This book is about the relationship between necessary reasoning and visual experience in Charles S. Peirce’s mathematical philosophy. It presents mathematics as a science that presupposes a special imaginative connection between our responsiveness to reasons and our most fundamental perceptual intuitions about space and time. Central to this view on the nature of mathematics is Peirce’s idea of diagrammatic reasoning. In practicing this kind of reasoning, one treats diagrams not simply as external auxiliary tools, but rather as immediate visualizations of the very process of the reasoning...

In 1940, two American mathematicians, Kasner and Newman, published a book titledMathematics and the Imaginationwhich was met with resounding success. The main claim of the book was that mathematics was a product of the imagination, arguably linking the two for the first time in the history of mathematics and even psychology. The ideas in Kasner and Newman’s work coincided largely with those of the eighteenth-century Italian philosopher Giambattista Vico (1688-1744), sometimes called the “philosopher of the imagination.” Vico put forth his views on how the imagination (which he called...