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  1. The Construction of Transfinite Equivalence Algorithms.Han Geurdes - manuscript
    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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  2. Who's Afraid of Mathematical Diagrams?Silvia De Toffoli - forthcoming - Philosophers' Imprint.
    Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...)
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  3. Denseness Results in the Theory of Algebraic Fields.Sylvy Anscombe, Philip Dittmann & Arno Fehm - 2021 - Annals of Pure and Applied Logic 172 (8):102973.
    We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.
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  4. Why Did Weyl Think That Emmy Noether Made Algebra the Eldorado of Axiomatics?Iulian D. Toader - 2021 - Hopos: The Journal of the International Society for the History of Philosophy of Science 11 (1):122-142.
    This paper argues that Noether's axiomatic method in algebra cannot be assimilated to Weyl's late view on axiomatics, for his acquiescence to a phenomenological epistemology of correctness led Weyl to resist Noether's principle of detachment.
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  5. How (and Why) the Conservation of a Circle is the Core (and Only) Dynamic in Nature.Ilexa Yardley - 2021 - Https://Medium.Com/the-Circular-Theory/.
    Solving Navier-Stokes and integrating it with Bose-Einstein. Moving beyond ‘mathematics’ and ‘physics.’ And, philosophy. Integrating 'point' 'line' 'circle.' (Euclid with 'reality.').
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  6. Choice-Free Stone Duality.Nick Bezhanishvili & Wesley H. Holliday - 2020 - Journal of Symbolic Logic 85 (1):109-148.
    The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this article, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski’s observation that the regular open sets of any topological space form a Boolean (...)
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  7. Genealogy of Algorithms: Datafication as Transvaluation.Virgil W. Brower - 2020 - le Foucaldien 6 (1):1-43.
    This article investigates religious ideals persistent in the datafication of information society. Its nodal point is Thomas Bayes, after whom Laplace names the primal probability algorithm. It reconsiders their mathematical innovations with Laplace's providential deism and Bayes' singular theological treatise. Conceptions of divine justice one finds among probability theorists play no small part in the algorithmic data-mining and microtargeting of Cambridge Analytica. Theological traces within mathematical computation are emphasized as the vantage over large numbers shifts to weights beyond enumeration in (...)
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  8. Algebraic and Topological Semantics for Inquisitive Logic Via Choice-Free Duality.Nick Bezhanishvili, Gianluca Grilletti & Wesley H. Holliday - 2019 - In Rosalie Iemhoff, Michael Moortgat & Ruy de Queiroz (eds.), Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science, Vol. 11541. Springer. pp. 35-52.
    We introduce new algebraic and topological semantics for inquisitive logic. The algebraic semantics is based on special Heyting algebras, which we call inquisitive algebras, with propositional valuations ranging over only the ¬¬-fixpoints of the algebra. We show how inquisitive algebras arise from Boolean algebras: for a given Boolean algebra B, we define its inquisitive extension H(B) and prove that H(B) is the unique inquisitive algebra having B as its algebra of ¬¬-fixpoints. We also show that inquisitive algebras determine Medvedev’s logic (...)
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  9. ‘Chasing’ the Diagram—the Use of Visualizations in Algebraic Reasoning.Silvia de Toffoli - 2017 - Review of Symbolic Logic 10 (1):158-186.
    The aim of this article is to investigate the roles of commutative diagrams (CDs) in a specific mathematical domain, and to unveil the reasons underlying their effectiveness as a mathematical notation; this will be done through a case study. It will be shown that CDs do not depict spatial relations, but represent mathematical structures. CDs will be interpreted as a hybrid notation that goes beyond the traditional bipartition of mathematical representations into diagrammatic and linguistic. It will be argued that one (...)
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  10. A Survey of Geometric Algebra and Geometric Calculus.Alan Macdonald - 2017 - Advances in Applied Clifford Algebras 27:853-891.
    The paper is an introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. No knowledge of physics is required. The section Further Study lists many papers available on the web.
