This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, (...) deductive reasoning, and quantifiers. Inductive reasoning and making conjectures are examined next, and once there are some statements to prove, techniques for proving conditional statements, disjunctions, biconditional statements, and quantified predicates are investigated. Terminology and proof techniques in set theory follow with discussions of the pick-a-point method and the algebra of sets. Cartesian products, equivalence relations, orders, and functions are all incorporated. Particular attention is given to injectivity, surjectivity, and cardinality. The text includes an introduction to topology and abstract algebra, with a comparison of topological properties to algebraic properties. This book can be used by itself for an introduction to proofs course or as a supplemental text for students in proof-based mathematics classes. The contents have been rigorously reviewed and tested by instructors and students in classroom settings. (shrink)

The first book to investigate the relevance of Theodor W. Adorno's work for theorizing the age of neoliberal capitalism. Through an engagement with Adorno's critical theory of society, Charles Prusik advances a novel approach to understanding the origins and development of neoliberalism. Offering a corrective to critics who define neoliberalism as an economic or political doctrine, Prusik argues that Adorno's dialectical theory of society can provide the basis for explaining the illusions and forms of domination that structure contemporary life. (...) Prusik explains the importance of Marx's critique of commodity fetishism in shaping Adorno's work and focuses on the related concepts of exchange, ideology, and natural history as powerful tools for grasping the present. Through an engagement with the ideas of neoliberal economic theory, Adorno and Neoliberalism criticizes the naturalization of capitalist institutions, social relations, ideology, and cultural forms. Revealing its origins in the crises of the Fordist period, Prusik develops Adorno's analyses of class, exploitation, monopoly, and reification to situate neoliberal policies as belonging to the fundamental antagonisms of capitalist society. (shrink)

This paper describes one style of functional analysis commonly used in the neurosciences called task-bound functional analysis. The concept of function invoked by this style of analysis is distinctive in virtue of the dependence relations it bears to transient environmental properties. It is argued that task-bound functional analysis cannot explain the presence of structural properties in nervous systems. An alternative concept of neural function is introduced that draws on the theoretical neuroscience literature, and an argument is given to show that (...) this alternative concept of may help to overcome the explanatory limitations of task-bound functional analysis. (shrink)

A recent article in this journal describes certain mathematical and philosophical controversies which occurred in Prussia during the middle decades of the 18th century. The article pays particular attention to the position of Christian Wolff and to the views of some of his followers. Both Wolff and the Wolffians are shown to have supported some of Leibniz's doctrines against those of the Newtonian camp. As a result, or perhaps in part as a premise, there is a strong tendency throughout the (...) article to identify Wolff himself with Leibniz. The reader will naturally conclude that Wolff (and, in turn, the Wolffians) must have been a faithful student and expositor of his master, Leibniz. My purpose in the present investigation is to contribute to an examination of the accuracy of this portrayal. (shrink)

Cortical color blindness, or cerebral achromatopsia, has been likened by some authors to ''blindsight'' for color or an instance of ''covert'' processing of color. Recently, it has been shown that, although such patients are unable to identify or discriminate hue differences, they nevertheless show a striking ability to process wavelength differences, which can result in preserved sensitivity to chromatic contrast and motion in equiluminant displays. Moreover, visually evoked cortical potentials can still be elicited in response to chromatic stimuli. We suggest (...) that these demonstrations reveal intact residual processes rather than the operation of covert processes, where proficient performance is accompanied by a denial of phenomenal awareness. We sought evidence for such covert processes by conducting appropriate tests on achromatopsic subject M.S. An ''indirect'' test entailing measurement of reaction times for letter identification failed to reveal covert color processes. In contrast, in a forced choice oddity task for color, M.S. was unable to verbally indicate the position of the different color, but was surprisingly adept at making an appropriate eye movement to its location. This ''direct'' test thus revealed the possible covert use of chromatic differences. (shrink)

Here are the chief riches of more than 3,000 years of Indian philosophical thought-the ancient Vedas, the Upanisads, the epics, the treatises of the heterodox and orthodox systems, the commentaries of the scholastic period, and the contemporary writings. Introductions and interpretive commentaries are provided.

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

Magnetic resonance imaging is a noninvasive imaging tool that utilizes a strong magnetic field and radio frequency waves to visualize in great detail organs, soft tissue, and bone. Unlike conventional x-rays, there is no exposure to ionizing radiation and at most field strengths the procedure is considered safe for nearly every age group. Because it is non-invasive and possesses excellent spatial resolution, the use of MRI as a research tool has increased exponentially over the past decade. Uses have ranged from (...) add-ons to a clinical study to studies of brain development in typically developing children. In addition, a major effort has been made in recent years to use MRI to study brain function. Because the clinical utility of fMRI has not yet been realized, fMRI is still considered highly exploratory, and we cannot yet identify incidental findings of a functional nature. (shrink)