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# Areas of Mathematics, Misc

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1. V. Michele Abrusci (1989). Some Uses of Dilators in Combinatorial Problems. Archive for Mathematical Logic 29 (2):85-109.

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2. Klaus Aehlig (2005). Induction and Inductive Definitions in Fragments of Second Order Arithmetic. Journal of Symbolic Logic 70 (4):1087 - 1107.
A fragment with the same provably recursive functions as n iterated inductive definitions is obtained by restricting second order arithmetic in the following way. The underlying language allows only up to n + 1 nested second order quantifications and those are in such a way, that no second order variable occurs free in the scope of another second order quantifier. The amount of induction on arithmetical formulae only affects the arithmetical consequences of these theories, whereas adding induction for arbitrary formulae (...)

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3. R. B. J. T. Allenby (1997). Numbers and Proofs. Copublished in North, South, and Central America by John Wiley & Sons Inc..
'Numbers and Proofs' presents a gentle introduction to the notion of proof to give the reader an understanding of how to decipher others' proofs as well as construct their own. Useful methods of proof are illustrated in the context of studying problems concerning mainly numbers (real, rational, complex and integers). An indispensable guide to all students of mathematics. Each proof is preceded by a discussion which is intended to show the reader the kind of thoughts they might have before any (...)
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4. Katie Atkinson & Trevor J. M. Bench-Capon (2007). Practical Reasoning as Presumptive Argumentation Using Action-Based Alternating Transition Systems. Artificial Intelligence 171 (10-15):855-874.
In this paper we describe an approach to practical reasoning, reasoning about what it is best for a particular agent to do in a given situation, based on presumptive justifications of action through the instantiation of an argument scheme, which is then subject to examination through a series of critical questions. We identify three particular aspects of practical reasoning which distinguish it from theoretical reasoning. We next provide an argument scheme and an associated set of critical questions which is able (...)
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5. John T. Baldwin (2004). Problems in Set Theory, Mathematical Logic, and the Theory of Algorithms. Bulletin of Symbolic Logic 10 (2):222-223.

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6. Arnold Beckmann (2002). Proving Consistency of Equational Theories in Bounded Arithmetic. Journal of Symbolic Logic 67 (1):279-296.
We consider equational theories for functions defined via recursion involving equations between closed terms with natural rules based on recursive definitions of the function symbols. We show that consistency of such equational theories can be proved in the weak fragment of arithmetic S 1 2 . In particular this solves an open problem formulated by TAKEUTI (c.f. [5, p.5 problem 9.]).

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7. Guessing the outcome of iterations of even most simple arithmetical functions could be an extremely hazardous experience. Not less harder, if at all possible, might be to prove the veracity of even a "sure" guess concerning iterations : this is the case of the famous 3x+1 conjecture. Our purpose here is to study and conceptualize some intuitive insights related to the ultimate (un)solvability of this conjecture.
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8. Edward G. Belaga & Maurice Mignotte (2006). Walking Cautiously Into the Collatz Wilderness: Algorithmically, Number Theoretically, Randomly. Discrete Mathematics and Theoretical Computer Science.
Building on theoretical insights and rich experimental data of our preprints, we present here new theoretical and experimental results in three interrelated approaches to the Collatz problem and its generalizations: algorithmic decidability, random behavior, and Diophantine representation of related discrete dynamical systems, and their cyclic and divergent properties.
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9. John L. Bell (2004). Whole and Part in Mathematics. Axiomathes 14 (4):285-294.
The centrality of the whole/part relation in mathematics is demonstrated through the presentation and analysis of examples from algebra, geometry, functional analysis,logic, topology and category theory.

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10. T. J. M. Bench-Capon (2012). Representing Popov V Hayashi with Dimensions and Factors. Artificial Intelligence and Law 20 (1):15-35.
Modelling reasoning with legal cases has been a central concern of AI and Law since the 1980s. The approach which represents cases as factors and dimensions has been a central part of that work. In this paper I consider how several varieties of the approach can be applied to the interesting case of Popov v Hayashi. After briefly reviewing some of the key landmarks of the approach, the case is represented in terms of factors and dimensions, and further explored using (...)

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11. Francesco Berto (2009). The Gödel Paradox and Wittgenstein's Reasons. Philosophia Mathematica 17 (2):208-219.
An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics match (...)

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12. G. S. Ceĭtin (ed.) (1971). Five Papers on Logic and Foundations. Providence, R.I.,American Mathematical Society.
Markov, A. A. On constructive mathematics.--Ceĭtin, G. S. Mean value theorems in constructive analysis.--Zaslavskiĭ, I. D. and Ceĭtlin, G. S. On singular coverings and properties of constructive functions connected with them.--Maslov, S. Ju. Certain properties of E. L. Post's apparatus of canonical calculi.--Zaslavskiĭ, I. D. Graph schemes with memory.
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14. Rodney Coleman (1979). An Introduction to Mathematical Stereology. University of Aarhus, Dept. Of Theoretical Statistics, Institute of Mathematics.
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15. Starting with 1985, we discovered the possible existence of electrons with net helicity in biomolecules as amino acids and their possibility to discern between the two quantum spin states. It is well known that the question of a possible fundamental role of quantum mechanics in biological matter constitutes still a long debate. In the last ten years we have given a rather complete quantum mechanical elaboration entirely based on Clifford algebra whose basic entities are isomorphic to the well known spin (...)
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16. D. Dantzig (1959). Review of G. Pólya, Mathematics and Plausible Reasoning, Vols. I and II. Synthese 11 (4):353-358.

