Much of the research done in cognitive science is designed to study the development of concepts – internal representations that define the idea-driven nature of modern human experience. And, in our experience, it’s difficult to mend the rift that’s been created between what we call thought and what we call reality. But a number of paths have opened up within various ’embodied mind’ theses that are intended to correct the mistaken duality. I’ve covered some of them in previous blogs. I’ll refer here, again, to biologist and philosopher Humberto Maturana, a proponent of a closely related idea known as enactivism. In a paper that appeared in Cybernetics and Human Knowing in 2002, Maturana refuted the idea that language is a collection of abstract representations that correspond to concrete things, and made the following observation:
Part of the difficulty in understanding the relation between language and existence rests on the view of language as a domain of representations and abstractions of entities that pertain to a different concrete domain. Yet language is not so, languaging occurs in the concreteness of the doings of the observer in his or her actual living in the praxis of living itself…Nothing exists outside the networks of conversations through which we bring forth all that exists, from ourselves to the cosmos that makes us possible…
Languaging is action. And this is how I have come to see mathematics, as action.
Despite the development of embodiment theories (one of the most well known being the Lafoff/Nunez book Where Mathematics Comes From) debates over the objective reality of mathematics versus the imaginative reality of mathematics are hardly resolved. But the distinction between these alternatives is blurred when mathematics is seen as action. All of this came to mind today when I read about the recently discovered and oldest known sundial.
A tire size stone, found marking a Bronze Age grave, had been carved out to mark time with the movement of the sun. It was determined that the stone could reflect half hour increments. The burial ground in the Ukraine, where the stone was found, dates back to the 12th or 13th century B.C. An article posted on livescience in early October describes the find and some of the details about the stone.
To verify that the stone was a sundial, archeologist Larisa Vodolazhskaya calculated the angles that would have been created by the sun and shadows at that latitude. She confirmed that the carvings on the slabs marked the hours accurately. “They are made for the geographic latitude at which the sundials were found,” she said. And these ancient carvings rely on a sophisticated grasp of geometry.
The circular depressions, placed in an elliptical pattern, are hour marks of an analemmatic sundial; the largest groove on the plate, Vodolazhskaya said, marks where the vertical, shadow-casting gnomon would have been placed at the winter solstice.
Meanwhile, a long carved line transected by a number of parallel grooves in the center of the slab would have acted as a linear scale for a more traditional horizontal sundial, where the hours are marked by a gnomon’s shadow falling along hour lines. In this case, the horizontal sundial actually had two gnomons, Vodolazhskaya said. One gnomon tracked the time in the morning hours and early afternoon, and the second covered from late morning to evening, measuring time in half-hour increments. Ancient sundials with half-hour marks are rare, though one was discovered earlier this year at the Valley of the Kings in Egypt.
A thorough analysis of the stone can be found in a paper by Vodolazhskaya. The figures in the paper illustrate the impressive precision of the stone’s markings. The sophistication of the geometry employed and the sundial’s ability to measure the passage of time in half hour increments are taken as evidence that ancient Egypt had some influence on the people who inhabited the northern coast of the Black Sea. What strikes me, however, is something that may be unique to an ancient ‘tool’ like this one. In some sense, the intricate geometric structure exhibited by the stone exists only when the stone and the sun are taken together. It emerges from the directed light of the sun together with human perception, thought, and craft. This would be true of any ancient sundial. It just happens that this one made me particularly aware of it, perhaps because of the fine tuning between this one and the latitude of the location where is was found. In the right light, the mathematics isn’t fully isolated from the action of the sun and the sundial maker.
The mathematics isn’t describing the relationship between sunlight, shadow and time, it is the relationship between sunlight, shadow and time. And this is why I thought again about Humberto Maturanna. I think that mathematics, like language, is a kind of shared action, occurring within and among things in the world. It exists in various bodies that may or may not find reason to express it formally.
If you haven’t seen it, the more self-contained mathematical action in the little film Nature by Numbers is worth a look.