Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

although we have an abstract system for doing math in high-dimensional spaces, mathematicians still have trouble visualizing them. Actually, let me be more frank—we can’t visualize them. Our brains just aren’t up to it. We aren’t wired that way. That cognitive limitation deals a serious blow to Poincaré’s program, at least in dimensions higher than three. His approach to nonlinear dynamics depends on visual intuition. If we can’t picture what’s going to happen in four or eighteen or a hundred dimensions, his approach can’t help us all that much. This has become a big obstacle to progress in the field of complex systems, where high-dimensional spaces are exactly what we need to understand if we want to make sense of the thousands of biochemical reactions taking place in a healthy living cell or explain how they go awry in cancer. If we are to have any hope of making sense of cell biology using differential equations, we need to be able to solve those equations with formulas (which Sofia Kovalevskaya showed we cannot) or picture them (which our limited brains won’t allow). So the mathematics of complex nonlinear systems is discouraging. It seems like it will always be hard, if not impossible, for anyone to make headway on the most difficult problems of our time, from the behavior of economies, societies, and cells to the workings of the immune system, genes, brains, and consciousness.