Search results for 'Variables (Mathematics' (try it on Scholar)

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  1.  13
    Arkady Plotnitsky (2006). A New Book of Numbers: On the Precise Definition of Quantum Variables and the Relationships Between Mathematics and Physics in Quantum Theory. [REVIEW] Foundations of Physics 36 (1):30-60.
    Following Asher Peres’s observation that, as in classical physics, in quantum theory, too, a given physical object considered “has a precise position and a precise momentum,” this article examines the question of the definition of quantum variables, and then the new type (as against classical physics) of relationships between mathematics and physics in quantum theory. The article argues that the possibility of the precise definition and determination of quantum variables depends on the particular nature of these relationships.
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  2. Karl Menger (1954). On Variables in Mathematics and in Natural Science. British Journal for the Philosophy of Science 5 (18):134-142.
    Attempting to answer the question "what is a variable?," menger discusses the following topics: (1) numerical variables and variables in the sense of the logicians, (2) variable quantities, (3) scientific variable quantities, (4) functions, And (5) variable quantities and functions in pure and applied analysis. (staff).
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  3. Alonzo Church (1957). Review: Karl Menger, On Variables in Mathematics and in Natural Science; Karl Menger, Variables, de Diverses Natures; Karl Menger, What Are Variables and Constants. [REVIEW] Journal of Symbolic Logic 22 (3):300-301.
     
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  4. A. P. Brogan (1917). Ichardson's and Landis's Fundamental Conceptions of Modern Mathematics - Variables and Quantities. [REVIEW] Journal of Philosophy 14 (2):49.
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  5. Gamal Cerda, Carlos Pérez, José I. Navarro, Manuel Aguilar, José A. Casas & Estíbaliz Aragón (2015). Explanatory Model of Emotional-Cognitive Variables in School Mathematics Performance: A Longitudinal Study in Primary School. Frontiers in Psychology 6.
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  6. Alonzo Church (1957). Menger Karl. On Variables in Mathematics and in Natural Science. The British Journal for the Philosophy of Science, Vol. 5 , Pp. 134–142.Menger Karl. Variables, de Diverses Natures. Bulletin des Sciences Mathématiques, Ser. 2 Vol. 78 , Pp. 229–234.Menger Karl. What Are Variables and Constants? Science, Vol. 123 , Pp. 547–548. [REVIEW] Journal of Symbolic Logic 22 (3):300-301.
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  7. R. O. Gandy (1968). Andrews P. B.. A Transfinite Type Theory with Type Variables. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam 1965, Xv + 143 Pp. [REVIEW] Journal of Symbolic Logic 33 (1):112-113.
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  8. Georg Kreisel (1970). Maslov S. Ú., Minc G. É., and Orévkov V. P.. Nérazréšimost′ Ν Konstruktivnom Isčislénii Prédikatov Nékotoryh Klassov Formul, Sodéržaščih Tol′Ko Odnoméstnyé Prédikatnyé Péréménnyé. Doklady Akadémii Nauk, Vol. 163 , Pp. 295–297.Maslov S. Ju., Minc G. E., and Orevkov V. P.. Unsolvability in the Constructive Predicate Calculus of Certain Classes of Formulas Containing Only Monadic Predicate Variables. Translation of the Preceding by Mendelson E.. Soviet Mathematics, Vol. 6 , Pp. 918–920. [REVIEW] Journal of Symbolic Logic 35 (1):143-144.
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  9. Raymond J. Nelson (1955). Slepian David. On the Number of Symmetry Types of Boolean Functions of N Variables. Canadian Journal of Mathematics, Vol 5 , Pp. 135–193. Reprinted in the Bell Telephone System Technical Publications, Monograph 2154. [REVIEW] Journal of Symbolic Logic 20 (1):70.
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  10.  1
    Lorena Segura & Juan Matías Sepulcre (2016). A Rational Belief: The Method of Discovery in the Complex Variable. Foundations of Science 21 (1):189-194.
    The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within (...)
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  11.  39
    Robert Goldblatt (2011). Quantifiers, Propositions, and Identity: Admissible Semantics for Quantified Modal and Substructural Logics. Cambridge University Press.
    Machine generated contents note: Introduction and overview; 1. Logics with actualist quantifiers; 2. The Barcan formulas; 3. The existence predicate; 4. Propositional functions and predicate substitution; 5. Identity; 6. Cover semantics for relevant logic; References; Index.
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  12. Michael Böttner & Wolf Thümmel (eds.) (2000). Variable-Free Semantics. Secolo.
