Mathematical Logic

08 Sep 2004 11:07

If, in 1901, a talented and sympathetic outsider had been called upon (say, by a granting-giving agency) to survey the sciences and name the branch which would be least fruitful in century ahead, his choice might well have settled upon mathematical logic, an exceedingly recondite field whose practitioners could all have fit into a small auditorium --- algebraists consumed by abstractive passion, or philosophers pursuing fantasies out of Leibniz and Ramon Llull, or (like Whitehead) both. It had no practical applications, and not even that much mathematics to show for itself: its crown was an exceedingly obscure definition of cardinal numbers. When, in 1910, it produced a work which the learned world was forced to notice --- the first volume of Whitehead and Russell's Principia Mathematica --- it was, so to speak, the academic Brief History of Time of its day, often mentioned, never used.

Our outsider would, of course, have been wrong. Mathematical logic was the inspiration for perhaps only half of twentieth-century philosohpy (that is, of honest philosophy; by volume, as Kolakowski says, Stalin was the century's most influential philosopher); many of our finest mathematicians, such as Norbert Wiener, John von Neumann and Andrei Kolmogorov cut their teeth on it, and notation (and notions) which began in the obscurities of Peirce and Peano are now to be found in every undergraduate math book. True, some early application --- one thinks particularly of Woodger's axiomatization of biology --- have, perhaps unfairly, gone nowhere, and McCulloch and Pitt's "A Logical Calculus of the Ideas Immanent in Nervous Activity" is more important for launching neural nets upon the world than for using Carnap's formalism. But in one extremely important field, however, it reigns supreme, and that is computation. Programming is, simply, mathematical logic in action; the melding of theory and practice is so complete that most practioners have no idea that their speech --- recursion, lexical scope, data abstraction, even those banes of C novices, pointers, referencing and dereferncing --- is prose. (Speaking of speech, Chomsky of course began as a logican, and his early work (air force and navy supported!) on formal languages is as much a part of logic as it is of linguistics or the theory of computation.) Of course, some of the computer's intellectual roots were more obviously useful --- but since these were the study of Brownian motion, and the physics of crystals and spectral lines, not much. (Its practical origins were military needs and vast quantities of government subsidies, which continue, but let's not disturb the myths about private enterprise any more than we must.)

I don't really know what the moral is, beyond the obvious one that useless knowledge isn't.

Things I want to understand better: Tarski's truth theory; the Russell-Whitehead "relation-arithmetic" and its descendants; model theory.

See also: Gödel's Theorem; Computation; Logical Positivism; Math I Ought to Learn