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  11. Poincaré on the Foundation of Geometry in the Understanding.Jeremy Shipley - 2017 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics: The CSHPM 2016 Annual Meeting in Calgary, Alberta. Springer. pp. 19-37.
    This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...)
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  12. Imagination in Mathematics.Andrew Arana - 2016 - In Amy Kind (ed.), Routledge Handbook on the Philosophy of Imagination. Routledge. pp. 463-477.
    This article will consider imagination in mathematics from a historical point of view, noting the key moments in its conception during the ancient, modern and contemporary eras.
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  13. Unequal Sample Sizes and the Use of Larger Control Groups Pertaining to Power of a Study.Marie Oldfield - 2016 - Dstl 1 (1).
    To date researchers planning experiments have always lived by the mantra that 'using equal sample sizes gives the best results' and although unequal groups are also used in experimentation, it is not the preferred method of many and indeed actively discouraged in literature. However, during live study planning there are other considerations that we must take into account such as availability of study participants, statistical power and, indeed, the cost of the study. These can all make allocating equal sample sizes (...)
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  14. The Nature of Local/Global Distinctions, Group Actions and Phases: A Sheaf=Theoretic Approach to Quantum Geometric Spectra.Elias Zafiris - 2015 - In Vera Bühlmann, Ludger Hovestadt & Vahid Moosavi (eds.), Coding as Literacy - Metalithicum IV. Basel: BIRKHÄUSER. pp. 172-186.
  15. Finite Methods in Mathematical Practice.Peter Schuster & Laura Crosilla - 2014 - In Godehard Link (ed.), Formalism and Beyond: On the Nature of Mathematical Discourse. De Gruyter. pp. 351-410.
    In the present contribution we look at the legacy of Hilbert's programme in some recent developments in mathematics. Hilbert's ideas have seen new life in generalised and relativised forms by the hands of proof theorists and have been a source of motivation for the so--called reverse mathematics programme initiated by H. Friedman and S. Simpson. More recently Hilbert's programme has inspired T. Coquand and H. Lombardi to undertake a new approach to constructive algebra in which strong emphasis is laid on (...)
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  16. Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics.Jean-Pierre Marquis - 2013 - Synthese 190 (12):2141-2164.
    In this paper, I introduce the idea that some important parts of contemporary pure mathematics are moving away from what I call the extensional point of view. More specifically, these fields are based on criteria of identity that are not extensional. After presenting a few cases, I concentrate on homotopy theory where the situation is particularly clear. Moreover, homotopy types are arguably fundamental entities of geometry, thus of a large portion of mathematics, and potentially to all mathematics, at least according (...)
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  17. Frontal Operators in Weak Heyting Algebras.Sergio A. Celani & Hernán J. San Martín - 2012 - Studia Logica 100 (1-2):91-114.
    In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [ 10 ]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation $${\tau(a) \leq b \vee (b \rightarrow a)}$$, for all $${a, b \in A}$$. These operators were studied from an algebraic, logical and topological point of view by Leo Esakia (...)
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  18. Notes on Groups and Geometry, 1978-1986.Steven H. Cullinane - 2012 - Internet Archive.
    Typewritten notes on groups and geometry.
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  19. Newton and Hamilton: In Defense of Truth in Algebra.Janet Folina - 2012 - Southern Journal of Philosophy 50 (3):504-527.
    Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, (...)
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  20. Algebraic Symbolism in Medieval Arabic Algebra.Jeffrey Oaks - 2012 - Philosophica 87:27-83.
  21. Fictionalism and Mathematical Objectivity.Iulian D. Toader - 2012 - In Metaphysics and Science. University of Bucharest Press. pp. 137-158.
    This paper, written in Romanian, compares fictionalism, nominalism, and neo-Meinongianism as responses to the problem of objectivity in mathematics, and then motivates a fictionalist view of objectivity as invariance.
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  22. Linear and Geometric Algebra.Alan MacDonald - 2011 - North Charleston, SC: CreateSpace.
    This textbook for the second year undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics. -/- Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much (...)