     
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  13.  38
    John Corcoran (2014). Corcoran Reviews Boute’s 2013 Paper “How to Calculate Proofs”. MATHEMATICAL REVIEWS 14:444-555.
    Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that term’s (...)
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  14.  67
    Richard Pettigrew (2008). Platonism and Aristotelianism in Mathematics. Philosophia Mathematica 16 (3):310-332.
    Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that (...)
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  15.  69
    Vladimir Kuznetsov (2009). Variables of Scientific Concept Modeling and Their Formalization. In В.И Маркин (ed.), Philosophy of mathematics: current problems. Proceedings of the second international conference (Философия математики: актуальные проблемы. Тезисы второй международной конференции). Макс Пресс 268-270.
    There are no universally adopted answers to the natural questions about scientific concepts: What are they? What is their structure? What are their functions? How many kinds of them are there? Do they change? Ironically, most if not all scientific monographs or articles mention concepts, but the scientific studies of scientific concepts are rare in occurrence. It is well known that the necessary stage of any scientific study is constructing the model of objects in question. Many years logical modeling was (...)
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  16.  26
    Murdoch J. Gabbay (2011). Foundations of Nominal Techniques: Logic and Semantics of Variables in Abstract Syntax. Bulletin of Symbolic Logic 17 (2):161-229.
    We are used to the idea that computers operate on numbers, yet another kind of data is equally important: the syntax of formal languages, with variables, binding, and alpha-equivalence. The original application of nominal techniques, and the one with greatest prominence in this paper, is to reasoning on formal syntax with variables and binding. Variables can be modelled in many ways: for instance as numbers (since we usually take countably many of them); as links (since they may (...)
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  17.  14
    Gregory Landini (1996). Logic in Russell's Principles of Mathematics. Notre Dame Journal of Formal Logic 37 (4):554-584.
    Unaware of Frege's 1879 Begriffsschrift, Russell's 1903 The Principles of Mathematics set out a calculus for logic whose foundation was the doctrine that any such calculus must adopt only one style of variables–entity (individual) variables. The idea was that logic is a universal and all-encompassing science, applying alike to whatever there is–propositions, universals, classes, concrete particulars. Unfortunately, Russell's early calculus has appeared archaic if not completely obscure. This paper is an attempt to recover the formal system, showing its (...)
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  18.  5
    Ronald Brown (2009). Memory Evolutive Systems. Axiomathes 19 (3):271-280.
    This is a review of the book ‘Memory Evolutive Systems; Hierarchy, Emergence, Cognition’, by A. Ehresmann and J.P. Vanbremeersch. I welcome the use of category theory and the notion of colimit as a way of describing how complex hierarchical systems can be organised, and the notion of categories varying with time to give a notion of an evolving system. In this review I also point out the relation of the notion of colimit to ideas of communication; the necessity of communications (...)
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  19.  10
    Roy Wagner (2009). Mathematical Variables as Indigenous Concepts. International Studies in the Philosophy of Science 23 (1):1-18.
    This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post-structuralist train of thought going from Mauss and L vi-Strauss to Baudrillard and Derrida. We import these authors' semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of (...)
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  20. I. P. Schagen (1990). Analysis of the Effects of School Variables Using Multilevel Models. Educational Studies 16 (1):61-73.
    Multilevel models allow data to be analysed which are hierarchical in nature; in particular, data which have been collected on pupils grouped into schools. Some of the associated variables may be measured at the pupil level, and others at the school level. The use of multilevel models produces estimates of variances between schools and pupils, as well as the effects of background variables in reducing or explaining these variances. One data set which has been analysed relates to the (...)
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  21.  8
    Edward H. Landis & Robert P. Richardson (1915). Numbers, Variables and Mr. Russell's Philosophy. The Monist 25 (3):321-364.
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  22. Bryan Pickel & Brian Rabern (forthcoming). The Antinomy of the Variable: A Tarskian Resolution. Journal of Philosophy.
    Kit Fine has reawakened a puzzle about variables with a long history in analytic philosophy, labeling it “the antinomy of the variable”. Fine suggests that the antinomy demands a reconceptualization of the role of variables in mathematics, natural language semantics, and first-order logic. The difficulty arises because: (i) the variables ‘x’ and ‘y’ cannot be synonymous, since they make different contributions when they jointly occur within a sentence, but (ii) there is a strong temptation to say that (...)
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  23. Simon Kochen & E. P. Specker (1967). The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics 17:59--87.
