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.