Areas of Mathematics

Mục tiêu của học phần: Sau khi nghiên cứu môn học, sinh viên vừa được trang bị các kiến thức cơ bản về bản chất và nguyên lý tính toán trong các nghiệp vụ tài chính, vừa biết vận dụng các kiến thức đó để xây dựng các bài toán tài chính trong những hoàn cảnh riêng với môi trường và các điều kiện khác nhau.

This book provides an overview of the various biometric technologies, decision-making algorithms and the subsequent market expansion opportunity. Further, this book proposes a Throughput Model, which draws from computer science, economic and psychology literatures to model perceptual, informational sources, judgmental processes and decision choice algorithms. This approach describes how biometrics might be implemented to reduce risks to individuals and organizations, especially when dealing with digital based mediums.

Il existe une distinction capitale entre expérience mentale et expérience logique ; cet ouvrage, publié en 1950 par Jean de La Harpe, professeur de philosophie à l’Université de Neuchâtel revient sur le problème de l’analyse psychologique des opérations fondamentales de la logique.

In diesem Buch werden Aspekte der Aussagenlogik und der Prädikaten­ logik der ersten Stufe behandelt. Eine Mathematisierung und Kalkü­ lisierung der Logik kann natürlich ganz verschieden ausfallen, je nach dem, von welchen Motiven man sich primär leiten läßt. Wir stellen drei Gesichtspunkte, die uns auch als die wesentlichsten erscheinen, in den Vordergrund: Die Formalisierung des Wahrheits­ begriffes, die Formalisierung des Beweisbarkeitsbegriffes und das Problem des Suchens nach Beweisen. Diese drei Aspekte führen zu drei verschiedenen Arten von Kalkülen. Die Betonung des (...) Wahrheitsbegriffes führte auf die untersuchung der Hilberttypkalküle von einem Standpunkt, wie er etwa auch im Buch von Rasiowa-Sikorski [Ra-Si] eingenommen wird. Hierbei wurde besonderen Wert auf die algebraischen Techniken gelegt, denn die Natur der Vollständigkeitsbeweise in diesen Kalkülen läßt sich u. E. eigentlich nur algebraisch verstehen. Etwas überspitzt könnte man formulieren, daß die Vollständigkeitsbeweise in Hilberttypkal­ külen Korollare zu Betrachtungen über Kongruenzrelationen in ge­ wissen Boole'schen Algebren sind. Bei den modelltheoretischen Be­ trachtungen haben wir uns kurz ge faßt und nur einige grundlegende Begriffe vorgestellt. (shrink)

Wenn es eine Aufgabe der Philosophie ist, die Herrschaft des Wortes über den menschlichen Geist zu brechen (Frege), dann ist zu fragen, ob die Prädikatenlogik die Wörter «für alle» und «es gibt» angemessen formalisiert. Übersetzt man «alle Menschen sind sterblich» mit «für alle x : wenn x ein Mensch ist, dann ist x sterblich», so hat man zwar die Syllogistik des Aristoteles überwunden, dafür tritt jetzt aber eine Variable x auf, die in der Aussage vorher nicht vorkam.

Ce livre est un recueil des articles que l'auteur a publiés dans la rubrique Logique et Calcul du magazine Pour la Science. La sélection a pour thème la logique dans toute sa diversité. Par exemple, elle peut concerner les mathématiques pures : L'infini est-il paradoxal en mathématiques, Libre arbitre et mécanique quantique ou Les limites logiques des mathématiques. Mais on la trouve aussi au coeur d'applications concrètes. Ainsi, L'étonnante loi de Benford, selon laquelle un nombre pris au hasard, par exemple (...) dans le journal, commence plus souvent par un " 1 " plutôt que par tout autre chiffre, permet de dépister les tricheurs. Ou encore, la logique qui aide à répondre à cette question d'actualité : La répartition idéale des biens existe-t-elle? Jean-Paul Delahaye, avec son style et ses talents reconnus de vulgarisateur, nous fait même réfléchir aux liens qui unissent La belle au bois dormant, les extraterrestres et la fin du monde! Idéal en cette année 2012. (shrink)