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  23. Zariski‐Type Topology for Implication Algebras.Manuel Abad, Diego Castaño & José P. Díaz Varela - 2010 - Mathematical Logic Quarterly 56 (3):299-309.
    In this work we provide a new topological representation for implication algebras in such a way that its one-point compactification is the topological space given in [1]. Some applications are given thereof.
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  24. Two Conjectures on the Arithmetic in ℝ and ℂ†.Apoloniusz Tyszka - 2010 - Mathematical Logic Quarterly 56 (2):175-184.
    Let G be an additive subgroup of ℂ, let Wn = {xi = 1, xi + xj = xk: i, j, k ∈ {1, …, n }}, and define En = {xi = 1, xi + xj = xk, xi · xj = xk: i, j, k ∈ {1, …, n }}. We discuss two conjectures. If a system S ⊆ En is consistent over ℝ, then S has a real solution which consists of numbers whose absolute values belong to (...)
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  25. A Simple Construction of Representable Relation Algebras with Non‐Representable Completions.Tarek Sayed Ahmed - 2009 - Mathematical Logic Quarterly 55 (3):237-244.
    We give a simple new construction of representable relation algebras with non-representable completions. Using variations on our construction, we show that the elementary closure of the class of completely representable relation algebras is not finitely axiomatizable.
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  26. Leo Corry. Modern Algebra and the Rise of Mathematical Structures. Viii + 431 Pp., Index. Second Revised Edition. Basel/Boston/Berlin: Birkhäuser Verlag, 2004. €69.55. [REVIEW]José Ferreirós - 2009 - Isis 100 (2):412-413.
  27. Amalgamation for Reducts of Polyadic Equality Algebras, a Negative Result.Tarek Sayed Ahmed - 2008 - Bulletin of the Section of Logic 37 (1):37-50.
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  28. Classes of Representable Algebras with the Amalgamation Property.Tarek Sayed Ahmed - 2008 - Bulletin of the Section of Logic 37 (2):115-121.
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  29. Elementary Constructive Theory of Henselian Local Rings.María E. Alonso, Henri Lombardi & Hervé Perdry - 2008 - Mathematical Logic Quarterly 54 (3):253-271.
    We give an elementary theory of Henselian local rings and construct the Henselisation of a local ring. All our theorems have an algorithmic content.
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  30. Episodes in the History of Modern Algebra. [REVIEW]Elena Corie Marchisotto - 2008 - Isis 99:424-425.
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  31. Ai, Me and Lewis (Abelian Implication, Material Equivalence and C I Lewis 1920).Robert K. Meyer - 2008 - Journal of Philosophical Logic 37 (2):169 - 181.
    C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure material equivalence. (...)
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  32. On Abstraction and the Importance of Asking the Right Research Questions: Could Jordan Have Proved the Jordan-Hölder Theorem?Dirk Schlimm - 2008 - Erkenntnis 68 (3):409-420.
    In 1870 Jordan proved that the composition factors of two composition series of a group are the same. Almost 20 years later Hölder (1889) was able to extend this result by showing that the factor groups, which are quotient groups corresponding to the composition factors, are isomorphic. This result, nowadays called the Jordan-Hölder Theorem, is one of the fundamental theorems in the theory of groups. The fact that Jordan, who was working in the framework of substitution groups, was able to (...)
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  33. Normal Subgroups of Nonstandard Symmetric and Alternating Groups.John Allsup & Richard Kaye - 2007 - Archive for Mathematical Logic 46 (2):107-121.
    Let ${\mathfrak{M}}$ be a nonstandard model of Peano Arithmetic with domain M and let ${n \in M}$ be nonstandard. We study the symmetric and alternating groups S n and A n of permutations of the set ${\{0,1,\ldots,n-1\}}$ internal to ${\mathfrak{M}}$ , and classify all their normal subgroups, identifying many externally defined such normal subgroups in the process. We provide evidence that A n and S n are not split extensions by these normal subgroups, by showing that any such complement if (...)
