The Interpretation of Probability in the Tractatus Logico-Philosophicus

In this paper, I propose an assessment of the interpretation of the mathematical notion of probability that Wittgenstein presents in TLP (1963: 5.15 – 5.156). I start by presenting his definition of probability as a relation between propositions. I claim that this definition qualifies as a logical interpretation of probability, of the kind defended in the same years by J. M. Keynes. However, Wittgenstein’s interpretation seems prima facie to be safe from two standard objections moved to logical probability, i. e. the mystic nature of the postulated relation and the reliance on Laplace’s principle of indifference. I then proceed to evaluate Wittgenstein’s idea against three criteria for the adequacy of an interpretation of probability: admissibility, ascertainability, and applicability. If the interpretation is admissible on Kolmogorov’s classical axiomatisation, the problem of ascertainability brings up a difficult dilemma. Finally, I test the interpretation in the application to three main contexts of use of probabilities. While the application to frequencies rests ungrounded, the application to induction requires some elaboration, and the application to rational belief depends on ascertainability.

O regime de juros compostos segundo a dialética ferramenta-objeto

ResumoEste trabalho faz parte de uma pesquisa de doutorado em andamento que trata sobre o tema educação financeira e suas potencialidades para o letramento financeiro do professor de matemática na licenciatura. Pretendemos descrever brevemente as fases de organização para a construção de conceitos matemáticos segundo a dialética ferramenta-objeto introduzida por Régine Douady. Nosso objetivo é apresentar um exemplo dessa organização no campo da matemática financeira, e mais precisamente, relativo ao regime de juros compostos. Esse tipo de organização enfatiza a importância de se alternar, no ensino, os aspectos ferramenta e objeto de uma dada noção matemática. No exemplo proposto foi desenvolvida uma sequência para articular a noção de juros compostos e sua mobilização numa nova situação. Palavras-chave: Didática da Matemática; Dialética ferramenta-objeto; Juros compostos.AbstractThis work is part of an ongoing doctoral research that deals with the theme of financial education and its potential for the financial literacy of the mathematics teacher in the degree. We intend to briefly describe the phases of organization for the construction of mathematical concepts according to the tool-object dialectic introduced by Régine Douady. Our goal is to present an example of this organization in the field of financial mathematics, and more precisely, regarding the compound interest regime. This type of organization emphasizes the importance of alternating, in teaching, the tool and object aspects of a given mathematical notion. In the proposed example, a sequence was developed to articulate the notion of compound interest and its mobilization in a new situation.Keywords: Didactics of Mathematics; Dialectic tool-object; Compound interest.

FORMAL REPRESENTATIONS OF DEPENDENCE AND GROUNDEDNESS

We study, in an abstract and general framework, formal representations of dependence and groundedness which occur in semantic theories of truth. Our goals are (a) to relate the different ways in which groundedness is defined according to the way dependence is represented and (b) to represent different notions of dependence as instances of a suitable generalisation of the mathematical notion of functional dependence.

A New Representation of the k-Gamma Functions

The products of the form z ( z + l ) ( z + 2 l ) … ( z + ( k − 1 ) l ) are of interest for a wide-ranging audience. In particular, they frequently arise in diverse situations, such as computation of Feynman integrals, combinatory of creation, annihilation operators and in fractional calculus. These expressions can be successfully applied for stated applications by using a mathematical notion of k-gamma functions. In this paper, we develop a new series representation of k-gamma functions in terms of delta functions. It led to a novel extension of the applicability of k-gamma functions that introduced them as distributions defined for a specific set of functions.

Tilings: mathematical models for quasicrystals

This chapter discusses tilings as mathematical models for quasicrystals. In a first approximation quasicrystals may be described as being space filling with copies of two or more types of tiles. This description gives a connection with the mathematical notion of tilings, which have been well studied. A brief introduction of tilings is presented in this chapter along with the method of substitution to create aperiodic tilings. The symmetry of the tilings is also treated in this chapter, as are model sets and random tilings. Quasiperiodic crystals often have approximants, that is, periodic structures that are close to the aperiodic ones. The relations between quasiperiodic crystals and approximants also is described in this chapter.

Paradoxes of Definability, Russell’s Paradox, the Liar

Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the singularity approach to the traditional paradoxes of definability (or denotation), associated with Berry, Richard, and König. The chapter goes on to argue that there are two settings for Russell’s paradox, one in terms of the mathematical notion of set, and the other in terms of the logico-semantic notion of extension. The chapter then applies the singularity approach to Russell’s paradox for extensions. The chapter moves on to the case of truth, and applies the singularity approach to various versions of the Liar paradox, paying particular attention to the so-called strengthened Liar.

Dynamic compensation, parameter identifiability, and equivariances

A recent paper by Karin, Swisa, Glaser, Dor, and Alon introduced the mathematical notion of dynamical compensation (DC) in biological circuits, arguing that DC helps explain important features of glucose homeostasis as well as other key physiological regulatory mechanisms. The present paper establishes a connection between DC and two well-known notions in systems biology: system equivalence and parameter (un)identifiability. This recasting leads to effective tests for verifying DC in mathematical models.

Vividness in Mathematics and Narrative

This chapter examines vividness in mathematics and narrative. It first gives two presentations of the mathematical notion of a group before explaining how to calculate highest common factors. It then considers the role of figures of speech, such as metaphor and irony, in mathematics and proceeds by citing a few passages from literature that highlight the use of concrete details to convey abstract thoughts; these include passages from Charles Dickens's Bleak House, Alan Hollinghurst's The Folding Star, and George Eliot's Middlemarch. The chapter argues that when working through a totally analogous process, exactly the same response can be created in certain mathematical texts.