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Littérature scientifique sur le sujet « Mathematics Proof theory. Logic, Symbolic and mathematical »

Auteur : Grafiati
Publié le 4 juin 2021
Mis à jour le 1 février 2022

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Articles de revues sur le sujet "Mathematics Proof theory. Logic, Symbolic and mathematical"

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Eklof, Paul C. « Fred Appenzeller. An independence result in quadratic form theory : infinitary combinatorics applied to ε-Hermitian spaces. The journal of symbolic logic, vol. 54 (1989), pp. 689–699. - Otmar Spinas. Linear topologies on sesquilinear spaces of uncountable dimension. Fundamenta mathematicae, vol. 139 (1991), pp. 119–132. - James E. Baumgartner, Matthew Foreman, and Otmar Spinas. The spectrum of the Γ-invariant of a bilinear space. Journal of algebra, vol. 189 (1997), pp. 406–418. - James E. Baumgartner and Otmar Spinas. Independence and consistency proofs in quadratic form theory. The journal of symbolic logic, vol. 56 (1991), pp. 1195–1211. - Otmar Spinas. Iterated forcing in quadratic form theory. Israel journal of mathematics, vol. 79 (1992), pp. 297–315. - Otmar Spinas. Cardinal invariants and quadratic forms. Set theory of the reals, edited by Haim Judah, Israel mathematical conference proceedings, vol. 6, Gelbart Research Institute for Mathematical Sciences, Bar-Ilan University, Ramat-Gan 1993, distributed by the American Mathematical Society, Providence, pp. 563–581. - Saharon Shelah and Otmar Spinas. Gross spaces. Transactions of the American Mathematical Society, vol. 348 (1996), pp. 4257–4277. » Bulletin of Symbolic Logic 7, no 2 (juin 2001) : 285–86. http://dx.doi.org/10.2307/2687785.

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Buss, Samuel, Ulrich Kohlenbach et Michael Rathjen. « Mathematical Logic : Proof Theory, Constructive Mathematics ». Oberwolfach Reports 8, no 4 (2011) : 2963–3002. http://dx.doi.org/10.4171/owr/2011/52.

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Buss, Samuel, Ulrich Kohlenbach et Michael Rathjen. « Mathematical Logic : Proof Theory, Constructive Mathematics ». Oberwolfach Reports 11, no 4 (2014) : 2933–86. http://dx.doi.org/10.4171/owr/2014/52.

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Buss, Samuel, Rosalie Iemhoff, Ulrich Kohlenbach et Michael Rathjen. « Mathematical Logic : Proof Theory, Constructive Mathematics ». Oberwolfach Reports 14, no 4 (18 décembre 2018) : 3121–83. http://dx.doi.org/10.4171/owr/2017/53.

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Buss, Samuel, Rosalie Iemhoff, Ulrich Kohlenbach et Michael Rathjen. « Mathematical Logic : Proof Theory, Constructive Mathematics ». Oberwolfach Reports 17, no 4 (13 septembre 2021) : 1693–757. http://dx.doi.org/10.4171/owr/2020/34.

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Arai, Toshiyasu. « Wilfried Buchholz. Notation systems for infinitary derivations. Archive for mathematical logic, vol. 30 no. 5–6 (1991), pp. 277–296. - Wilfried Buchholz. Explaining Gentzen's consistency proof within infinitary proof theory. Computational logic and proof theory, 5th Kurt Gödel colloquium, KGC '97, Vienna, Austria, August 25–29, 1997, Proceedings, edited by Georg Gottlob, Alexander Leitsch, and Daniele Mundici, Lecture notes in computer science, vol. 1289, Springer, Berlin, Heidelberg, New York, etc., 1997, pp. 4–17. - Sergei Tupailo. Finitary reductions for local predicativity, I : recursively regular ordinals. Logic Colloquium '98, Proceedings of the annual European summer meeting of the Association for Symbolic Logic, held in Prague, Czech Republic, August 9–15, 1998, edited by Samuel R. Buss, Petr Háajek, and Pavel Pudlák, Lecture notes in logic, no. 13, Association for Symbolic Logic, Urbana, and A K Peters, Natick, Mass., etc., 2000, pp. 465–499. » Bulletin of Symbolic Logic 8, no 3 (septembre 2002) : 437–39. http://dx.doi.org/10.2178/bsl/1182353905.