  24. Han Geurdes, The Construction of Transfinite Equivalence Algorithms.
    Context: Consistency of mathematical constructions in numerical analysis and the application of computerized proofs in the light of the occurrence of numerical chaos in simple systems. Purpose: To show that a computer in general and a numerical analysis in particular can add its own peculiarities to the subject under study. Hence the need of thorough theoretical studies on chaos in numerical simulation. Hence, a questioning of what e.g. a numerical disproof of a theorem in physics or a prediction in numerical (...)
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  25.  79
    Jody Azzouni (1997). Applied Mathematics, Existential Commitment and the Quine-Putnam Indispensability Thesis. Philosophia Mathematica 5 (3):193-209.
    The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via (...), rather than by substitution-instances) need not imply any ontological commitments. (shrink)
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  26.  32
    Philipp Mitteroecker & Simon M. Huttegger (2009). The Concept of Morphospaces in Evolutionary and Developmental Biology: Mathematics and Metaphors. Biological Theory 4 (1):54-67.
    Formal spaces have become commonplace conceptual and computational tools in a large array of scientific disciplines, including both the natural and the social sciences. Morphological spaces are spaces describing and relating organismal phenotypes. They play a central role in morphometrics, the statistical description of biological forms, but also underlie the notion of adaptive landscapes that drives many theoretical considerations in evolutionary biology. We briefly review the topological and geometrical properties of the most common morphospaces in the biological literature. In contemporary (...)
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  27.  6
    Harald Martens & Achim Kohler (2009). Mathematics and Measurements for High-Throughput Quantitative Biology. Biological Theory 4 (1):29-43.
    Bioscientists generate far more data than their minds can handle, and this trend is likely to continue. With the aid of a small set of versatile tools for mathematical modeling and statistical assessment, bioscientists can explore their real-world systems without experiencing data overflow. This article outlines an approach for combining modern high-throughput, low-cost, but non-selective biospectroscopy measurements with soft, multivariate biochemometrics data modeling to overview complex systems, test hypotheses, and making new discoveries. From preliminary, broad hypotheses and goals, many relevant (...)
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  28.  40
    William W. Tait (2006). Proof-Theoretic Semantics for Classical Mathematics. Synthese 148 (3):603 - 622.
    We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is (...)
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  29.  9
    Kajsa Bråting (2012). Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-Nineteenth Century. Foundations of Science 17 (4):301-320.
    In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993 ) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Björling’s general view of real- and complexvalued functions. We argue that Björling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is (...)
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  30.  4
    Karl Egil Aubert (1982). The Role of Mathematics in the Exploration of Reality. Inquiry 25 (3):353 – 359.
    In his well?known paper from 1954, Herbert A. Simon sets out to demonstrate that it is possible, in principle, to make public predictions within the social sciences that will be confirmed by the events. However, Simon's proof by means of the Brouwer fixed?point theorem not only rests on an illegitimate use of continuous variables, it is also founded on the questionable assumption that facts ? even on the level of possibilities ? can be established by purely mathematical means. The (...)
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  31.  36
    Gregory Landini (1998). Russell's Hidden Substitutional Theory. Oxford University Press.
    This book explores an important central thread that unifies Russell's thoughts on logic in two works previously considered at odds with each other, the Principles of Mathematics and the later Principia Mathematica. This thread is Russell's doctrine that logic is an absolutely general science and that any calculus for it must embrace wholly unrestricted variables. The heart of Landini's book is a careful analysis of Russell's largely unpublished "substitutional" theory. On Landini's showing, the substitutional theory reveals the unity of (...)
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  32. R. Knox & T. Smibert (eds.) (2013). A Treatise on Man and the Development of His Faculties. Cambridge University Press.
    The Belgian polymath Lambert Adolphe Jacques Quetelet pioneered social statistics. Applying his training in mathematics to the physical and psychological dimensions of individuals, he identified the 'average man' as characterised by the mean values of measured variables that follow a normal distribution. He believed that comparing the features of individuals against this average would allow scientists to better explore the processes that determine normal and abnormal qualities. Quetelet's methods influenced many, among them Florence Nightingale, and his simple measure for (...)
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  33.  12
    Henri Poincaré (1952). Science and Method. New York]Dover Publications.
    " Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to (...)
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  34. B. Brogaard (2006). The 'Gray's Elegy' Argument, and the Prospects for the Theory of Denoting Concepts. Synthese 152 (1):47 - 79.
    Russell’s new theory of denoting phrases introduced in “On Denoting” in Mind 1905 is now a paradigm of analytic philosophy. The main argument for Russell’s new theory is the so-called ‘Gray’s Elegy’ argument, which purports to show that the theory of denoting concepts (analogous to Frege’s theory of senses) promoted by Russell in the 1903 Principles of Mathematics is incoherent. The ‘Gray’s Elegy’ argument rests on the premise that if a denoting concept occurs in a proposition, then the proposition is (...)
