Michelle Karnes. University of Chicago Press, 2011.

“In Imagination, Meditation, and Cognition in the Middle Ages, Michelle Karnes revises the history of medieval imagination with a detailed analysis of its role in the period’s meditations and theories of cognition. Karnes here understands imagination in its technical, philosophical sense, taking her cue from Bonaventure, the thirteenth-century scholastic theologian and philosopher who provided the first sustained account of how the philosophical imagination could be transformed into a devotional one. Karnes examines Bonaventure’s meditational works, the Meditationes vitae Christi, the Stimulis amoris, Piers Plowman, and Nicholas Love’s Myrrour, among others, and argues that the cognitive importance that imagination enjoyed in scholastic philosophy informed its importance in medieval meditations on the life of Christ. Emphasizing the cognitive significance of both imagination and the meditations that relied on it, she revises a long-standing association of imagination with the Middle Ages. In her account, imagination was not simply an object of suspicion but also a crucial intellectual, spiritual, and literary resource that exercised considerable authority.”

University of Chicago Press, May 15, 2010 – Science – 416 pages

“Nineteenth-century chemists were faced with a particular problem: how to depict the atoms and molecules that are beyond the direct reach of our bodily senses. In visualizing this microworld, these scientists were the first to move beyond high-level philosophical speculations regarding the unseen. In Image and Reality, Alan Rocke focuses on the community of organic chemists in Germany to provide the basis for a fuller understanding of the nature of scientific creativity.

Arguing that visual mental images regularly assisted many of these scientists in thinking through old problems and new possibilities, Rocke uses a variety of sources, including private correspondence, diagrams and illustrations, scientific papers, and public statements, to investigate their ability to not only imagine the invisibly tiny atoms and molecules upon which they operated daily, but to build detailed and empirically based pictures of how all of the atoms in complicated molecules were interconnected. These portrayals of “chemical structures,” both as mental images and as paper tools, gradually became an accepted part of science during these years and are now regarded as one of the central defining features of chemistry. In telling this fascinating story in a manner accessible to the lay reader, Rocke also suggests that imagistic thinking is often at the heart of creative thinking in all fields.

Image and Reality is the first book in the Synthesis series, a series in the history of chemistry, broadly construed, edited by Angela N. H. Creager, John E. Lesch, Stuart W. Leslie, Lawrence M. Principe, Alan Rocke, E.C. Spary, and Audra J. Wolfe, in partnership with the Chemical Heritage Foundation.”

I’ve been reading Benoit Mandelbrot’s recent autobiography, The Fractalist. For anyone who hasn’t run across fractals, the famous mandelbrot set, or Julia sets, Wikipedia has a good starting point. You can also enjoy this video:

If you’re still here, Mandelbrot was the mathematician who coined the term fractal and brought their visual beauty to life. His work has no doubt resonated far beyond academic mathematics precisely because it is so graphic and engrossing. Although I’m still in the early days of his autobiography, Mandelbrot has already introduced a couple themes he sees as central to the course of his life: his early predisposition to think geometrically, and his varied interests — what some people would call unfocused, but which he sees as different angles on a common topic.

As Mandelbrot grew up, he moved from Poland to France to escape growing anti-semitism, and then spent much of his adolescent years hiding in rural France during the Nazi occupation of World War II. Somehow Mandelbrot learned to think about mathematics in terms of shapes and geometries: he credits some early textbooks that were sufficiently introductory they hadn’t eliminated pictures in favor of equations and algebra. Already as he prepared to enter college, he ran up against this same obstacle: the common mathematical culture that values the ability to manipulate equations over using diagrams. Fortunately, Mandelbrot had sufficient presence of mind to be able to translate difficult algebraic problems (such as triple integrals) into geometric terms quickly, so that he still excelled on major scholastic exams.

But it seems to me that here we already have a common split in the imagination of mathematicians and the pedagogy of mathematics. On the one hand, symbols, equations, and algebra. On the other hand, pictures, visualization, and geometry. Both are central to the history and development of mathematics over millennia, yet there is a consistent prejudice in at least the last hundred years (probably longer) toward the “pure” formalism of symbols over the “dirty” concreteness of images. The tension is epitomized in the simple story that no triangle we could draw on paper is ever a “true” triangle of mathematics. In real life, when I draw a triangle it has irregularities instead of perfectly straight edges, the three angles probably don’t actually sum to 180 degrees, and the figure has a finite resolution limited by the grain of the paper. The triangle as a mathematical object is distinct from these material features, and is properly known only abstractly.

Is a visual approach to mathematics necessary? One way to think of this, which I think tilts the issue in favor of symbolism, is to ask whether images would be part of a final, ultimate theory of mathematics. If we went forward in time a thousand years, or ten thousand, and collected all the theorems of mathematics available, would we need pictures to state or prove any of these theorems? If the answer is no, then one could argue that any use of visualization today is more like a crutch than a permanent feature of mathematics.

