Although many agree that all teaching rests on a theory of knowledge, this is an in-depth exploration of the philosophy of mathematics for education, building on the work of Lakatos and Wittgenstein.

Presents general information about meteorology, weather, and climate and includes more than thirty activities to help study these topics, including making a ...

The titile essay of this collection suggests that Bertrand Russell's lifelong preoccupation: the disentanglement, with ever-increasing precision, of what is subjective or intellectualy cloudy from what is objective or capable of logical demonstration. The first five essays he calls 'entirely popular': they include two on the revolutionary changes in mathematics in the last hundred years, and one on the value of science in human culture. The last five, 'somewhat more technical', are concerned with particular problems of philosophy: the ultimate nature (...) of matter, the connection between the sense-data and physics, the problem of casuality and different ways of knowing. In these one can see the Russell method in operation, intellectual analysis dissecting the problem to its bare bones. (shrink)

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre (...) Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Simonton examines the idea of the genius through his own theory called chance-configuration theory.

How can we account for the sudden appearance of such dazzling artists and scientists as Mozart, Shakespeare, Darwin, or Einstein? How can we define such genius? What conditions or personality traits seem to produce exceptionally creative people? Is the association between genius and madness really just a myth? These and many other questions are brilliantly illuminated in The Origins of Genius. Dean Simonton convincingly argues that creativity can best be understood as a Darwinian process of variation and selection. The artist (...) or scientist generates a wealth of ideas, and then subjects these ideas to aesthetic or scientific judgment, selecting only those that have the best chance to survive and reproduce. Indeed, the true test of genius is the ability to bequeath an impressive and influential body of work to future generations. Simonton draws on the latest research into creativity and explores such topics as the personality type of the genius, whether genius is genetic or produced by environment and education, the links between genius and mental illness, the high incidence of childhood trauma, especially loss of a parent, amongst Nobel Prize winners, the importance of unconscious incubation in creative problem-solving, and much more. Simonton substantiates his theory by examining and quoting from the work of such eminent figures as Henri Poincare, W. H. Auden, Albert Einstein, Marie Curie, Charles Darwin, Niels Bohr, and many others. For anyone intrigued by the spectacular feats of the human mind, The Origins of Genius offers a revolutionary new way of understanding the very nature of creativity. (shrink)

This ambitious book by one of the most original and provocative thinkers in science studies offers a sophisticated new understanding of the nature of scientific, mathematical, and engineering practice and the production of scientific knowledge. Andrew Pickering offers a new approach to the unpredictable nature of change in science, taking into account the extraordinary number of factors—social, technological, conceptual, and natural—that interact to affect the creation of scientific knowledge. In his view, machines, instruments, facts, theories, conceptual and mathematical structures, disciplined (...) practices, and human beings are in constantly shifting relationships with one another—"mangled" together in unforeseeable ways that are shaped by the contingencies of culture, time, and place. Situating material as well as human agency in their larger cultural context, Pickering uses case studies to show how this picture of the open, changeable nature of science advances a richer understanding of scientific work both past and present. Pickering examines in detail the building of the bubble chamber in particle physics, the search for the quark, the construction of the quarternion system in mathematics, and the introduction of computer-controlled machine tools in industry. He uses these examples to address the most basic elements of scientific practice—the development of experimental apparatus, the production of facts, the development of theory, and the interrelation of machines and social organization. (shrink)

Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein. It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity. In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence (...) of unconscious mental processes in mathematical invention and other forms of creativity. Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life. The roots of creativity for Hadamard lie not in consciousness, but in the long unconscious work of incubation, and in the unconscious aesthetic selection of ideas that thereby pass into consciousness. His discussion of this process comprises a wide range of topics, including the use of mental images or symbols, visualized or auditory words, "meaningless" words, logic, and intuition. Among the important documents collected is a letter from Albert Einstein analyzing his own mechanism of thought. (shrink)

Analysis and case studies show that including different orientations toward the natural world makes for more effective scientific practice and science education.