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The Inka Empire stretched over much of the length and breadth of the South American Andes, encompassed elaborately planned cities linked by a complex network of roads and messengers, and created astonishing works of architecture and artistry and a compelling mythology—all without the aid of a graphic writing system. Instead, the Inkas' records consisted of devices made of knotted and dyed strings—called khipu—on which they recorded information pertaining to the organization and history of their empire. Despite more than a century of research on these remarkable devices, the khipu remain largely undeciphered.

In this benchmark book, twelve international scholars tackle the most vexed question in khipu studies: how did the Inkas record and transmit narrative records by means of knotted strings? The authors approach the problem from a variety of angles. Several essays mine Spanish colonial sources for details about the kinds of narrative encoded in the khipu. Others look at the uses to which khipu were put before and after the Conquest, as well as their current use in some contemporary Andean communities. Still others analyze the formal characteristics of khipu and seek to explain how they encode various kinds of numerical and narrative data.

There is no question that native cultures in the New World exhibit many forms of mathematical development. This Native American mathematics can best be described by considering the nature of the concepts found in a variety of individual New World cultures. Unlike modern mathematics in which numbers and concepts are expressed in a universal mathematical notation, the numbers and concepts found in native cultures occur and are expressed in many distinctive ways. Native American Mathematics, edited by Michael P. Closs, is the first book to focus on mathematical development indigenous to the New World.

Spanning time from the prehistoric to the present, the thirteen essays in this volume attest to the variety of mathematical development present in the Americas. The data are drawn from cultures as diverse as the Ojibway, the Inuit (Eskimo), and the Nootka in the north; the Chumash of Southern California; the Aztec and the Maya in Mesoamerica; and the Inca and Jibaro of South America. Among the strengths of this collection are this diversity and the multidisciplinary approaches employed to extract different kinds of information. The distinguished contributors include mathematicians, linguists, psychologists, anthropologists, and archaeologists.

A rethinking of Cantor and infinitary mathematics by the creator of Vopěnka's principle.
 
The dominant current of twentieth-century mathematics relies on Georg Cantor’s classical theory of infinite sets, which in turn relies on the assumption of the existence of the set of all natural numbers, the only justification for which—a theological justification—is usually concealed and pushed into the background.

This book surveys the theological background, emergence, and development of classical set theory, warning us about the dangers implicit in the construction of set theory, and presents an argument about the absurdity of the assumption of the existence of the set of all natural numbers. It instead proposes and develops a new infinitary mathematics driven by a cautious effort to transcend the horizon bounding the ancient geometric world and mathematics prior to set theory, while allowing mathematics to correspond more closely to the real world surrounding us. Finally, it discusses real numbers and demonstrates how, within a new infinitary mathematics, calculus can be rehabilitated in its original form employing infinitesimals.

Non-well-founded structures arise in a variety of ways in the semantics of both natural and formal languages. Two examples are non-well-founded situations and non-terminating computational processes. A natural modelling of such structures in set theory requires the use of non-well-founded sets. This text presents the mathematical background to the anti-foundation axiom and related axioms that imply the existence of non-well-founded sets when used in place of the axiom of foundation in axiomatic set theory.

Nonmonotonic reasoning is a subfield of Artificial Intelligence trying to find more realistic formal models of reasoning than classical logic. In common sense reasoning one often draws conclusions that have to be withdrawn when further information is obtained. The set of conclusions thus does not grow monotonically with the given information. It is this phenomenon that nonmonotonic reasoning methods try to formalize. This volume gives an overview on recent results in the field and points to relevant literature for further study. This up-to-date survey of research in the area of nonmonotonic reasoning includes a concise description of the most influential nonmonotonic logics (e.g. circumscription, autoepistemic logic and default logic), a presentation of recent research in abduction, as well as an overview of semantics for logic programs with default negation. The primary goal of this volume is to make recent results in the field more accessible. An extensive bibliography is included.

A revolutionary notebook that challenges us to play outside (and with) the lines.
 
A standard notebook displays page after page of horizontal lines. But what if we break the pattern? What if the ruled pages grew unruly? In this Nonstandard Notebook, lines twist, fragment, curve, and crisscross in beautiful formations. Each sheet is a distinctive work of imagination, asking us to draw, doodle, and journal in the same spirit.
 
Page after page, as we journey from lines to parabolas to waves, deep questions arise—about form, art, and mathematics. How do we harness the infinite? Why do patterns permeate nature? What are the limitations and possibilities of human vision? Nonstandard Notebook explores these questions and more through its provocative and inspirational images, each displayed with the mathematics that generated it. We see how straight lines can form fractal crenellations, how circles can disrupt and unify, and how waves and scaling can form complex landscapes (or even famous faces). Created by mathematicians, educators, and math popularizers Tim Chartier and Amy Langville, and with a foreword from Ben Orlin (bestselling author of Math with Bad Drawings), Nonstandard Notebook shows that rules—both the rules of mathematics and the rules of a notebook—do not mark the end of creativity, but the beginning.

A classic on mind, matter, and the unified world model

C. G. Jung's work in his later years suggested that the seemingly divergent sciences of psychology and modern physics might, in fact, be approaching a unified world model in which the dualism of matter and psyche would be resolved into “one world” or Unus Mundus. Jung believed that the natural integers are the archetypal patterns that regulate the unitary realm of psyche and matter, and that number serves as a special instrument for man's becoming conscious of this unity.

Written in a clear style and replete with illustrations which help make the mathematical ideas visible, Number and Time is a piece of original scholarship which introduces a view of how "mind" connects with "matter" at the most fundamental level.