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Avigad, Jeremy. « Forcing in Proof Theory ». Bulletin of Symbolic Logic 10, no 3 (septembre 2004) : 305–33. http://dx.doi.org/10.2178/bsl/1102022660.

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AbstractPaul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbert-style proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing model-theoretic arguments.
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Gentilini, Paolo. « Proof theory and mathematical meaning of paraconsistent C-systems ». Journal of Applied Logic 9, no 3 (septembre 2011) : 171–202. http://dx.doi.org/10.1016/j.jal.2011.04.001.

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NEGRI, SARA, et JAN VON PLATO. « Proof systems for lattice theory ». Mathematical Structures in Computer Science 14, no 4 (août 2004) : 507–26. http://dx.doi.org/10.1017/s0960129504004244.

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A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.
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RABE, FLORIAN. « A logical framework combining model and proof theory ». Mathematical Structures in Computer Science 23, no 5 (1 mars 2013) : 945–1001. http://dx.doi.org/10.1017/s0960129512000424.

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Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.
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Thèses sur le sujet "Mathematics Proof theory. Logic, Symbolic and mathematical"

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Duff, Karen Malina. « What Are Some of the Common Traits in the Thought Processes of Undergraduate Students Capable of Creating Proof ? » Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1856.pdf.

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Peske, Wendy Ann. « A topological approach to nonlinear analysis ». CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2779.

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A topological approach to nonlinear analysis allows for strikingly beautiful proofs and simplified calculations. This topological approach employs many of the ideas of continuous topology, including convergence, compactness, metrization, complete metric spaces, uniform spaces and function spaces. This thesis illustrates using the topological approach in proving the Cauchy-Peano Existence theorem. The topological proof utilizes the ideas of complete metric spaces, Ascoli-Arzela theorem, topological properties in Euclidean n-space and normed linear spaces, and the extension of Brouwer's fixed point theorem to Schauder's fixed point theorem, and Picard's theorem.
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Brierley, William. « Undecidability of intuitionistic theories ». Thesis, McGill University, 1985. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=66016.

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Eliasson, Jonas. « Ultrasheaves ». Doctoral thesis, Uppsala : Matematiska institutionen, Univ. [distributör], 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-3762.

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Konecny, Jan. « Isotone fuzzy Galois connections and their applications in formal concept analysis ». Diss., Online access via UMI:, 2009.

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Thesis (Ph. D.)--State University of New York at Binghamton, Thomas J. Watson School of Engineering and Applied Science, Department of Systems Science and Industrial Engineering, 2009.
Includes bibliographical references.
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Schwartzkopff, Robert. « The numbers of the marketplace : commitment to numbers in natural language ». Thesis, University of Oxford, 2015. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.711821.

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Freire, Rodrigo de Alvarenga. « Os fundamentos do pensamento matematico no seculo XX e a relevancia fundacional da teoria de modelos ». [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/281061.

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Orientador: Walter Alexandre Carnielli
Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciencias Humanas
Made available in DSpace on 2018-08-12T22:46:52Z (GMT). No. of bitstreams: 1 Freire_RodrigodeAlvarenga_D.pdf: 761227 bytes, checksum: 3b1a0de92aa93b50f2bfc602bf6173bc (MD5) Previous issue date: 2009
Resumo: Esta Tese tem como objetivo elucidar, ao menos parcialmente, a questão do significado da Teoria de Modelos para uma reflexão sobre o conhecimento matemático no século XX. Para isso, vamos buscar, primeiramente, alcançar uma compreensão da própria reflexão sobre o conhecimento matemático, que será denominada de Fundamentos do Pensamento Matemático no século XX, e da própria relevância fundacional. Em seguida, analisaremos, dentro do contexto fundacional estabelecido, o papel da Teoria de Modelos e da sua interação com a Álgebra, em geral, e, finalmente, empreenderemos um estudo de caso específico. Nesse estudo de caso mostraremos que a Teoria de Galois pode ser vista como um conteúdo lógico, e buscaremos compreender o significado fundacional desse enquadramento modelo-teórico para uma parte da Álgebra clássica.
Abstract: The aim of the present Thesis is to bring some light to the question about the status and relevance of Model Theory to a reflection about the mathematical knowledge in the twentieth century. To pursue this target, we will, first of all, try to reach a comprehension of the reflection about the mathematical knowledge, itself, what will be designated as Foundations of Mathematical Thought in the twentieth century, and of the foundational relevance, itself. In the sequel, we will provide an analysis, of the role of Model Theory and its interaction with Algebra, in general, within the established foundational setting and, finally, we will discuss a specific study case. In this study case we will show that Galois Theory can be seen as a logical content, and we will try to understand the foundational meaning of this model-theoretic framework for some part of classical Algebra.
Doutorado
Logica
Doutor em Filosofia
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Sentone, Francielle Gonçalves. « Paradoxos geométricos em sala de aula ». Universidade Tecnológica Federal do Paraná, 2017. http://repositorio.utfpr.edu.br/jspui/handle/1/2701.

