This thesis presents an investigation into the structure of proof in non-standard analysis using proof-planning. The theory of non-standard analysis, developed by Robinson in the 1960s, offers a more algebraic way of looking at proof in analysis. Proof-planning is a technique for reasoning about proof at the meta-level. In this thesis, we use it to encapsulate the patterns of reasoning that occur in non-standard analysis proofs. We first introduce in detail the mathematical theory and the proof-planning architecture. We then present our research methodology, describe the formal framework, which includes an axiomatisation, and develop suitable evaluation criteria. We then present our development of proof-plans for theorems involving limits, continuity and differentiation. We then explain how proof-planning applies to theorems which combine induction and non-standard analysis. Finally we give a detailed evaluation of the results obtained by combining the two attractive approaches of proof-planning and non-standard analysis, and draw conclusions from the work.

L’Analyse Non Standard (l’ANS) est un formalisme mathématique particulier inventé dans les années 1960 par le mathématicien A. Robinson. Ce formalisme permet de renouer avec les infinitésimaux de Leibniz qui avaient été abandonnés au XIXème siècle pour satisfaire aux exigences nouvelles de la « rigueur ». Sa pertinence a été contestée par divers mathématiciens parmi les plus grands et a donné naissance à une polémique dans les milieux mathématiques français ; les partisans de l’ANS en sont sortis vaincus et ne se sont plus guère exprimés après 1995. Un quart de siècle plus tard l’ANS est considérée au plan international comme une pratique tout à fait légitime et certains mathématiciens, à leur tour parmi les plus grands, en préconisent l’usage.Pourquoi cette mauvaise réception d’idées nouvelles en mathématiques dans un pays réputé pour son excellence dans ce domaine ?Il est normal que des idées révolutionnaires, voire simplement nouvelles, rencontrent de la résistance et suscitent un débat. Toutefois on observe que ce débat qui commençait à prendre de l’importance au début des années 1980 a été étouffé dans les années 1990 par ceux qui avaient en charge les institutions de la communauté mathématique. Pourquoi ce refus du débat ?La thèse soutenue est que, à cette époque, une des fonctions que l’idéologie dominante assigne aux mathématiques est de dire le vrai ; par exemple les théories économiques libérales prétendent à la scientificité parce que fortement mathématisées. Ne dit-on pas c’est mathématique pour affirmer d’une chose qu’elle est inéluctable. Une dispute trop visible sur la nature de la rigueur mathématique aurait risqué de brouiller cette image du mathématicien. Dans le même ordre d’idées, à la même époque, la communauté mathématique (et plus généralement scientifique) avait refusé de débattre avec un de ses membres les plus brillants, A. Grothendieck, de la responsabilité sociale du savant.Cette question de la réception de l’ ANS illustre la thèse bien connue que si une science se développe en partie pour résoudre des problèmes qu’elle se pose à elle même, ici donner un statut logique irréprochable à la pratique des infinitésimaux, cette motivation interne ne suffit pas à elle seule à expliquer tous les aspects de son développement. Les savants doivent tenir compte de la société dans laquelle ils vivent. Il est intéressant de faire ce constat dans le domaine des mathématiques dites pures, c’est à dire qui se prétendent en dehors de toute contrainte et ne travailler que pour l’honneur de l’esprit humain, pour reprendre la célèbre formule de Jacobi. / Non Standard Analysis (ANS) is a particular mathematical formalism invented in the 1960s by the mathematician A. Robinson. This formalism allows to reconnect with the infinitesimals of Leibniz which had been abandoned in the nineteenth century to satisfy the new requirements of rigor. Its relevance has been challenged by various mathematicians among the greatest and gave birth to a controversy in the French mathematical circles ; the supporters of the ANS came out defeated and hardly spoke after 1995. A quarter of a century later, ANS is considered internationally as a perfectly legitimate practice and some mathematicians, including famous ones, advocate its use.Why this bad reception of new ideas in a country renowned for its excellence in the field of mathematical research?It is natural for revolutionaries, or simply news ideas, to be at the origin of resistance and debate. However, we observe that this debate, which was starting and gaining importance in the early 1980’s, was stifled by those who were in charge of the institutions of the mathematical community. Why this refusal of debate?My thesis is that, at this time, one of the functions that the dominant ideology assigned to mathematics was to « say the truth »; for example liberal economic theories claim to scientificity because they are highly mathematized. It is commun to say « it is mathematical » to say that something is unavoidable. A dispute too visible about the nature of mathematical rigor could blur this image of the mathematician. In the same vein, at the same time, the mathematical (and more generally scientific) community had refused to debate with one of its most brilliant members, A. Grothendieck, on the social responsibility of the scientist.This question of the reception of the ANS illustrates the well-known thesis that if a science develops partly to solve problems that it poses to itself, in our case to give an irreproachable logical status to the practice of infinitesimals, this internal motivation is not enough on its own to explain all aspects of its development. Scholars must consider the society in which they live. It is interesting to make this observation in the so-called field of pure mathematics, which claim to be free from all constraints and work only « for the honor of the human mind » to use Jacobi's famous formula.