L'ouvrage invite à penser l'épistémologie non classique des mathématiques comme une construction épistémique des mathématiques quasi empirique qu'Imré Lakatos a réalisé au moyen des présupposés ontologiques (comme la redécouverte du temps, le passage du déterminisme à l'indéterminisme, etc.) et qu'il réussit grâce à l'implémentation du principe de complexité dialectique de l'esprit mathématique. L'auteur étudie la thèse selon laquelle, depuis le début du XXe siècle, la science est aussi bien objective que subjective. Et c'est à partir de celle-ci que le principe (...) de "la complexité dialectique de l'esprit mathématique" est mis au jour pour dévoiler le constructivisme de Lakatos. (shrink)

In the 21st century, digitalization is a global challenge of mankind. Even for the public, it is obvious that our world is increasingly dominated by powerful algorithms and big data. But, how computable is our world? Some people believe that successful problem solving in science, technology, and economies only depends on fast algorithms and data mining. Chances and risks are often not understood, because the foundations of algorithms and information systems are not studied rigorously. Actually, they are deeply rooted in (...) logics, mathematics, computer science and philosophy. Therefore, this book studies the foundations of mathematics, computer science, and philosophy, in order to guarantee security and reliability of the knowledge by constructive proofs, proof mining and program extraction. We start with the basics of computability theory, proof theory, and information theory. In a second step, we introduce new concepts of information and computing systems, in order to overcome the gap between the digital world of logical programming and the analog world of real computing in mathematics and science. The book also considers consequences for digital and analog physics, computational neuroscience, financial mathematics, and the Internet of Things (IoT). (shrink)

Originally published in 1964. This book is concerned with general arguments, by which is meant broadly arguments that rely for their force on the ideas expressed by all, every, any, some, none and other kindred words or phrases. A main object of quantificational logic is to provide methods for evaluating general arguments. To evaluate a general argument by these methods we must first express it in a standard form. Quantificational form is dealt with in chapter one and in part of (...) chapter three; in the remainder of the book an account is given of methods by which arguments when formulated quantificationally may be tested for validity or invalidity. Some attention is also paid to the logic of identity and of definite descriptions. Throughout the book an attempt has been made to give a clear explanation of the concepts involved and the symbols used; in particular a step-by-step and partly mechanical method is developed for translating complicated statements of ordinary discourse into the appropriate quantificational formulae. Some elementary knowledge of truth-functional logic is presupposed. (shrink)

Originally published in 1967. An introduction to the literature of nonstandard logic, in particular to those nonstandard logics known as many-valued logics. Part I expounds and discusses implicational calculi, modal logics and many-valued logics and their associated calculi. Part II considers the detailed development of various many-valued calculi, and some of the important metathereoms which have been proved for them. Applications of the calculi to problems in the philosophy are also surveyed. This work combines criticism with exposition to form a (...) comprehensive but concise survey of the field. (shrink)

This volume contains a facsimile reprint of the 1883 Boston edition of Studies in Logic by Members of the Johns Hopkins University, edited by Charles S. Peirce. In relation to this work there are three mutually related aspects of Peirce’s thought which deserve to be particularly emphasized: the community structure of science as propagated and practiced by Peirce; his consideration of the fundamental relationship between logic and semiotics; and his emphatic plea for a historisation of science and, hence, of semiotics. (...) Peirce’s Studies in Logic is preceded in this volume by a portrait of Peirce as scientist, mathematician, historian, logician and philosopher by Max. H. Fisch, and a history of semiotics and Charles S. Peirce by Achim Eschbach. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and (...) made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

The term "fuzzy logic," as it is understood in this book, stands for all aspects of representing and manipulating knowledge based on the rejection of the most fundamental principle of classical logic---the principle of bivalence. According to this principle, each declarative sentence is required to be either true or false. In fuzzy logic, these classical truth values are not abandoned. However, additional, intermediate truth values between true and false are allowed, which are interpreted as degrees of truth. This opens a (...) new way of thinking---thinking in terms of degrees rather than absolutes. For example, it leads to the definition of a new kind of sets, referred to as fuzzy sets, in which membership is a matter of degree. The book examines the genesis and development of fuzzy logic. It surveys the prehistory of fuzzy logic and inspects circumstances that eventually lead to the emergence of fuzzy logic. The book explores in detail the development of propositional, predicate, and other calculi that admit degrees of truth, which are known as fuzzy logic in the narrow sense. Fuzzy logic in the broad sense, whose primary aim is to utilize degrees of truth for emulating common-sense human reasoning in natural language, is scrutinized as well. The book also examines principles for developing mathematics based on fuzzy logic and provides overviews of areas in which this has been done most effectively. It also presents a detailed survey of established and prospective applications of fuzzy logic in various areas of human affairs, and provides an assessment of the significance of fuzzy logic as a new paradigm. (shrink)

