It’s one of the curses of scientific discoveries that they are greeted, remarkably often, with misunderstanding or outright dismissal. Charles Darwin’s geology teacher, the famed Adam Sedgwick, wrote to his student after the publication of On the Origin of Species, “I have read your book with more pain than pleasure. Parts of it I admired greatly, parts I laughed at till my sides were almost sore; other parts I read with absolute sorrow, because I think them utterly false and grievously mischievous.” Sylvia Nasar, writing about John Nash’s Nobel Prize–winning work on game theory, remarked that his ideas “seemed initially too simple to be truly interesting, too narrow to be widely applicable, and, later on, so obvious that its discovery by someone was deemed all but inevitable.” Scientific revolutions are rarely unopposed.
Shannon’s work, too, was coldly received in some quarters. The first major criticism, and the one with the most edge to it, came from the mathematician Joseph L. Doob. Brought from the Midwest to New York at the age of three, he was marked out as a bright student early and enrolled in New York’s Ethical Culture Fieldston School. The school was unique in New York society at the time: its founder’s belief that the poor deserved the highest-caliber education was radical, but the school’s academic reputation also drew the well-heeled. Over the twentieth century, it would produce such alumni as Marvin Minsky, a pioneer of artificial intelligence and future colleague of Shannon’s, and J. Robert Oppenheimer, the father of the atomic bomb.
After excelling there, Doob departed for Harvard, where, it is said, he grew so frustrated with the plodding pace of math instruction that he took sophomore and junior calculus simultaneously—and aced both. Unlike many of his fellow students, Doob never harbored any doubts about his future as a mathematician.
His confidence showed in the scale of the work he attempted: his 1953 book on probability theory, an 800-page tome, was greeted as the most influential work on the topic since the nineteenth century. His assurance displayed itself in another way too: Doob was a fierce critic of anything he regarded as flabby thinking. Doob was open about the fact that he was, perhaps too frequently, looking for trouble. Asked why he became interested in mathematics in the first place, he answered:
I have always wanted to understand what I was doing, and why I was doing it, and I have often been a pest because I have objected when what I heard or read was not to be taken literally. The boy who noticed that the emperor wasn’t dressed and objected loudly has always been my model. Mathematics seemed to match my psychology, a mistake reflecting the fact that somehow I did not take into account that mathematics is created by humans.
His sharp words, friends recalled, often came mixed with humor. Once, he and a colleague, Robert Kaufman, fell into a heated argument about whether students ought to be required to read classic literature. “Robert was all for it, and Joe was doing everything to provoke him. In disgust, Robert said: ‘Oh my God!’ and Joe calmly replied, ‘Please don’t exaggerate, just call me Professor.’ ”
Above all, Doob professed loyalty to the “austere and often abstruse” world of pure mathematics. If applied mathematics concerns itself with concrete questions, pure mathematics exists for its own sake. Its cardinal questions are not “How do we encrypt a telephone conversation?” but rather “Are there infinitely many twin primes?” or “Does every true mathematical statement have a proof?” The divorce between the two schools has ancient origins. Historian Carl Boyer traces it to Plato, who regarded mere computation as suitable for a merchant or a general, who “must learn the art of numbers or he will not know how to array his troops.” But the philosopher must study higher mathematics, “because he has to arise out of the sea of change and lay hold of true being.” Euclid, the father of geometry, was a touch snobbier: “There is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to give the student threepence, ‘since he must make gain of what he learns.’ ”
Closer to our times, the twentieth-century mathematician G. H. Hardy would write what became the ur-text of pure math. A Mathematician’s Apology is a “manifesto for mathematics itself,” which pointedly borrowed its title from Socrates’s argument in the face of capital charges. For Hardy, mathematical elegance was an end in itself. “Beauty is the first test,” he insisted. “There is no permanent place in the world for ugly mathematics.” A mathematician, then, is not a mere solver of practical problems. He, “like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” By contrast, run-of-the-mill applied mathematics was “dull,” “ugly,” “trivial,” and “elementary.”
