- H. Benis Sinaceur (IHPST, Paris, France)
- Facets and Levels of Mathematical Abstraction
Abstract: Mathematical abstraction is the process of considering and manipulating operations,
rules, methods and concepts divested from their reference to real world phenomena and circumstances, and also deprived
from the content connected to particular applications. There is no one single way of performing mathematical abstraction.
The term “abstraction” does not name a unique procedure but a general process,
which goes many ways that are mostly simultaneous and intertwined; in
particular, the process does not amount only to logical subsumption. I will consider comparatively how philosophers
consider abstraction and how mathematicians perform it, with the aim to bring to light the fundamental thinking processes
at play, and to illustrate by significant examples how much intricate and multi-leveled may be the combination of typical
mathematical techniques which include axiomatic method, invariance principles, equivalence relations
and functional correspondences.
- H. Graves (Algos Associates, Fort Worth, TX, USA)
- A Practical Doctrine for Mathematical Applications
Abstract: A doctrine, in the sense of Jon Beck, is outlined for representing
and reasoning about mathematical applications. The doctrine is a two category whose objects are axioms sets
and whose morphisms are functors. The use of this doctrine for developing and reasoning about axiom sets corresponds
closely to informal practice, but differs from textbook development. An application axiom set is specified by a signature
and formulae in the language of the signature. Each application axiom set uses a base language with term constructions
from topos theory. The axioms are Horn rule axioms. These rule axioms sets generate a topos as their deductive closure.
A First Order Logic is used to express the axioms, but extends standard presentations in
that terms are allowed to have decidable preconditions for being well-formed. Constructions such as composition of maps are
defined as functions terms. The axiom sets are represented as tuples within the 2-category doctrine.
The 2-category is a meta logic for operating on axiom sets and maps between them. The doctrine is also a specification
for a class of software tools for developing and analysing axiom sets that represent applications.
- R. Kahle (CENTRIA and DM, FCT, Universidade Nova de Lisboa, Caparica, Portugal)
- The logical cone. A new account to counterfactuals
Abstract: We give a new account to counterfactuals, based on the notion: which facts,
relevant for the consequent, may depend on the fact in the antecedent?
- E. F. Karavaev (Dept. of Logic,
St. Petersburg State University, Russia)
- One way to determine the intervals in hybrid temporal logic
Abstract: This presentation discusses the opportunity of improvement of technical means of hybrid
temporal logic through the introduction of time intervals. In the procedure of constructing intervals the
author of the presentation follows ideas and development expressed and carried out by A.A. Markov in
his article published in 1932. So the ‘Priorean paradigm’ of understanding of the logic (temporal
qualification of judgments and the idea of hybrid logic) is complemented by a building of time metric
based on the relation ‘earlier than’. It seems that the described improvement of the machinery of temporal
logic allows, in particular, to perfect the approaches to the modelling of planning and strategic management.
-
V. Kreinovich (University of Texas at El Paso, USA),
O. Kosheleva (University of Texas at El Paso, USA)
- Logic of Scientific Discovery:
How Physical Induction Affects What Is Computable
Abstract: Most of our knowledge about a physical world comes from physical induction:
if a hypothesis is confirmed by a sufficient number of observations, we conclude that this hypothesis is universally true.
We show that a natural formalization of this property affects what is computable when processing measurement
and observation results, and we explain how this formalization is related to Kolmogorov complexity and randomness.
We also consider computational consequences of an alternative idea also coming form physics: that no physical law
is absolutely true, that every physical law will sooner or later need to be corrected. It turns out
that this alternative approach enables us to use measurement results go beyond what is usually computable.
- A. Lecomte (Laboratoire SFL - CNRS - Université
Paris 8, Saint-Denis, France)
- An Interaction Framework for Dialogue
Abstract: Many considerations lead to the idea that interaction is at the heart of language.
Conversations are at the basis of the development of language. Only a few formal works are attempting to show
how this feature is preeminent. Most works in formal semantics for instance are oriented toward model-theoretic aspects:
linguistic expressions are just seen as describing situations. Even if this is not of course completely wrong,
it seems that most of them do not directly take their references from the environment,
but from reduction steps in the interaction of two participants in a dialogue. Theoretical Computer Science
has given a paradigm for such an interaction, starting from the reduction of λ-terms which corresponds
to proof normalization. New approaches extend this view by entering into the picture not only proofs but also
counter-proofs, thus leading to Girard’s Ludics, which provides several tools to express dialogue’s dynamicity.
