# The Invention of Infinity: Mathematics and Art in the Renaissance

## Article excerpt

J. V. FIELD

The Invention of Infinity: Mathematics

and Art in the Renaissance

Oxford: Oxford University Press, 1997.

264 pp.; 100 b/w ills., 32 color, 200 line

drawings. \$35

In Two New Sciences, a dialogue published toward the end of his life in 1638, Galileo addressed the philosophical problem of mathematical infinity, essentially taking the Aristotelian position. Simplicio says to Salviati:

"From this [the problems posed by the mathematics of indivisibles] immediately arises a doubt that seems to me unresolvable. It is that we certainly do find lines of which one may say that one is greater than another; whence, if both contained infinitely many points, there would have to be admitted to be found in the same category a thing greater than an infinite, since the infinitude of points of the greater line will exceed the infinitude of points of the lesser. Now the occurrence of an infinite greater than the infinite seems to me a concept not to be understood in any sense.

To this Salviati replies, "These are some of those difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and bounded things."' Like Simplicio, most of us feel intuitively that the very exercise of quantifying infinity is paradoxical, since quantities seemingly must be tied to actual, limited objects rather than theoretical, unlimited objects. And yet it is precisely in the leap from an empirical insistence on the geometry of objects in the world to the analytic insistence that mathematics need not be tied to visualizable objects that allowed for a mathematics of the infinite. In 1641, Evangelista Torricelli presented the sizable community of European mathematicians with a proof showing that a solid could have infinite length but finite volume, his "acute hyperbolic solid." Imagining that "infinity could be measured using a solid of infinite length but finite volume"2 seemed then, and still seems now, quite impossible (how could it be possible to measure the volume of something that extends to infinity?), and yet this turn toward the imagined hypothetical in which the finite becomes measured by the infinite proved to be a crucial moment in the history of analytic geometry.

For art historians, however, to whom the arcane details of early modern mathematics may seem inscrutable at best and uninteresting at worst, the notion that infinity can or cannot be represented may be better associated with the invention and exploration of linear perspective from the 15th century onward. It was Erwin Panofsky who initially suggested that perspective made possible the idea of representing the ostensibly unrepresentable infinite extension of space: "For it is not only the effect of perspectival construction, but indeed its intended purpose, to realize in the representation of space precisely that homogeneity and boundlessness foreign to the direct experience of space. In a sense, perspective transforms psychophysiological space into mathematical space."3 Panofsky's famous essay on perspective was one of the first and most important attempts to show the epistemological links between art, science, and philosophy in the Renaissance and as such represents one of the best examples of interdisciplinary work that benefits each of the disciplines it draws from. It is a testament to the strength of interdisciplinary endeavors that in the fields of Renaissance and early modern studies, historians of science, comparative literature specialists, and art historians often seem to have a great deal to say to one another about the epistemological status of the image. Authors such as Barbara Stafford, Paula Findlen, and Eileen Reeves cross the boundaries of these disciplines on a regular basis and to great advantage, opening up the discussion of image making and its relation to science.

J. V. Field's book forges another inroad into the cross-disciplinary hybrid, this time from the direction of mathematics and its history. …