Is It Boole that Makes The World Go Round?May 16, 2000
Philip J. Davis
The Advent of the Algorithm: The Idea that Rules the World. By David Berlinski, Harcourt Brace, New York, 2000, 368 pages, $28.00.
David Berlinski writes with the mind of a logician, the perspective of a historian, and the soul of a romancer. He is simultaneously an idol worshiper and a creator of idols and myths. His pages run the gamut from clear and lucid expositions of difficult logical material to snippets of biography, modulated autobiography, and imagined conversations with the Great. His paragraphs, often decorated with irrelevant baroque encrustations, share with us bits of titillating scandal or pepper us with wise-guy one-liners that carry the bite of Lytton Strachey. His judgments are sweeping, colorful, pronounced from the high pulpit of a cathedral of his own making. He is addictive-much more than peanuts: He teaches, he amuses, he invites reflection, he infuriates. In short, there is no one in the whole wide world who writes science for a popular audience with so much literary panache. He is unique.
So what have we here? Basically, Berlinski has given us a bird's-eye tutorial on mathematical logic, beginning with Aristotle, proceeding to Leibnitz, but concentrating on the developments of the past century and a half. Here the names of Boole, Frege, Peano, Russell and Whitehead, Hilbert, Gödel, Post, Church, Turing loom large. Have I missed a few? Cantor? Tarski? Chaitin?
Readers will learn the fundamentals of the propositional calculus and make their way through to Gödel's incompleteness theorem as the crowning glory (or embarrassment) of the whole logical enterprise. Church's lambda calculus is included as a bonus. To top off the text, there are some words about entropy, complexity, DNA, and the question of whether human life in all its complexity can, fulfilling the Dream of Leibnitz, be reduced to computation in the sense of Church-Turing.
Here, then, are the bare bones, mathematically speaking, of Berlinski's accomplishment. However, to leave it at that would be to underestimate the human flesh that he has loaded onto these bones. After all, since the appearance of Nagel and Newman's Gödel's Proof (1958), and perhaps even earlier, the remarkable accomplishments of Gödel have been accessible to a general educated readership. What Berlinski, the logician, the philosopher, the novelist, has produced is a personal assessment of mathematical logic as viewed from with-in his own life space and not altogether confined to the usual terse notation.
Does Berlinski's juxtaposition of severe mathematical abstraction with legends, metaphorically focused short-short stories, aperçus on the state of the Princeton University of years ago, glimpses of his own love life, fuse into a consistent whole? Even as mathematical logic is fading---or, as one authority (logician Juliet Floyd) wrote me, "is in a crisis of prestige, torn between making itself credible to the mathematics community and turning itself into theoretical computer science"---has Berlinski invented a new historical genre that will preserve the brilliance of its glory days? Is he, perhaps, the E.L. Doctorow of the mathematical expositors? Or has he reintroduced a kind of medieval writing in which scientific theory, practice, philosophy, theology, personal illumination, and belles lettres are hardly to be distinguished from one another? I'm not sure. But I know this: It's a style that's been selling, and his imitators are surely lurking around the corner.
One measure of a book's impact is its ability to raise questions in a reader's mind and eyebrows on a reader's forehead. Here and there, The Advent of the Algorithm has raised both in me. Berlinski seems to imply that the advent of the algorithm in its formal sense should be dated to the work of Frege (1848-1925). But didn't Babbage (1792-1871) have it all, implicit in his mechanistic plans, a mathematical body language of a sort, even though not written in the strange (and later abandoned) squiggles of Frege's symbolic logic?
Then, too, Berlinski seems to imply that the actual development of digital computer hardware rested heavily on the preexistence of a highly developed mathematical logic. I read it otherwise---that the impetus came from the desire to ease the processing of social and business data and to solve intractable equations of mathematical physics. Toward these ends, the enabling fact was technological development. Granted that John von Neumann was the first to give a complete logical treatment of the electrical engineering aspect of the computer and that in the late '40s the computer was often called a "logical engine," I see symbolic logic acting more as a comfort to the electrical circuit engineers and as an infusion into their diagrams of a more prestigious intellectual content.
Moving far beyond the propositional calculus, the Gödel incompleteness theorem, one of the high points of logic, paradoxically emerges as downbeat, asserting, in one of its incarnations, the impossibility of writing a superprogram that will decide whether any other program will exit in our lifetimes. Gödelologists will tell me whether the theorem also implies the nonexistence of a universal antidote for computer viruses. So where's the developmental help from incompleteness?
Yes, the history of mathematical logic and its relationship to mathematics, physics, and computer science is full of ambiguities and paradoxes. Why, for example, when the Gödel theorem has been touted in some quarters as the most significant mathematical result of the century, when it is a major production number of the present book, has it been almost irrelevant to the work of mathematical researchers? Why is an intense course in logic not required for math majors, and why have a few weeks of propositional calculus been reduced to a non-mandatory recommendation for CS students?
I certainly agree with Berlinski's subtitle: We are living in the age of the algorithm, and the databases that provide input and receive back output have become arbiters of whatever is to be regarded as true. Newton's and Einstein's laws are algorithmizable; ballplayers' salaries are determined algorithmically from their stats, much as the EC came up with its famous specifications of what is and what is not a cucumber. And this kind of thing has only just begun.
But is it really logic that now---through algorithms---makes the world go round? Not beer, as the song has it? Not "chance, love, and logic," as the great American logician Charles Sanders Peirce would have it and as one might even infer from the personal framework that surrounds Berlinski's presentation? There may be something tradition-bound about his romanticized positioning of the technical aspects of mathematical logic---a piece of nostalgia, perhaps, deriving from his graduate school days under Alonzo Church. There are signs that the algorithm, though fundamental to all digital communication (in the sense in which, for example, elementary particles of physics are fundamental to a Hawaiian wedding luau), may be having to share the stage center of technological and instructional emphasis. Here are a few straws in the wind: Brown computer scientist Peter Wegner thinks that in the future the emphasis will shift from algorithmic models of computation to interactive models. Yes, in an age in which a single computer has become a basic "elementary particle" of information interaction, to be combined with myriads of other individual computers and acted on non-algorithmically by the whole of the exterior environment, human and non, the metaphor "logical engine" already has the quaintness of the village pump. As MIT computer scientist Lynn A. Stein puts it: "The conventional metaphor [for computation] will be replaced by the notion of a community of interacting entities."
Change has also been anticipated in pure mathematics. Fields medalist David Mumford speculates that the spotlight will turn away from theorems and onto structures. Insofar as traditional mathematics---definition, theorem, proof---is algorithmic in spirit, this presages a significant shift.
Berlinski is aware of the possibilities of historic change when he writes: "By now the ideas elaborated by Gödel, Church, Turing, and Post have passed entirely into the body of mathematics, where themes and dreams and definitions are all immured, but the essential idea of an algorithm blazes forth from any digital computer, the unfolding of genius having passed inexorably from Gödel's incompleteness theorem to Space Invaders VII on an arcade Atari, a progression suggesting something both melancholy and exuberant about our culture."
In the meantime, algorithmization will remain a vital element; its shift from software to hardware, an interesting development, will hardly be noticed by a public focused on bottom-line outputs and interactive goodies. But the practice of algorithmics may, in the not-too-distant future, be confined to a small elite of specialists, numerical analysts and implementers of models of all sorts, who will supply their traditional clientele with the necessary products.
To develop these speculations would require another book from Berlinski, which fans, such as myself, would urge him to write, encrustations and all.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.