135 research outputs found
Ethics and economics in Karl Menger: how did social sciences cope with Hilbertism
This paper deals with the contributions made to the social sciences by the mathematician Karl Menger (1902-1985), the son of the more famous economist, Carl Menger. Mathematician and a logician, he focused on whether it was possible to explain the social order in formal terms.1 He stressed the need to find the appropriate means with which to treat them, avoiding recourse to historical descriptions, which are unable to yield social laws. He applied Hilbertism to economics and ethics in order to build an axiomatic and formalized model of the individual behavior and the dynamics of social groups.
A geometry of information, I: Nerves, posets and differential forms
The main theme of this workshop (Dagstuhl seminar 04351) is `Spatial
Representation: Continuous vs. Discrete'. Spatial representation has two
contrasting but interacting aspects (i) representation of spaces' and (ii)
representation by spaces. In this paper, we will examine two aspects that are
common to both interpretations of the theme, namely nerve constructions and
refinement. Representations change, data changes, spaces change. We will
examine the possibility of a `differential geometry' of spatial representations
of both types, and in the sequel give an algebra of differential forms that has
the potential to handle the dynamical aspect of such a geometry. We will
discuss briefly a conjectured class of spaces, generalising the Cantor set
which would seem ideal as a test-bed for the set of tools we are developing.Comment: 28 pages. A version of this paper appears also on the Dagstuhl
seminar portal http://drops.dagstuhl.de/portals/04351
Representing continuous t-norms in quantum computation with mixed states
A model of quantum computation is discussed in (Aharanov et al 1997 Proc.
13th Annual ACM Symp. on Theory of Computation, STOC pp 20–30) and
(Tarasov 2002 J. Phys. A: Math. Gen. 35 5207–35) in which quantum gates
are represented by quantum operations acting on mixed states. It allows one to
use a quantum computational model in which connectives of a four-valued logic
can be realized as quantum gates. In this model, we give a representation of
certain functions, known as t-norms (Menger 1942 Proc. Natl Acad. Sci. USA
37 57–60), that generalize the triangle inequality for the probability distributionvalued
metrics. As a consequence an interpretation of the standard operations
associated with the basic fuzzy logic (H´ajek 1998 Metamathematics of Fuzzy
Logic (Trends in Logic vol 4) (Dordrecht: Kluwer)) is provided in the frame of
quantum computatio
Fuzzy Sets and Formal Logics
The paper discusses the relationship between fuzzy sets and formal logics as well as the influences fuzzy set theory had on the development of particular formal logics. Our focus is on the historical side of these developments. © 2015 Elsevier B.V. All rights reserved.partial support by the Spanish projects EdeTRI (TIN2012-39348- C02-01) and 2014 SGR 118.Peer reviewe
Partial maps with domain and range: extending Schein's representation
The semigroup of all partial maps on a set under the operation of composition admits a number of operations relating to the domain and range of a partial map. Of particular interest are the operations R and L returning the identity on the domain of a map and on the range of a map respectively. Schein [25] gave an axiomatic characterisation of the semigroups with R and L representable as systems of partial maps; the class is a finitely axiomatisable quasivariety closely related to ample semigroups (which were introduced—as type A semigroups—by Fountain, [7]). We provide an account of Schein's result (which until now appears only in Russian) and extend Schein's method to include the binary operations of intersection, of greatest common range restriction, and some unary operations relating to the set of fixed points of a partial map. Unlike the case of semigroups with R and L, a number of the possibilities can be equationally axiomatised
From Classical Logic to Fuzzy Logic and Quantum Logic: A General View
The aim of this article is to offer a concise and unitary vision upon the algebraic connections between classical logic and its generalizations, such as fuzzy logic and quantum logic. The mathematical concept which governs any kind of logic is that of lattice. Therefore, the lattices are the basic tools in this presentation. The Hilbert spaces theory is important in the study of quantum logic and it has also been used in the present paper
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