A área conhecida como Análise Não Standard consiste na aplicação dos métodos da Teoria dos Modelos e da Teoria dos Ultrafiltros para a obtenção de extensões peculiares de sistemas matemáticos infinitos. As novas estruturas construídas segundo esse procedimento satisfazem ao Princípio da Transferência, uma propriedade de suma importância e influência a qual afirma que as mesmas sentenças de primeira ordem com quantificadores limitados são verdadeiras para o sistema original e a sua extensão. Concebida em 1961 por Abraham Robinson e aprimorada por vários matemáticos nos anos subsequentes, tal área de pesquisa provou ser bastante proveitosa e esclarecedora para diversas outras partes da Matemática, como a Topologia, a Teoria das Probabilidades, a Análise Funcional e a Análise Complexa. Manifesta-se uma reavaliação da Teoria dos Domínios Ordenados seguida de um tratamento completo e gradual das fundações da Análise Não Standard assumindo a perspectiva dos Monomorfismos Não Standard, onde adota-se como metateoria a teoria dos conjuntos de Neumann-Bernays-Gödel com o Axioma da Escolha. A fim de impulsionar a assimilação da metodologia abordada, o estudo explora as propriedades do corpo não arquimediano dos números hiper-reais de maneira intuitiva e informal, utilizando-se destas para revelar demonstrações alternativas e relativamente diretas de alguns dos principais resultados do Cálculo Diferencial e Integral, como o Teorema do Valor Intermediário, o Teorema de Bolzano-Weierstrass, o Teorema do Ponto Crítico, o Teorema da Função Inversa e o Teorema Fundamental do Cálculo. / The field known as Non-standard Analysis consists in the application of the methods of Model Theory and Ultrafilter Theory to the attainment of peculiar extensions of infinite mathematical systems. The new structures produced under that procedure satisfy the Transfer Principle, a property of the utmost importance and influence which states that the same first-order sentences with bounded quantifiers are true for the original system and its extension. Conceived in 1961 by Abraham Robinson and improved by a number of mathematicians in the following years, such area of research has proved to be very fruitful and illuminating to many other parts of Mathematics, such as Topology, Probability Theory, Functional Analysis and Complex Analysis. The work presents a reexamination of the Theory of Ordered Domains followed by a thorough and gradual treatment of the foundations of Non-standard Analysis under the perspective of Non-standard Monomorphisms, where Neumann-Bernays-Gödels set theory with the Axiom of Choice is adopted as metatheory. In order to boost the assimilation of the methodology put forward, the study explores the properties of the non-archimedean field of hyperreal numbers in an intuitive and informal fashion, employing them to reveal alternative and relatively direct proofs of some of the main results of Differential and Integral Calculus, such as the Intermediate Value Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem, the Inverse Function Theorem and the Fundamental Theorem of Calculus.

