We describe a possible physical device that computes a function that cannot be computed by a Turing machine. The device is physical in the sense that it is compatible with General Relativity. We discuss some objections, focusing on those which deny that the device is either a computer or computes a function that is not Turing computable. Finally, we argue that the existence of the device does not refute the Church–Turing thesis, but nevertheless may be a counterexample to Gandy's thesis.

1. The Physical Church-Turing Thesis. Physicists often interpret the Church-Turing Thesis as saying something about the scope and limitations of physical computing machines. Although this was not the intention of Church or Turing, the Physical Church Turing thesis is interesting in its own right. Consider, for example, Wolfram’s formulation: One can expect in fact that universal computers are as powerful in their computational capabilities as any physically realizable system can be, that they can simulate any physical system . . . (...) No physically implementable procedure could then shortcut a computationally irreducible process. (Wolfram 1985) Wolfram’s thesis consists of two parts: (a) Any physical system can be simulated (to any degree of approximation) by a universal Turing machine (b) Complexity bounds on Turing machine simulations have physical significance. For example, suppose that the computation of the minimum energy of some system of n particles takes at least exponentially (in n) many steps. Then the relaxation time of the actual physical system to its minimum energy state will also take exponential time. (shrink)

There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument (...) for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation. (shrink)

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more (...) mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

I discuss the philosophical implications that the rising new science of quantum computing may have on the philosophy of computer science. While quantum algorithms leave the notion of Turing-Computability intact, they may re-describe the abstract space of computational complexity theory hence militate against the autonomous character of some of the concepts and categories of computer science.

Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's (...) Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and--in particular--the effectively-computable functions on string representations of numbers. (shrink)

In recent years two different axiomatic characterizations of the intuitive concept of effective calculability have been proposed, one by Sieg and the other by Dershowitz and Gurevich. Analyzing them from the perspective of Carnapian explication, I argue that these two characterizations explicate the intuitive notion of effective calculability in two different ways. I will trace back these two ways to Turing’s and Kolmogorov’s informal analyses of the intuitive notion of calculability and to their respective outputs: the notion of computorability and (...) the notion of algorithmability. I will then argue that, in order to adequately capture the conceptual differences between these two notions, the classical two-step picture of explication is not enough. I will present a more fine-grained three-step version of Carnapian explication, showing how with its help the difference between these two notions can be better understood and explained. (shrink)

What are the limits of physical computation? In his ‘Church’s Thesis and Principles for Mechanisms’, Turing’s student Robin Gandy proved that any machine satisfying four idealised physical ‘principles’ is equivalent to some Turing machine. Gandy’s four principles in effect define a class of computing machines (‘Gandy machines’). Our question is: What is the relationship of this class to the class of all (ideal) physical computing machines? Gandy himself suggests that the relationship is identity. We do not share this view. We (...) will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing-machine computable. (shrink)

This thesis focuses on models of conceptual change in science and philosophy. In particular, I developed a new bootstrapping methodology for studying conceptual change, centered around the formalization of several popular models of conceptual change and the collective assessment of their improved formal versions via nine evaluative dimensions. Among the models of conceptual change treated in the thesis are Carnap’s explication, Lakatos’ concept-stretching, Toulmin’s conceptual populations, Waismann’s open texture, Mark Wilson’s patches and facades, Sneed’s structuralism, and Paul Thagard’s conceptual revolutions. (...) -/- In order to analyze and compare the conception of conceptual change provided by these different models, I rely on several historical reconstructions of episodes of scientific conceptual change. The historical episodes of scientific change that figure in this work include the emergence of the morphological concept of fish in biological taxonomies, the development of scientific conceptions of temperature, the Church-Turing thesis and related axiomatizations of effective calculability, the history of the concept of polyhedron in 17th and 18th century mathematics, Hamilton’s invention of the quaternions, the history of the pre-abstract group concepts in 18th and 19th century mathematics, the expansion of Newtonian mechanics to viscous fluids forces phenomena, and the chemical revolution. I will also present five different formal and informal improvements of four specific models of conceptual change. I will first present two different improvements of Carnapian explication, a formal and an informal one. My informal improvement of Carnapian explication will consist of a more fine-grained version of the procedure that adds an intermediate, third step to the two steps of Carnapian explication. I will show how this novel three-step version of explication is more suitable than its traditional two-step relative to handle complex cases of explications. My second, formal improvement of Carnapian explication will be a full explication of the concept of explication itself within the theory of conceptual spaces. By virtue of this formal improvement, the whole procedure of explication together with its application procedures and its pragmatic desiderata will be reconceptualized as a precise procedure involving topological and geometrical constraints inside the theory of conceptual spaces. My third improved model of conceptual change will consist of a formal explication of Darwinian models of conceptual change that will make vast use of Godfrey-Smith’s population-based Darwinism for targeting explicitly mathematical conceptual change. My fourth improvement will be dedicated instead to Wilson’s indeterminate model of conceptual change. I will show how Wilson’s very informal framework can be explicated within a modified version of the structuralist model-theoretic reconstructions of scientific theories. Finally, the fifth improved model of conceptual change will be a belief-revision-like logical framework that reconstructs Thagard’s model of conceptual revolution as specific revision and contraction operations that work on conceptual structures. -/- At the end of this work, a general conception of conceptual change in science and philosophy emerges, thanks to the combined action of the three layers of my methodology. This conception takes conceptual change to be a multi-faceted phenomenon centered around the dynamics of groups of concepts. According to this conception, concepts are best reconstructed as plastic and inter-subjective entities equipped with a non-trivial internal structure and subject to a certain degree of localized holism. Furthermore, conceptual dynamics can be judged from a weakly normative perspective, bound to be dependent on shared values and goals. Conceptual change is then best understood, according to this conception, as a ubiquitous phenomenon underlying all of our intellectual activities, from science to ordinary linguistic practices. As such, conceptual change does not pose any particular problem to value-laden notions of scientific progress, objectivity, and realism. At the same time, this conception prompts all our concept-driven intellectual activities, including philosophical and metaphilosophical reflections, to take into serious consideration the phenomenon of conceptual change. An important consequence of this conception, and of the analysis that generated it, is in fact that an adequate understanding of the dynamics of philosophical concepts is a prerequisite for analytic philosophy to develop a realistic and non-idealized depiction of itself and its activities. (shrink)