Search results for 'Mathematical notation' (try it on Scholar)

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  1.  13
    W. E. Underwood (1980). Symbolic Configurations and Two-Dimensional Mathematical Notation. Semiotics:523-532.
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  2.  13
    Yanjie Zhao (1997). What is Mathematical Notation. Semiotics:257-273.
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  3.  13
    W. E. Underwood (1980). Symbolic Configurations and Two-Dimensional Mathematical Notation. Semiotics:523-532.
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  4.  12
    Yanjie Zhao (1997). What is Mathematical Notation. Semiotics:257-273.
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  5.  1
    W. Douglas Maurer (1999). The Influence of the Computer Upon Mathematical Notation. Semiotica 125 (1-3):165-168.
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  6.  2
    Toshiyasu Arai (2002). Buchholz Wilfried. Notation Systems for Infinitary Derivations. Archive for Mathematical Logic, Vol. 30 No. 5–6 (1991), Pp. 277–296. Buchholz Wilfried. 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 Gottlob Georg, Leitsch Alexander, and Mundici Daniele, Lecture Notes in Computer Science, Vol. 1289, Springer, Berlin, Heidelberg, New York, Etc., 1997 ... [REVIEW] Bulletin of Symbolic Logic 8 (3):437-439.
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  7.  8
    Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.
  8. Robert Feys (1969). Dictionary of Symbols of Mathematical Logic. Amsterdam, North-Holland Pub. Co..
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  9. William James Meyers (1975). A Mathematical Theory of Parenthesis, Free Notations. Państwowe Wydawn. Naukowe.
     
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  10. P. B. Andrews (2002). An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Kluwer Academic Publishers.
    This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)
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  11. Gregory Landini (2012). Frege's Notations: What They Are and How They Mean. Palgrave Macmillan.
     
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  12.  1
    Susanna Saracco (forthcoming). Theoretical Childhood and Adulthood: Plato’s Account of Human Intellectual Development. Philosophia:1-19.
    The Platonic description of the cognitive development of the human being is a crucial part of his philosophy. This account emphasizes not only the existence of phases of rational growth but also the need that the cognitive progress of the individuals is investigated further. I will reconstruct what rational growth is for Plato in light of the deliberate choice of the philosopher to leave incomplete his schematization of human intellectual development. I will argue that this is a means chosen by (...)
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  13. Michael Kohlhase, Adaptation of Notations in Living Mathematical Documents.
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  14.  59
    Michael Kohlhase, Notations for Living Mathematical Documents.
    Notations are central for understanding mathematical discourse. Readers would like to read notations that transport the meaning well and prefer notations that are familiar to them. Therefore, authors optimize the choice of notations with respect to these two criteria, while at the same time trying to remain consistent over the document and their own prior publications. In print media where notations are fixed at publication time, this is an over-constrained problem. In living documents notations can be adapted at reading (...)
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  15. Jan Von Plato (2007). In the Shadows of the Löwenheim-Skolem Theorem: Early Combinatorial Analyses of Mathematical Proofs. Bulletin of Symbolic Logic 13 (2):189-225.
    The Löwenheim-Skolem theorem was published in Skolem's long paper of 1920, with the first section dedicated to the theorem. The second section of the paper contains a proof-theoretical analysis of derivations in lattice theory. The main result, otherwise believed to have been established in the late 1980s, was a polynomial-time decision algorithm for these derivations. Skolem did not develop any notation for the representation of derivations, which makes the proofs of his results hard to follow. Such a formal (...) is given here by which these proofs become transparent. A third section of Skolem's paper gives an analysis for derivations in plane projective geometry. To clear a gap in Skolem's result, a new conservativity property is shown for projective geometry, to the effect that a proper use of the axiom that gives the uniqueness of connecting lines and intersection points requires a conclusion with proper cases (logically, a disjunction in a positive part) to be proved. The forgotten parts of Skolem's first paper on the Löwenheim-Skolem theorem are the perhaps earliest combinatorial analyses of formal mathematical proofs, and at least the earliest analyses with profound results. (shrink)
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  16.  38
    Madeline M. Muntersbjorn (1999). Naturalism, Notation, and the Metaphysics of Mathematics. Philosophia Mathematica 7 (2):178-199.
    The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without (...)
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  17.  5
    I. A. Akchurin, M. F. Vedenov & Iu V. Sachkov (1966). Methodological Problems of Mathematical Modeling in Natural Science. Russian Studies in Philosophy 5 (2):23-34.