A área conhecida como Análise Não Standard consiste na aplicação dos métodos da Teoria dos Modelos e da Teoria dos Ultrafiltros para a obtenção de extensões peculiares de sistemas matemáticos infinitos. As novas estruturas construídas segundo esse procedimento satisfazem ao Princípio da Transferência, uma propriedade de suma importância e influência a qual afirma que as mesmas sentenças de primeira ordem com quantificadores limitados são verdadeiras para o sistema original e a sua extensão. Concebida em 1961 por Abraham Robinson e aprimorada por vários matemáticos nos anos subsequentes, tal área de pesquisa provou ser bastante proveitosa e esclarecedora para diversas outras partes da Matemática, como a Topologia, a Teoria das Probabilidades, a Análise Funcional e a Análise Complexa. Manifesta-se uma reavaliação da Teoria dos Domínios Ordenados seguida de um tratamento completo e gradual das fundações da Análise Não Standard assumindo a perspectiva dos Monomorfismos Não Standard, onde adota-se como metateoria a teoria dos conjuntos de Neumann-Bernays-Gödel com o Axioma da Escolha. A fim de impulsionar a assimilação da metodologia abordada, o estudo explora as propriedades do corpo não arquimediano dos números hiper-reais de maneira intuitiva e informal, utilizando-se destas para revelar demonstrações alternativas e relativamente diretas de alguns dos principais resultados do Cálculo Diferencial e Integral, como o Teorema do Valor Intermediário, o Teorema de Bolzano-Weierstrass, o Teorema do Ponto Crítico, o Teorema da Função Inversa e o Teorema Fundamental do Cálculo. / The field known as Non-standard Analysis consists in the application of the methods of Model Theory and Ultrafilter Theory to the attainment of peculiar extensions of infinite mathematical systems. The new structures produced under that procedure satisfy the Transfer Principle, a property of the utmost importance and influence which states that the same first-order sentences with bounded quantifiers are true for the original system and its extension. Conceived in 1961 by Abraham Robinson and improved by a number of mathematicians in the following years, such area of research has proved to be very fruitful and illuminating to many other parts of Mathematics, such as Topology, Probability Theory, Functional Analysis and Complex Analysis. The work presents a reexamination of the Theory of Ordered Domains followed by a thorough and gradual treatment of the foundations of Non-standard Analysis under the perspective of Non-standard Monomorphisms, where Neumann-Bernays-Gödels set theory with the Axiom of Choice is adopted as metatheory. In order to boost the assimilation of the methodology put forward, the study explores the properties of the non-archimedean field of hyperreal numbers in an intuitive and informal fashion, employing them to reveal alternative and relatively direct proofs of some of the main results of Differential and Integral Calculus, such as the Intermediate Value Theorem, the Bolzano-Weierstrass Theorem, the Extreme Value Theorem, the Inverse Function Theorem and the Fundamental Theorem of Calculus.

The theory presented here stemmed from years of teaching analysis at pre-university level first using the concept of infinitesimal as defined in nonstandard analysis by Robinson, then the concept of ultrasmall as defined in our joint work with Hrbacek and Lessmann presented in the appendix. This research led to the question : If one has finite yet ultralarge quantities, is it possible to avoid infinite quantities ? The alternative set theory of Vopěnka is a theory of finite sets including classes that can be infinite. The theory of objects is a merger of certain axioms of Vopěnka with axioms that determine levels of visibility as in relative analysis. We took as first principle : $x\subseteq y\Rightarrow x\sqsubseteq y$, which specifies that if object $x$ is included in object $y$, then $x$ "appears" at the level of $y$. This statement would be false with infinite quantities and is in fact a characterisation of finite sets : this is a well-known theorem of nonstandard analysis. The introduction of this principle as starting point is making a strong point that all objects will be finite - in the usual sense of the word. The other founding axiom is Gordon and Andreev's axiom schema : If $\Phi$ is a formula, and if $\Phi(\emptyset)$ is true and that $\Phi(x)$ and $\Phi(y)$ imply $\Phi(x\cup\{y\})$, then $\Phi(x)$ is true for all $x$. An emphasis is made on the concept of contextual formulae. This concept is one of our contributions to relative analysis of Hrbacek and determines an equivalence to well-formed formulae. We show that the resulting system is relatively consistent with Hrbacek's FRIST and Péraire's RIST which are conservative extensions of ZFC. The theory of objects extends set theory of Zermelo and Fraenkel without choice and with negation of the infinity axiom. Integers and rationals are defined and endowed with an ultraproximity relation. A draft of a construction of "numeric grains" is presented : these numbers could prove to have properties sufficiently similar to real numbers to allow to perform analysis.

