Embracing the Crisis in the Foundations of Mathematics

The crisis in the foundations of mathematics is a conceptual crisis. I suggest that we embrace the crisis and adopt a pluralist position towards foundations. There are many foundations in mathematics. However, ‘many foundations’ (for one building) is an oxymoron. Therefore, we shift vocabulary to say that mathematics, as one discipline, is composed of many different theories. This entails that there are no absolute mathematical truths, only truths within a theory. There is no unified, consistent ontology, only ontology within a theory.

Models to illustrate the foundations of mathematics /

Mode of access: Internet.

HILBERT BETWEEN THE FORMAL AND THE INFORMAL SIDE OF MATHEMATICS

Abstract: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that Hilbert encountered in his foundational studies consisted in securing a link between formalization and intuition. We will also analyze a related problem, that we will call "Frege's Problem", form the time of the foundation of geometry and investigate the role of the Axiom of Completeness in its solution.

Foundations of stochastic geometry and theory of random sets

The first section of this chapter starts with the Buffon problem, which is one of the oldest in stochastic geometry, and then continues with the definition of measures on the space of lines. The second section defines random closed sets and related measurability issues, explains how to characterize distributions of random closed sets by means of capacity functionals and introduces the concept of a selection. Based on this concept, the third section starts with the definition of the expectation and proves its convexifying effect that is related to the Lyapunov theorem for ranges of vector-valued measures. Finally, the strong law of large numbers for Minkowski sums of random sets is proved and the corresponding limit theorem is formulated. The chapter is concluded by a discussion of the union-scheme for random closed sets and a characterization of the corresponding stable laws.

Theoretical Foundations of Developing Modeling Instruction Curriculum for College Biology Courses

Modeling Instruction (MI) has been successfully implemented in high school science classes. Moreover, MI curriculum for introductory physics has also been developed at a university level. Noticing the gap, the author will provide theoretical foundations to support the statement that MI curriculum should be developed for college biology courses.

The Tausk controversy on the foundations of quantum mechanics: Physics, philosophy, and politics

In 1966 the Brazilian physicist Klaus Tausk (b. 1927) circulated a preprint from the International Centre for Theoretical Physics in Trieste, Italy, criticizing Adriana Daneri, Angelo Loinger, and Giovanni Maria Prosperi`s theory of 1962 on the measurement problem in quantum mechanics. A heated controversy ensued between two opposing camps within the orthodox interpretation of quantum theory, represented by Leon Rosenfeld and Eugene P. Wigner. The controversy went well beyond the strictly scientific issues, however, reflecting philosophical and political commitments within the context of the Cold War, the relationship between science in developed and Third World countries, the importance of social skills, and personal idiosyncrasies.

Communicating with the community about grieving: A description and review of the foundations of a broken leg analogy of grieving

With loss permeating the lives of all people throughout the entire life span and its potential for serious long-term deleterious effects, providing effective care for those experiencing situations of loss would be an important element of any preventive community-based mental health promotion approach. The low use of mental health services by those confronted with loss, and the lack of such services in many areas, makes it imperative to enhance the competency of the broader community to provide appropriate care during times of loss. Contributory to such care is a sound knowledge of the literature concerning loss and grieving. The broken leg analogy of grief presented in this article integrates the traditional and emerging grief theories and empirical evidence into an analogy of grieving that can be easily communicated to the general community, is relevant to many losses. and vet upholds the differing positions concerning grieving held by various schools of psychiatric thought.

Mathematical education and interdisciplinarity : promoting public awareness of mathematics in the Azores

Mathematical literacy in Portugal is very unsatisfactory in what concerns international standards. Even more disturbingly, the Azores archipelago ranks as one of the worst regions of Portugal in this respect. We reason that the popularisation of Mathematics through interactive exhibitions and activities can contribute actively to disseminate mathematical knowledge, increase awareness of the importance of Mathematics in today’s world and change its negative perception by the majority of the citizens. Although a significant investment has been undertaken by the local regional government in creating several science centres for the popularisation of Science, there is no centre for the popularisation of Mathematics. We present our first steps towards bringing Mathematics to unconventional settings by means of hands-on activities. We describe in some detail three activities. One activity has to do with applying trigonometry to measure distances in Astronomy, which can also be applied to Earth objects. Another activity concerns the presence of numerical patterns in the Azorean flora. The third activity explores geometrical patterns in the Azorean cultural heritage. It is our understanding that the implementation of these and other easy-to-follow and challenging activities will contribute to the awareness of the importance and beauty of Mathematics.

The foundations of zoology / by William Keith Brooks.

1899-1907

The foundations of zoology / by William Keith Brooks.

1899-1907