Logic And Foundations Of Mathematics

Representation of Boolean algebras in Topologized Distributive lattices

by Takuya Igami

This paper has not been completed yet. Especially, the review of literatures and bibliography should be added later.

In this paper, I gave some topologized distributive lattices. This lattice expresses directly properties of Priestley... more In this paper, I gave some topologized distributive lattices. This lattice expresses directly properties of Priestley space. Though this new lattice is very easy to handle with, and has very rich properties, it should be still related to recent research of topological algebras and canonical extension.

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

"Redes, lógicas no clásicas y neuronas. De los límites de la matematización más allá de la Física"

by Vicente Caballero de la Torre

En el presente artículo se exponen las líneas maestras de la "Teoría de Grafos" y aquellos problemas de... more En el presente artículo se exponen las líneas maestras de la "Teoría de Grafos" y aquellos problemas de corte formal que la misma teoría muestra como modelo para explicar el funcionamiento del cerebro. Las redes -concepto que dicha teoría intenta sistematizar y comprender matemáticamente- son de sumo interés para cualquiera que pretenda arrojar una cierta luz sobre el perfil que en la actualidad están tomando fenómenos tan diversos y actuales como el terrorismo, la cibernética y, por supuesto, las últimas investigaciones neurocientíficas.

"Advice to the Humanities scholar: Do some math!"

by William J. Urban

presentation delivered April 8, 2011

The Representation of Geographic Object Semantics Using Inclusion Rules

by Kristin Stock

Stock, K.M. (1998). The Representation of Geographic Object Semantics Using Inclusion Rules. Paper presented at GIS/LIS 98 Annual Conference and Exposition held at Fort Worth, Texas, 8-12 November 1998.

A number of attempts have been made to provide standard terms or definitions for real world entities to aid in datamore A number of attempts have been made to provide standard terms or definitions for real world entities to aid in data
sharing. However, the information communities model of the OpenGIS Consortium recognizes that individual user
groups will have their own set of definitions and language, and that translations between these will be necessary
(OpenGIS Consortium, 1996). In order for translations to be successful, a method for capturing the semantics of
database elements is required. Simple definitions have been shown to be inadequate (Mark, 1993; Kuhn, 1994).
An alternative method for the representation of element semantics uses inclusion rules, and is based on psychological
theory of concept attainment, and particularly on a model proposed by Klausmeier, Ghatala and Frayer (1974). The
method identifies inclusion rules as being the basis for concept attainment, and combines these rules into predicates
to represent element semantics. The method allows cross referencing between predicates that define element
semantics, so reduces reliance on the expression style of individual participants in the data sharing activity.
In addition to providing a method for representation of element semantics, the inclusion rules method allows the
relationships between element semantics to be determined. This determination is necessary in order for semantic
translation to occur.
An example using topographic elements indicates that rules can be defined and predicates formed to represent
element semantics with limited dependence on individual expression. The example then shows that predicates can
be used to translate element semantics to allow data sharing between heterogeneous communities.

Harmony, Normality and Stability

by Nils Kürbis

As the title says: my account of proof-theoretic harmony, normality and stability!

Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules... more Gentzen mentions that it should be possible to specify a function that maps introduction rules onto elimination rules in systems of natural deduction. This paper specifies such a function. I specify two kinds of rules, one in which it is more natural to assume an introduction rule to be given and elimination rules are determined from it, and another kind in which it is an elimination rule which is given and the introduction rules are determined from it. The process also works the other way round, so that it doesn't really matter which rules are supposed to be given first. The process is very general and applies to a large class of logics. The paper begins with a discussion of the philosophical importance of this in connection with the notion of harmony. I discuss Dummett's ideas on harmony and stability, which is supposed to be stronger than harmony. Dummett suggests that normalisability is a formal criterion of harmony. However, he seems to aim at something else, and this criterion does not give an independent formally precise notion of stability. I propose formally precise definitions of harmony and stability, which are distinct from normalisability. My aim is not exegetical, and according to my definitions, classical as well as intuitionist logics count as governed by stable (and hence harmonious) rules of inference.