La théorie présentée ici est issue d'années d'enseignement de l'analyse au niveau pré-universitaire en utilisant d'abord le concept d'infiniment petit, tel que défini dans l'analyse nonstandard de Robinson, puis ensuite d'ultrapetit, tel que défini dans notre travail en collaboration avec Hrbacek et Lessmann et présenté en annexe. A la suite de ces recherches, s'est posée la question : Si l'on a à disposition des quantités finies mais ultragrandes, est-il possible de se passer de quantités dites infinies ? La théorie alternative des ensembles de Vopěnka est une théorie avec des ensembles finis et des classes qui, elles, peuvent être infinies. La théorie des objets est le résultat d'un mélange de certains axiomes de Vopěnka avec des axiomes déterminant des niveaux de visibilité tels que dans l'analyse relative. On s'est donné comme premier principe : $x\subseteq y\Rightarrow x\sqsubseteq y$ qui spécifie que si l'objet $x$ est inclus dans l'objet $y$, alors $x$ "paraît" au niveau de $y$. Cette affirmation serait fausse avec des quantités infinies ; elle est néanmoins une caractérisation des ensembles finis : cela est bien connu en analyse nonstandard. L'introduction de ce principe comme point de départ est donc une affirmation forte que les objets devront être finis au sens habituel de ce terme. L'autre axiome fondateur ici est le schéma d'axiomes d'induction de Gordon et Andreev : Si $\Phi$ est une formule, et si $\Phi(\emptyset)$ est vrai et que $\Phi(x)$ et $\Phi(y)$ impliquent $\Phi(x\cup\{y\})$, alors $\Phi(x)$ est vrai pour tout $x$. Un accent particulier est mis sur le concept de formules dites contextuelles. Ce concept est une de nos contributions à l'analyse relative de Hrbacek et détermine les formules bien formées. On montre que le système qui en résulte est relativement cohérent avec la théorie FRIST de Hrbacek et la théorie RIST de Péraire qui sont elles-mêmes des extensions conservatives de ZFC. La théorie des objets est une extension de la théorie des ensembles de Zermelo et Fraenkel sans axiome du choix et négation de l'axiome de l'infini. Les nombres entiers et rationnels sont définis et ces derniers sont munis de relations d'ultraproximité. Une ébauche d'une construction de "grains numériques" est présentée : ces nombres pourraient avoir des propriétés suffisamment semblables aux nombres réels pour permettre de faire de l'analyse. / The theory presented here stemmed from years of teaching analysis at pre-university level first using the concept of infinitesimal as defined in nonstandard analysis by Robinson, then the concept of ultrasmall as defined in our joint work with Hrbacek and Lessmann presented in the appendix. This research led to the question : If one has finite yet ultralarge quantities, is it possible to avoid infinite quantities ? The alternative set theory of Vopěnka is a theory of finite sets including classes that can be infinite. The theory of objects is a merger of certain axioms of Vopěnka with axioms that determine levels of visibility as in relative analysis. We took as first principle : $x\subseteq y\Rightarrow x\sqsubseteq y$, which specifies that if object $x$ is included in object $y$, then $x$ "appears" at the level of $y$. This statement would be false with infinite quantities and is in fact a characterisation of finite sets : this is a well-known theorem of nonstandard analysis. The introduction of this principle as starting point is making a strong point that all objects will be finite - in the usual sense of the word. The other founding axiom is Gordon and Andreev's axiom schema : If $\Phi$ is a formula, and if $\Phi(\emptyset)$ is true and that $\Phi(x)$ and $\Phi(y)$ imply $\Phi(x\cup\{y\})$, then $\Phi(x)$ is true for all $x$. An emphasis is made on the concept of contextual formulae. This concept is one of our contributions to relative analysis of Hrbacek and determines an equivalence to well-formed formulae. We show that the resulting system is relatively consistent with Hrbacek's FRIST and Péraire's RIST which are conservative extensions of ZFC. The theory of objects extends set theory of Zermelo and Fraenkel without choice and with negation of the infinity axiom. Integers and rationals are defined and endowed with an ultraproximity relation. A draft of a construction of "numeric grains" is presented : these numbers could prove to have properties sufficiently similar to real numbers to allow to perform analysis.

