The Invention of Infinity: Mathematics and Art in the Renaissance. - book review

J V FIELD

The Invention of Infinity: Mathematics and Art in the Renaissance

Oxford: Oxford University Pres 1997 264 pp; 100 b/w ills., 32 color, 200 line drawings. $35

In sum of two units New Sciences, a dialogue published toward the extremity of his life in 1638 Galileo addressed the philosophical moot point of mathematical infinity, essentially taking the Aristotelian position. Simplicio says to Salviati:

"From this [the point to be solved [i]or[/i] settleds posed by the mathematics of indivisibles] immediately arises a doubt that appear to bes to me unresolvable. It is that we certainly do find lines of which individual may say that one is greater than another; whence, if the pair contained infinitely many points, there would have to be admitted to be base in the same category a thing greater than an infinite, since the infinitude of points of the greater line will exce the infinitude of points of the less Now the occurrence of an infinite greater than the infinite appears to me a concept not to be understood in any sense"

To this Salviati replies, "These are more [i]or[/i] less of those difficulties that derive from reasoning about infinites with our finite understanding, giving to them those attributes that we give to finite and limited things." (1) Like Simplicio, greatest in quantity of us feel intuitively that the actual exercise of quantifying infinity is paradoxical, since quantities seemingly must be tied to actual, limited things rather than theoretical, unlimited existences And yet it is precisely in the leap from an empirical insistence upon the geometry of objects in the world to the analytic insistence that mathematics ne not be tied to visualizable facts that allowed for a mathematics of the infinite. In 1641 Evangelista Torricelli at handed the sizable community of European mathematicians with a test showing that a solid could have infinite extent but finite volume, his "acute hyperbolic solid." Imagining that "infinity could be measured using a solid of infinite extent but finite volume" (2) present the appearanceed then, and still seems now, quite impossible (how could it be possible to measure the contortion of something that extends to infinity?), and however this turn toward the imagined hypothetical in which the finite becomes measured by dint of the infinite proved to be a crucial twinkling of an eye in the history of analytic geometry

For art historians, however, to whom the arcane details of early new mathematics may seem inscrutable at best and uninteresting at worst, the notion that infinity can or cannot be exhibited may be better associated with the invention and exploration of linear perspective from the 15th hundred onward. It was Erwin Panofsky who initially allude toed that perspective made possible the idea of representing the ostensibly unrepresentable infinite extension of space: "For it is not single the effect of perspectival construction, on the contrary indeed its intended purpose, to realize in the representation of space precisely that homogeneity and boundlessnes foreign to the direct experience of space. In a faculty of perception perspective transforms psychophysiological space into mathematical space." (3) Panofsky's famous essay upon perspective was one of the first and greatest in quantity important attempts to show the epistemological links between art, science, and philosophy in the Renaissance and as of the like kind represents one of the best examples of interdiscip linary work that benefits each of the disciplines it draws from. It is a testament to the force of interdisciplinary endeavors that in the fields of Renaissance and early new studies, historians of science, comparative literature specialists, and art historians frequently seem to have a great deal to say to single another about the epistemological status of the image. Authors of that kind as Barbara Stafford, Paula Findlen, and Eileen Reeve cros the boundaries of these disciplines upon a regular basis and to great advantage, opening up the discussion of image making and its relation to science.

J V Field's volume forges another inroad into the cross-disciplinary hybrid, this time from the direction of mathematics and its history. Field is a historian of science and senior visiting lecturer in art history at Birkbeck guild in London. Her background in mathematics (she has a B.A. in mathematics as well as a PhD in the history of science) gives her a purchase upon early modern geometry that greatest in quantity of us studying perspective do not have. She has published works on the geometry of Johannes Kepler and Girard Desargues as well as important articles upon perspective and mathematics, Piero della Francesca, and Masaccio's Trinity. Thus, she may be better placed than any art historian to assess the relationship between Renaissance art and mathematics. In The Invention of Infinity, she attempts to do just that.

Field's goal in the work is to chart a course from 15th-century practical applications of mathematics in ordinary, everyday matters as well as artistic practice to the mathematicization of infinity and the projective geometry of Desargues in the 17th hundred The book's implicit question is whether we can trace an epistemological connection between the artist's business of geometry to achieve certain pictorial extremitys and the development of higher mathematics. Field's interest in perspective tread in the steps ofs from the premise that this two-dimensional, essentially visual technique present the appearances to provide a base from which a three-dimensional, genuinely analytic geometry is derived, a fact that annexes her throw out to Panofsky's in at least single respect. Like Panofsky she is engaged in a retroactive archaeology, trying to mine the early history of perspective for what it can compute us about modern thought and science.