- L. I. Manevitch (Institute of Chemical Physics, Moscow, Russia)
- Asymptotic thinking as a philosophical principle
Abstract: It was noted that asymptotic thinking, being efficient mathematical tool, similarly to
symmetry analysis is also a philosophical principle reflecting the essential features of the cognition
process. While the clearest manifestation of this principle is achieved in dynamical systems, its
numerous applications also relate to physics, biology and humanities.
-
S. I. Nikolenko (St. Petersburg Department
of V. A. Steklov Institute of Mathematics RAS, Russia),
S. Koltsov (National Research University
Higher School of Economics, St. Petersburg, Russia),
O. Koltsova (National Research University
Higher School of Economics, St. Petersburg, Russia)
- Measuring Topic Quality in Latent Dirichlet Allocation
Abstract: Topic modeling is an important direction of study for modern text mining;
unsupervised mining of collections of topics is intended to produce understanding and capture the essence of issues
a dataset is devoted to. However, existing techniques of topic evaluation in topic models such as latent Dirichlet allocation
(LDA) are still lacking in their ability to represent human interpretability and worth for qualitative studies.
In this work, we propose a novel topic quality metric that more closely corresponds to human judgement than existing ones.
We support this claim with the results of an experimental study where test subjects rate LDA topics on how interpretable
they are.
- A. B. Patkul (Department of Ontology and Epistemology,
St. Petersburg State University, Russia)
- The Problem of the Logic’s Destruction
and Grounding in Phenomenology
Abstract: The article is dedicated to the problem of the critical grounding
of the formal logic (as well traditional as contemporary symbol logic) in phenomenological philosophy.
The examples of such critical grounding of this discipline in E. Husserl and M. Heidegger are inquired here in more
particularly way. The following theses are ascertained: Husserl believes that the logic is based in the
constitutive activity of transcendental consciousness. And Heidegger thinks that logic has its origin in
the human understanding of the being and therefore in the usage of copular verb in the sense of
“presence-at-hand”. Moreover, the logic is read out from nature as domain, the being of which is the
presence, and the logic’s application on the other domains of the being is its unjustified extrapolation
in Heidegger’s opinion. And conclusion the question is put in the article, whether such
phenomenological “destruction” of logic is only negation of it or a possible way to its grounding
outside of the traditional shape of logic.
- V. Perminov (Faculty of Philosophy,
M. V. Lomonosov Moscow State University, Russia)
- Praxeological Substantiation of Logic
Abstract: Logical norms can be understood as a formal criterion of truth which makes
it possible to eliminate untrue statements on the base of their form. Background of logic is the insight of full
(absolute) truth, different from relative truth of science. For more adequate understanding of the
nature of logic, we should rehabilitate old apriorism, having given to it new, praxiologic substantiation.
- E. Rivello (Scuola Normale Superiore, Pisa, Italy)
- Eliminating the Ordinals from Proofs.
An Analysis of Transfinite Recursion
Abstract: Transfinite ordinal numbers enter mathematical practice mainly via the method
of definition by transfinite recursion. Outside of axiomatic set theory, there is a significant mathematical tradition
in works recasting proofs by transfinite recursion in other terms, mostly with the intention
of eliminating the ordinals from the proofs. Leaving aside the different motivations
which lead each specific case, we investigate the mathematics of this action of proof transforming
and we address the problem of formalising the philosophical notion of elimination
which characterises this move.
- V. A. Shaposhnikov (Faculty of Philosophy,
Lomonosov Moscow State University, Russia)
- The Applicability Problem
and a Naturalistic Perspective on Mathematics
Abstract: The paper outlines a philosophical account of the interplay between pure and applied mathematics. This account is argued to harmonize well with the naturalistic philosophy of mathematics. The autonomy of mathematics is considered as a transitional form between theological and naturalistic views of mathematics. From the naturalistic standpoint, it is natural to understand pure mathematics through applied mathematics but not vice versa. The proposed approach to mathematics is interpreted as a revival of Aristotle’s philosophy of mathematics and owes a lot to James Franklin. Wigner’s puzzle of applicability is explained away as a survival of the positivist philosophy of mathematics.