Alternatively, we might object to this “ultimate” notion of mathematics. Where did we get this crystal ball from anyway? How convincing is an argument that entirely depends on predicting what math will look like in a thousand years? By comparison, a thousand years ago mathematicians didn’t even have the concept of algebra as a system of abstract variables that we do today. (This was one of the great innovations of Descartes in the 16th century.) Maybe in 3013 mathematicians will prove theorems using a five dimensional flight simulator running on some bioengineered brain in a box.

One thing we do know is that visualization has mattered to at least some mathematicians and some areas of mathematics consistently over time since Euclid. This suggests that geometric imagination is here to stay. Under this view, symbolic and geometric approaches are complementary heuristics: imperfect ways of thinking that have characteristic strengths and weaknesses that can compensate for each other when used in tandem. This cooperation between styles can operate at the level of a mathematical problem, a subfield or topic area, the career of a mathematician, or even mathematics as a whole discipline. The most interesting question in this situation is how different modes of mathematical imagination can work together, and hopefully Mandelbrot’s autobiography will have some ideas about this I can write about here soon.

Here is an image of a protein folding landscape. It’s a pretty complicated one to start with, but I found it looking through my archive and you have to start somewhere. The landscape itself is recognizable in the top middle of the diagram, with the topographic lines. The height of the landscape is colored from lowest — dark blue — to highest — bright red. The 3D landscape is projected onto the flat XY plane below. (Imagine taking a 3D topographic map of a mountain range and smushing it flat on a table.)

Arrayed alongside the energy landscape are pictures of what proteins at certain places in the landscape look like. These images are really composites. Imagine taking a long photographic exposure of someone dancing on a stage. You would see faint versions of their arms, legs, and body as they moved around all superimposed in the same image. In the places where they spent more time — either because they lingered there or came back multiple times — the image would be stronger and less transparent. In this case, we’re looking at the superposition of images of a single protein over time. The protein itself is just a thin cylinder colored red at one end and blue at the other, with white in the middle. But depending on how much the protein moves, we can either see one coherent pattern (Basin b3) or a big jumble (Basin b2, although notice the red part of the protein doesn’t move much). So the analogy with the photograph is close but not perfect, since every snapshot of the protein is drawn as opaque in the diagram in order to highlight the presence and absence of variation in its shape over time.

The energy landscape diagram therefore offers us a way of visualizing and imagining the relationship between a global property of the protein — its energy level — and what three dimensional structure it has. I’ll be surveying more images of protein landscapes to describe how the diagram functions as a way of imagining new hypotheses about how proteins fold as well as summarizing and testing these hypotheses.

From: Carlo Camilloni, Daniel Schaal, Kristian Schweimer, Stephan Schwarzinger, and Alfonso De Simone. “Energy Landscape of the Prion Protein Helix 1 Probed by Metadynamics and NMR.” Biophys J, 2012 vol. 102 (1) pp. 158-167.

BLACKBURN Press, Dec 30, 2004 – Science – 200 pages

“In The Art of Scientific Investigation, originally published in 1950, W.I.B. Beveridge explores the development of the intuitive side in scientists. The author’s object is to show how the minds of humans can best be harnessed to the processes of scientific discovery. This book therefore centers on the “human factor”; the individual scientist. The book reveals the basic principles and mental techniques that are common to most types of investigation. Professor Beveridge discusses great discoveries and quotes the experiences of numerous scientists. “The virtue of Mr. Beveridge’s book is that it is not dogmatic. A free and universal mind looks at scientific investigation as a creative art. . . .” The New York Times

Harvard University Press, Nov 1, 1998 – Science – 382 pages

“New scientific ideas are subjected to an extensive process of evaluation and validation by the scientific community. Until the early 1980s, this process of validation was thought to be governed by objective criteria, whereas the process by which individual scientists gave birth to new scientific ideas was regarded as inaccessible to rational study. In this book Gerald Holton takes an opposing view, illuminating the ways in which the imagination of the scientist functions early in the formation of a new insight or theory. In certain crucial instances, a scientist adopts an explicit or implicit presupposition, or thema, that guides his work to success or failure and helps determine whether the new idea will draw acclaim or controversy. Using firsthand accounts gleaned from notebooks, interviews, and correspondence of such twentieth-century scientists as Einstein, Fermi, and Millikan, Holton shows how the idea of the scientific imagination has practical implications for the history and philosophy of science and the larger understanding of the place of science in our culture. The new introduction, “How a Scientific Discovery Is Made: The Case of High-Temperature Superconductivity,” reveals the scientific imagination at work in current science, by disclosing the role of personal motivations that are usually hidden from scientific publications, and the lessons of the case for science policy today.”