‘Interview’, Probability and Statistics: 5 Questions

Intervie

The 1900 Turn in Bertrand Russell’s Logic, the Emergence of his Paradox, and the Way Out

Russell’s initial project in philosophy (1898) was to make mathematics rigorous reducing it to logic. Before August 1900, however, Russell’s logic was nothing but mereology. First, his acquaintance with Peano’s ideas in August 1900 led him to discard the part-whole logic and accept a kind of intensional predicate logic instead. Among other things, the predicate logic helped Russell embrace a technique of treating the paradox of infinite numbers with the help of a singular concept, which he called ‘denoting phrase’. Unfortunately, a new paradox emerged soon: that of classes. The main contention of this paper is that Russell’s new conception only transferred the paradox of infinity from the realm of infinite numbers to that of class-inclusion. Russell’s long-elaborated solution to his paradox developed between 1905 and 1908 was nothing but to set aside of some of the ideas he adopted with his turn of August 1900: (i) With the Theory of Descriptions, he reintroduced the complexes we are acquainted with in logic. In this way, he partly restored the pre-August 1900 mereology of complexes and simples. (ii) The elimination of classes, with the help of the ‘substitutional theory’, and of propositions, by means of the Multiple Relation Theory of Judgment, completed this process

Introduction to special issue: Aesthetics in mathematics

Mathematicians often appreciate the beauty and elegance of particular theorems, proofs, and definitions, attaching importance not only to the truth but also to the aesthetic merit of their work. As Henri Poincaré [1930, p. 59] put it, mathematical beauty is a ‘real aesthetic feeling that all true mathematicians recognise’. Others went further, regarding mathematical beauty as a key motivation driving the formulation of mathematical proofs and even as a criterion for choosing one proof over another. As Hermann Weyl famously and provocatively declared, ‘My work always tried to unite the true with the beautiful, but when I had to choose one or the other, I usually chose the beautiful’ (cited [Chandrasekhar, 1987, p. 52])....Talk of the beauty of mathematical theorems, proofs, and definitions may thus be commonplace

The Problem of Intuition in Mathematics in the Thoughts and Creativity of Selected Polish Mathematicians in the Context of the Nineteenth-Century Breakthrough in Mathematics

In the article, I examine the presence and importance of intuitive cognition in mathematics. I show the occurrence of mathematical intuition in four contexts: discovery, understanding, justification, and acceptance or rejection. I will deal with examples from the history of mathematics, when new mathematical theories were being created (the end of the nineteenth and the beginning of the twentieth century will be particularly important, including the period of establishing the Polish mathematical school). I will also refer to the research (mainly) of Polish philosophers and mathematicians in this field. The goal of the article is also an attempt to understand the breakthrough that took place in mathematics at the turn of the nineteenth century. The analysis also shows, by highlighting the specifics of intuition and mathematical creativity, the difficulties that arise when acquiring new concepts and mathematical arguments. Research goes in the direction of deepening research on the very phenomenon of intuition in cognition, by pointing to the universal nature of mathematical intuition.In the article, I examine the presence and importance of intuitive cognition in mathematics. I show the occurrence of mathematical intuition in four contexts: discovery, understanding, justification, and acceptance or rejection. I will deal with examples from the history of mathematics, when new mathematical theories were being created (the end of the nineteenth and the beginning of the twentieth century will be particularly important, including the period of establishing the Polish mathematical school). I will also refer to the research (mainly) of Polish philosophers and mathematicians in this field. The goal of the article is also an attempt to understand the breakthrough that took place in mathematics at the turn of the nineteenth century. The analysis also shows, by highlighting the specifics of intuition and mathematical creativity, the difficulties that arise when acquiring new concepts and mathematical arguments. Research goes in the direction of deepening research on the very phenomenon of intuition in cognition, by pointing to the universal nature of mathematical intuition

Evolution of Cooperation in Public Goods Games with Stochastic Opting-Out

This paper investigates the evolution of strategic play where players drawn from a finite well-mixed population are offered the opportunity to play in a public goods game. All players accept the offer. However, due to the possibility of unforeseen circumstances, each player has a fixed probability of being unable to participate in the game, unlike similar models which assume voluntary participation. We first study how prescribed stochastic opting-out affects cooperation in finite populations. Moreover, in the model, cooperation is favored by natural selection over both neutral drift and defection if return on investment exceeds a threshold value defined solely by the population size, game size, and a player's probability of opting-out. Ultimately, increasing the probability that each player is unable to fulfill her promise of participating in the public goods game facilitates natural selection of cooperators. We also use adaptive dynamics to study the coevolution of cooperation and opting-out behavior. However, given rare mutations minutely different from the original population, an analysis based on adaptive dynamics suggests that the over time the population will tend towards complete defection and non-participation, and subsequently, from there, participating cooperators will stand a chance to emerge by neutral drift. Nevertheless, increasing the probability of non-participation decreases the rate at which the population tends towards defection when participating. Our work sheds light on understanding how stochastic opting-out emerges in the first place and its role in the evolution of cooperation.Comment: 30 pages, 4 figures. This is one of the student project papers arsing from the Mathematics REU program at Dartmouth 2017 Summer. See https://math.dartmouth.edu/~reu/ for more info. Comments are always welcom