This category needs an editor. We encourage you to help if you are qualified.
Volunteer, or read more about what this involves.
Related categories
Siblings:
22 found
Search inside:
(import / add options)   Sort by:
  1. Jeremy Avigad, Notes on a Formalization of the Prime Number Theorem.
    On September 6, 2004, using the Isabelle proof assistant, I verified the following statement: (%x. pi x * ln (real x) / (real x)) ----> 1 The system thereby confirmed that the prime number theorem is a consequence of the axioms of higher-order logic together with an axiom asserting the existence of an infinite set. All told, our number theory session, including the proof of the prime number theorem and supporting libraries, constitutes 673 pages of proof scripts, or roughly 30,000 (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  2. Jeremy Avigad (2006). Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals. 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 (...)
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  3. Michael Beeson (1976). The Unprovability in Intuitionistic Formal Systems of the Continuity of Effective Operations on the Reals. Journal of Symbolic Logic 41 (1):18-24.
  4. Alexander Berenstein (2004). Dividing in the Algebra of Compact Operators. 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.
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  5. Xavier Caicedo & Roberto Cignoli (2001). An Algebraic Approach to Intuitionistic Connectives. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  6. Sergio A. Celani & Hernán J. San Martín (2012). Frontal Operators in Weak Heyting Algebras. 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 (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  7. Roger M. Cooke & Michiel Lambalgen (1983). The Representation of Takeuti's *20c ||_ -Operator. Studia Logica 42 (4):407 - 415.
    Gaisi Takeuti has recently proposed a new operation on orthomodular lattices L, ⫫: $\scr{P}(L)\rightarrow L$ . The properties of ⫫ suggest that the value of ⫫ $(A)(A\subseteq L)$ corresponds to the degree in which the elements of A behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular lattices L and the existence of two-valued homomorphisms on L.
    Remove from this list | Direct download (3 more)  
     
    My bibliography  
     
    Export citation  
  8. Maarten de Rijke & Yde Venema (1995). Sahlqvist's Theorem for Boolean Algebras with Operators with an Application to Cylindric Algebras. Studia Logica 54 (1):61-78.
    For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  9. Janet Folina (2012). Newton and Hamilton: In Defense of Truth in Algebra. 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, (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  10. Josep M. Font & Ventura Verdú (1993). The Lattice of Distributive Closure Operators Over an Algebra. Studia Logica 52 (1):1 - 13.
    In our previous paper Algebraic Logic for Classical Conjunction and Disjunction we studied some relations between the fragmentL of classical logic having just conjunction and disjunction and the varietyD of distributive lattices, within the context of Algebraic Logic. The central tool in that study was a class of closure operators which we calleddistributive, and one of its main results was that for any algebraA of type (2,2) there is an isomorphism between the lattices of allD-congruences ofA and of all distributive (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  11. 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 (...)
    Remove from this list |
    Translate to English
    | Direct download  
     
    My bibliography  
     
    Export citation  
  12. Burghard Herrmann (1997). Characterizing Equivalential and Algebraizable Logics by the Leibniz Operator. 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, (...)
    Remove from this list | Direct download (7 more)  
     
    My bibliography  
     
    Export citation  
  13. Jose Ferreiros Jeremy Gray (ed.) (2006). The Architecture of Modern Mathematics.
    Remove from this list |
    Translate to English
    |
     
    My bibliography  
     
    Export citation  
  14. Ladislav Kvasz (2006). The History of Algebra and the Development of the Form of its Language. 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 <span class='Hi'>Euler</span>, 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 (in the sense of Wittgenstein's Tractatus) of the symbolic language of (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  15. Jean-Pierre Marquis (2013). Mathematical Forms and Forms of Mathematics: Leaving the Shores of Extensional Mathematics. 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 (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  16. Jean-pierre Marquis (1997). Abstract Mathematical Tools and Machines for Mathematics. 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 (...)
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation  
  17. Robert K. Meyer (2008). Ai, Me and Lewis (Abelian Implication, Material Equivalence and C I Lewis 1920). 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. (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  18. Alejandro Petrovich (1996). Distributive Lattices with an Operator. Studia Logica 56 (1-2):205 - 224.
    It was shown in [3] (see also [5]) that there is a duality between the category of bounded distributive lattices endowed with a join-homomorphism and the category of Priestley spaces endowed with a Priestley relation. In this paper, bounded distributive lattices endowed with a join-homomorphism, are considered as algebras and we characterize the congruences of these algebras in terms of the mentioned duality and certain closed subsets of Priestley spaces. This enable us to characterize the simple and subdirectly irreducible algebras. (...)
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  19. Charles C. Pinter (1975). Algebraic Logic with Generalized Quantifiers. Notre Dame Journal of Formal Logic 16 (4):511-516.
  20. G. Spencer-Brown (1972). Laws of Form. New York,Julian Press.
    Remove from this list |
     
    My bibliography  
     
    Export citation  
  21. Alasdair Urquhart (1979). Distributive Lattices with a Dual Homomorphic Operation. Studia Logica 38 (2):201 - 209.
    The lattices of the title generalize the concept of a De Morgan lattice. A representation in terms of ordered topological spaces is described. This topological duality is applied to describe homomorphisms, congruences, and subdirectly irreducible and free lattices in the category. In addition, certain equational subclasses are described in detail.
    Remove from this list | Direct download (5 more)  
     
    My bibliography  
     
    Export citation  
  22. Andrzej Wroński (2004). The Distance Function in Commutative ℓ-Semigroups and the Equivalence in Łukasiewicz Logic. 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.
    Remove from this list | Direct download (6 more)  
     
    My bibliography  
     
    Export citation