Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918 he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show (...) that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal. (shrink)

Varied evidence confirms more strongly than less varied evidence, ceteris paribus. This epistemological Variety of Evidence Thesis enjoys widespread intuitive support. We put forward a novel explication of one notion of varied evidence and the Variety of Evidence Thesis within Bayesian models of scientific inference by appealing to measures of entropy. Our explication of the Variety of Evidence Thesis holds in many of our models which also pronounce on disconfirmatory and discordant evidence. We argue that our models pronounce rightly. Against (...) a backdrop of failures of the Variety of Evidence Thesis, the intuitive case for the Variety of Evidence Thesis emerges strengthened. Our models do however not support the general case for the thesis since our explication of it fails to hold in certain cases. The parameter space of this failure is explored and an explanation for the failure is offered. (shrink)

Bob Hale has defended a new conception of properties that is broadly Fregean in two key respects. First, like Frege, Hale insists that every property can be defined by an open formula. Second, like Frege, but unlike later definabilists, Hale seeks to justify full impredicative property comprehension. The most innovative part of his defense, we think, is a “definability constraint” that can serve as an implicit definition of the domain of properties. We make this constraint formally precise and prove that (...) it fails to characterize the domain uniquely. Thus, we conclude, there is no easy road to impredicative definabilism. (shrink)

The paper explores Hermann Weyl’s turn to intuitionism through a philosophical prism of normative framework transitions. It focuses on three central themes that occupied Weyl’s thought: the notion of the continuum, logical existence, and the necessity of intuitionism, constructivism, and formalism to adequately address the foundational crisis of mathematics. The analysis of these themes reveals Weyl’s continuous endeavor to deal with such fundamental problems and suggests a view that provides a different perspective concerning Weyl’s wavering foundational positions. Building on a (...) philosophical model of scientific framework transitions and the special role that normative indecision or ambivalence plays in the process, the paper examines Weyl’s motives for considering such a radical shift in the first place. It concludes by showing that Weyl’s shifting stances should be regarded as symptoms of a deep, convoluted intrapersonal process of self-deliberation induced by exposure to external criticism. (shrink)

This essay discusses the problem of the two cultures. According to the author the problem arises because science is the source of a new way of conceiving reality and man, different from the mental conception entertained by nonscientific persons. The article suggests methodological guidelines for the philosopher interested in understanding the humanistic mentality of the scientists. The approach proposed is inductive-genetic. The aim is to help the philosopher explore science in its developmental becoming so that he may become aware of (...) the reasons for the characteristic mentality of scientists as regards humanistic issues. The essay concludes with a discussion of the new perspectives disclosed to scientific-humanistic dialogue. (shrink)

People commonly use terms like ‘random’, ‘by chance’, or ‘accidentally’ when they describe occurrences that sidestep the normal course of events, with no apparent causal link to ongoing activities. Such intrusive events are typically perceived as happening all of a sudden. This was demonstrated in seven experiments (N = 1299) by asking people to identify statements they believed belonged to stories about chance events, and by comparing chance vs. non-chance events from their own life and from the lives of others. (...) Suddenness also appeared to be more strongly associated with beginnings than with endings of an episode. Thus, statements about sudden beginnings of a career or a relationship were more readily believed to happen ‘by chance’ than ‘not by chance’. But life-changing events did not become more accidental simply by being placed in the beginning of a ‘life chapter’. These studies highlight a central but neglected feature of perceived randomness. (shrink)