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CAPES
Apresentamos neste trabalho alguns paradoxos lógico-matemáticos, como o paradoxo de Galileu, e também alguns paradoxos geométricos, como os paradoxos de Curry, de Hooper e de Banach-Tarski. Empregamos os paradoxos de Curry e de Hooper para motivar o estudo de conceitos de Geometria e de Teoria dos Números, tais como área, semelhança de triângulos, o Teorema de Pitágoras, razões trigonométricas no triângulo retângulo, o coeficiente angular da reta e a sequência de Fibonacci, e organizamos atividades lúdicas para a sala de aula no Ensino Fundamental e no Ensino Médio.
We present in this work some logical-mathematical paradoxes, as Galileo's paradox, and also some geometric paradoxes, such as Curry's paradox, Hooper's paradox and the Banach-Tarski paradox. We employ the Curry and Hooper paradoxes to motivate the study of concepts of Geometry and Number Theory, such as area, triangle similarity, Pythagorean Theorem, trigonometric ratios in the right triangle, angular coefficient of the line, and Fibonacci sequence, and we organize recreation activities for the classroom in Elementary and High School.
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Van, Staden Anna Maria. « The role of logical principles in proving conjectures using indirect proof techniques in mathematics ». Thesis, 2012. http://hdl.handle.net/10210/6769.

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M.Ed.
Recently there has been renewed interest in proof and proving in schools worldwide. However, many school students and even teachers of mathematics have only superficial ideas on the nature of proof. Proof is considered the heart of mathematics as individuals explore, make conjectures and try to convince themselves and others about the truth or falsity of their conjectures. There are basically two categories of deductive proof, namely proof by direct argument and indirect proofs. The aim of this study was to examine the structural features common to most of the mathematical proofs for formalised mathematical systems, with the emphasis on indirect proof techniques. The main question was to investigate which mathematical activities and logical principles at secondary school level are necessary for students to become proficient with proof writing. A great deal of specialised language is associated with reasoning. Such words as axiom, theorem, proof, and conjecture are just some of the terms that students must understand as they engage in the proof-making task. The formal aspect of mathematics at secondary school is extremely important. It is inevitable that students become involved with hypothetical arguments. They use among others, proofs by contradiction. Furthermore, necessary and sufficient conditions are related to theorems and their converses. It is therefore apparent that the study of logic is necessary already at secondary school level in order to practise mathematics satisfactorily. An analysis of the mathematics syllabus of the Department of Education has indicated that students should use indirect techniques of proof. According to this syllabus students should be familiar with logical arguments. The conclusion which is reached, gives evidence that students’ background in logic is completely lacking and inadequate. As a result they cannot cope adequately with argumentation and this causes a poor perception of what mathematics entails. Although proof writing can never be reduced to a mechanical process, considerable anxiety and uncertainty can be eliminated from the process if students are exposed to the principles of elementary logic and techniques. Mathematics educators and education researchers have reported students’ difficulties with mathematical proof and point out the conflict between the nature of this essential mathematical activity and current approaches to teaching it. This recent interest has led to an increased effort to teach proof in innovative ways.
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Van, der Vyver Thelma. « Proof systems for propositional modal logic ». Diss., 1997. http://hdl.handle.net/10500/16280.