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17. N. G. de Bruijn (1995). On the Roles of Types in Mathematics. In Philippe De Groote (ed.), The Curry-Howard Isomorphism. Academia 27-54.

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18. Recently, several authors have claimed to have found graph-theoretic counterexamples to the Principle of the Identity of Indiscernibles. In this paper, I argue that their counterexamples presuppose a certain view of what unlabeled graphs are, and that this view is optional at best.

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19. S. S. Demidov (1988). On an Early History of the Moscow School of Theory of Functions. Philosophia Mathematica (1):29-35.

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21. James Franklin (2006). Review of N. Wildberger, Divine Proportions: Rational Trigonometry to Universal. [REVIEW] Mathematical Intelligencer 28 (3):73-74.
Reviews Wildberger's account of his rational trigonometry project, which argues for a simpler way of doing trigonometry that avoids irrationals.

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22. Alexandre Guay & Brian Hepburn (2009). Symmetry and its Formalisms: Mathematical Aspects. Philosophy of Science 76 (2):160-178.
This article explores the relation between the concept of symmetry and its formalisms. The standard view among philosophers and physicists is that symmetry is completely formalized by mathematical groups. For some mathematicians however, the groupoid is a competing and more general formalism. An analysis of symmetry that justifies this extension has not been adequately spelled out. After a brief explication of how groups, equivalence, and symmetries classes are related, we show that, while it’s true in some instances that groups are (...)

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23. Vyvyan Howard (2004). Unbiased Stereology. Garland Science/Bios Scientific Publishers.
The Advanced Methods series is intented for advanced undergraduates, postgraduates and established research scientists. Titles in the series are designed to cover current important areas of research in life sciences, and include both theoretical background and detailed protocols. The aim is to give researchers sufficient theory, supported by references, to take the given protocols and adapt them to their particular experimental systems. Unbiased Stereology , Second Edition expands the comprehensive practical first edition guide to 3-D measurements in microscopy using stereological (...)
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25. Adam Kolany (1997). Consequence Operations Based on Hypergraph Satisfiability. Studia Logica 58 (2):261-272.
Four consequence operators based on hypergraph satisfiability are defined. Their properties are explored and interconnections are displayed. Finally their relation to the case of the Classical Propositional Calculus is shown.

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26. P. J. M. (1966). Colloquium on the Foundations of Mathematics, Mathematical Machines and Their Applications. Review of Metaphysics 19 (4):821-821.

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27. There's Something about Gödel is a bargain: two books in one. The first half is a gentle but rigorous introduction to the incompleteness theorems for the mathematically uninitiated. The second is a survey of the philosophical, psychological, and sociological consequences people have attempted to derive from the theorems, some of them quite fantastical.The first part, which stays close to Gödel's original proofs, strikes a nice balance, giving enough details that the reader understands what is going on in the proofs, without (...)

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28. John Mighton (2004). Review of D. Aeheson, 1089 and All That. [REVIEW] Mathematical Intelligencer 26 (2):70.

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30. A new advanced systems theory concerning the emergent nature of the Social, Consciousness, and Life based on Mathematics and Physical Analogies is presented. This meta-theory concerns the distance between the emergent levels of these phenomena and their ultra-efficacious nature. The theory is based on the distinction between Systems and Meta-systems (organized Openscape environments). We first realize that we can understand the difference between the System and the Meta-system in terms of the relationship between a ‘Whole greater than the sum of (...)

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31. Brunetto Piochi (1983). Logical Matrices and Non-Structural Consequence Operators. Studia Logica 42 (1):33 - 42.
In the present paper, we study some properties of matrices for non-structural consequence operators. These matrices were introduced in a former work (see [3]). In sections 1. and 2., general definitions and theorems are recalled; in section 3. a correspondence is studied, among our matrices and Wójcicki's ones for structural operators. In section 4. a theorem is given about operators, induced by submatrices or epimorphic images, or quotient matrices of a given one.Such matrices are used to characterize lattices of non-structural (...)

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32. Burkard Polster (2004). Q. Walker & Co..
Q.E.D. presents some of the most famous mathematical proofs in a charming book that will appeal to nonmathematicians and math experts alike. Grasp in an instant why Pythagoras’s theorem must be correct. Follow the ancient Chinese proof of the volume formula for the frustrating frustum, and Archimedes’ method for finding the volume of a sphere. Discover the secrets of pi and why, contrary to popular belief, squaring the circle really is possible. Study the subtle art of mathematical domino tumbling, and (...)
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33. Michael D. Potter (2000). Reason's Nearest Kin: Philosophies of Arithmetic From Kant to Carnap. Oxford University Press.
This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.

