Henri Poincaré was not just one of the most inventive, versatile, and productive mathematicians of all time--he was also a leading physicist who almost won a Nobel Prize for physics and a prominent philosopher of science whose fresh and surprising essays are still in print a century later. The first in-depth and comprehensive look at his many accomplishments, Henri Poincaré explores all the fields that Poincaré touched, the debates sparked by his original investigations, and how his discoveries still contribute to (...) society today. Math historian Jeremy Gray shows that Poincaré's influence was wide-ranging and permanent. His novel interpretation of non-Euclidean geometry challenged contemporary ideas about space, stirred heated discussion, and led to flourishing research. His work in topology began the modern study of the subject, recently highlighted by the successful resolution of the famous Poincaré conjecture. And Poincaré's reformulation of celestial mechanics and discovery of chaotic motion started the modern theory of dynamical systems. In physics, his insights on the Lorentz group preceded Einstein's, and he was the first to indicate that space and time might be fundamentally atomic. Poincaré the public intellectual did not shy away from scientific controversy, and he defended mathematics against the attacks of logicians such as Bertrand Russell, opposed the views of Catholic apologists, and served as an expert witness in probability for the notorious Dreyfus case that polarized France. Richly informed by letters and documents, Henri Poincaré demonstrates how one man's work revolutionized math, science, and the greater world. (shrink)
Some maladaptive thought processes are characterized by reflexive and habitual patterns of cognitive and emotional reactivity. We review theoretical and empirical work suggesting that mindfulness—a state of nonjudgmental awareness of the present moment—can facilitate the discontinuation of such automatic mental operations. We propose a framework that suggests a series of more specific mechanisms supporting the de-automatizing function of mindfulness. Four related but distinct elements of mindfulness (awareness, attention, focus on the present, and acceptance) can each contribute to de-automatization through subsequent (...) processes, including discontinuing automatic inference, enhancing cognitive control capacity, facilitating metacognitive insight, and preventing suppression or thought distortion. De-automatizing can, in turn, allow enhancement of adaptive self-control ability and increased well-being. (shrink)
Mathematicians use the word ‘deep’ to convey a high appreciation of a concept, theorem, or proof. This paper investigates the extent to which the term can be said to have an objective character by examining its first use in mathematics. It was a consequence of Gauss's work on number theory and the agreement among his successors that specific parts of Gauss's work were deep, on grounds that indicate that depth was a structural feature of mathematics for them. In contrast, French (...) mathematicians had a less structural, more problem-oriented approach to mathematics and did not speak of depth so readily. (shrink)
This edited volume, aimed at both students and researchers in philosophy, mathematics and history of science, highlights leading developments in the overlapping areas of philosophy and the history of modern mathematics. It is a coherent, wide ranging account of how a number of topics in the philosophy of mathematics must be reconsidered in the light of the latest historical research and how a number of historical accounts can be deepened by embracing philosophical questions.
Blair proposes that fluid intelligence, working memory, and executive function form a unitary construct: fluid cognition. Recently, our group has utilized a combined correlational–experimental cognitive neuroscience approach, which we argue is beneficial for investigating relationships among these individual differences in terms of neural mechanisms underlying them. Our data do not completely support Blair's strong position. (Published Online April 5 2006).
The primrose path and prisoner's dilemma paradigms may require cognitive (executive) control: The active maintenance of context representations in lateral prefrontal cortex to provide top-down support for specific behaviors in the face of short delays or stronger response tendencies. This perspective suggests further tests of whether altruism is a type of self-control, including brain imaging, induced affect, and dual-task studies.
No mathematician did more to change mathematics in the second half of the twentieth century than Alexandre Grothendieck. This would have been true even if he had been a quiet figure with a liking for playing the piano and walking in the hills but, as this book makes very clear, he was far from that, and his character and his way of working enhanced his impact. Above all, there was his abrupt departure from the world of mathematics in 1970 and (...) his occasional interventions in it since.This review was submitted a few days before Grothendieck's death in November 2014—Editor.As a teenager, Grothendieck had lived with his mother protected from the Nazis by courageous Huguenots in the village of Le Chambon-sur-Lignon. Immediately after the war, he went to the University of Montpellier where he studied mathematics and soon made his way to Nancy and was introduced to Dieudonné and Schwartz, who were members of Bourbaki. His first original work was in the theory of Banach spaces where, in the wo .. (shrink)
The lives of few mathematicians offer the drama that is presented by the life of L. E. J. Brouwer, correctly identified on the cover of this book as a topologist, intuitionist, and philosopher, and before we go any further, it will be worth indicating why.It is not just that Brouwer would rank high among mathematicians for his work in topology alone: he set standards for rigour and created a theory of dimension for topological spaces, and his fixed-point theorem is of (...) great importance. Nor is it just that his philosophy of intuitionism created a new and vibrant branch of logic, that is, arguably, the only viable alternative to naïve, classical logic outside the enclave of professional logicians. Rather, it is that there is a tension, popularly taken to amount to a contradiction, in the fundamental ideas behind his topology and his intuitionism. Added to that, he took a highly principled stand on mathematical issues that led him into confrontations with major figures and a certain degre .. (shrink)
General intelligence is largely based on two distinguishable mental abilities: crystallized intelligence (gC) and fluid reasoning ability (gF). The target article authors' P-FIT model emphasizes a network of regions throughout the brain as the neural basis for fluid reasoning and/or working memory. However, it provides little significant insight into the neural basis of gC, or how or why gC is more stable than gF across the life span.