Mathematics as Sign: Writing, Imagining, Counting. Brian Rotman. xii + 170 pp. Stanford University Press, 2000. $45 cloth, $16.95 paper.
Alfred North Whitehead wrote: "It is a mistake to think that the Greeks discovered the elements of mathematics, and that we have added the advanced parts of the subject. The opposite is more nearly the case; they were interested in the higher parts of the subject and never discovered its elements. . . . Elementary mathematics is one of the most characteristic creations of modern thought." A cynic might go a step further and suggest that all those modern mathematicians toiling in the nether depths of the subject are not down there to build foundations but to undermine them. The obvious example is G?del's theorem, which is a statement of what mathematics can't do, a proof that the foundations can never be entirely secure.
With all this deconstruction already accomplished by the modernists, one can't help wondering: What's a poor postmodernist mathematician to do? What's left to undermine? Brian Rotman has an answer to suggest, but it is a sadly shallow follow-up to the excavations of G?del's generation.
Rotman takes up two main themes. First comes a semiotic analysis of mathematical writing. He observes that proofs and other mathematical narratives are full of imperatives, such as "drop a perpendicular from point P onto line L" or "count the elements of set S." Who is expected to carry out such orders? Often they set forth rather challenging tasks (as in "Sum the series 1 + 1/2 + 1/4 + 1/8 . . ."). The reader of this instruction surely cannot perform it in any literal sense, because it would require infinitely many additions. Rotman therefore argues that mathematics is a kind of thought experiment or waking dream, in which an imaginary surrogate of the mathematician carries out the required infinite labors. In the process, the mathematician gets split into three parts:
Mathematical reasoning is thus an irreducibly tripartite activity in which the Person (Dreamer awake) observes the Subject (Dreamer) imagining a proxy?the Agent (Imago)?of him- or herself, and, on the basis of the likeness between Subject and Agent, comes to be persuaded that what the Agent experiences is what the Subject would experience were he or she to carry out the unidealized versions of the activities in question.
As a model of what mathematicians do when they do mathematics, this is an interesting proposal. I would like to know more about it. For example, how much difference does it make if the Agent is an imaginary machine instead of an imaginary person? And how can we add social interactions to the model? If several Persons or Subjects are dreaming jointly at the blackboard, are their Agents also socializing? Unfortunately, Rotman takes up none of these questions. Instead he veers off into a long polemic against mathematical Platonism (the doctrine that mathematical objects are discovered rather than invented). This is a very old and tired controversy, not much enlivened by being recast in the vocabulary of semiotics.
Rotman's second theme has to do with infinity and the continuum. Over the centuries, many mathematicians have been made queasy by these concepts, but Rotman admits to dizziness at somewhat lower heights than others. He not only questions the endless progression of integers climbing to infinity but even has doubts about some large but still finite numbers. We are comfortable with small, "local" numbers, he says, because we can reach quantities such as 3 or 42 or 281,421,906 by counting things. Big numbers, like (10^{10})^{10}, deserve a different status, because no process of counting in the physical universe will ever get that far. We'd run out of fingers to count on.
Rotman sketches an arithmetic based on this distinction between countable iterates and uncountable transiterates. "Within the countable numbers, arithmetic works the way you learned it in elementary school, except that each operation will inevitably specify numbers that can't be reached by counting. For example, if you count upward, you'll reach a countable number x such that x plus x is not countable. Before you get there, you'll have reached a countable number y such that y times y is not countable." Further questions arise: Are addition and multiplication commutative when the result is a transiterate? What happens to the rational numbers when the numerator or denominator can be a transiterate? This is just the kind of waking dream that a mathematician's Agent might enjoy. But, again, Rotman declines to explore his invention (he would not call it a discovery!) in greater depth. Instead, the book concludes with a meditation on "nomad mathematics" in opposition to "royal or State mathematics." Here we are deep in postmodernist territory, and I'm afraid that even my Agent doesn't know what to make of it.?Brian Hayes