SHORTLISTED FOR THE 2017 ROYAL SOCIETY SCIENCE BOOK PRIZE Even small children know there are infinitely many whole numbers - start counting and you'll never reach the end. But there are also infinitely many decimal numbers between zero and one. Are these two types of infinity the same? Are they larger or smaller than each other? Can we even talk about 'larger' and 'smaller' when we talk about infinity? In Beyond Infinity, international maths sensation Eugenia Cheng reveals the inner workings of infinity. What happens when a new guest arrives at your infinite hotel - but you already have an infinite number of guests? How does infinity give Zeno's tortoise the edge in a paradoxical foot-race with Achilles? And can we really make an infinite number of cookies from a finite amount of cookie dough? Wielding an armoury of inventive, intuitive metaphor, Cheng draws beginners and enthusiasts alike into the heart of this mysterious, powerful concept to reveal fundamental truths about mathematics, all the way from the infinitely large down to the infinitely small.

This exploration of the scientific limits of knowledge challenges our deep-seated beliefs about our universe, our rationality, and ourselves. “A must-read for anyone studying information science.” —Publishers Weekly, starred review Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own intuitions about the world—including our ideas about space, time, and motion, and the complex relationship between the knower and the known. Yanofsky describes simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve: • perfectly formed English sentences that make no sense • different levels of infinity • the bizarre world of the quantum • the relevance of relativity theory • the causes of chaos theory • math problems that cannot be solved by normal means • statements that are true but cannot be proven Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there.

In the ten years preceding publication, the known solar system more than doubled in size. For the first time in almost two centuries an entirely new population of planetary objects was found. This 'Kuiper Belt' of minor planets beyond Neptune revolutionised our understanding of the solar system's formation and finally explained the origin of the enigmatic outer planet Pluto. This is the fascinating story of how theoretical physicists decided that there must be a population of unknown bodies beyond Neptune and how a small band of astronomers set out to find them. What they discovered was a family of ancient planetesimals whose orbits and physical properties were far more complicated than anyone expected. We follow the story of this discovery, and see how astronomers, theoretical physicists and one incredibly dedicated amateur observer came together to explore the frozen boundary of the solar system.

WHAT IS THE FORCE . That drove them on foot across the land bridge from Asia to the New World long before recorded history? That keeps Joshua Warden on the wagon train headed to the California gold fields after losing so much? That is found in a shovel full of prairie soil-demanding that Amos Krebbs put down roots? That directs Stephano Romano's return to the sea? That moved Harry Krebbs from the Kansas prairie and focuses his eyes and those of his astronaut son, Alan, on the far horizon . and beyond? That keeps the feet of industrialist Gilbert Krebbs and Congressman Bobby Dobbson firmly planted in the center? That drives a Monarch butterfly across a thousand miles of desert and mountains to a tiny spot on the California coast, where she has never been, but where she must return? That guides the sleek gray fish and the badly wounded Steve Romano to seek refuge in the Sea of Cortez? That fuels William Henry Stitt and his great grandson in their unending quest for knowledge? WHAT IS THE FORCE?

The most extensive, definitive work on the television classic "The Outer Limits", lavishly illustrated with photographs from the author's own collection.

A. The Outer Limits of the Continental Shelf The 1982 United Nations Convention on the Law of the Sea (herein- ter the “Convention”) marks the beginning of a new era in the law of 1 the sea. The negotiations for this treaty at the Third United Nations Conference for the Law of the Sea (hereinafter “UNCLOS III”), lasted for nine years, from 1973 to 1982. The Convention regulates the principal aspects of international oceans affairs. It establishes and fixes the limits of maritime zones, provides for the rights and duties of states in these zones, establishes the law app- cable in the international seabed area on the basis of the principle of common heritage of mankind, imposes obligations on states to protect the marine environment, and provides for the means of dispute sett- ment. One of the most contentious and divisive issues at UNCLOS III were the outer limits of the continental shelf. Previously, in the 1958 Con- 2 vention on the Continental Shelf (hereinafter the “1958 Convention”), no limits were established for the continental shelf. States were allowed to claim areas of continental shelves based on their capacity to exploit the mineral resources of the shelf. The legal framework in the 1958 Convention would obviously conflict with the principle of the common heritage of mankind. Delegates realized that limits have to be est- lished, but up to where and on the basis of which principles, was a c- tentious question.

Throughout its long history, and not just as the key aesthetic category for the Romantic Movement, the sublime has created the necessary link between aesthetic and moral judgment, offering the prospect of transcending the limits of measurement, even imagination. The best of science makes genuine claims to the sublime. For in science, as in art, every day brings the entirely new, the extreme, and the unrepresentable. How does one depict negative mass, for example, or the folding of a protein that is contagious? Can one capture emergent phenomena as they emerge? Science is continually faced with describing that which is beyond. This book, through contributions from nine prominent scholars, tackles that challenge. The explorations within Beyond the Finite range from the images taken by the Hubble Telescope to David Bohm's quantum romanticism, from Kant and Burke to a "downward spiraling infinity" of the 21st century sublime, all lucid yet transcendent. Squarely positioned at the interface between science and art, this volume's chapters capture a remarkable variety of perspectives, with neuroscience, chemistry, astronomy, physics, film, painting and music discussed in relation to the sublime experience, topics surely to peak the interest of academics and students studying the sublime in various disciplines.

New Knowledge and Changing Circumstances in the Law of the Sea focuses on the challenges posed to the existing legal framework, in particular the United Nations Convention on the Law of the Sea, and the various ways in which States are addressing these challenges.