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In classical propositional logic (CPL) logical reasoning is formalised as logical entailment and can be computed by means of tableau and resolution proof procedures. Unfortunately CPL is not expressive enough and using first order logic (FOL) does not solve the problem either since proof procedures for these logics are not decidable. Modal propositional logics (MPL) on the other hand are both decidable and more expressive than CPL. It therefore seems reasonable to apply tableau and resolution proof systems to MPL in order to compute logical entailment in MPL. Although some of the principles in CPL are present in MPL, there are complexities in MPL that are not present in CPL. Tableau and resolution proof systems which address these issues and others will be surveyed here. In particular the work of Abadi & Manna (1986), Chan (1987), del Cerro & Herzig (1988), Fitting (1983, 1990) and Gore (1995) will be reviewed.
Computing
M. Sc. (Computer Science)
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Livres sur le sujet "Mathematics Proof theory. Logic, Symbolic and mathematical"

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1957-, Taylor John, dir. 100% mathematical proof. Chichester : Wiley, 1996.

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Introduction to mathematical proof : A transition to advanced mathematics. Boca Raton : CRC Press, Taylor & Francis Group, 2015.

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Rowan, Garnier, dir. Understanding mathematical proof. Boca Raton : Taylor & Francis, 2014.

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B, Maddox Randall, dir. A transition to abstract mathematics : Learning mathematical thinking and writing. 2e éd. Amsterdam : Academic Press, 2009.

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Wansing, H. Proof theory of modal logic. Dordrecht : Springer, 1996.

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Proof, logic, and conjecture : The mathematician's toolbox. New York : W.H. Freeman, 1998.

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Resolution proof systems : An algebraic theory. Dordrecht : Kluwer Academic Publishers, 1996.

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1977-, Nguyen Phuong, dir. Perspectives in logic : Logical foundations of proof complexity. Cambridge : Cambridge University Press, 2010.

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Benson, Donald C. The moment of proof : Mathematical epiphanies. New York : Oxford University Press, 1999.

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Handbook of mathematical induction : Theory and applications. Boca Raton, FL : CRC Press, 2011.

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Chapitres de livres sur le sujet "Mathematics Proof theory. Logic, Symbolic and mathematical"

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Grattan-Guinness, Ivor. « Turing’s mentor, Max Newman ». Dans The Turing Guide. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198747826.003.0052.

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The interaction between mathematicians and mathematical logicians has always been much slighter than one might imagine. This chapter examines the case of Turing’s mentor, Maxwell Hermann Alexander Newman (1897–1984). The young Turing attended a course of lectures on logical matters that Newman gave at Cambridge University in 1935. After briefly discussing examples of the very limited contact between mathematicians and logicians in the period 1850–1930, I describe the rather surprising origins and development of Newman’s own interest in logic. One might expect that the importance to many mathematicians of means of proving theorems, and their desire in many contexts to improve the level of rigour of proofs, would motivate them to examine and refine the logic that they were using. However, inattention to logic has long been common among mathematicians. A very important source of the cleft between mathematics and logic during the 19th century was the founding, from the late 1810s onwards, of the ‘mathematical analysis’ of real variables, grounded on a theory of limits, by the French mathematician Augustin-Louis Cauchy. He and his followers extolled rigour—most especially, careful definitions of major concepts and detailed proofs of theorems. From the 1850s onwards, this project was enriched by the German mathematician Karl Weierstrass and his many followers, who introduced (for example) multiple limit theory, definitions of irrational numbers, and an increasing use of symbols, and then from the early 1870s by Georg Cantor with his set theory. However, absent from all these developments was explicit attention to any kind of logic. This silence continued among the many set theorists who participated in the inauguration of measure theory, functional analysis, and integral equations. The mathematicians Artur Schoenflies and Felix Hausdorff were particularly hostile to logic, targeting the famous 20th-century logician Bertrand Russell. (Even the extensive dispute over the axiom of choice focused mostly on its legitimacy as an assumption in set theory and its use of higher-order quantification: its ability to state an infinitude of independent choices within finitary logic constituted a special difficulty for ‘logicists’ such as Russell.) Russell, George Boole, and other creators of symbolic logics were exceptional among mathematicians in attending to logic, but they made little impact on their colleagues.
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Smullyan, Raymond M. « Tarski’s Theorem for Arithmetic ». Dans Gödel's Incompleteness Theorems. Oxford University Press, 1992. http://dx.doi.org/10.1093/oso/9780195046724.003.0005.

