Infinity

“The infinite!” wrote the great David Hilbert, “No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification....” Gathering essays from 13 noted scholars, Infinity explores this concept through the prism of mathematics and history as well as physics, cosmology, and theology.

Mathematical ideas of infinity

The book is organized into five main sections. The first features an overview that explores how investigations of infinity have expanded across various disciplines. Infinity as a concept arises in many different contexts—mathematics, physics, metaphysics, theology, psychology, and the arts—and it is interesting to trace the parallels between infinity when applied, say, to mathematical sets as opposed to notions of the divine.

The next two sections focus on the topic of infinity in mathematics. The first of these begins with an historical survey of the mathematical areas in which infinity has cropped up, running from the Pythagoreans to the P vs. NP problem in computer science. It’s followed by a contrarian essay arguing that the notion of infinity is somehow inconsistent; it takes an extreme finitist tack, and presents a case that even the existence of very large sets is questionable.

The next section on mathematics kicks off with an argument for a maximally rich universe of set theory. It seeks to undermine the so-called “generic-multiverse position,” which suggests that there is no one true model in which something like the continuum hypothesis is definitely true or false. It is followed by an essay drawing on deep results from recursion theory and nonstandard model theory to argue that there really is no coherent distinction between free will and determinism. The final piece in this section takes a novel approach to questions about the consistency or inconsistency of set theory.

Physics, cosmology, and beyond

In the book’s fourth section, we ponder whether infinity can be found in the real universe studied by physics and cosmology. According to the opening essay, if we view the space of our three-dimensional universe as curved into the hypersurface of a hypersphere, we are able to have a space that is both finite and unbounded. A second piece argues that even if we adopt the Big Bang scenario—in which the universe in some sense sprang into being at some finitely removed past time, it is possible for the universe to be infinitely large. Furthermore, even though it appears as if a Big Bang occurred, it may also be that our past timeline is, in fact, infinite. Even more intriguing, we can have an infinite number of inflationary “pocket universes” coexisting, and these pocket universes may themselves be infinite. A related idea is that of the multiverse, and a separate essay notes that this concept, while popular in some quarters today, lacks a certain aesthetic appeal. The last piece in this section distinguishes between two kinds of infinities in cosmology—“infinitely distant” regions such as one might find beyond the boundary of the observable universe, and “infinitely divergent” regions called singularities, where the standard structure of spacetime breaks down—and we see how cosmologists manage to formulate theories and draw conclusions about their physical characteristics.

The final section of the book explores further historical, philosophical, and theological aspects of infinity, based on Western metaphysics and the Judaeo-Christian tradition. Among the essays is one arguing that the starting point of the metaphysical notion of infinity is the notion of absolute indeterminacy. Two other essays pursue potential connections between theology and set theory.

Impressive in scope, Infinity offers a snapshot of thinking about a concept that has always intrigued humanity—and always will.