The cognitive development of mathematics - Spatial and timing measures associated with algorithms

The thesis considers in turn measures of algorithms, measures of programs, and measures of computations. We define an algorithm's measure as the average of the space time requirements of its associated computations, and here, as with measures for programs, a question of optimisation arises: that of finding the algorithm fora function which has the least such measure. The problem of optimisation for algorithmic measure in its general form, proves to be unsolvable, but we show that an effective optimisation procedure does exist with regard to algorithms for finite functions, and give in detail the solution of the special case for functions with a domain of two elements. Further, we reduce the determination of the optimum algorithm for infinite functions to that of calculating the value of a primitive recursive function for any sufficiently large t. Taking a different viewpoint, we investigate the existence of lower bounds to the measures of algorithms for certain functions. A similar analysis is applied to the spatial measure defined for programs, program length, and we discuss some of the philosophical ramifications of program brevity. In addition, a pseudo-spatial measure, the number of instructions a program contains is considered. By using reduction theorems which map onto one another corresponding instructions in equivalent programs, we are able to adapt to this pseudo-spatial measure our results on program length. We then examine a problem of secondary optimisation, which involves a minimisation of both algorithmic and program measure. Measures of computations have been analysed by Myhill, Trakhtenbrot, Smullyan, Ritchie, Cleave, Rabin, Arbib and Blum. An essential element in Blum's definition of computational measure are the 'measuring predicates' and we investigate certain relations between their spatial and timing. Requirements (as where an upper bound on one such quantity places a lower bound on another). The relevant literature is discussed in brief, and we take up a number of points which arise. Most of the arguments and results in the thesis are formulated in terms of Turing machines, but they are applicable to other means of representing algorithms. In the final chapter we investigate how the definition of Turing machines may be extended so as to provide a more authentic model of actual computers both in regard to space-time measure, and to the domain of functions they encompass. <p

Logic, ontology, and arithmetic : a study of the development of Bertrand Russell’s Mathematical Philosophy from The Principles of Mathematics to Principia Mathematica

O presente trabalho tem por objeto de análise o desenvolvimento da Filosofia Matemática de Bertrand Russell desde os Principles of Mathematics até ­ e inlcuindo ­ a primeira edição de Principia Mathematica, tendo como fio condutor as mudanças no pensamento de Russell com respeito a três tópicos interligados, a saber: (1) a concepção de Russell da Lógica enquanto uma ciência (2) os compromissos ontológicos da Lógica e (3) a tese Russelliana de que a Matemática Pura ­ a Aritmética particular ­ é nada mais do que um ramo da Lógica. Esses três tópicos interligados formam um fio condutor que seguimos na tese para avaliar qual interpretação fornece o melhor relato da evidência textual disponível em Principia Mathematica e nos manuscritos produzidos por Russell no período relevante. A posição geral defendida é que a interpretação de Gregory Landini apresenta argumentos decisivos contra a ortodoxia de comentadores que atribuem à Principia uma hierarquia de tipos ramificada de entidades confusamente formulada, e mostramos que os três pontos apontados acima que formam o fio condutor da tese corroboram fortemente a interpretação de Landini. Os resultados que apontam para a conclusão geral de nossa investigação estão apresentados na tese dividida em duas partes. A primeira parte discute o desenvolvimento da lógica de concepção de Russell e do projeto Logicista desde sua gênese e nos Principles of Mathematics até Principia Mathematica. Esta primeira parte define o contexto para a segunda, que discute a Lógica Russeliana e o Logicismo em sua versão madura apresentada em Principia. Mostramos que, ao fim e ao cabo, o a teoria Lógica e a forma da tese Logicista apresentada em Principia é o resultado do longo processo iniciado com descoberta da Teoria dos Símbolos Incompletos que levou Russell a gradualmente reduzir os compromissos ontológicos de sua concepção da Lógica enquanto uma ciência, culminando na teoria apresentada na Introdução de Principia, na qual ele procura formular uma hierarquia dos tipos que evita o compromisso ontológico com classes, proposições e também com as assim chamadas funções proposicionais e que esse mesmo processo levou Russell a uma concepção da tese de Logicista de acordo com a qual a Matemática é uma ciência cujos compromissos ontológicos não incluem qualquer espécie de objetos (no sentido Fregeano) sejam eles particulares concretos ou abstratos.The present work has as its object of analysis the development of Bertrand Russell’s Mathematical Philosophy from the Principles of Mathematics up to ­ and including ­ the first edition of Principia Mathematica, having as a guiding thread the changes in Russell’s thought with respect to three interconnected topics, namely: (1) Russell’s conception of Logic as a science (2) the ontological commitments of Logic and (3) Russell’s thesis that Pure Mathematics ­ in particular Arithmetic ­ is nothing more than a branch of Logic. These three interconnected topics form a common thread that we follow in the dissertation to assess which interpretation offers the best account of the available textual evidence in Principia Mathematica and in the manuscripts produced by Russell in the relevant period. The general position held is that Gregory Landini’s interpretation presents decisive arguments against the orthodoxy of commentators who attribute to Principia a confusingly formulated hierarchy of ramfified types of entities, and we show that the three points indicated out above that form the main thread of the thesis strongly corroborate Landini’s interpretation. The results that point to the general conclusion of our investigation are stated in the dissertation divided into two parts. The first part discusses the development of Russell’s conception of Logic and the of Logicist project from its genesis and in Principles of Mathematics up to Principia Mathematica. This first part sets the context for a second, which discusses a Russellian Logic and Logicism in its mature version presented in Principia. We show that, in the end, the Logic theory and the form of the Logicist thesis presented in Principia is the result of a long process that started with the discovery of the theory of Incomplete Symbols which led Russell to reduce the ontological commitments of his conception of Logic as a science, culminating in the theory of types presented in Principia’s Introduction, in which Russell seeks to formulate a hierarchy of types that avoids the ontological commitment to classes, propositions and also with so­called propositional functions, and that this same process led Russell to a conception of the Logicist thesis according to Mathematics is a science with no ontological commitments to any kind of objects (in the Fregean sense) whether these are conceived as concrete or abstract particulars