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In the last chapter, we dealt with mathematical languages in considerable generality. We shall now turn to some particular mathematical languages. One of our goals is to reach Gödel’s incompleteness theorem for the particular system known as Peano Arithmetic. We shall give several proofs of this important result; the simplest one is based partly on Tarski’s theorem, to which we first turn. The first concrete language that we will study is the language of first order arithmetic based on addition, multiplication and exponentiation. [We also take as primitive the successor function and the less than or equal to relation, but these are inessential.] We shall formulate the language using only a finite alphabet (mainly for purposes of a convenient Gödel numbering); specifically we use the following 13 symbols. . . . 0’ ( ) f, υ ∽ ⊃ ∀ = ≤ # . . . The expressions 0, 0′, 0″, 0‴, · · · are called numerals and will serve as formal names of the respective natural numbers 0, 1, 2, 3, · · ·. The accent symbol (also called the prime) is serving as a name of the successor function. We also need names for the operations of addition, multiplication and exponentiation; we use the expressions f′, f″, f‴ as respective names of these three functions. We abbreviate f′ by the familiar “+”; we abbreviate f’’ by the familiar dot and f‴ by the symbol “E”. The symbols ~ and ⊃ are the familiar symbols from prepositional logic, standing for negation and material implication, respectively. [For any reader not familiar with the use of the horseshoe symbol, for any propositions p and q, the propositions p ⊃ q is intended to mean nothing more nor less than that either p is false, or p and q are both true.] The symbol ∀ is the universal quantifier and means “for all.” We will be quantifying only over natural numbers not over sets or relations on the natural numbers. [Technically, we are working in first-order arithmetic, not second-order arithmetic.] The symbol “=” is used, as usual, to denote the identity relation, and “≤” is used, as usual, to denote the “less than or equal to” relation.
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Shin, Sun-Joo. « Situation-Theoretic Account of Valid Reasoning with Venn Diagrams ». Dans Logical Reasoning with Diagrams. Oxford University Press, 1996. http://dx.doi.org/10.1093/oso/9780195104271.003.0009.

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Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.
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Takahashi, Tadashi. « Proving in Mathematics Education – On the Proof using ATP ». Dans Theory and Practice : An Interface or A Great Divide ?, 558–63. WTM-Verlag Münster, 2019. http://dx.doi.org/10.37626/ga9783959871129.0.105.

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The aim of the mathematics education is the acquisition of “knowledge/skill of the mathematics” and “the mathematical thinking”. Proving is a chain of the logic in mathematics and is “mathematical thinking” itself. So, proving is the domain that is important from a point of view that can evaluate the acquisition of enough “mathematical thinking”. There is a variety of sense of values in the present situation of the proof using the ATP (Automated theorem proving). We should establish a clear vision as mathematics education in this situation. That is, in mathematics education, we should build sense of values for proof using the ATP newly. To that end, we fix contents of the mathematics, and it is necessary to prove them by using ATP. We would like to assume the aim the theorems of Euclid's Elements. Because the contents are the basics of the mathematical thinking. The proving is an important aim in the mathematics education, it is necessary to clarify new value by using the ATP as mathematics education.
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Pollack, Robert. « How to believe a machine-checked proof ». Dans Twenty Five Years of Constructive Type Theory. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780198501275.003.0013.

