Introduction: the 1930S Revolution

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Introduction: The 1930s Revolution

The theory of computability was launched in the 1930s by a group of young math- ematicians and logicians who proposed new, exact, characterizations of the idea of algorithmic computability. The most prominent of these young iconoclasts were Kurt Gödel, Alonzo Church, and Alan Turing. Others also contributed to the new field, most notably Jacques Herbrand, Emil Post, Stephen Kleene, and J. Barkley Rosser. This seminal research not only established the theoretical basis for computability: these key thinkers revolutionized and reshaped the mathematical world—a revolution that culminated in the Information Age. Their motive, however, was not to pioneer the discipline that we now know as theoretical computer science, although with hindsight this is indeed what they did. Nor was their motive to design electronic digital computers, although Turing did go on to do so (in fact producing the first complete paper design that the world had seen for an electronic stored-program universal computer). Their work was rather the continuation of decades of intensive investigation into that most abstract of subjects, the foundations of mathematics—investigations carried out by such great thinkers as Leopold Kronecker, Richard Dedekind, Gottlob Frege, Bertrand Russell, David Hilbert, L. E. J. Brouwer, Paul Bernays, and John von Neumann. The concept of an algorithm, or an effective or computable procedure, was central during these decades of foundational study, although for a long time no attempt was made to characterize the intuitive concept formally. This changed when Hilbert’s foundation- alist program, and especially the issue of decidability, made it imperative to provide an exact characterization of the idea of a computable function—or algorithmically calculable function, or effectively calculable function, or decidable predicate. Dif- ferent authors used different terminology for this central intuitive concept, which, they realized, stood in need of precise analysis. Computability: Turing, Gödel, Church, and Beyond examines not only the historic breakthroughs made in the 1930s by these three great thinkers, but also the legacy of their work in the modern world. The 1930s began with Gödel’s publication of his completeness theorem for first-order logic, and a year later, in S

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viii Introduction

1931, he published his famous incompleteness results. The latter concerned formal systems of arithmetic involving (what we now call) primitive recursive axioms and inference rules. A first major step had been taken in the develop- ment of modern computability theory. In 1934, in a series of lectures at the Princeton Institute for Advanced Study, Gödel went on to present the concept of a general recursive function (which he created by refining a suggestion that Herbrand had communicated to him in 1931). Notes of these landmark lectures, taken at the time by Church’s students Kleene and Rosser, were published in Martin Davis’s classic volume The Undecidable, under the title “On undecidable propositions of formal mathematical systems.” In 1936, Kleene published his own formulation of the theory of general recursive functions—the version standardly used today—and in 1938 Kleene extended his analysis to cover partial recursive functions as well. Church began his work on what he called the lambda calculus early in the 1930s, and was soon joined by his Princeton students Kleene and Rosser. When Gödel visited Princeton in 1934, Church suggested to him that the new concept of “lambda- definability” be taken as a precise, formal definition of the intuitive idea of effective calculability, but Gödel famously rejected Church’s suggestion, calling it “thoroughly unsatisfactory.” Nevertheless, Church did go on to propose this identification pub- licly, in a 1935 presentation and then the following year in a journal article. He also made the alternative suggestion that the intuitive idea of an effectively calculable function could be identified with the concept of a recursive function. These two suggestions are mathematically equivalent and are now known collectively as “Church’s thesis.” The name “Church-Turing thesis” is also used, in recognition of Turing’s slightly later work, also published in 1936. In his strikingly original paper of 1936, Turing characterized the intuitive idea of computability in terms of the activity of an abstract automatic computing machine. This machine, now called simply the “Turing machine,” figures in modern theoretical computer science as the most fundamental model of computation. (Post, in 1947, recast the Turing machine into the mathematical formulation usually used today, a finite collection of 4-tuples.) In 1936, Turing also described one of the most important scientific ideas of the twentieth century, his “universal computing machine”—a single Turing machine that, by making use of what we now call “programs” stored in its memory, can compute everything that is computable in the intuitive sense (or so Turing persuasively argued). Turing’s thesis, that the universal Turing machine can compute anything and everything computable in the intuitive sense, is equivalent to each of Church’s formulations of Church’s thesis. Now, numerous equivalent formulations of the Church-Turing thesis are known. Arguably, Turing’s formulation is the most fundamental, since it links the intuitive idea of computability with the S concept of a computing machine.

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The 1930s Revolution ix

Within a dozen or so years, Turing’s universal machine had been realized elec- tronically—the world’s first stored-program electronic digital computers had arrived. The transformation from abstract conception to physical machine was brought about by Tom Flowers, Max Newman, Freddie Williams, Tom Kilburn, Maurice Wilkes, John von Neumann, Presper Eckert, John Mauchly, Harry Huskey, and others—and certainly not forgetting Turing himself, in his postwar guise as computer designer. Turing’s electronic computer, the Automatic Computing Engine, or ACE, was in the event not the first of the new machines to operate, but it was by a wide margin the fastest. The early work by Turing, Gödel, Church, and the other founding fathers of computability has blossomed into what is today a vast bouquet of fields of study, as diverse as recursive function theory, computer engineering and computer program- ming, numerical analysis, computational linguistics, virtual reality, artificial intelli- gence, artificial life, computational theories of mind, complexity theory, machine learning, quantum computing, and modern cryptography—and much more besides. This multidisciplinary book is about what all these fields rest upon: the foundations of computability. Several chapters of the book are historical in perspective, focusing closely on the work of the founding fathers of computability. Soare and Sieg review the emergence of the various characterizations of computability in the 1930s, detailing Gödel’s role in these developments. Kripke clarifies and develops an important argument put forward by Turing in support of his 1936 analysis of computability. Copeland and Shagrir examine the theoretical underpinnings of the historic analyses of computability, and also elucidate what Gödel and Turing wrote about computability and the mind. More recent work in the foundations of computability is discussed by Davis, whose topic is the solution of Hilbert’s tenth problem; and by Putnam, who discusses the impact of Gödel’s work on modern mathematical logic and recursion theory; and Soare, who describes the legacy of Turing and Post in modern interactive and online computing. Aharonov, Vazirani, and Aaronson focus on recent issues in complexity theory and quantum computing, and Feferman and Posy discuss developments in the theory of computability over the real numbers. A number of chapters address the relationship between computability and the mind. Copeland and Shagrir, Putnam, and Sieg discuss the issue of whether Turing- computability places an upper bound on the powers of the human mind, while Aaronson raises the question whether the mind is limited to solving problems whose solutions are obtainable in polynomial time. Other chapters target issues in the philosophy of mathematics and the philosophy of science. Posy discusses the relationship between computability and constructivity in mathematics, and Aaronson the relationship between computational complexity and mathematical proof. Kripke, Soare, and Shapiro investigate issues concerning the Church-Turing thesis itself, and S

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x Introduction

Kripke explains the significance that Gödel’s 1931 theorem IX holds for Hilbert’s Entscheidungsproblem. Aharonov and Vazirani ask how the natural world can be studied at all, given that quantum mechanics exhibits exponential complexity, and they describe a foundation for experimental science that is based on the mathematical concept of interactive proof. The computational revolution of the 1930s can fairly be said to have fuelled the rocketing expansion of the horizons of knowledge that characterizes our modern scientific era. Turing, Gödel, and Church, although relatively unsung heroes, are pivotal figures in the story of modern science.

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