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Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 8 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
How to combine the benefits of strict and soft typing
 IN GREG MICHAELSON, PHIL TRINDER, AND HANSWOLFGANG LOIDL, EDITORS, TRENDS IN FUNCTIONAL PROGRAMMING. INTELLECT
, 2000
"... We discuss the properties of strictly typed languages on the one hand and soft typing on the other one and identify disadvantages of these approaches to type checking in the context of powerful type languages. To overcome the problems we develop an approach that combines ideas of both strict and sof ..."
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Cited by 3 (3 self)
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We discuss the properties of strictly typed languages on the one hand and soft typing on the other one and identify disadvantages of these approaches to type checking in the context of powerful type languages. To overcome the problems we develop an approach that combines ideas of both strict and soft typing. This approach is based on the concept of complete typing that is guaranteed to accept every welltyped program. The main component of a complete type checker is defined.
Strong Determinism vs. Computability
 The Foundational Debate, Complexity and Constructivity in Mathematics and
, 1995
"... Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed ..."
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Cited by 3 (1 self)
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Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially noncomputational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1
The Automatic Solution of Recurrence Relations  I. Linear Recurrences of Finite Order with Constant Coefficients
"... We describe algorithmic techniques for the efficient solution of a wide class of linear recurrences of finite order with constant coefficients. We give ..."
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Cited by 2 (1 self)
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We describe algorithmic techniques for the efficient solution of a wide class of linear recurrences of finite order with constant coefficients. We give
Combining strict and soft typing in functional programming
 INFORMATIK’99 – INFORMATIK BERWINDET GRENZEN
, 1999
"... We discuss the properties of strictly typed languages on the one hand and soft typing of the other and identify disadvantages of these approaches to type checking in the context of powerful type languages. To overcome the problems we develop an approach that combines ideas of strict and soft typing. ..."
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Cited by 1 (0 self)
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We discuss the properties of strictly typed languages on the one hand and soft typing of the other and identify disadvantages of these approaches to type checking in the context of powerful type languages. To overcome the problems we develop an approach that combines ideas of strict and soft typing. This approach is based on the concept of complete typing that is guaranteed to accept every welltyped program. The main component of a complete type checker is defined.
Undecidability Everywhere?
, 1996
"... We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51 ..."
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We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...
A Sketch of Complete Type Inference for Functional Programming
 IN PROC. OF THE INTERNATIONAL WORKSHOP ON FUNCTIONAL AND (CONSTRAINT) LOGIC PROGRAMMING (WFLP 2001
, 2001
"... Complete type inference for functional programming is an approach to incorporate static type inference into dynamically typed languages that is based on the following idea: For every program or program expression that can be evaluated without a runtime type error, types denoting all valid input ..."
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Complete type inference for functional programming is an approach to incorporate static type inference into dynamically typed languages that is based on the following idea: For every program or program expression that can be evaluated without a runtime type error, types denoting all valid input values (in case of functions) and all corresponding output/result values are inferred. A type error is just raised for program expressions that must provably fail for every input. In this
Physical unknowables
, 2008
"... Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unk ..."
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Different types of physical unknowables are discussed. Provable unknowables are derived from reduction to problems which are known to be recursively unsolvable. Recent series solutions to the nbody problem and related to it, chaotic systems, may have no computable radius of convergence. Quantum unknowables include the random occurrence of single events, complementarity and value indefiniteness.
UNDECIDABLE PROBLEMS: A SAMPLER
, 2012
"... After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics. ..."
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After discussing two senses in which the notion of undecidability is used, we present a survey of undecidable decision problems arising in various branches of mathematics.
The equational compatibility problem for the real line
"... Abstract. Walter Taylor proved recently that there is no algorithm for deciding of a finite set of equations whether it is topologically compatible with the real line in the sense that it has a model with universe R and with basic operations which are all continuous with respect to the usual topolog ..."
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Abstract. Walter Taylor proved recently that there is no algorithm for deciding of a finite set of equations whether it is topologically compatible with the real line in the sense that it has a model with universe R and with basic operations which are all continuous with respect to the usual topology of the real line. Taylor’s account used operation symbols suitable for the theory of rings with unit together with three unary operation symbols intended to name trigonometric functions supplemented finally by a countably infinite list of constant symbols. We refine Taylor’s work to apply to single equations using operation symbols for the theory of rings with unit supplemented by two unary operation symbols and at most one additional constant symbol.