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Suppose I say “Here is a machine-checked proof of Fermat's last theorem (FLT)”. How can you use my putative machine-checked proof as evidence for belief in FLT? I start from the position that you must have some personal experience of understanding to attain belief, and to have this experience you must engage your intuition and other mental processes which are impossible to formalise. By machine-checked proof I mean a formal derivation in some given formal system; I am talking about derivability, not about truth. Further, I want to talk about actually believing an actual formal proof, not about formal proofs in principle; to be interesting, any approach to this problem must be feasible. You might try to read my proof, just as you would a proof in a journal; however, with the current state of the art, this proof will surely be too long for you to have confidence that you have understood it. This paper presents a technological approach for reducing the problem of believing a formal proof to the same psychological and philosophical issues as believing a conventional proof in a mathematics journal. The approach is not entirely successful philosophically as there seems to be a fundamental difference between machine-checked mathematics, which depends on empirical knowledge about the physical world, and informal mathematics, which needs no such knowledge (see section 3.2.2). In the rest of this introduction I outline the approach and mention related work. In following sections I discuss what we expect from a proof, add details to the approach, pointing out problems that arise, and concentrate on what I believe is the primary technical problem: expressiveness and feasibility for checking of formal systems and representations of mathematical notions. The problem is how to believe FLT when given only a putative proof formalised in a given logic. Assume it is a logic that you believe is consistent, and appropriate for FLT. The “thing” I give you is some computer files. There may be questions about the physical and abstract representations of the files (how to read them physically and how to parse them as a proof), and correctness of the hardware and software to do these things; ignore them until section 3.1.
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Whitty, Robin, et Robin Wilson. « Introducing Turing’s mathematics ». Dans The Turing Guide. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198747826.003.0048.

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Alan Turing’s mathematical interests were deep and wide-ranging. From the beginning of his career in Cambridge he was involved with probability theory, algebra (the theory of groups), mathematical logic, and number theory. Prime numbers and the celebrated Riemann hypothesis continued to preoccupy him until the end of his life. As a mathematician, and as a scientist generally, Turing was enthusiastically omnivorous. His collected mathematical works comprise thirteen papers, not all published during his lifetime, as well as the preface from his Cambridge Fellowship dissertation; these cover group theory, probability theory, number theory (analytic and elementary), and numerical analysis. This broad swathe of work is the focus of this chapter. But Turing did much else that was mathematical in nature, notably in the fields of logic, cryptanalysis, and biology, and that work is described in more detail elsewhere in this book. To be representative of Turing’s mathematical talents is a more realistic aim than to be encyclopaedic. Group theory and number theory were recurring preoccupations for Turing, even during wartime; they are represented in this chapter by his work on the word problem and the Riemann hypothesis, respectively. A third preoccupation was with methods of statistical analysis: Turing’s work in this area was integral to his wartime contribution to signals intelligence. I. J. Good, who worked with Turing at Bletchley Park, has provided an authoritative account of this work, updated in the Collected Works. By contrast, Turing’s proof of the central limit theorem from probability theory, which earned him his Cambridge Fellowship, is less well known: he quickly discovered that the theorem had already been demonstrated, the work was never published, and his interest in it was swiftly superseded by questions in mathematical logic. Nevertheless, this was Turing’s first substantial investigation, the first demonstration of his powers, and was certainly influential in his approach to codebreaking, so it makes a fitting first topic for this chapter. Turing’s single paper on numerical analysis, published in 1948, is not described in detail here. It concerned the potential for errors to propagate and accumulate during large-scale computations; as with everything that Turing wrote in relation to computation it was pioneering, forward-looking, and conceptually sound. There was also, incidentally, an appreciation in this paper of the need for statistical analysis, again harking back to Turing’s earliest work.
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Martin-Löf, Per. « An intuitionistic theory of types ». Dans Twenty Five Years of Constructive Type Theory. Oxford University Press, 1998. http://dx.doi.org/10.1093/oso/9780198501275.003.0010.

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The theory of types with which we shall be concerned is intended to be a full scale system for formalizing intuitionistic mathematics as developed, for example, in the book by Bishop 1967. The language of the theory is richer than the language of first order predicate logic. This makes it possible to strengthen the axioms for existence and disjunction. In the case of existence, the possibility of strengthening the usual elimination rule seems first to have been indicated by Howard 1969, whose proposed axioms are special cases of the existential elimination rule of the present theory. Furthermore, there is a reflection principle which links the generation of objects and types and plays somewhat the same role for the present theory as does the replacement axiom for Zermelo-Fraenkel set theory. An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis. Mathematical objects and their types. We shall think of mathematical objects or constructions. Every mathematical object is of a certain kind or type. Better, a mathematical object is always given together with its type, that is, it is not just an object, it is an object of a certain type.
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Barwise, Jon, et John Etchemendy. « Visual Information and Valid Reasoning ». Dans Logical Reasoning with Diagrams. Oxford University Press, 1996. http://dx.doi.org/10.1093/oso/9780195104271.003.0005.