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  34. Polyadic and Cylindric Algebras of Sentences.Mohamed Amer & Tarek Sayed Ahmed - 2006 - Mathematical Logic Quarterly 52 (5):444-449.
    In this note we give an interpretation of cylindric algebras as algebras of sentences of first order logic. We show that the isomorphism types of such algebras of sentences coincide with the class of neat reducts of cylindric algebras. Also we show how this interpretation sheds light on some recent results. This is done by likening Henkin's Neat Embedding Theorem to his celebrated completeness proof.
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  35. Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals.Jeremy Avigad - 2006 - In Jose Ferreiros Jeremy Gray (ed.), The architecture of modern mathematics.
    Philosophical concerns rarely force their way into the average mathematician’s workday. But, in extreme circumstances, fundamental questions can arise as to the legitimacy of a certain manner of proceeding, say, as to whether a particular object should be granted ontological status, or whether a certain conclusion is epistemologically warranted. There are then two distinct views as to the role that philosophy should play in such a situation. On the first view, the mathematician is called upon to turn to the counsel (...)
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  36. From Valla to Viète: The Rhetorical Reform of Logic and its Use in Early Modern Algebra.Giovanna Cifoletti - 2006 - Early Science and Medicine 11 (4):390-423.
    Lorenzo Valla's rhetorical reform of logic resulted in important changes in sixteenth-century mathematical sciences, and not only in mathematical education and in the use of mathematics in other sciences, but also in mathematical theory itself. Logic came to be identified with dialectic, syllogisms with enthymemes and necessary truth with the limit case of probable truth. Two main ancient authorities mediated between logical and mathematical concerns: Cicero and Proclus. Cicero's 'common notions' were identified with Euclid's axioms, so that mathematics could be (...)
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  37. The Architecture of Modern Mathematics: Essays in History and Philosophy.José Ferreirós Domínguez & Jeremy Gray (eds.) - 2006 - Oxford, England: Oxford University Press.
    This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.
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  38. The History of Algebra and the Development of the Form of its Language.Ladislav Kvasz - 2006 - Philosophia Mathematica 14 (3):287-317.
    This paper offers an epistemological reconstruction of the historical development of algebra from al-Khwrizm, Cardano, and Descartes to Euler, Lagrange, and Galois. In the reconstruction it interprets the algebraic formulas as a symbolic language and analyzes the changes of this language in the course of history. It turns out that the most fundamental epistemological changes in the development of algebra can be interpreted as changes of the pictorial form of the symbolic language of algebra. Thus the paper develops further the (...)
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  39. Dividing in the Algebra of Compact Operators.Alexander Berenstein - 2004 - Journal of Symbolic Logic 69 (3):817-829.
    We interpret the algebra of finite rank operators as imaginaries inside a Hilbert space. We prove that the Hilbert space enlarged with these imaginaries has built-in canonical bases.
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  40. The Distance Function in Commutative ℓ-Semigroups and the Equivalence in Łukasiewicz Logic.Andrzej Wroński - 2004 - Studia Logica 77 (2):241 - 253.
    The equivalence connective in ukasiewicz logic has its algebraic counterpart which is the distance function d(x,y) =|x–y| of a positive cone of a commutative -group. We make some observations on logically motivated algebraic structures involving the distance function.
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  41. Boolean Algebras With A Distinguished Automorphism.M. Abad, J. P. D.\'{\I.}az Varela & M. Zander - 2003 - Reports on Mathematical Logic:101-112.
    In this paper we investigate a subvariety $\BA$ of tense algebras, which we call Boolean algebras with a distinguished automorphism. This variety provides a unifying framework for the algebras studied by Monteiro in [4] and by Moisil in [5,6]. Among others we prove that $\BA$ is generated by its finite members and we characterize the locally finite subvarieties of $\BA$.
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  42. Pedro Nunes : His Lost Algebra and Other Discoveries. [REVIEW]Giovanna Cifoletti - 2003 - Isis 94:369-371.
