The importance of imagination in mathematics has often been remarked by mathematicians. The great English mathematician J. J. Sylvester (1814-1897) thus wrote:

As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to simulate the faculty of invention.

"A plea for the mathematician", Nature 1 (1869), 261

This is an interdisciplinary project that brings together philosophers, logicians, historians and mathematicians to work on problems and themes of common interest. The general aim is to improve our understanding of the role of ‘imaginary’ or ‘ideal’ elements in mathematics. The more particular aims are (i) to identify and distinguish the historically important attempts to justify the use of imaginary and ideal methods, (ii) to clarfy the differences between these and (iii) to gain a more systematic understanding of the similarities and differences between imaginary/ideal methods, on the one hand, and more ordinary (or real) methods on the other. The investigators are grateful to the Alexander von Humboldt Stiftung (TransCoop Program) and the University of Notre Dame for generous financial support.