Stable Harmony

by Nils Kürbis

A talk I gave at Logica 2007. Not sure where the title came from ...

Negation: A Problem for the Proof-Theoretic Justification of Deduction

by Nils Kürbis

I won the Jacobsen Essay Price of the University of London for this essay!

What is Wrong with Classical Negation?

by Nils Kürbis

The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on... more The focus of this paper are the meaning-theoretical arguments against classical logic that Dummett bases on consideration about the meanings of negation. Using Dummettian principles, I shall outline three such arguments, of increasing strength, and show that they are unsuccessful by giving responses to each argument on behalf of the classical logician. What is crucial is that in responding to these arguments a classicist need not challenge any of the basic assumptions of Dummett’s outlook on the theory of meaning. In particular, I shall grant Dummett his general bias towards verificationism or justificationism, encapsulated in the slogan ‘meaning is use’. The second general assumption I see no need to question is Dummett’s particular breed of molecularism. Some of Dummett’s assumptions will have to be given up, if classical logic is to be vindicated in his meaning-theoretical framework. A major result of this paper will be that the meaning of negation cannot be defined by rules of inferences in the Dummettian framework.

El comienzo de la Lógica Matemática

by Carlos Martín Collantes

EL COMIENZO DE LA LÓGICA MATEMÁTICA

by Carlos Martín Collantes

Una introducción al análisis categorista de la lógica

by Luis Estrada-González

Si ese viejo profesor no creyera que la teoría de categorías es una vieja moda francesa, diría que este artículo es... more Si ese viejo profesor no creyera que la teoría de categorías es una vieja moda francesa, diría que este artículo es una buena aproximación a lo que todo lógico educado debería saber de teoría de topos.

Freedom and Mathematical Science: The problem of freedom in the introduction of ideal entities in mathematics

by Ioannis Vandoulakis

I. Strangas, A. Hanos (Eds) Proceedings of the Greek Association of Research in History and Philosophy of Law. Circle: Concepts of Freedom and Law. Vol. 3, 189-227, Athens-Thessalonica: Sakkulas Publishers, Paris-Torino-Budapest: L’Harmattan, 2006 [in Greek with extended summary in French]

On A.A. Markov’s attitude towards Brouwer’s intuitionism

by Ioannis Vandoulakis

Abstracts of the 14th Congress of Logic, Methodology and Philosophy of Science, Nancy, July 19-26, 2011, 159

The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his... more The paper examines Andrei A. Markov’s critical attitude towards L.E.J. Brouwer’s intuitionism, as is expressed in his notes to the Russian translation of Heyting’s Intuitionism, published in Moscow in 1965. It is argued that Markov’s algorithmic approach was shaped under the impact of the mathematical style and values prevailing in the Petersburg mathematical school, which is characterized by the proclaimed primacy of applications and the search for rigor and effective solutions.

Puzzles as a creative form of play in metaverse

by Cristiano Tonéis

The present study invites us for an ontological reflection about the close relationship between the... more The present study invites us for an ontological reflection about the close relationship between the logical-mathematical reasoning developed through the existence and experience in the puzzles into metaverse and creativity or creative thinking. For that reason, we seek to focus on the epistemological character of the puzzles in the metaverse and intrinsic creativity of metaverse as a natural process to be developed and not a privilege of few persons. Encourage creative processes through the digital universe - metaverse - and the use of heuristic construction for problem solving and openness to lateral thinking. Encourage creativity, creative and original thinking, which lead us to a new way of playing, the way of puzzles.

Hilbert, completeness and geometry

by Giorgio Venturi

This paper is an extended verion of a talk given in Mialano on 21st June 2011, for the workshop "Due giornate di studio sulla filosofia della matematica", organized by the SELP, and sponsored by the AILA.

This paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to... more This paper aims to show how the mathematical content of Hilbert’s Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem will bring out how Hilbert’s conception of mathematics is not as innovative as his conception of the axiomatic method. The role that the demonstrative tools play in Hilbert’s foundational reflections will also drive us to deal with the problem of the purity of methods, explicitly addressed by Hilbert. In this respect Hilbert’s position is very innovative and deeply linked to his modern conception of the axiomatic method. In the end we will show that the role played by the Axiom of Completeness for geometry is the same as the Axiom of Induction for arithmetic and of Church-Turing thesis for computability theory. We end this paper arguing that set theory is the right context in which applying the axiomatic method to mathematics and we postpone to a sequel of this work the attempt to offer a solution similar to Hilbert’s for the completeness of set theor

Finite Methods in Mathematical Practice

by Laura Crosilla

Co-authored with Peter Schuster, draft only

Research Proposal Summary

by Aviv Keren

¿Lógica Combinatoria o Teoría Estándar de Conjuntos?

by Lorenzo Peña

Arbor nº 520 (abril 1989), pp. 33-73. ISSN 0210-1963

KEYWORDS.- set theory, combinatory logic, comprising relation, reflexivity, unlevelling.

RESUMEN.- more KEYWORDS.- set theory, combinatory logic, comprising relation, reflexivity, unlevelling.

RESUMEN.-
Filosóficamente la teoría estándar de conjuntos es insatisfactoria, al no existir ninguna noción presistemática como la que quieren aducir sus adeptos; esa concepción resulta sospechosamente híbrida. La teoría no puede ser verdadera, salvo interpretada de un modo no estándar.
Afortunadamente, hay alternativas preferibles. Entre ellas una teoría combinatoria de conjuntos basada en una lógica no clásica. Esta lógica combinatoria gira en torno a la relación primitiva de abarcar, que puede también guardar reflexivamente un ente consigo mismo. Los conjuntos son, así, abarcamientos que no se sujetan al constreñimiento de la desnivelación.

Computational Structuralism and Frege's Constraint

by Paula Quinon

This document is the very first version of the final draft. There are still places to be precised and developed. All comments are welcomed.

A new version with improved technical part (too much sketchy in the current version) will be uploaded within few days.

Please, contact me before quoting, and always refer to the newest versions on academia.edu

A preliminary version of this paper has been presented at CLMPS Nancy 2011. http://lu.academia.edu/paulaquinon/Talks/57460/Freges_Constraint_and_Computational_Structuralism

The main aim of this paper is to reconsider a principle, used in some accounts of the foundations of mathematics,... more The main aim of this paper is to reconsider a principle, used in some accounts of the foundations of mathematics, called *Frege's Constraint* -- in particular, to reconsider the use of this principle in studying natural numbers. Frege’s Constraint states that any adequate foundation for a mathematical theory must explicitly take into account applications of the entities forming the intended interpretation of the theory. According to the traditional version of Frege’s Constraint -- as inspired by Frege's writings and developed by neo-Fregeans -- arithmetic should be based on the cardinality function encoded in Hume's Principle (HP). This paper finds it consistent with such a foundational strategy to claim it necessary to give an account of other important applications of the elements of the intended interpretation -- such as the computability of basic arithmetic functions in the case of arithmetic. I show that the neo-Fregean version of Frege’s Constraint cannot do this: it is unable to guarantee that those arithmetic functions intended to be computable actually *are* computable within the intended interpretation of Frege's Arithmetic (HP2).

The final section considers the consequences of adopting a different approach to Frege's Constraint: one which claims that the main property of natural numbers is their ability to act as the subjects of computations. I show that, with this alternative approach, one can make sense of the cardinality constraint (natural numbers serve to state and compare sizes of finite sets) and also justify the basic structural property of natural numbers (the omega-ordering of the intended interpretation of arithmetic).