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34. David A. Rabson, John F. Huesman & Benji N. Fisher (2003). Cohomology for Anyone. Foundations of Physics 33 (12):1769-1796.
Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have (...)

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35. The “centre-periphery” relationship historically structured scientific exchanges between metropolis and province, between the fount of empire and its outposts. But the exchange, if regarded merely as a one-way flow of scientific information, ignores both the politics of knowledge and the nature of its appropriation. Arguably, imperial structures do not entirely determine scientific practices and the exchange of knowledge. Several factors neutralise the over-determining influence of politics—and possibly also the normative values of science—on scientific practice.In examining these four examples of Indian (...)

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36. A new proof is presented of Tsotsos' result that the VISUAL MATCH problem is NP-complete when no (high-level) constraints are imposed on the search space. Like the proof given by Tsotsos, it is based on the polynomial reduction of the NP-complete problem KNAPSACK to VISUAL MATCH. Tsotsos' proof, however, involves limited-precision real numbers, which introduces an extra degree of complexity to his treatment. The reduction of KNAPSACK to VISUAL MATCH presented here makes no use of limited-precision numbers, leading to a (...)

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37. Moses Richardson (1958). Fundamentals of Mathematics. New York, Macmillan.
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38. This volume draws together contributions from researchers whose work builds on the theory developed by Martin-Lof over the last twenty-five years.

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39. Dennis Sentilles (1975). A Bridge to Advanced Mathematics. Baltimore,Williams & Wilkins.

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40. Yaroslav Sergeyev (2013). Solving Ordinary Differential Equations by Working with Infinitesimals Numerically on the Infinity Computer. Applied Mathematics and Computation 219 (22):10668–10681.
There exists a huge number of numerical methods that iteratively construct approximations to the solution y(x) of an ordinary differential equation (ODE) y′(x) = f(x,y) starting from an initial value y_0=y(x_0) and using a finite approximation step h that influences the accuracy of the obtained approximation. In this paper, a new framework for solving ODEs is presented for a new kind of a computer – the Infinity Computer (it has been patented and its working prototype exists). The new computer is (...)

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41. Infinity and infinite sets, as traditionally defined in mathematics, are shown to be logical absurdities. To maintain logical consistency, mathematics ought to abandon the traditional notion of infinity. It is proposed that infinity should be replaced with the concept of “indefiniteness”. This further implies that other fields drawing on mathematics, such as physics and cosmology, ought to reject theories that postulate infinities of space and time. It is concluded that however indefinite our calculations of space and time become, the Universe (...)
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42. Stewart Shapiro (ed.) (1985). Intentional Mathematics. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.

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43. David Sherry (2011). Thermoscopes, Thermometers, and the Foundations of Measurement. Studies in History and Philosophy of Science Part A 42 (4):509-524.
Psychologists debate whether mental attributes can be quantified or whether they admit only qualitative comparisons of more and less. Their disagreement is not merely terminological, for it bears upon the permissibility of various statistical techniques. This article contributes to the discussion in two stages. First it explains how temperature, which was originally a qualitative concept, came to occupy its position as an unquestionably quantitative concept (§§1–4). Specifically, it lays out the circumstances in which thermometers, which register quantitative (or cardinal) differences, (...)

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44. David Sherry (2009). Reason, Habit, and Applied Mathematics. Hume Studies 35 (1/2):57-85.
Hume describes the sciences as "noble entertainments" that are "proper food and nourishment" for reasonable beings (EHU 1.5-6; SBN 8).1 But mathematics, in particular, is more than noble entertainment; for millennia, agriculture, building, commerce, and other sciences have depended upon applying mathematics.2 In simpler cases, applied mathematics consists in inferring one matter of fact from another, say, the area of a floor from its length and width. In more sophisticated cases, applied mathematics consists in giving scientific theory a mathematical form (...)

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45. David Sherry (1985). A Concordance for Wittgenstein's Remarks on the Foundations of Mathematics. History and Philosophy of Logic 6 (1):211-213.

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46. Stephen G. Simpson (2010). The Gödel Hierarchy and Reverse Mathematics. In Kurt Gödel, Solomon Feferman, Charles Parsons & Stephen G. Simpson (eds.), Kurt Gödel: Essays for His Centennial. Association for Symbolic Logic
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47. Kim Solin (2012). Dual Choice and Iteration in an Abstract Algebra of Action. Studia Logica 100 (3):607-630.
This paper presents an abstract-algebraic formulation of action facilitating reasoning about two opposing agents. Two dual nondeterministic choice operators are formulated abstract-algebraically: angelic (or user) choice and demonic (or system) choice. Iteration operators are also defined. As an application, Hoare-style correctness rules are established by means of the algebra. A negation operator is also discussed.

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48. Edward Russell Stabler (1953). An Introduction to Mathematical Thought. Cambridge, Mass., Addison-Wesley.

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49. Mark Steiner (1995). Review of S. Sternberg, Group Theory and Physics. [REVIEW] Philosophia Mathematica 3 (3):313-316.