    The constantly accelerating progress of contemporary natural science is indissolubly associated with the development and use of mathematics and with the processes of mathematical modeling of the phenomena of nature. The essence of this diverse and highly fertile interaction of mathematics and natural science and the dialectics of this interaction can only be disclosed through analysis of the nature of theoretical notions in general. Today, above all in the ranks of materialistically minded researchers, it is generally accepted that (...)
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  18. Stefano Mazzanti (2013). Iteration on Notation and Unary Functions. Mathematical Logic Quarterly 59 (6):415-434.
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  19.  5
    John Corcoran & David Levin (1972). Conceptual Notation and Related Articles. Monograph Collection (Matt - Pseudo).
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  20.  2
    Noriya Kadota (1993). On Wainer's Notation for a Minimal Subrecursive Inaccessible Ordinal. Mathematical Logic Quarterly 39 (1):217-227.
    We show the following results on Wainer's notation for a minimal subrecursive inaccessible ordinal τ: First, we give a constructive proof of the collapsing theorem. Secondly, we prove that the slow-growing hierarchy and the fast-growing hierarchy up to τ have elementary properties on increase and domination, which completes Wainer's proof that τ is a minimal subrecursive inaccessible. Our results are obtained by showing a strong normalization theorem for the term structure of the notation. MSC: 03D20, 03F15.
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  21. Helmut Pfeiffer & H. Pfeiffer (1992). A Notation System for Ordinal Using Ψ‐Functions on Inaccessible Mahlo Numbers. Mathematical Logic Quarterly 38 (1):431-456.
    G. Jäger gave in Arch. Math. Logik Grundlagenforsch. 24 , 49-62, a recursive notation system on a basis of a hierarchy Iαß of α-inaccessible regular ordinals using collapsing functions following W. Buchholz in Ann. Pure Appl. Logic 32 , 195-207. Jäger's system stops, when ordinals α with Iα0 = α enter. This border is now overcome by introducing additional a hierarchy Jαß of weakly inaccessible Mahlo numbers, which is defined similarly to the Jäger hierarchy. An ordinal μ is called (...)
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  22.  12
    Wilfried Buchholz (1991). Notation Systems for Infinitary Derivations. Archive for Mathematical Logic 30 (5-6):277-296.
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  23.  5
    Gerhard Jäger (1984). Ρ-Inaccessible Ordinals, Collapsing Functions and a Recursive Notation System. Archive for Mathematical Logic 24 (1):49-62.
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  24.  4
    Douglas S. Bridges (1978). A Note on Morse's Lambda‐Notation in Set Theory. Mathematical Logic Quarterly 24 (8):113-114.
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  25. Zdzisław Pawlar (1963). Organization of Addressless Computers Working in Parenthesis Notation. Mathematical Logic Quarterly 9 (16‐17):243-249.
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  26.  1
    S. Dennis, M. S. Humphreys & J. Wiles (1996). Mathematical Constraints on a Theory of Human Memory - Response. Behavioral and Brain Sciences 19 (3):559-560.
    Colonius suggests that, in using standard set theory as the language in which to express our computational-level theory of human memory, we would need to violate the axiom of foundation in order to express meaningful memory bindings in which a context is identical to an item in the list. We circumvent Colonius's objection by allowing that a list item may serve as a label for a context without being identical to that context. This debate serves to highlight the value of (...)
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  27.  12
    Jon Williamson (2010). In Defence of Objective Bayesianism. OUP Oxford.
    Objective Bayesianism is a methodological theory that is currently applied in statistics, philosophy, artificial intelligence, physics and other sciences. This book develops the formal and philosophical foundations of the theory, at a level accessible to a graduate student with some familiarity with mathematical notation.
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  28. Fulya Horozal, Florian Rabe & Michael Kohlhase, Extending OpenMath with Sequences.
    Sequences play a great role in mathematical communication. In mathematical notation, we use sequence ellipsis (. . . ) to denote "obvious" sequences like 1, 2, . . . , 7, and in conceptualizations sequence constructors like (i 2+1) i∈N. Furthermore, sequences have a prominent role as argument sequences of flexary functions. While the former cases can adequately be represented and reasoned about as domain objects in Open- Math and MathML, argument sequences are at the language level, (...)
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  29.  48
    Igor L. Aleksander & B. Dunmall (2003). Axioms and Tests for the Presence of Minimal Consciousness in Agents I: Preamble. Journal of Consciousness Studies 10 (4):7-18.
