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Mathematics holds a central place in the traditional liberal arts. The four mathematical disciplines of the quadrivium-arithmetic, geometry, music, and astronomy-reveal their enduring significance in this work, which offers the first unified, textbook treatment of these four subjects. Drawing on fundamental sources including Euclid, Boethius, and Ptolemy, this presentation respects the proper character of each discipline while revealing the relations among these liberal arts, as well as their connections to later mathematical and scientific developments. This book makes the quadrivium newly accessible in a number of ways. First, the careful choice of material from ancient sources means that students receive a faithful, integral impression of the classical quadrivium without being burdened or confused by an unwieldy mass of scattered results. Second, the terminology and symbols that are used convey the real insights of older mathematical approaches without introducing needless archaism. Finally, and perhaps most importantly, the book is filled with hundreds of exercises. Mathematics must be learned actively, and the exercises structured to complement the text, and proportioned to the powers of a learner to offer a clear path by which students make quadrivial knowledge their own. Many readers can profit from this introduction to the quadrivium. Students in high school will acquire a sense of the nature of mathematical proof and become confident in using mathematical language. College students can discover that mathematics is more than procedure, while also gaining insight into an intellectual current that influenced authors they are already reading: authors such as Plato, Aristotle, Augustine, Thomas Aquinas, and Dante. All will find a practical way to grasp a body of knowledge that, if long neglected, is never out of date.

Intensional logic, as understood here, is based on the broad presupposition that so-called "intensional contexts" in natural language can be explained semantically by the idea of multiple reference. The text reviews tense, modality, and conditionals, then presents developments in intensional theory, including partiality and generalized quantifiers. JOHAN van BENTHEM is professor of mathematical logic at the University of Amsterdam.

This book is meant to be a primer, that is, an introduction, to probability logic, a subject that appears to be in its infancy. Probability logic is a subject envisioned by Hans Reichenbach and largely created by Adams. It treats conditionals as bearers of conditional probabilities and discusses an appropriate sense of validity for arguments such conditionals, as well as ordinary statements as premisses.

This is a clear well-written text on the subject of probability logic, suitable for advanced undergraduates or graduates, but also of interest to professional philosophers. There are well-thought-out exercises, and a number of advanced topics treated in appendices, while some are brought up in exercises and some are alluded to only in footnotes. By this means, it is hoped that the reader will at least be made aware of most of the important ramifications of the subject and its tie-ins with current research, and will have some indications concerning recent and relevant literature.

Serious and accessible—finally the special relativity course book that both physics majors and lifelong learners deserve.
 
Special relativity challenges one’s physical intuition of space, time, matter, and energy in a way that few other topics in physics do. Yet the subject is often treated as an extra in undergraduate courses—something to be picked up in a few random lectures and presented as a combination of geometric and logical puzzles (seemingly with the premise of getting the novice student to concede that Einstein was a genius and that the universe is weird). But special relativity is absolutely fundamental to modern physics. It is the canvas on which electromagnetism, particle physics, field theory, and ultimately general relativity are based. For physics students, developing a relativistic intuition isn’t just a luxury: it’s a requirement.
 
Physicist and popular author Dave Goldberg provides a rigorous but conversational introduction to fill this void in spacetime education. Employing the standard calculus a sophomore or junior university student in science, engineering, or computer science will have encountered, Goldberg connects relativity to a student’s work ahead, acquainting them with topics like tensors, the development of new physical theories, and how relativity directly relates to other disciplines. But more than this, Goldberg welcomes lifelong learners who may have encountered special relativity in popular accounts, but are seeking a mathematical challenge to understand an elegant physical theory.

This book presents the theory of adjoint sensitivity analysis for high frequency applications through time-domain electromagnetic simulations in MATLAB®. This theory enables the efficient estimation of the sensitivities of an arbitrary response with respect to all parameters in the considered problem. These sensitivities are required in many applications including gradient-based optimization, surrogate-based modeling, statistical analysis, and yield analysis.

Modal logic originated in philosophy as the logic of necessity and possibility. Nowadays it has reached a high level of mathematical sophistication and found many applications in a variety of disciplines, including theoretical and applied computer science, artificial intelligence, the foundations of mathematics, and natural language syntax and semantics. This volume represents the proceedings of the first international workshop on Advances in Modal Logic, held in Berlin, Germany, October 8-10, 1996. It offers an up-to-date perspective on the field, with contributions covering its proof theory, its applications in knowledge representation, computing and mathematics, as well as its theoretical underpinnings. "This collection is a useful resource for anyone working in modal logic. It contains both interesting surveys and cutting-edge technical results" --Edwin D. Mares The Bulletin of Symbolic Logic, March 2002

Conceived by Johan van Benthem and Yde Venema, arrow logic started as an attempt to give a general account of the logic of transitions. The generality of the approach provided a wide application area ranging from philosophy to computer science. The book gives a comprehensive survey of logical research within and around arrow logic. Since the natural operations on transitions include composition, inverse and identity, their logic, arrow logic can be studied from two different perspectives, and by two (complementary) methodologies: modal logic and the algebra of relations. Some of the results in this volume can be interpreted as price tags. They show what the prices of desirable properties, such as decidability, (finite) axiomatisability, Craig interpolation property, Beth definability etc. are in terms of semantic properties of the logic. The research program of arrow logic has considerably broadened in the last couple of years and recently also covers the enterprise to explore the border between decidable and undecidable versions of other applied logics. The content of this volume reflects this broadening. The editors included a number of papers which are in the spirit of this generalised research program.

A publication of the University of Michigan’s Science Series, The Asymptotic Developments of Functions Defined by Maclaurin Series by Walter Burton Ford is an inquiry into the problem of functions defined by Maclaurin series. Here, Ford introduces his own theorem of asymptotic developments, as well as other mathematical theorems, and applies them to mathematical problems. This book was published with the hope of stimulating further research in the field.

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century.

Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. 

For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. 

Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing.