcu- lability is captured fully by a particular precise mathematical concept. Church’s Thesis, for example, claims in its original form that the effectively calculable num- ber theoretic functions are exactly those functions whose values are computable in G¨odel’s equational calculus, i.e., the general recursive functions. There is general agreement that Turing gave the most convincing analysis of effective calculability in his 1936 paper On computable numbers — with an appli- cation to the Entscheidungsproblem. It is Turing’s distinctive philosophical con- tribution that he brought the computing agent into the center of the analysis and that was for Turing a human being, proceeding mechanically.1 Turing’s student Gandy followed in his [1980] the outline of Turing’s work in his analysis of ma- chine computability. Their work is not only closely examined in this essay, but also thoroughly recast. In the end, the detailed conceptual analysis presented be- low yields rigorous characterizations that dispense with theses, reveal human and machine computability as axiomatically given mathematical concepts and allow their systematic reduction to Turing computability. 1.2 Overview The core of section 2 is devoted to decidability and calculability. Dedekind intro- duced in his essay Was sind und was sollen die Zahlen? the general concept of a “(primitive) recursive” function and proved that these functions can be made explicit in his logicist framework. Beginning in 1921, these obviously calculable functions were used prominently in Hilbert’s work on the foundations of math- ematics, i.e., in the particular way he conceived of finitist mathematics and its role in consistency proofs. Hilbert’s student Ackermann discovered already be- fore 1925 a non-primitive recursive function that was nevertheless calculable. In 1931, Herbrand, working on Hilbert’s consistency problem, gave a very general and open-ended characterization of “finitistically calculable number-theoretic func- tions” that included also the Ackermann function. This section emphasizes the 1The Shorter Oxford English Dictionary makes perfectly clear that mechanical, when applied to a person or action, means “performing or performed without thought; lacking spontaneity or originality; machine-like; automatic, routine.” 528 Wilfried Sieg broader intellectual context and points to the rather informal and epistemologi- cally motivated demand that, in the development of logic and mathematics, certain notions (for example, proof) should be decidable by humans and others should not (for example, theorem). The crucial point is that the core concepts were deeply intertwined with mathematical practice and logical tradition before they came to- gether in Hilbert’s consistency program or, more generally, in meta-mathematics. In section 3, entitled Recursiveness and Church’s Thesis, we