    This paper relates to a formal statement of the mechanisms that are thought minimally necessary to underpin consciousness. This is expressed in the form of axioms. We deem this to be useful if there is ever to be clarity in answering questions about whether this or the other organism is or is not conscious. As usual, axioms are ways of making formal statements of intuitive beliefs and looking, again formally, at the consequences of such beliefs. The use of this style (...)
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  30. Thomas J. Scheff (2005). The Structure of Context: Deciphering "Frame Analysis". Sociological Theory 23 (4):368-385.
    This article proposes that Goffman's "Frame Analysis" can be interpreted as a step toward unpacking the idea of context. His analysis implies a recursive model involving frames within frames. The key problem is that neither Goffman nor anyone else has clearly defined what is meant by a frame. I propose that it can be represented by a word, phrase, or proposition. A subjective context can be represented as an assembly of these items, joined together by operators such as and, since, (...)
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  31.  9
    Chen Chung Chang (1966). Continuous Model Theory. Princeton, Princeton University Press.
    CONTINUOUS MODEL THEORY CHAPTER I TOPOLOGICAL PRELIMINARIES. Notation Throughout the monograph our mathematical notation does not differ drastically from ...
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  32.  30
    Pavel Tichy (1986). Constructions. Philosophy of Science 53 (4):514-534.
    The paper deals with the semantics of mathematical notation. In arithmetic, for example, the syntactic shape of a formula represents a particular way of specifying, arriving at, or constructing an arithmetical object (that is, a number, a function, or a truth value). A general definition of this sense of "construction" is proposed and compared with related notions, in particular with Frege's concept of "function" and Carnap's concept of "intensional isomorphism." It is argued that constructions constitute the proper subject (...)
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  33. Hans Moravec (1995). Roger Penrose's Gravitonic Brains: A Review of Shadows of the Mind by Roger Penrose. [REVIEW] Psyche 2 (1).
    Summarizing a surrounding 200 pages, pages 179 to 190 of Shadows of the Mind contain a future dialog between a human identified as "Albert Imperator" and an advanced robot, the "Mathematically Justified Cybersystem", allegedly Albert's creation. The two have been discussing a Gödel sentence for an algorithm by which a robot society named SMIRC certifies mathematical proofs. The sentence, referred to in mathematical notation as Omega(Q*), is to be precisely constructed from on a definition of SMIRC's algorithm. (...)
     
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  34.  19
    Front Page, Earliest Uses of Symbols of Set Theory and Logic.
    The study of logic goes back more than two thousand years and in that time many symbols and diagrams have been devised. Around 300 BC Aristotle introduced letters as term-variables, a "new and epoch-making device in logical technique." (W. & M. Kneale The Development of Logic (1962, p. 61). The modern era of mathematical notation in logic began with George Boole (1815- 1864), although none of his notation survives. Set theory came into being in the late 19th (...)
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  35.  19
    Thomas J. Scheff (2007). A Concept of Social Integration. Philosophical Psychology 20 (5):579 – 593.
    Clear definitions of alienation and solidarity are needed as a step toward an explicit theory of social integration. The idea of alienation has played a key role in the development of sociology, but it's meaning has never been clear. Both theories and empirical studies confound relational-dispositional, cognitive-emotional and/or interpersonal-societal components. This essay proposes definitions that follow from the work of Erving Goffman and others. Goffman's idea of "co-presence" implies a model of solidarity as mutual awareness to the point of merging (...)
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  36.  12
    B. P. Larvor (2007). Michel Serfati. La Révolution Symbolique: La Constitution de l'Ecriture Symbolique Mathématique. Preface by Jacques Bouverasse. Paris: Éditions Petra, 2005. Pp. Ix + 427. ISBN 2-84743-006-7. [REVIEW] Philosophia Mathematica 15 (1):122-126.
    It is difficult to imagine mathematics without its symbolic language. It is especially difficult to imagine doing mathematics without using mathematical notation. Nevertheless, that is how mathematics was done for most of human history. It was only at the end of the sixteenth century that mathematicians began to develop systems of mathematical symbols . It is startling to consider how rapidly mathematical notation evolved. Viète is usually taken to have initiated this development with his Isagoge (...)
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  37. Thomas Mormann (2005). Mathematical Metaphors in Natorp’s Neo-Kantian Epistemology and Philosophy of Science. In Falk Seeger, Johannes Lenard & Michael H. G. Hoffmann (eds.), Activity and Sign. Grounding Mathematical Education. Springer
    A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...)
