Dante, art, vision, and mathematics
We adopted a dog a couple of months ago, and there have been moments when I have watched a change in his attention or a change in his behavior, and wondered how his awareness might be structured. When we drive with him, he usually sits in front of the back seats of our Honda Element, appearing to be looking out the front windshield (although I think he’s too low to see anything). I look back at him occasionally when I’m driving and he often meets my glance. I’ve wondered whether it would be possible for him to meet my glance in the rearview mirror, the way my children do. This question led me to think about the structure of our awareness and whether, or how, it might be related to mathematics. In the completely familiar trick of light that causes a reflection, we find meaning in seeing exactly the same visual image (which the brain already works to compute) from two different perspectives. My children may not know anything about the reflective properties of light, but they know how to respond to a reflection. What is it that the brain is doing when it learns to occasionally expect the presence of equivalent visual stimuli from more than one source? When my daughter speaks to me through the rear view mirror of our car, her brain is doing facial recognition processing as well as with something that has to do with sense space, direction, and light. Perhaps the mathematical investigation of projections is the symbolic representation of some of this neuronal action.
I was reminded of this question, and the impact that a new visual/spatial configuration might have on how we understand what we perceive, when Davide Castelvecchi recently blogged about the resemblance between Dante’s early 14th century description of the cosmos and the one we’ve built with modern mathematics.
The cosmic microwave background, or CMB, shows us a slice of the universe as it looked more than 13.7 billion years ago, and the structure of that universe bears a striking resemblance to that of Dante’s heaven—at least according to some commentators. It is as if the poet had presaged some of the most striking developments of modern mathematics and cosmology six centuries before they emerged.
The parallel rests, to some extent, at what it means to “look” at space that has a non-Euclidean geometry. This is far more complex than what it means to look at a reflected image, but for me they were related. Castelvecchi does a nice job of describing a perspective that contradicts our experience but, in fact, is geometrically possible and, more to the point, is used to understand the structure of the perceived universe. Dante journeys through a series of concentric spheres centered at the earth and extending out to the stars. Past the stars is another sphere that encloses the entire physical universe and crossing its boundary he is in the spiritual realm.
The otherworld however also has a geometric structure, and it is completely symmetrical to that of the physical world, with nine concentric spheres, which are inhabited by angels and the souls of the most virtuous dead. But instead of growing ever larger, these spheres grow ever smaller. And at the center, Dante says, sits God, occupying a single point and emanating a blinding light.
Thus Dante’s entire universe—both physical and spiritual—consists of two sets of concentric spheres, one centered at Earth, the other at God. If you were to point a laser vertically up toward the sky from any point on Earth, you’d be pointing it straight at that single point where Dante places God.
This is not the first time that the mathematics of Dante’s universe has been discussed. Mathematician David Pierce wrote on the same topic back in 2008 in response to a series of quotations about infinite spheres. And while still an undergrad (in 1979), Mark Peterson did a really nice job of describing how Dante’s universe looks like a 3-sphere. Mark Peterson is author of the book Galileo’s Muse in which he makes the claim that it was the art of the Renaissance (not the science) that led to way to modern science.
All of these insights circle around the idea that mathematics gives us a precise way to access things that it may be possible to just “see,” the way an artist might, or a savant. And I am of the opinion that the pursuit of these ideas may help unlock some of the secrets, not just of nature, but of perception itself. One of the references in the Castelvecchi blog is the book Poetry of the Universe by the late Robert Osserman.
Early in the book Osserman refers to mathematics as ‘mind goggles’ that make it possible to clarify images beneath the surface of our world. He says that his book:
is a celebration of the human imagination – the facility to make the kind of mental leaps without which the impact of the outer world on our senses would be mostly noise. Mathematical imagination and imagery, closely linked, provide the vision that allows us to see the hidden but exquisite structure below the surface. (emphasis my own).