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1978]: Two Applications of Logic to Mathematics
 Tokyo: Iwanami Shoten and Princeton
"... The present work was typeset by AMSL ATEX, the TEX macro systems of the ..."
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The present work was typeset by AMSL ATEX, the TEX macro systems of the
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
Can Newtonian systems, bounded in space, time, mass and energy compute all functions?
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and anal ..."
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and analysis of simple examples that are models of simple subtheories of physical theories. Then we illustrate the methodology, by presenting a simple example for Newtonian Kinematics, and a critique that leads to a substantial extension of the methodology. The example proves that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the example implies that simple Newtonian kinematic systems that are bounded in space, time, mass and energy can compute all possible sets and functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Next, we use the example to extend the methodology. The marble run shows that a formal theory for computation by physical systems needs strong conditions on the notion of experimental procedure and, specifically, on methods for the construction of equipment. We propose to extend the methodology by defining languages to express experimental procedures and the construction of equipment. We conjecture that the functions computed by experimental computation in Newtonian Kinematics are “equivalent” to those computed by algorithms, i.e. the partial computable functions.
History of "Church's theses" and a manifesto on converting physics into a rigorous algorithmic discipline
, 1999
"... Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of ..."
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Church's thesis claims that any "reasonable computer" may be simulated by a Turing machine. The "strong" thesis says that the simulation may be performed with only polynomial slowdown. This document is both a history of "Church's thesis"  and particularly of the notion that it is a statement about physics  and an opinionated philosophical statement.
Recursivity in quantum mechanics
 Trans. Amer. Math. Soc
, 1983
"... ABSTRACT. The techniques of effective descriptive set theory are applied to the mathematical formalism of quantum mechanics in order to see whether it actually provides effective algorithms for the computation of various physically significant quantities, e.g. matrix elements. Various Hamiltonians a ..."
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ABSTRACT. The techniques of effective descriptive set theory are applied to the mathematical formalism of quantum mechanics in order to see whether it actually provides effective algorithms for the computation of various physically significant quantities, e.g. matrix elements. Various Hamiltonians are proven to be recursive (effectively computable) and shown to generate unitary groups which act recursively on the Hilbert space of physical states. In particular, it is shown that the niparticle Coulombic Hamiltonian is recursive, and that the time evolution of nparticle quantum Coulombic systems is recursive. Introduction. Computable analysis [1] and effective descriptive set theory [3] study mathematical processes to see whether they can be done effectively, e.g. by computer programs. Kreisel [2] has raised the possibility of applying these branches of mathematics to mathematical physics. Since the goal of physics is to be able to predict phenomena, it is of interest to see whether physical theories provide effective
On the calculating power of Laplace’s demon (Part I)
, 2006
"... We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a s ..."
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We discuss several ways of making precise the informal concept of physical determinism, drawing on ideas from mathematical logic and computability theory. We outline a programme of investigating these notions of determinism in detail for specific, precisely articulated physical theories. We make a start on our programme by proposing a general logical framework for describing physical theories, and analysing several possible formulations of a simple Newtonian theory from the point of view of determinism. Our emphasis throughout is on clarifying the precise physical and metaphysical assumptions that typically underlie a claim that some physical theory is ‘deterministic’. A sequel paper is planned, in which we shall apply similar methods to the analysis of other physical theories. Along the way, we discuss some possible repercussions of this kind of investigation for both physics and logic. 1
On the calculating power of Laplace’s demon
"... Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameter ..."
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Abstract. We discuss some of the choices that arise when one tries to make the idea of physical determinism more precise. Broadly speaking, ‘ontological ’ notions of determinism are parameterized by one’s choice of mathematical ideology, whilst ‘epistemological ’ notions of determinism are parameterized by the choice of an appropriate notion of computability. We present some simple examples to show that these choices can indeed make a difference to whether a given physical theory is ‘deterministic’ or not. Keywords: Laplace’s demon, physical determinism, philosophy of mathematics, notions of computability. 1
BOOK REVIEWS COMPUTABILITY: COMPUTABLE FUNCTIONS, LOGIC AND THE FOUNDATIONS OF MATHEMATICS
, 1989
"... There are now many books which cover roughly the same material as this one. It has two distinguishing features. First, it contains many wellchosen, sometimes long, quotations from wellknown works about the foundations of mathematics; some of these provide motivation, others (about Church's th ..."
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There are now many books which cover roughly the same material as this one. It has two distinguishing features. First, it contains many wellchosen, sometimes long, quotations from wellknown works about the foundations of mathematics; some of these provide motivation, others (about Church's thesis and about various kinds of constructive mathematics) exhibit the relevance of the formal work to foundational problems. Secondly, almost no technical knowledge is assumed: in each section the groundwork is carefully presented and explained; in particular, this is done for Turing machines, primitive and partial recursive functions, propositional logic, formal systems of arithmetic and the incompleteness theorems. There is a detailed and clear treatment of Robinson's system Q showing its power and its weakness. Verifications and further developments are often left to the reader; there is a teacher's handbook which I have not seen. There are good historical remarks and suggestions for further reading. But the prospective user should be warned that a number of expected topics are treated only by mention. These include: connections with actual computers and programming languages; problems of complexity; relative recursiveness and degrees of unsolvability; the existence of simple, creative and recursively inseparable sets.