Mary Leng. Mathematics and Reality. Oxford: Oxford University Press, 2010. ISBN 978-0-19-928079-7. Pp. x + 278

Abstract

The position of scientific instrumentalism, found with variations in works from Vaihinger's Philosophie des Als Ob to van Fraassen's Scientific Image, has four components as follows. There is no good justification for believing that atoms and other unobservables exist; so it should be presumed that they do not. There is no serious hope of being able to explain away the commitment of current science to atoms as merely apparent. There is no serious hope of being able to develop an alternative science uncommitted to atoms. It is appropriate to suspend disbelief in atoms when doing science, so long as one recognizes when doing philosophy that atoms are merely useful fictions. The analogous mathematical instrumentalism consists of parallel components concerning numbers and other abstracta. In the mathematical case, skepticism is called nominalism. Until fairly recently instrumentalism was rather unpopular among nominalists. Hartry Field, for instance, in his Science without Numbers dismisses the option of ‘merely taking back in one's philosophical moments what one asserts in doing science’ in a sentence or two near the beginning of the book. Of late, however, a number of nominalists, notably Steve Yablo, have been moving in what is recognizably an instrumentalist direction. A fully worked out, book-length presentation of mathematical instrumentalism has, however, heretofore been lacking. Mary Leng's Mathematics and Reality fills this gap in the literature, providing a full-scale exposition and defense of a version of mathematical instrumentalism.