.../... / “Standards make a positive contribution to the world we live in. They facilitate trade, spreadknowledge, disseminate innovative advances in technology, and share good management andconformity assessment practices”7. There are a multitude of standard and standard consortiaorganizations producing market relevant standards, specifications, and technical reports in thedomain of Information Communication Technology (ICT). With the number of ICT relatedstandards and specifications numbering in the thousands, it is not readily apparent to users howthese standards inter-relate to form the basis of technical interoperability. There is a need todevelop and document a process to identify how standards inter-relate to form a basis ofinteroperability in multiple contexts; at a general horizontal technology level that covers alldomains, and within specific vertical technology domains and sub-domains. By analyzing whichstandards inter-relate through normative referencing, key standards can be identified as technicalcenters of gravity, allowing identification of specific standards that are required for thesuccessful implementation of standards that normatively reference them, and form a basis forinteroperability across horizontal and vertical technology domains. This Thesis focuses on defining a methodology to analyze ICT standards to identifynormatively referenced standards that form technical centers of gravity utilizing Data Mining(DM) and Social Network Analysis (SNA) graph technologies as a basis of analysis. As a proofof concept, the methodology focuses on the published International Standards (IS) published bythe International Organization of Standards/International Electrotechnical Committee; JointTechnical Committee 1, Sub-committee 36 Learning Education, and Training (ISO/IEC JTC1 SC36). The process is designed to be scalable for larger document sets within ISO/IEC JTC1 that covers all JTC1 Sub-Committees, and possibly other Standard Development Organizations(SDOs).Chapter 1 provides a review of literature of previous standard analysis projects and analysisof components used in this Thesis, such as data mining and graph theory. Identification of adataset for testing the developed methodology containing published International Standardsneeded for analysis and form specific technology domains and sub-domains is the focus ofChapter 2. Chapter 3 describes the specific methodology developed to analyze publishedInternational Standards documents, and to create and analyze the graphs to identify technicalcenters of gravity. Chapter 4 presents analysis of data which identifies technical center of gravitystandards for ICT learning, education, and training standards produced in ISO/IEC JTC1 SC 36.Conclusions of the analysis are contained in Chapter 5. Recommendations for further researchusing the output of the developed methodology are contained in Chapter 6.

The thesis deals with analysis of the quality of corporate social responsibility reports sampled from in Czech Republic seated organizations. First chaps concentrate on themes of sustainable development and corporate social responsibility (CSR). The thesis treats of individual instruments in the field of CSR (standards, approaches or codes). The focus is concentrated on G3 Standard of Global Reporting Initiative, which is crucial for the whole analysis. GRI Standard is internationally reputable instrument for CSR reporting. Before the analysis itself the thesis offers short view of researches in the sphere of sustainable development. CSR reporting is still not a common practice among organizations doing business in Czech Republic. There are some exceptions. What is the quality of CSR reports of firms who are in Czech Republic corporate social responsible? Fundamental part of the thesis is concentrated in the analysis of five CSR reports according to the statement of requirements of GRI Standard. From the analysis results some recommendation for firms who want to report abreast of GRI Standard.