-
S. Soloviev (IRIT, Université Toulouse 3)
- Context-dependent invertibility, isomorphism and subtyping in type theory:
possible linguistic applications
Abstract: The aim of this note is to attract attention to context-dependent invertibility
of terms in type theory and related notions of context-dependent isomorphisms and coercions,
a relatively understudied topic that may be of interest to applications,
especially in linguistics where meaning naturally depends on context.
-
A. Spaskov (Institute of Philosophy, Academy of Sciences of Belarus, Minsk, Belarus),
O. Kozina (Moscow Social-Psychological University, Russia)
- Number and Time
Abstract: The paper deals with the nature of mathematical concepts,
genesis of natural numbers and the temporal structure of consciousness. We analyze the arithmetic model
of time and propose a new geometrical model of three-dimensional time, which is based on the hypothesis
of independ-ent time dimensions corresponding to external linear and internal cyclic time.
- Sh. Steinert-Threlkeld (Department of Philosophy,
Stanford University, CA, USA)
- On the Decidability of Iterated Languages
Abstract: A special kind of substitution on languages called iteration is presented
and studied. We show that each of the star-free, regular, and deterministic context-free languages are closed under
iteration and that it is decidable whether a given regular language or a DCFL is an iteration of two languages.
We also determine the state complexity of iteration of regular languages.
Connections to the van Benthem / Keenan ‘Frege Boundary’ are discussed.
- V. Stepanov (Dorodnicyn Computing Centre RAS, Moscow, Russia)
- Truth Theory for Logic of Self-Reference Statements
as a Quaternion Structure
Abstract: Article is aimed at giving to linguists the tool
which they can use for studying
of the mechanism of references of one statements on others, including on itself.
For this purpose the quantifier of the self-reference is entered and approximation of
the self-reference quantifier on sequences of statements of language is given.
The language model on discrete dynamic systems is defined.
In a dynamic model of self-reference statements developed by the author it is revealed that for language
with the propositional connectives of biconditional (<—>) and negation (˜)
the truth table for biconditional for type formulae of True, Liar, TruthTeller and (TruthTeller<—>Liar),
is Cayley table for the Klein four-group V. It suggests that the truth space for values of self-reference statements
is described by quaternion algebra H.
- D. Tiskin (Dept. of Logic,
St. Petersburg State University, Russia)
- Transparent Evaluation and Pronouns in Attitude Reports
Abstract: The paper outlines an account for Simon Charlow’s data concerning de se
and de re readings of pronouns and anaphors in attitude contexts. Using Arnim von Stechow’s binding technique
as well as the insights about the internal structure of pronouns (due to Rose-Marie Déchaine and MartinaWiltschko)
and about transparent readings of predicates (due to Yasutada Sudo) I treat de se readings as primitive
and de re ones as derived. An additional assumption about the semantics of reflexives
is used to explain why, as shown by Charlow, a de se anaphor cannot be bound by a de re subject.
Next, I show another direction within the problem of anaphora
one might proceed in with the treatment of pronouns found in Déchaine and Wiltschko’s paper.
Finally, comparing my proposal with its predecessors, I touch upon the issue of the extent
to which a semantic theory should be philosophically laden.
- J. A. Wislicki (Faculty of Polish Studies, University of Warsaw, Poland)
- Semantics of quotation. Against the functional approach to quotation
Abstract: The aim of this paper is twofold. First, it is to argue against the functional approach to
quotation, according to which enquotation is a syntactic map that delivers expressions of the
metalanguage. Second, it is to define a semantic operation that allows to express the meaning of
quotation without getting involved into semantic inconsistencies. I discuss the semantic expressive
power of the most influential functional theories of quotation and show the problems that arise in that
kind of approach. Then I draw an important connection between Reichenbach's idea of the so-called
'arrow quotes' and the account of quotational context given by Pagin and Westerstahl. The core ideas of
both proposals become a bottom line of the semantic account that allows to calculate the meaning of
quotation via composition principles without getting involved into semantic inconsistencies.
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