In popular conception, mathematics is the ultimate resolvable discipline, immune to the epistemological murkiness that so bedevils other fields of knowledge in this relativistic age. Yet Philip Davis, emeritus professor of mathematics at Brown University, has pointed out recently that mathematics also is “a multi-semiotic enterprise” prone to ambiguity and definitional drift.

Earlier this year, Davis gave a lecture to the mathematics department at USC titled “How Do We Know When a Problem Is Solved?” Often, he told the audience, we cannot tell, for “the formulation and solution of problems change throughout history, throughout our own lifetimes, and even through our rereadings of texts.”

Part of the difficulty resides in the notion of what we mean by a solution, or as Davis put it: “What kind of answer will you accept?” —

Margaret Wertheim—Definitional Drift: Math Goes Postmodern (LA Times)

Interesting… the author saved “Dare we say it: Math is becoming postmodern” for the very end of her essay, but the headline writer gave it all away up front. Because I went into this article expecting to read about mathematics as a postmodern phenomenon, I was disappointed to find that claim only hinted at, not fully supported.

This isn’t a criticism of the essay, but rather an observation of the different rhetorics of newspaper headlines (which are designed to grab the reader) and the classical essay (which is designed to build slowly to a conclusion that rewards the committed reader).

Dennis,

The idea of post-modernism in math is actually something that predates the current usage of the term in literary thought. The ideas of “what do we mean by solved” and “can we examine the underlying syntactic and semantic structure of this problem” are actually the basis for the evolution of formal set theory and the axiomatization of mathematics. From George Boole (1815-1864) to David Hilbert (1862-1943) to Betrand Russel and Alfred Whitehead (1870-1970), the development of mathematical axioms and self-examination are legendary. The deepest paradoxes and problems in math are based on these works! (Russel’s paradox and Kurt Godel’s incompleteness theorem spring to mind.) The term post-modern has simply never been applied to higher mathematics before.

That confirms my suspicion, Josh. In the 80s, chaos theory was very “big” in literaty and humanities studies, as a kind of refuge against the supposed certainty of science. Who was the mathematician, probably in the late 19thC, who proposed a global weather prediction system, in which telegraphs would wire the temperature from various places around the world, and imagined rows and rows of mathematicians doing calculations to predict the weather based on those reports? Does that vague description ring any bells? Anyway, since there is no way to gather enough data to make accurate predictions about the complex motions of bodies of air, we see plenty of instances in which even the correct, accurate use of mathematics doesn’t result in a single universally “correct” answer, even if statistically we can say one method ends up being more accurate than another.

Edward Lorenz coined the phrase “the butterfly effect” by describing the immense sensitivity of weather patterns to small (1 in 10,000 for his work) changes: “A butterfly flaps its wings in Paris, and the weather changes in New York.” I’ve got one of his books in my office :) The problem with the popularization of chaos theory (the character of Ian Malcom in Jurrasic Park) is that the name (chaos) detracts from the idea (hidden order within systems). The popular articles from the 80s could benefit from a good fisking from some math geeks :) Also remember that every field has its own crackpots, and chaos theory has _plenty_ of them!

I think Lorenz is too recent to be the person of whom I was thinking, but I’ll figure it out sometime…