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  38.  99
    Imre Lakatos (ed.) (1976). Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press.
    Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or (...)
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  39.  6
    Walter A. Carnielli, Itala M. L. D'ottaviano & Brazilian Conference on Mathematical Logic (1999). Advances in Contemporary Logic and Computer Science Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. [REVIEW] Monograph Collection (Matt - Pseudo).
    This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society (co-sponsored by the Centre for Logic, Epistemology and the History of Science, State University of Campinas, Sao Paulo) in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from (...)
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  40.  82
    Holly Andersen (forthcoming). Complements, Not Competitors: Causal and Mathematical Explanations. British Journal for the Philosophy of Science.
    A finer-grained delineation of a given explanandum reveals a nexus of closely related causal and non- causal explanations, complementing one another in ways that yield further explanatory traction on the phenomenon in question. By taking a narrower construal of what counts as a causal explanation, a new class of distinctively mathematical explanations pops into focus; Lange’s characterization of distinctively mathematical explanations can be extended to cover these. This new class of distinctively mathematical explanations is illustrated with the (...)
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  41. Helen De Cruz & Johan De Smedt (2013). Mathematical Symbols as Epistemic Actions. Synthese 190 (1):3-19.
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols (...)
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  42.  86
    Ingo Brigandt (2013). Systems Biology and the Integration of Mechanistic Explanation and Mathematical Explanation. Studies in History and Philosophy of Biological and Biomedical Sciences 44 (4):477-492.
    The paper discusses how systems biology is working toward complex accounts that integrate explanation in terms of mechanisms and explanation by mathematical models—which some philosophers have viewed as rival models of explanation. Systems biology is an integrative approach, and it strongly relies on mathematical modeling. Philosophical accounts of mechanisms capture integrative in the sense of multilevel and multifield explanations, yet accounts of mechanistic explanation (as the analysis of a whole in terms of its structural parts and (...)
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  43.  12
    Herbert B. Enderton (1972). A Mathematical Introduction to Logic. New York,Academic Press.
    A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, (...)
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  44.  72
    W. V. Quine (1951). Mathematical Logic. Cambridge, Harvard University Press.
    INTRODUCTION MATHEMATICAL logic differs from the traditional formal logic so markedly in method, and so far surpasses it in power and subtlety, ...
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  45.  42
    Charles S. Chihara (1990). Constructibility and Mathematical Existence. Oxford University Press.
    This book is concerned with `the problem of existence in mathematics'. It develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. It explores the philosophical implications of such an approach through an examination of the writings of Field, Burgess, Maddy, Kitcher, and others.
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  46.  61
    Brian Rabern (2016). The History of the Use of ⟦.⟧-Notation in Natural Language Semantics. Semantics and Pragmatics 9 (12).
    In contemporary natural languages semantics one will often see the use of special brackets to enclose a linguistic expression, e.g. ⟦carrot⟧. These brackets---so-called denotation brackets or semantic evaluation brackets---stand for a function that maps a linguistic expression to its "denotation" or semantic value (perhaps relative to a model or other parameters). Even though this notation has been used in one form or another since the early development of natural language semantics in the 1960s and 1970s, Montague himself didn't make (...)
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  47.  19
    J. B. Paris (1994). The Uncertain Reasoner's Companion: A Mathematical Perspective. Cambridge University Press.
    Reasoning under uncertainty, that is, making judgements with only partial knowledge, is a major theme in artificial intelligence. Professor Paris provides here an introduction to the mathematical foundations of the subject. It is suited for readers with some knowledge of undergraduate mathematics but is otherwise self-contained, collecting together the key results on the subject, and formalising within a unified framework the main contemporary approaches and assumptions. The author has concentrated on giving clear mathematical formulations, analyses, justifications and consequences (...)
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  48.  41
    Haskell B. Curry (1963/1977). Foundations of Mathematical Logic. Dover Publications.
    Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
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  49.  48
    Seungbae Park (2016). Against Mathematical Convenientism. Axiomathes 26 (2):115-122.
    Indispensablists argue that when our belief system conflicts with our experiences, we can negate a mathematical belief but we do not because if we do, we would have to make an excessive revision of our belief system. Thus, we retain a mathematical belief not because we have good evidence for it but because it is convenient to do so. I call this view ‘mathematical convenientism.’ I argue that mathematical convenientism commits the consequential (...)
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  50.  21
    Heinz-Dieter Ebbinghaus (1996). Mathematical Logic. Springer.
    This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most (...)
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