It was the pure mathematicians who looked down on Von Neumann’s work on game theory, calling it, among other things, “just the latest fad” and “déclassé.” The same group would level a similar judgment against John Nash—just as Doob would against Claude Shannon.
As America’s leading probability theorist, Doob was well positioned to review Shannon’s work. His critique appeared in the pages of Mathematical Review in 1949. After briefly summarizing the contents of Shannon’s paper, he dismissed them with a sentence that would irritate Shannon’s supporters for years: “The discussion is suggestive throughout, rather than mathematical, and it is not always clear that the author’s mathematical intentions are honorable.” By the genteel standards of an academic review, this was lacerating, the equivalent of pistols at dawn.
Nearly forty years later, interviewer Anthony Liversidge raised the issue of the Doob critique with Shannon:
LIVERSIDGE: When The Mathematical Theory of Communication was published, there was an indignant review by a certain mathematician, accusing you of mathematical dishonesty because your results weren’t proved, he said, with mathematical rigor. Did you think that plain silly, or did you think, “Well, maybe I should work hard to meet his criticisms?”
SHANNON: I didn’t like his review. He hadn’t read the paper carefully. You can write mathematics line by line with each tiny inference indicated, or you can assume that the reader understands what you are talking about. I was confident I was correct, not only in an intuitive way but in a rigorous way. I knew exactly what I was doing, and it all came out exactly right.
Shannon rarely, if ever, felt the need to defend himself; the Doob critique, in other words, had clearly gotten to him. What’s more, Shannon was fully aware of his having vaulted over some of the intervening mathematics in the interest of practicality. Importantly, he noted in the middle of “A Mathematical Theory of Communication” that “the occasional liberties taken with limiting processes in the present analysis can be justified in all cases of practical interest.” This made sense: his primary readers were communications engineers; practical intentions mattered as much as, if not more than, purely mathematical ones. For Doob to critique the precision of his math felt, to many of Shannon’s acolytes, a bit like examining the Mona Lisa and finding fault with the frame.
Ironically, Doob’s claim that the paper was not mathematical enough ran up against the opposite complaint from engineers. As the mathematician Solomon Golomb put it, “When Shannon’s paper appeared, some communications engineers found it to be too mathematical (there are twenty-three theorems!) and too theoretical.” The problem, in hindsight, might not have been Doob’s misunderstanding of the mathematics at work; rather, he didn’t understand that Shannon’s math was a means to an end. “In reality,” said Golomb, “Shannon had almost unfailing instinct for what was actually true and gave outlines of proofs that other mathematicians . . . would make fully rigorous.” In the words of one of Shannon’s later collaborators, “Distinguished and accomplished as Doob was, the gaps in Shannon’s paper which seemed large to Doob seemed like small and obvious steps to Shannon. Doob might not realize this for, how often if ever, would he have encountered a mind like Shannon’s?”
The information theorist Sergio Verdú offered a similar assessment of Shannon’s paper: “It turned out that everything he claimed essentially was true. The paper was weak in what we now call ‘converses’ . . . but in fact, that adds to his genius rather than detracting from it, because he really knew what he was doing.” In a sense, leaving the dots for others to connect was a calculated gamble on Shannon’s part: had he gone through that painstaking work himself, the paper would have been much longer and appeared much later, both factors that would have likely diminished its reception. By the end of the 1950s, other engineers and mathematicians in both the United States and the Soviet Union had followed Shannon’s lead—and translated Shannon’s creative and rigorous explanations into both the language of pure mathematicians and the language of engineers.
Criticism like Doob’s may have stung—but there was a measure of respect in the fact that a mathematician of Doob’s stature had read Shannon’s work at all. Doob and Shannon also settled their differences in 1963. Shannon was invited by the American Mathematical Society to deliver the prestigious Josiah Willard Gibbs Lecture, a signal honor within the field. The person who introduced him on that night—and, as the society’s president, surely had a hand in the invitation—was none other than Joseph L. Doob.