Philosophy & Rhetoric

1. Fermat's last video

Modern mathematics is nearly characterized by the use of rigorous proofs. This practice, the result of literally thousands of years of refinement, has brought to mathematics a clarity and reliability unmatched by any other science.

(Jaffe and Quinn 1993, 1)

The above passage illustrates how mathematicians have come to esteem rigorous argument as the most important feature of their subject. In turn, the example of mathematics has, over a long period, made rigorous argument an ideal in many other areas of inquiry—from the sciences to theology. How interesting it is, then, that some mathematicians have begun to question that ideal. Gian-Carlo Rota, for example, has said that the axiomatic method of presentation in mathematics has now reached "the zenith of fanaticism" and that the identification of mathematics with this method "is having a corrosive effect on the way mathematics is viewed by scientists in other disciplines" (1997, 189). William Thurston has blamed the formality of contemporary mathematics for "the whole system of generally poor communication" within the subject. According to Thurston, ordinary conversation between mathematicians can often convey a new solution in a matter of minutes, whereas a formal proof delivered in a seminar will take an hour, and a written article may take several hours or even days to understand. In Thurston's view, this disparity arises precisely because conversationalists go far beyond formal mathematical language:

They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention. With these channels of communication, they are in a much better position to convey what's going on, not just in their logical and linguistic facilities, but in their other mental facilities as well. [End Page 154]

(Thurston 1994, 166)

Sadly, however, mathematicians can only converse like this in private. To reach the rest of their audience, they have to publish in journals. Journal articles are limited to formal language and so must convey results more slowly. Formality, though less efficient, is more publishable.

Yet there is hope. Another mathematician, Andrew Odlyzko, has suggested that the unfortunate limitations of printed journals can now be overcome with computers. Computers can allow mathematicians to distribute something more than the traditional style of printed proof. Odlyzko suggests an electronic scrapbook that would include a video of each author lecturing about their results (Odlyzko 1995, 90-91). Fermat's Last Video? The point is that if these suggestions come to pass, publication of mathematical results will no longer involve stripping away informal features and replacing them with formal language, as if rigorous argument was all that mattered. Instead, the informal features that help mathematicians to understand new results so quickly in ordinary conversation will be given equal air time. The ideal, for publication at least, will be less formal.

Is this just a gimmick? Is it something that will go away, if we shut our eyes, leaving the centuries old ideal of formal proof intact? What I wish to discuss here are some of the reasons why mathematics, as well as other branches of knowledge, may adopt greater informality and may cease to idealise rigorous argument.

2. From parents to premisses

I will start by examining how and why rigorous, formal argument became an ideal in the first place. This is a long story, going back to the origins of Western literature; so I am forced to be brief.¹ I will start with the matter of how argument became an ideal. The first thing we should recognise is that even in the West's earliest literary works—the Iliad and the Odyssey—there are plenty of rudimentary arguments, particularly in the speeches put into the mouths of the major characters. Nevertheless, for sustained or rigorous argument, we must look a little harder. Havelock has put forward the following passage as an example of unusual rigour in early literature (Havelock 1966b, 209). The passage is taken from the start of Hesiod's poem, Works and Days, a piece of wisdom literature almost as old as...