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  35. Han Geurdes (2010). CHSH and Local Causlaity. Adv Studies Theoretical Physics 4 (20):945.
    Mathematics equivalent to Bell's derivation of the inequalities, also allows a local hidden variables explanation for the correlation between distant measurements.
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  36.  19
    Peter Gibbins (1987). Particles and Paradoxes: The Limits of Quantum Logic. Cambridge University Press.
    Quantum theory is our deepest theory of the nature of matter. It is a theory that, notoriously, produces results which challenge the laws of classical logic and suggests that the physical world is illogical. This book gives a critical review of work on the foundations of quantum mechanics at a level accessible to non-experts. Assuming his readers have some background in mathematics and physics, Peter Gibbins focuses on the questions of whether the results of quantum theory require us to abandon (...)
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  37.  2
    Jelena Teodorović (2012). Student Background Factors Influencing Student Achievement in Serbia. Educational Studies 38 (1):89-110.
    This paper describes student‐level findings of the first large‐scale comprehensive school effectiveness study of the primary education in Serbia. Twenty‐five student‐level variables were examined in a three‐level HLM model using a study sample of almost 5000 students, over 250 classrooms and over 100 schools. Differences between the students were in large part responsible for differences in achievement scores in mathematics and Serbian language. Parental education, Roma minority status, developmental or family problems, gender, student motivation, parental involvement in student work (...)
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  38.  43
    Henri Poincaré (1914). Science and Method. Dover Publications.
    " Vivid . . . immense clarity . . . the product of a brilliant and extremely forceful intellect." — Journal of the Royal Naval Scientific Service "Still a sheer joy to read." — Mathematical Gazette "Should be read by any student, teacher or researcher in mathematics." — Mathematics Teacher The originator of algebraic topology and of the theory of analytic functions of several complex variables, Henri Poincare (1854–1912) excelled at explaining the complexities of scientific and mathematical ideas to (...)
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  39.  50
    Mathieu Marion (1995). Wittgenstein and Finitism. Synthese 105 (2):141 - 176.
    In this paper, elementary but hitherto overlooked connections are established between Wittgenstein's remarks on mathematics, written during his transitional period, and free-variable finitism. After giving a brief description of theTractatus Logico-Philosophicus on quantifiers and generality, I present in the first section Wittgenstein's rejection of quantification theory and his account of general arithmetical propositions, to use modern jargon, as claims (as opposed to statements). As in Skolem's primitive recursive arithmetic and Goodstein's equational calculus, Wittgenstein represented generality by the use of free (...)
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  40.  19
    Michael Stöltzner, The Dynamics of Thought Experiments - Comment to Atkinson.
    Commenting on Atkinson 's paper I argue that leading to a successful real experiment is not the only scale on which a thought experiment's value is judged. Even the path from the original EPR thought experiment to Aspect's verification of the Bell inequalities was long-winded and involved considerable input from the sides of technology and mathematics. Von Neumann's construction of hidden variables was, moreover, a genuinely mathematical thought experiment that was successfully criticized by Bell. Such thought experiments are also (...)
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  41.  57
    James H. Davenport & Michael Kohlhase, Unifying Math Ontologies: A Tale of Two Standards.
    One of the fundamental and seemingly simple aims of mathematical knowledge management (MKM) is to develop and standardize formats that allow to “represent the meaning of the objects of mathematics”. The open formats OpenMath and MathML address this, but differ subtly in syntax, rigor, and structural viewpoints (notably over calculus). To avoid fragmentation and smooth out interoperability obstacles, effort is under way to align them into a joint format OpenMath/MathML 3. We illustrate the issues that come up in such an (...)
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  42.  8
    Ewa Graczynska & Francis Pastijn (1981). Proofs of Regular Identities. Bulletin of the Section of Logic 10 (1):35-37.
    This is an abstract of the paper to be submitted to Houston Journal of Mathematics. Our nomenclature and notation will be basically those of [3]. We shall consider algebras of type : T ! N, where T is a nonempty set, and N the set of all positive integers. By V we denote the set of all variables occurring in a polynomial symbol p. An identity p = q is called strongly non-regular if it is of the form p (...)
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  43.  3
    Farideh Salili & Kit‐Tai Hau (1994). The Effect of Teachers’ Evaluative Feedback on Chinese Students’ Perception of Ability: A Cultural and Situational Analysis. Educational Studies 20 (2):223-236.
