It is well known that mathematical ideas and techniques have long dominated the physical sciences. And in the twentieth century social and biological scientists began to mathematize their disciplines. But these are not, at least to me, the most intellectually interesting uses of mathematical ideas outside of mathematics. I have always been much more intrigued by appeals to mathematical ideas by humanists, by which I mean those who seek to understand our common humanity and our place in the cosmos.It seems to me that the best way to explore these non-scientific uses of mathematical ideas is to present them in context. Since the historical scope of What is a Number? is vast, beginning in the sixth century B.C.E. and ending in the present, I have focused primarily, but not exclusively, on three periods and themes: ancient attempts to understand the fundamental nature of reality; medieval arguments supporting a faith-based view of the cosmos, and refuting portions of the ancient one; and modern appeals to mathematical concepts to promote a conception of reality or an aesthetic theory.The mathematics of each of these periods, and so the conclusions reached by those appealing to its ideals, have a distinctive flavor. At the risk of over simplifying, ancient Greek mathematics was based on the ratio of whole numbers and on geometric perfection. This led to a universe having harmony as its fundamental organizing principle, to art based on ratio and proportion, and to a theory of matter based on geometric beauty. Although some mathematical and observational discoveries were contrary to this aesthetic, its real challenges came in the Middle Ages with attempts to reconcile these ancient ideas with presumed theological truths. Medieval theologians and philosophers had to perform a delicate balancing act—retain those features of ancient natural philosophy that agreed with their worldview (e.g., the finite, geometrically perfect cosmos) and reject those that did not (e.g., the deduction that geometric perfection in the heavens implies that time must be cyclic). To achieve this balance many theologians turned to mathematical principles and embraced previously suspect mathematical concepts, such as irrational numbers or infinite collections.These examinations reveal several interesting things. One is that the mathematical ideas themselves were not so well understood as those employing them might have hoped. Another is that the way someone conceives of a certain concept permits them to demonstrate a proposition just as well as its contrary—by using different conceptions of a number the ancients demonstrated that time is cyclic and medieval theologians that it is not. And , that it is possible to appeal to a presumed mathematical truth for reaching different conclusions—knowledge of the behavior of parallel lines in Euclidean geometry and an examination of shadows on the earth were used to prove both that the earth is flat and that it is round.I find that these obscure medieval, theological arguments, which indeed included discussions of the properties of angels, allowed for many of the artistic and philosophical breakthroughs of the Renaissance and subsequently of the modern era. Questioning the ancients eventually undermined the theory of vision that had been described by Euclid and allowed for the invention of the single-point perspective method of painting. Allowing for existing mathematical infinitudes led some to the conclusion that there should be no bound to God’s benevolence and creative powers and so the cosmos must be infinite and contain an infinitude of inhabited worlds. Accepting irrational numbers allowed for thinking about mathematical results separate from physical ones and led to a view of mathematics as a purely axiomatic system.This last point eventually yielded the discoveries, in the nineteenth century, that geometry need not necessarily be Euclidean, so alternate geometric truths are possible, and, in the twentieth century, that mathematical truth itself is difficult to identify. However, these discoveries have neither reduced the appeals by humanists to mathematical ideas nor demoralized mathematicians. Writers and artists have explored both fractured, and plastic, space and time; theologians and cosmologists have readily embraced infinitude and multiple dimensions.What I find to be especially interesting is that a sort of mysticism has returned to mathematics. Mathematicians are often praised more highly for their insights than for their proofs. And these insights are judged not by how well they mesh with the ancient worldview based on harmony and geometric perfection, but by a new aesthetic—elegance.