The philosophical approach of this volume is mainly structuralist, using logical tools to investigate the formal structure of various kinds of objects in our world, as characterised by language and as systematised by philosophy. This volume mainly analyses the structural properties of collections or pluralities (with applications to the philosophy of set theory), homogeneous objects like water, and the semantics and philosophy of events. This book thereby complements algebraic work that has been done on other philosophical entities, i.e. propositions, properties, relations, or situations. Located in the triangle of language, logic and philosophy, this volume is unique in combining the resources of different ¹elds in an interdisciplinary enterprise. Half of the fourteen chapters of this volume are original papers, complementing the collection of the author's previously published essays on the subject.

The authors report the results of some half dozen years of research into when and how children acquire numerical skills. They provide a new set of answers to these questions, and overturn much of the traditional wisdom on the subject.

The award-winning former editor of Science News shows that one of the most fascinating and controversial ideas in contemporary cosmology—the existence of multiple parallel universes—has a long and divisive history that continues to this day.

We often consider the universe to encompass everything that exists, but some scientists have come to believe that the vast, expanding universe we inhabit may be just one of many. The totality of those parallel universes, still for some the stuff of science fiction, has come to be known as the multiverse.

The concept of the multiverse, exotic as it may be, isn’t actually new. In The Number of the Heavens, veteran science journalist Tom Siegfried traces the history of this controversial idea from antiquity to the present. Ancient Greek philosophers first raised the possibility of multiple universes, but Aristotle insisted on one and only one cosmos. Then in 1277 the bishop of Paris declared it heresy to teach that God could not create as many universes as he pleased, unleashing fervent philosophical debate about whether there might exist a “plurality of worlds.”

As the Middle Ages gave way to the Renaissance, the philosophical debates became more scientific. René Descartes declared “the number of the heavens” to be indefinitely large, and as notions of the known universe expanded from our solar system to our galaxy, the debate about its multiplicity was repeatedly recast. In the 1980s, new theories about the big bang reignited interest in the multiverse. Today the controversy continues, as cosmologists and physicists explore the possibility of many big bangs, extra dimensions of space, and a set of branching, parallel universes. This engrossing story offers deep lessons about the nature of science and the quest to understand the universe.

Ranging from math to literature to philosophy, Uncountable explains how numbers triumphed as the basis of knowledge—and compromise our sense of humanity.

Our knowledge of mathematics has structured much of what we think we know about ourselves as individuals and communities, shaping our psychologies, sociologies, and economies. In pursuit of a more predictable and more controllable cosmos, we have extended mathematical insights and methods to more and more aspects of the world. Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity. Yet, in the process, are we losing sight of the human? When we apply mathematics so broadly, what do we gain and what do we lose, and at what risk to humanity?

These are the questions that David and Ricardo L. Nirenberg ask in Uncountable, a provocative account of how numerical relations became the cornerstone of human claims to knowledge, truth, and certainty. There is a limit to these number-based claims, they argue, which they set out to explore. The Nirenbergs, father and son, bring together their backgrounds in math, history, literature, religion, and philosophy, interweaving scientific experiments with readings of poems, setting crises in mathematics alongside world wars, and putting medieval Muslim and Buddhist philosophers in conversation with Einstein, Schrödinger, and other giants of modern physics. The result is a powerful lesson in what counts as knowledge and its deepest implications for how we live our lives.