    The research explored the effect of teachers’ evaluative feedback on students’ perception of ability. The subjects were 758 Chinese students from elementary schools, high schools, and a university in Hong Kong. Subjects rated ability and effort of two hypothetical students who achieved identical results in a mathematics test but received different feedback from the teacher. They also rated the teacher's perception of their own and their friends’ ability and effort in a similar situation after a real mathematics test. The results (...)
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  44.  16
    Dimiter Skordev (1984). On a Modal-Type Language for the Predicate Calculus. Bulletin of the Section of Logic 13 (3):111-116.
    In order to avoid the use of individual variables in predicate calculus, several authors proposed language whose expressions can be interpreted, in general, as denotations of predicates . The present author also proposed a language of this kind [5]. The absence of individual variables makes all these languages rather different from the traditional language of predicate calculus and from the usual language of mathematics. The translation procedures from the ordinary predicate languages into the predicate languages without individual (...) and vice versa have a technical character. It is desirable to make possible carrying out such procedures by performing some transformations in a suitable new language which comprises both a sublanguage similar to the ordinary predicate languages and another one not using individual variables. We shall propose now a language of the desired kind. The expressions of this language could be interpreted, in general, as denoting predicates which depend on parameters on a given object domain – this predicate has one argument and depends on one parameter). The proposed language will have the modal features noted in [3] and [5]. It is worthy of mention that the language proposed at the end of [4] although containing expressions interpreted as predicates and expressions with parameters does not allow simultaneous availability of argument places and parameters. (shrink)
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  45.  61
    Rob Clifton (2004). Quantum Entanglements: Selected Papers. Oxford University Press.
    Rob Clifton was one of the most brilliant and productive researchers in the foundations and philosophy of quantum theory, who died tragically at the age of 38. Jeremy Butterfield and Hans Halvorson collect fourteen of his finest papers here, drawn from the latter part of his career (1995-2002), all of which combine exciting philosophical discussion with rigorous mathematical results. Many of these papers break wholly new ground, either conceptually or technically. Others resolve a vague controversy intoa precise technical problem, which (...)
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  46.  5
    Paolo Mancosu (1991). Generalizing Classical and Effective Model Theory in Theories of Operations and Classes. Annals of Pure and Applied Logic 52 (3):249-308.
    Mancosu, P., Generalizing classical and effective model theory in theories of operations and classes, Annas of Pure and Applied Logic 52 249-308 . In this paper I propose a family of theories of operations and classes with the aim of developing abstract versions of model-theoretic results. The systems are closely related to those introduced and already used by Feferman for developing his program of ‘explicit mathematics’. The theories in question are two-sorted, with one kind of variable for individuals and the (...)
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  47.  14
    Martin H. Krieger (1991). Theorems as Meaningful Cultural Artifacts: Making the World Additive. Synthese 88 (2):135 - 154.
    Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such as accelerations, (...)
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  48.  29
    S. Shapiro (2011). The Company Kept by Cut Abstraction (and its Relatives). Philosophia Mathematica 19 (2):107-138.
    This article concerns the ongoing neo-logicist program in the philosophy of mathematics. The enterprise began life, in something close to its present form, with Crispin Wright’s seminal [1983]. It was bolstered when Bob Hale [1987] joined the fray on Wright’s behalf and it continues through many extensions, objections, and replies to objections . The overall plan is to develop branches of established mathematics using abstraction principles in the form: Formula where a and b are variables of a given type (...)
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  49.  9
    Andrea Formisano, Eugenio G. Omodeo & Alberto Policriti (2005). The Axiom of Elementary Sets on the Edge of Peircean Expressibility. Journal of Symbolic Logic 70 (3):953 - 968.
    Being able to state the principles which lie deepest in the foundations of mathematics by sentences in three variables is crucially important for a satisfactory equational rendering of set theories along the lines proposed by Alfred Tarski and Steven Givant in their monograph of 1987. The main achievement of this paper is the proof that the 'kernel' set theory whose postulates are extensionality. (E), and single-element adjunction and removal. (W) and (L), cannot be axiomatized by means of three-variable sentences. (...)
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  50.  1
    G. Mints (1993). A Normal Form for Logical Derivations Implying One for Arithmetic Derivations. Annals of Pure and Applied Logic 62 (1):65-79.
    We describe a short model-theoretic proof of an extended normal form theorem for derivations in predicate logic which implies in PRA a normal form theorem for the arithmetic derivations . Consider the Gentzen-type formulation of predicate logic with invertible rules. A derivation with proper variables is one where a variable b can occur in the premiss of an inference L but not below this premiss only in the case when L is () or () and b is its eigenvariable. (...)
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