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Some chapters focus on the pioneering work by Turing, Godel, and Church, including the Church-Turing thesis and Godel's response to Church's and Turing's proposals. Other chapters cover more recent technical developments, including computability over the reals, Godel's influence on mathematical logic and on recursion theory and the impact of work by Turing and Emil Post on our theoretical understanding of online and interactive computing; and others relate computability and complexity to issues in the philosophy of mind, the philosophy of science, and the philosophy of mathematics.

Soare, Umesh V.


Philosophical theories and arguments from a comprehensive range of Indian philosophical traditions including the Nyaya, Saiva, Vedanta, Jain, Buddhist, materialist and skeptical traditions, as well as some 20th century thought are covered. Discourse of race and postcolonial thought in continental philosophy remains widely overlooked and in need of development. And about what happens when you get a group of physically fit young women really, really angry. Edelman links the family and child poverty crisis to the fragile and ephemeral commitment of government to assist the needy. This ambivalent status is what gives the philosophy of mathematics its special interest.


We could come across procedures in the future that we want to count as computations that we do not right now: Dorit Aharonov and Umesh Vazirani discuss one such possibility, quantum computation, in their chapter. To the extent that today's concept of computability is settled, as the widespread acceptance of the Church-Turing thesis suggests, Shapiro urges us to see that settling as a sharpening, the result of human decisions, rather than the discovery of the "true" contours of the concept of computability. A brief word is in order concerning the opposing positions of Kripke's and Shapiro's articles.

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Kripke's alleged proof of the Church-Turing thesis hinges on what he calls "Hilbert's thesis", that every computation can be fully expressed in a first-order way. If pre-theoretic computation is subject to open texture, then no particular expression of it fully captures its content, and hence no first-order expression does so. Thus the open texture of computability would undermine the cogency of Kripke's proof by contradicting Hilbert's thesis.

Thus, as Kripke recognizes, Hilbert's thesis will be a locus of disagreement with his proof. A point more relevant to Shapiro's argument is what to make of all the proved equivalences between different models of computation: anything computable by a Turing machine is computable by a partial recursive function, is computable in the lambda calculus, and so on. One might hold that these equivalences are evidence that we have found the "real" concept of computability, because they indicate the "inevitability" of their analyses.

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No matter how we try to sharpen our concept of computability, we keep finding the same class of computations, extensionally speaking. Thus the original concept was precise enough to fix this class of computations, and this is evidence that these equivalent sharpenings are "right". The normativity at play here between concept and analysis demands clarification, but such clarification is needed to spell out open texture as well, since open texture is at root a negative view that no single analysis of a mathematical concept can be, or alternately can be shown to be, adequate for capturing the full content of that concept.

Questions of computability have often been linked to questions about the nature of the human mind, since one may wonder if the mind is a computational machine. In a paper previously published in but included in this volume, Putnam argues that if the mind more precisely, the linguistic and scientific faculties of the mind, in Chomsky's terms were representable by a Turing machine, then we could never know by mathematical or empirical means which Turing machine it is.

The articles of B. Turing, by contrast, identifies mechanized reasoning as "discipline" that human reasoners exceed in exercising what he calls "initiative", deployed when discipline fails in tackling problems. As Sieg notes, Turing suggested that new theories resulting from such "initiative" could themselves be mechanized once settled. The resulting conception of the dynamics of mathematical theory change is also the focus of Copeland and Shagrir's article.

Copeland and Shagrir emphasize what they call Turing's "multi-machine theory of mind", in which human minds are Turing machines at each stage of development, but which machine they are changes during a lifetime rather than remaining fixed from birth to death. These changes occur when minds break the "discipline" of one mechanization and learn new processes, resulting in a new machine.

While computational models of mind are not in vogue at present, Turing's view seems worthy of interest to contemporary investigators in cognitive science and the philosophy of mind. Solomon Feferman's article is an engrossing discussion of computation over the real numbers, a key component of contemporary science and engineering in their use of numerical methods for simulations, modeling, optimization, and statistical analysis. Both are claimed by their creators to be well-suited as foundations for scientific computation and its numerical methods.

Feferman notes that these two models of computation are incompatible, in that each classifies functions over the reals as computable that the other does not.

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His focal question is whether they can both be reasonable models of computation on the reals. To answer this, Feferman observes that one needs the right perspective, one given by thinking of computation over an arbitrary structure, since the two competing models identify the structure of the reals differently algebraically and topologically, respectively. He then explicates three theories of computation over an arbitrary structure: the first due to Harvey Friedman; the second to John Tucker and Jeffery Zucker; and the third to himself, adapting work of Richard Platek and Yiannis Moschovakis.

He affirms the adequacy of each of these three theories as suitable for implementing the particular models of both BSS and of Braverman and Cook, thus answering positively his focal question. Thus one who desires to implement scientific computations may choose either of these two models of computation, and the decision to choose between them must be made on other grounds. Feferman notes that, in practice, users of scientific computations simply use approximations to the reals and real-valued functions, using what computer scientists call "floating-point arithmetic" think, for instance, of Runga-Kutta methods for solving ordinary differential equations.

The BSS and Braverman and Cook models, by contrast, do not take the reals as approximations, but rather as continuous structures. They are thus arguably more precise from a foundational point of view than the methods used by most practicing numerical analysts today. The developers of the BSS and Braverman and Cook models are concerned with the pragmatics of computation, but on grounds of computational complexity rather than the pragmatics of implementation in daily work.

Feferman notes that there is another approach to computing over the reals, introduced by Errett Bishop's constructive analysis. Bishop works with approximations to the reals, and is thus closer to normal practice in scientific computation than the other models of computation considered here. Feferman narrates recent work in interpreting Bishop's work by logicians working in computability theory, and notes that further analysis of Bishop's constructive mathematics in this direction would be worthwhile because there are indications that such work could permit information on the computational complexity of Bishop's model to be mined.

Robert Soare describes the legacy of Turing and Post in modern interactive and online computing. The volume covers other ground as well.


Both Solomon Feferman and Carl Posy discuss developments in the theory of computability over the real numbers. Feferman draws a comparison between the different theories of computation on reals from a conceptual basis. Posy discusses the relationship between computability and constructivity in mathematics. Copeland and Shagrir, and Wilfried Sieg address the relationship between computability and the mind. They discuss the issue of whether Turing-computability places an upper bound on the powers of the human mind.

The book closes with two exciting chapters on complexity theory, which is the next generation in the philosophy of computing. Scott Aaronson argues that complexity theory is very relevant to many philosophical issues.