Due to the historical scope of this book there are several themes that weave through the entire narration. Take, for example, the attempts of various writers to understand the concept of infinity. In chapter 3 I delineate three different notions of infinity: poetic, metaphysical, and quantitative. Aristotle, whose ideas about infinity dominated western thought for at least a millennium, used geometric principles to explain his understanding of quantitative infinity. This analysis led Aristotle to reject the belief that there could exist an infinitude, either physical or mathematical. Aristotle combined his rejection of existing infinitudes with ideas from his physics to conclude that space could be endlessly subdivided; there is no shortest distance. By examining the nature of motion Aristotle then concluded that time could also be endlessly subdivided; so there is no shortest increment of time.One consequence of Aristotle’s view of space and time is that if there were to exist an indivisible piece of matter (like an atom without distinguishable parts) then its motion would be impossible. This conclusion troubled many medieval theologians because of its implications for angels. One conception of angels was that they were made of an incorruptible substance. But an incorruptible substance cannot be made up of other substances so an angel cannot have distinguishable parts—in Aristotle’s terms an angel is indivisible. Since Aristotle had concluded that any such entity could not move, and Christian cosmology employed angels in the motions of the heavenly bodies, some theologians sought to undermine Aristotle’s argument and began with his conception of space and time. In doing this, because of the almost unquestioned correspondence between the physical and mathematical worlds, these theologians sought to demonstrate that a geometric continuum could not be endlessly subdivided. More specifically, they sought to demonstrate that a line segment was made up of only finitely many points.In the fourteenth century this topic was widely discussed. One argument attempting to demonstrate that a line segment contains only a finite number of points invoked God’s omniscience: If a line segment can be endlessly subdivided then even God could not know all of its points. Not wanting to delve into purely theological issues, this argument was refuted by a result from mathematics: If a line segment consists of only finitely many points then, by looking at the side and diagonal of a square, it is possible to conclude that the square root of two is a ratio of whole numbers (contrary to its known irrationality). At least one theologian got around this last objection by re-imagining the nature of a line segment, but others sought less radical counters. One of these made the acceptance of existing infinitudes more acceptable: If a line segment consisted of only finitely many points then an omniscient God would see gaps between them. But there are no gaps in a line segment so it must consist of infinitely many adjacent points.What these arguments show is that scholastics struggled with conflicts between their theology, the concept of an all-powerful God, their understanding of, for lack of a better word, physics, and the then-current conception of mathematical objects. Moreover, these discussions, and the attempts to defend positions of faith with mathematical ideas, assisted in the acceptance of previously rejected mathematical concepts—in the brief example above, irrational numbers and existing infinitudes.What is a Number? could mistakenly be taken as critique of mathematics or of the typical mathematicians’ view that, despite twentieth century discoveries in logic, mathematical certainty can be achieved. Even worse, this book could be seen as being critical of humanists who might import mathematical ideas into their work. The book is neither. Instead, What is a Number? attempts to refute the received wisdom that mathematical concepts are divorced from other intellectual pursuits—revealing them instead to be dynamic, even malleable, and influential in many humanists’ enquiries. The realization that these concepts are imprecisely defined, or that mathematical truth is a contingent thing, only further supports the view that mathematical ideas are as closely allied with those of the humanist as the scientist.This last claim is perhaps striking because it is contrary to the contemporary view of mathematics. Mathematics is usually regarded as a handmaiden of the sciences, where in the last three centuries it has led to significant advances. But for over two millennia mathematical ideas and concepts have influenced humanistic thought. Mathematical ideas have been used not only by humanists who profess a rational predisposition but also by modern artists, existentialist philosophers, and mystical theologians. Indeed, mathematical ideals seem to be inseparable from many of those that define our humanness.