If you were asked to think of a scientist, and next to imagine an artist, then probably two quite different persons will appear in your mind's eye. However, when you think of it more, it is clear that art and science are actually quite closely related. There is a mathematical theory of music, architects have to deal with the reality of physical laws, painters apply geometric properties while the laws of optics rule their observation, etc. In particular in the Renaissance the science of art and the art of science went hand in hand.
Martin Kemp in his book The Science of Art (Yale University Press, 1990) emphasized how (optical) science has influenced art since Brunelleschi's (1377-1446) invention of perspective. This book wants to invert this idea and argues that science developed by using artistic `inventions'. The artist does not always paint what is real. For example parallel lines may not be parallel in a painting. The artist gives form to a symbol that represents an idea. This is in line with Ernst Cassirer's theory of symbolism. This suggests that here a conceptual vision of the world changed and modern science arose by moving from the concrete to the abstract. The visual is an essential element in the development of science. A picture is a (sometimes simplified) observable form of what can be imagined. Like Leonardo Da Vinci has an `pictorial style', Hilbert's `mathematical style' is his `general theory of forms' and Hermann Weyl's Symmetry (Princeton University Press, 1952) can be seen as an `art guide' to science. The Renaissance did not only change the way artists represented reality, but it also triggered new visions that were decisive for the way in which mathematics developed.
The argumentation for this point of view is given in eight essays by different authors. Some contributions are more mathematical, others are more philosophical. The subtitle of this book From Perspective Drawing to Quantum Randomness reflects both the period and the span of subjects that are covered.
This is the table of contents.
I. Ways of Perspective
- 1. From perspective drawing to the eight dimension (John Stillwell)
- 2. Seeing reality in perspective: the art of optics and the science of painting (Nader El-Birzi)
- 3. The role of perspective in the transformation of European culture (Dalibor Vesely)
- 4. Visual differential geometry and Beltrami's hyperbolic plane (Tristan Needham)
- 5. All done by mirrors: symmetries, quaternions, spinors, and Clifford algebras (Simon Altmann)
II. The Complex Route
- 6. Artists and gambles on the way to quantum physics (Annarita Angelini & Rossella Lupacchini)
- 7. Radices sophisticae, racines imaginaires: the origins of complex numbers in the late Renaissance (Veronica Gavagna)
- 8. Random, complex, and quantum (Artur Ekert)
The first part concentrates on the impact of the discovery of perspective on the way mathematics has evolved. Before Brunelleschi, the natural perspective was a matter of optics. It is the way how light travels in `the real world', i.e., not on the canvas. But when the rules of perspective were discovered, the rules became internal to the painting. The painting got its own internal mathematical rules. That became a `science of painting' (El-Birzi). Although it also introduced a new approach to the science of optics and to science in general (Vesely). On a canvas the classical geometric rules do not always work. For example parallel lines meet in the vanishing point. This stimulated the discovery of projective geometry. The latter allowed for a geometric study of algebra and hence the analysis of what kind of algebra is possible in higher dimensions, which leads to complex numbers, quaternions, octonions,... (Stillwell). On the geometric trail, Lobachevsky and Bolyai gave an axiomatic definition of non-Euclidean geometry, but Eugenio Beltrami's role is under-estimated. Differential geometry illustrates his important contribution: hyperbolic geometry can be represented on a surface with negative curvature (Needham). In this context it is only a side remark, but only few people are aware that Beltrami was also the first to introduce the singular value decomposition. Although symmetry has always been an artistic element, there is yet a distinctive lack of symmetry in paintings. Portraits show the right cheek predominantly, motion is better appreciated if it is suggested to move from left to right, and there is a preference for an upward diagonal composition. All this left-right and upside-down symmetries also show up in mathematical objects like quaternions, spinors, etc. So there is some art in science as well (Altmann).
Renaissance is not only the period where perspective became available, but it is also the birth period of complex numbers and probability. While the first part sketched the role of perspective, the second part focusses on the impact of complex numbers and probability on modern quantum physics. Cardano (1501-1576) was the originator of both probability (he was a notorious gambler) and the complex numbers. The latter showed up when applying formulas for the roots of a cubic. Sometimes `imaginary' roots resulted when the square roots had to be taken from negative numbers, however, when applying certain `sign rules' in the computations, arithmetic still worked (Gavagna). Randomness and complex numbers meet in quantum theory where a complex amplitude for probability of an event is needed, where the term making the difference with the classical theory reflects the quantum interference (Ekert). In their own contribution, the editors go though all the aspects of the second part, based on Cardano's work.
There are many books that link art and mathematics, but this one is unique in that it brings a distinct message: the ideas that revolutionized painting in the Renaissance can be applied to the evolution of mathematics. Artists produce paintings and scientists develop new theories on a similar basis. What perspective was for painters, is what complex numbers and probability are for quantum physics. One should not read the book for the rules of perspective, for mathematics, and perhaps not even for the history of mathematics. Neither is it a treatise on aesthetics. Of course all this is present, and some of the contributions are rather mathematical and historical, but the main objective is the philosophical-epistomologocal analysis of the interplay, sometimes touching on the metaphysics. It's a philosophical book written for mathematicians. So, if you are a mathematician, be prepared to interpolate some time for reflection while reading the book.