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A BURGESSIAN CRITIQUE OF NOMINALISTIC TENDENCIES IN CONTEMPORARY MATHEMATICS AND ITS HISTORIOGRAPHY
, 2012
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MEANING IN CLASSICAL MATHEMATICS: IS IT AT ODDS WITH INTUITIONISM?
, 2011
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Towards a unified treatment of induction, I: the general recursion theorem, unfinished draft manuscript
, 1996
"... The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called pa ..."
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The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g: A → N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−)ar(r), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullbackpreserving functor T, a structure similar to the von Neumann hierarchy is developed which analyses the free Talgebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α: A. TA, namely that any pullback diagram of the form
Raymond Llull's contributions to computer science
, 2005
"... Llull has been described, anachronistically, as the first computer scientist. Nevertheless he did make significant contributions to the discipline that is now seen as computer science. Some of these contributions are also relevant to mathematics. The latter has become wellknown, principally beca ..."
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Llull has been described, anachronistically, as the first computer scientist. Nevertheless he did make significant contributions to the discipline that is now seen as computer science. Some of these contributions are also relevant to mathematics. The latter has become wellknown, principally because his ideas were taken up by Leibniz (16461716) and his followers. We discuss these contributions. In addition we draw attention to Llull's use of letters for entities and argue that he was the first to use the idea of substituting a value for a variable.
Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules Explaining Content Effects in the Wason Selection Task
"... „Cum deus calculat et cogitationem exercet, fit mundus“ (When God calculates and develops thought, he creates the world) G. W. Leibniz, 1765 [1996, 25] The Wason Selection Task is „probably the most intensive studied task in the psychology of reasoning […], which has raised more doubts over human ra ..."
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„Cum deus calculat et cogitationem exercet, fit mundus“ (When God calculates and develops thought, he creates the world) G. W. Leibniz, 1765 [1996, 25] The Wason Selection Task is „probably the most intensive studied task in the psychology of reasoning […], which has raised more doubts over human rationality The copyright of this thesis rests with the author. (c) Momme v. Sydow, Göttingen, 2006 than any other psychological task“ M. Oaksford and N. Chater, 1998, 173, 174 Abstract v Research on the Wason selection task (WST) has raised fundamental doubts about the rationality of human hypothesis testing and added to the development of both domainspecific and domaingeneral theories of reasoning. This work proposes a rational but domainspecific synthesis aimed at integrating converging lines of
Existence and Propositional Attitudes: A Fregean Analysis
"... It is a commonly held view that Frege's doctrine of senses and references is not compatible with the idea that there are de re beliefs. The present paper is meant to challenge that view. Moreover, it seeks to show that, instead of forcing Frege's semantic framework to answer questions rais ..."
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It is a commonly held view that Frege's doctrine of senses and references is not compatible with the idea that there are de re beliefs. The present paper is meant to challenge that view. Moreover, it seeks to show that, instead of forcing Frege's semantic framework to answer questions raised by twentiethcentury philosophy of language, we had better find those questions to which it might be a proper answer. It is argued that the proper treatment of Frege's views requires the acknowledgment of the central role of individualistic epistemology in his thought. Once that feature is recognized, Frege's doctrine of senses and references can be considered a theory, or at least a sketch of a theory, of cognition, which has interesting connections with Kant's and Husserl's views.
Reflections on the Foundations of MetaProgramming: Is a Type Theory Needed?
"... bryöecrc.de Metaprogramming is an important programming technique, which is widely applied in logic pro gramming cf. e.g. [1,2,3,4,5,6]. However, in spite of several studies cf. e.g. [28,29,30] the semantics of metaprograms and their formalization in logic remain open issues. Formalizations in ..."
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bryöecrc.de Metaprogramming is an important programming technique, which is widely applied in logic pro gramming cf. e.g. [1,2,3,4,5,6]. However, in spite of several studies cf. e.g. [28,29,30] the semantics of metaprograms and their formalization in logic remain open issues. Formalizations in classical logic, such as that subjacent to the Gödel language [31], are in the spirit of the classical formalization of firstorder programs cf. e.g. [8,9]. They interpret metapredicates and predicate variables as higherorder symbols. How attractive they might be, the formalizations in classical logic are not conform to a programmer's intuition. Moreover, although such approaches rather con vincingly formalize constructs such as Prolog "call " and "clause " l they generally fail to account for constructs permitting a dynamic creation of predicate symbols and atoms, such as Prolog "univ" (=..) , "functor", and "arg ". The formalizations of metaprogramming based on manysorted logics, e.g. [10], are more intuitive. Nevertheless, they also fail to explain the dynamic creation of symbols. We argue that the inadequacy of classical logic to formalize metaprogramming is related to the theory of types of this logic, i.e. to the principles that were introduced first by Russel in [ l l]. 1 We argue that the standard models of metaprograms viewed as higherorder theories in