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  43. Inequivalent Representations of Geometric Relation Algebras.Steven Givant - 2003 - Journal of Symbolic Logic 68 (1):267-310.
    It is shown that the automorphism group of a relation algebra ${\cal B}_P$ constructed from a projective geometry P is isomorphic to the collineation group of P. Also, the base automorphism group of a representation of ${\cal B}_P$ over an affine geometry D is isomorphic to the quotient of the collineation group of D by the dilatation subgroup. Consequently, the total number of inequivalent representations of ${\cal B}_P$ , for finite geometries P, is the sum of the numbers ${\mid Col(P)\mid\over (...)
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  44. Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer. [REVIEW]Leo Corry - 2002 - Isis 93:126-127.
    Charles W. Curtis is a prominent mathematician who has made important contributions to the field of representation theory. His textbooks in this field have been classics for a long time. In Pioneers of Representation Theory he has set out to present the historical development of the main ideas of the discipline, from the work of Georg Ferdinand Frobenius in the 1890s up to 1960. In addition to Frobenius, the book focuses mainly on three other “pioneers”: William Burnside, Issai Schur, and (...)
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  45. Armand Borel. Essays in the History of Lie Groups and Algebraic Groups. Xiii + 184 Pp., Bibl., Indexes. Providence, R.I.: American Mathematical Society; London: London Mathematical Society, 2001. [REVIEW]Patti Hunter - 2002 - Isis 93 (4):719-719.
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  46. On Free Annotated Algebras.Renato A. Lewin, Irene F. Mikenberg & Marı́a G. Schwarze - 2001 - Annals of Pure and Applied Logic 108 (1-3):249-259.
    In Lewin et al. 359–386) the authors proved that certain systems of annotated logics are algebraizable in the sense of Block and Rigozzi 396). Later in Lewin et al. the study of the associated quasi-varieties of annotated algebras is initiated. In this paper we continue the study of the these classes of algebras, in particular, we report some recent results about the free annotated algebras.
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  47. An Algebraic Approach to Intuitionistic Connectives.Xavier Caicedo & Roberto Cignoli - 2001 - Journal of Symbolic Logic 66 (4):1620-1636.
    It is shown that axiomatic extensions of intuitionistic propositional calculus defining univocally new connectives, including those proposed by Gabbay, are strongly complete with respect to valuations in Heyting algebras with additional operations. In all cases, the double negation of such a connective is equivalent to a formula of intuitionistic calculus. Thus, under the excluded third law it collapses to a classical formula, showing that this condition in Gabbay's definition is redundant. Moreover, such connectives can not be interpreted in all Heyting (...)
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  48. Mathematische Schriften. Volume 2: 1672-1676. Algebra by Gottfried Wilhelm Leibniz. [REVIEW]Marc Parmentier - 1998 - Isis 89:723-724.
  49. Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator.Burghard Herrmann - 1997 - Studia Logica 58 (2):305-323.
    In [14] we used the term finitely algebraizable for algebraizable logics in the sense of Blok and Pigozzi [2] and we introduced possibly infinitely algebraizable, for short, p.i.-algebraizable logics. In the present paper, we characterize the hierarchy of protoalgebraic, equivalential, finitely equivalential, p.i.-algebraizable, and finitely algebraizable logics by properties of the Leibniz operator. A Beth-style definability result yields that finitely equivalential and finitely algebraizable as well as equivalential and p.i.-algebraizable logics can be distinguished by injectivity of the Leibniz operator. Thus, (...)
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  50. Abstract Mathematical Tools and Machines for Mathematics.Jean-Pierre Marquis - 1997 - Philosophia Mathematica 5 (3):250-272.
    In this paper, we try to establish that some mathematical theories, like K-theory, homology, cohomology, homotopy theories, spectral sequences, modern Galois theory (in its various applications), representation theory and character theory, etc., should be thought of as (abstract) machines in the same way that there are (concrete) machines in the natural sciences. If this is correct, then many epistemological and ontological issues in the philosophy of mathematics are seen in a different light. We concentrate on one problem which immediately follows (...)
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