Saturday, May 25, 2013

The Opposite of Mathematics

I recently stumbled across a book whose premise intrigued me: Circles Disturbed: The Interplay of Mathematics and Narrative. This is an edited volume that explores the different ways mathematics and narrative interact. The power and fascination of this combination (collision? intersection? union? mating?) arise from how opposed these ways of looking at the world are, and how powerful – both have been proposed as being the metaphysical foundation layer of reality. They can’t both be, but maybe in setting them against each other in various ways one can get a glimpse of something more fundamental than either. 

Mathematical structures exist in a timeless Platonic eternity, they alone (perhaps God also) are not subject to time and change. Mathematics is also thought to be universal, so much so that we believe if we ever meet an alien race we will be able to talk to them by starting with the Pythagorean theorem. Narrative on the other hand is essentially both temporal and specific; this particular and peculiar sequence of events happened to these people, at this time. Narratives are rooted in a particular culture and often don’t translate well to others. Mathematics is a narrow path; but part of the appeal of stories is the sense that anything could happen. Narrative and mathematics thus seem to be so different as to barely inhabit the same universe, but obviously they do intersect in the human mind if nowhere else.

This particular collision of worldviews has some personal resonance for me. I spent my formative years around an artificial intelligence lab, and one of my academic efforts back then was to try to dislodge some of the stale ideas that I felt were holding back the field, which tended to take mathematical form (in most non-humanities fields the pressure to at least seem to have a rigorous formal foundation for your work is intense). One strategy was to displace some of this overly-mathematical conceptual infrastructure with ideas from narrative theory.

The writers in this volume, for the most part either professional mathematicians or historians / philosophers of mathematics, are not quite as brash or as confrontational as I was back then. They are mostly looking at mathematical practice, that is, the very human situated activities that somehow enable the transcendental structures of mathematics to reveal themselves. Since mathematics is a human activity, and a demanding one, naturally it makes effective use of the full range of human mental tools, including such un-mathematical things as rhetoric, narrative, analogy, and figurative language. Mathematics and narrative intersect in other ways, such as stories with mathematical protagonists, or explorations of the formal structure of stories. (This particular volume doesn’t pay too much attention to the small subgenre of mathematical science fiction, but see these other collections).

An even more personal resonance: one of the chapters contains a story involving someone I knew: Tom Trobaugh, who was in grad school with me and tragically committed suicide. I knew him mostly as a musician, I had no idea that he was doing mathematics at a professional level, but apparently he did during his life and oddly continued to do so after his death. The chapter in question takes off from a rather bizarre episode where Trobaugh appeared to one of his co-authors in a dream, after his suicide, and provided a key insight:
The first author must state that his coauthor and close friend, Tom Trobaugh, quite intelligent, singularly original, and inordinately generous, killed himself consequent to endogenous depression. Ninety-four days later, in my dream, Tom's simulacrum remarked, "The direct limit characterization of perfect complexes shows that they extend, just as one extends a coherent sheaf." Awaking with a start, I knew this idea had to be wrong, since some perfect complexes have a non-vanishing K0 obstruction to extension. I had worked on this problem for 3 years, and saw this approach to be hopeless. But Tom's simulacrum had been so insistent, I knew he wouldn't let me sleep undisturbed until I had worked out the argument and could point to the gap. This work quickly led to the key results of this paper. To Tom, I could have explained why he must be listed as a coauthor.
The article as a whole is a detailed meditation on the nature of agency, rhetoric, and literary form in mathematics, especially in the envisioned automated forms of it that have been a gleam in the eye for most of the 20th century.

My own writing and thinking isn’t very mathematical in the usual sense, but it often seems to hover uncomfortably between structure and narrative. My background leads me to always gravitate towards conceptual abstraction, and I assume the value here (if any) is in the Big Ideas, not the specific happenings of my life. But in order to get thoughts to flow into the blog format they often require a temporal hook, if usually a trivial one like a holiday or event or random experience, such as a chance encounter with a book. My hunch is that all writing has to bridge this gap between the personal and the universal, the temporal and the timeless, and there a thousand different ways to do it. The blog is a relatively new genre of writing that encourage new recombinations; being able to experiment with them is one of the attractions of the form.

2 comments:

meaningness said...

How extraordinary!

I was at Tom Trobaugh's funeral. Or I guess memorial service is the word, because his body wasn't there, only his ghost.

It was highly memorable for me. One thing I found seriously spooky was that he played the piano at his own memorial. This was via the digital Bösendorfer memorizing player piano which had just been invented.

I had no idea that his ghost went on, after that strange feat, to do important mathematical work.

I'm perfectly familiar with this sort of thing on the Tibetan Buddhist side of my life, where dead people frequently visit each other in dreams and inspire important new works.

I hadn't really considered it as part of the MIT side of my life.

Thanks for passing on the story.

mtraven said...

The walls between different parts of life tend to become permeable...especially in the internet age. Maybe the boundary between life and death as well, at least in the TB world.

I must have been at that memorial too.