The well-known Ratio Studiorum of 1599 states that logical instruction should follow F. Toletus or P. Fonseca. The latter authored the famous Institutionum Dialecticarum Libri Octo, the former a similar manual, Introductio in Dialecticam Aristotelis. As is often observed, the contrast between the Aristotelian and present symbolic logics is perhaps most striking in their analysis of relational statements. Both authors recognize the relational logical form as independent from the traditional subject-predicate form and see the need to recognize relational inferential rules. (...) They differ in their specific rules, however, so neither of the authors has captured the system of relational syllogism in its entirety. (shrink)

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on (...) the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike. (shrink)

In addition to essays which have appeared before, this collection includes two new works, "Synthetic a Priori" and "Realistic Postscript." Clearing away the last remnants of his former phenomenalism, Bergmann explicitly proclaims a realistic ontology. Characters are things just as truly as individuals are. Non-obtaining facts exist in a mode of possibility. Bergmann extends his analysis of the act, which he acknowledges to be central to his philosophy, to acts with physical or non-mental intentions. In the light of his own (...) views, he examines the ontologies of Frege, Husserl, Moore, Wittgenstein and Strawson. The confrontation with Husserl, who moved in the opposite direction, from realism to idealism, is particularly interesting.—J. B. B. (shrink)

This paper is a historical companion to a previous one, in which it was studied the so-called abstract Galois theory as formulated by the Portuguese mathematician José Sebastião e Silva ). Our purpose is to present some applications of abstract Galois theory to higher-order model theory, to discuss Silva's notion of expressibility and to outline a classical Galois theory that can be obtained inside the two versions of the abstract theory, those of Mark Krasner and of Silva. Some comments are (...) made on the universal theory of structures. (shrink)

1 Mathematical practice: a short overview This volume is a collection of essays that discuss the relationships between the practices deployed by logicians and mathematicians, either as individuals or as members of research communities, and the results from their research. We are interested in exploring the concept of 'practices' in the formal sciences. Though common in the history, philosophy and sociology of science, this concept has surprisingly thus far been little reflected upon in logic...

"This volume contains the final versions of a collection of papers presented at the Annual Conference of the European Association for Computer Science Logic, CSL '93, held at Swansea, UK in September 1993. The 21 full papers included were selected from a total of 62 submissions and essentially contribute to the whole area of computer science logic research. They are devoted to such topics as set constraints, lambda calculi, process algebras, program semantics, intuitionistic logics, fixed-point logics, the equivalence problem, Horn (...) clauses, quantifiers, and proof tranformations."--PUBLISHER'S WEBSITE. (shrink)

This volume presents the proceedings of the Computer Science Logic Workshop CSL '92, held in Pisa, Italy, in September/October 1992. CSL '92 was the sixth of the series and the first one held as Annual Conference of the European Association for Computer Science Logic (EACSL). Full versions of the workshop contributions were collected after their presentation and reviewed. On the basis of 58 reviews, 26 papers were selected for publication, and appear here in revised final form. Topics covered in the (...) volume include: Turing machines, linear logic, logic of proofs, optimization problems, lambda calculus, fixpoint logic, NP-completeness, resolution, transition system semantics, higher order partial functions, evolving algebras, functional logic programming, inductive definability, semantics of C, classes for a functional language, NP-optimization problems, theory of types and names, sconing and relators, 3-satisfiability, Kleene's slash, negation-complete logic programs, polynomial-time oracle machines, and monadic second-order properties. (shrink)

James Walmsley in “Categoricity and Indefinite Extensibility” argues that a realist about some branch of mathematics X (e.g. arithmetic) apparently cannot use the categoricity of an axiomatisation of X to justify her belief that every sentence of the language of X has a truth-value. My discussion note first corrects Walmsley’s formulation of his claim. It then shows that his argument for it hinges on the implausible idea that grasping that there is some model of the axioms amounts to grasping that (...) there is a unique such model. (shrink)