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Psychologists have long been interested in the relationship between visualization and the mechanisms of human reasoning. Mathematicians have been aware of the value of diagrams and other visual tools both for teaching and as heuristics for mathematical discovery. As the chapters in this volume show, such tools are gaining even greater value, thanks in large part to the graphical potential of modern computers. But despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics. In particular, we are all taught to look askance at proofs that make crucial use of diagrams, graphs, or other nonlinguistic forms of representation, and we pass on this disdain to our students. In this chapter, we claim that visual forms of representation can be important, not just as heuristic and pedagogic tools, but as legitimate elements of mathematical proofs. As logicians, we recognize that this is a heretical claim, running counter to centuries of logical and mathematical tradition. This tradition finds its roots in the use of diagrams in geometry. The modern attitude is that diagrams are at best a heuristic in aid of finding a real, formal proof of a theorem of geometry, and at worst a breeding ground for fallacious inferences. For example, in a recent article, the logician Neil Tennant endorses this standard view: . . . [The diagram] is only an heuristic to prompt certain trains of inference; . . . it is dispensable as a proof-theoretic device; indeed, . . . it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array (Tennant [1984]). . . . It is this dogma that we want to challenge. We are by no means the first to question, directly or indirectly, the logocentricity of mathematics arid logic. The mathematicians Euler and Venn are well known for their development of diagrammatic tools for solving mathematical problems, and the logician C. S. Peirce developed an extensive diagrammatic calculus, which he intended as a general reasoning tool.
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Johnson, Steven D., et Jon Barwise. « Toward The Rigorous Use Of Diagrams In Reasoning About Hardware ». Dans Logical Reasoning with Diagrams. Oxford University Press, 1996. http://dx.doi.org/10.1093/oso/9780195104271.003.0015.

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The logician’s conventional notion of proof has grown increasingly anachronistic through the twentieth century as computing capabilities have advanced. classical proof theory provides a partial model of actual mathematical reasoning. when we move away from mathematics toward reasoning in engineering and computation, its limitations are even more pronounced. the standard idea of a formal system seems frozen in the information technology of frege’s time; it is decidedly quaint in the presence of today’s desk-top computer. contrary to formalists’ dogma, experience suggests that pictures, diagrams, charts, and graphs are important tools in human reasoning, not mere illustrations as traditional logic would have us believe. nor is the computer merely an optimized turing machine. the computer’s graphical capabilities have advanced to the point that diagrams can be manipulated in sophisticated ways, and it is time to exploit this capability in the analysis of reasoning, and in the design of new reasoning aids. in this chapter, we propose a new understanding of the role of various sorts of diagrams in the specification and design of computational hardware. this proposal stems from a larger project, initiated by barwise and etchemendy [1991a], the goals of which are to develop a mathematical basis from which to understand the substantive logical relationships between diagrams and sentences, and to develop a new generation of automated reasoning tools from that basis. microelectronic cad systems are among the supreme examples of visualized reasoning environments. their tools are highly oriented toward diagrams, are quite sophisticated, and are comparatively well integrated. these systems also integrate logical and physical design, providing a strong coherence between specification and implementation views. formalized reasoning meshes poorly with these working frameworks. although it provides needed rigor for today’s highly complex design challenges, its preoccupation with formulas at the expense of diagrams is simply too cumbersome. we should attempt to draw lessons from these advanced design environments, making the reasoning rigorous without subverting their character. this chapter is built around a simple design example, a synchronizing circuit. our purposes are, first, to illustrate heterogeneous use of pictoral “formalisms” in design, and second, to expose basic questions for the logical analysis that follows. we will develop a mathematical basis in which the example can be analyzed. these are admittedly modest beginnings, but we hope that they start to put to rest the idea that only formulas can be used in formal reasoning about hardware.
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