The Advent of the Algorithm by David Berlinski
The Advent of the Algorithm: The Idea that Rules the World
by David Berlinski
Harcourt. 334 pp. $28.00
In his most recent book, David Berlinski, the author of A Tour of the Calculus (1995), is by turns a mathematician, a storyteller, a would-be poet, and a philosopher. His subject is surely topical: the algorithm is the mathematical idea that lies at the heart of all programming code, which is what drives those great engines of modernity, digital computers. But Berlinski’s book goes far beyond popularizing an important, if abstruse, innovation. Touching upon disciplines as varied as physics, mathematical logic, and psychology, it carries the reader from anecdote to technical explanation to philosophical speculation to lyrical musing and back again with barely a pause.
Berlinski, known to COMMENTARY readers for his scintillating essays on some of the most contentious issues in modern science, is at his best here when writing about mathematics. We learn at the outset that the algorithm is a kind of artifact—a finite procedure that manipulates symbols in discrete steps and according to fixed instructions. The rules of arithmetic are themselves kinds of algorithms, cranking out sums and products through the repetition of basic operations (add the righthand column, carry the tens) until an answer is produced.
Basically, then, an algorithm is a formal scheme for computing anything that can be computed, from a multiplication table to the entropy of a model gas. The concept seems intuitive enough: just as you cannot bake a cake without a recipe, you cannot perform a computation without an algorithm. But is an algorithm a recipe? A code? A set of rules? How can we formalize the statement, “carry the tens,” in a way so simple that even a computer can understand it?
In fact, as Berlinski tries to show, the algorithm, understood rigorously, is the ultimate abstraction. Its execution is purely mechanical, requiring no intelligence or insight, and it itself is blind to the meaning of the symbols with which it operates (in the case of computers, l’s and O’s). Yet what it does is meaningful, at least to the person who needs to know that two plus two equals four and not five. According to Berlinski, this split between “form” and “meaning” was first appreciated by logicians, grappling for a way to rescue their discipline in light of Kurt Gödel’s famous proof that any consistent formal system—like the rules of inference in mathematics and logic—can generate true statements that it itself cannot demonstrate to be true.
In order really to understand the algorithm, then, we need first to understand mathematical logic. And so Berlinski proceeds carefully to guide the reader through the elements of formal logic, beginning with the simple Aristotelian system of familiar syllogisms (all dogs are mammals; all mammals are animals; therefore all dogs are animals) and gradually expanding its scope through the insights of a series of brilliant mathematicians, each introduced to us along with his work: Gottfried Leibniz, the echt court philosopher who dreamed of a universal logical language; Giuseppe Peano, the Italian farmer’s son who rewrote the rules of arithmetic; the dyspeptic Gottlob Frege, founder of modern symbolic logic; and so forth.
Berlinski’s descriptive portraits of these great men are as evocative as his explanations of their thought are lucid and meticulous. At each point he takes us up and down what he calls the “inferential staircase,” a sort of checklist by which we can verify the steps in his reasoning. But he does not stop there. To make sure we have grasped the basic points, Berlinski enlists the aid of a cast of interlocutors, reproducing conversations with lawyers, doctors, his literary agent, a cardinal—even summoning the ghosts of Frege and the great Argentinian writer Jorge Luis Borges. Every now and then he leaves off logic entirely to give us a parable about the quest for ultimate truth—or an excerpt from his own amorous biography.
The end result is an entertaining, sometimes distracting, almost always enlightening book. The laws of logic are made accessible, if perhaps not intuitive; and despite Berlinski’s many detours, there is no doubt that by the time his survey is complete, we have learned something valuable about mathematical logic. Indeed, to come back after all this to the algorithm is a little like meeting the wizard of Oz, enshrined in a dazzling Emerald City built by the real heroes of Berlinski’s narrative, the mathematicians Kurt Gödel, Alonzo Church, Alan Turing, and Emil Post.
Berlinski makes it clear, in fact, that he has not brought us all this way just to see what makes computers buzz and whir. Not that computers are unimportant; the centrality of the algorithm to information technology, and hence to the infrastructure of modern society, is what makes it relevant to every one of us. E-mail, word processing, the Internet—all are in some direct sense the product of algorithms.
In another sense, though, the digital computer is just the wizard’s machinery, a bunch of wires and transistors that execute the algorithm’s commands and produce the results. What is of greater interest to Berlinski is the way the algorithm itself, the idea of effective computability, has changed the way we, and especially scientists, look at the world. It is his contention that it has led to philosophical and scientific sloppiness.
As natural scientists use algorithms in models that explain how the world works, Berlinski writes, they forget that these are just mechanical toys whose operations are meaningless unless and until we interpret them. Because algorithms are manmade, they cannot themselves be part of the laws of nature. Yet scientists often think, and write, as if algorithms do explain how gases work, or how a tree grows from the DNA in a seed, or how a human being, seeing a tree, knows what it is—as if, in short, they explain consciousness itself.
But is this what physicists or psychologists who use algorithms to model complex phenomena really believe? Consider Berlinski’s discussion of the computational theory of mind—in a nutshell, the idea that thinking itself is a kind of computation performed by the brain, not a function of what used to be called the soul or a mysterious immaterial secretion of neural tissue. One way of testing whether a cognitive function may be computational is by implementing it in a “connectionist” network: a collection of simple data-processing units hooked up in a way that is supposed to resemble a network of neurons in the brain.
Berlinski shows us how such networks are built and alludes to some of the things they can do, like sorting photographs of human faces based on the presence of some prominent feature (like shifty eyes). He also exposes their limitations, the most important of which (for him) is that they are utterly incapable of interpreting their own output. A network trained to recognize shifty-eyed characters, for example, does not know what shiftiness signifies; it merely extracts a set of visual features, leaving their meaning to be grasped by the human programmer.
Berlinski cannot envision how any neural network, no matter how huge or with how many data-processing units, can overcome this basic shortcoming: the gap between computation and meaning is simply too large. But there is more to the computational theory of mind than that. Yes, artificial neural networks do not do anything truly amazing—they simply perform some statistical operations—but, as far as most cognitive scientists are concerned, the computational theory does not force us to view the brain as one giant, ignorant neural network. More likely, they say, it is an array of little, ignorant neural networks, each performing a specialized task. Stepping back from such isolated, individual networks to interacting networks of networks allows us to see how the “consciousness gap” might be bridged—by layer upon layer of organization.
The picture of the mind that emerges resembles a sort of enormous flowchart: at the highest level are the broadest and most general “divisions” of cognition (vision, for instance), each of them composed of various subdivisions (recognition and tracking), departments (color, shape, texture), committees, subcommittees, and so forth, right down to the lowliest algorithm in charge of figuring out whether a line segment is horizontal. The computer scientist Marvin Minsky has dubbed this confederation of blind agents the “society of mind,” capturing in an ingenious phrase the notion that purpose can emerge from a cadre of seemingly purposeless operators.
Berlinski, one suspects, is not prepared to make any such conceptual leap from a single blind operator to an organization of blind operators, and no doubt he has his reasons; it is disappointing, though, that he does not take on the computational theory of mind at its strongest point (a complaint that might be entered about his treatment of other scientific fields as well). Still, that does not detract from the admirable job he has done in showing us how the algorithm works, tracing its history, and situating it within the context of human knowledge and creativity (the latter understood broadly to include parables and occasionally salacious anecdotes). The Advent of the Algorithm is not an academic book by any stretch, but neither is it a typical example of popular science-writing, being designed as much to excite the imagination as to inform. It does both, and then some.