Richard Hamming: An Outspoken Techno-RealistNovember 16, 1998
If there were an Academy of Computation in Athens, according to the author, carved on the lintel would surely be one of Richard Hamming's most well known statements: "The purpose of computation is insight, not numbers."
Philip J. Davis
Richard Hamming, "Mathematics on a Distant Planet." American Mathematical Monthly, Vol. 105, No. 7, August-September, 1998, pages 640-650. (Based on a luncheon talk given at the Northern California Section meeting of the Mathematical Association of America, San Francisco, February 22, 1997.)
Having just read and been moved by this posthumous article, which could be described as "Dick Hamming's mathematical credo," I should like to use my space to comment on the article and then to take off from one point made at its end.
Richard Hamming (1915-1998) got a BS from the University of Chicago and, in 1942, a PhD in mathematics from the University of Illinois. In 1943 he was at the Los Alamos Lab, and in 1946 he joined Bell Telephone Labs, where he remained until his retirement, thirty years later. (Alta Vista, searching the Web for "Hamming codes," produced 787 citations, and this is surely an inadequate measure of the importance of Hamming codes to information technology.) Thereafter, and until recently, he was a professor of computer science at the Naval Post-graduate School in Monterey, California.
I knew Hamming only slightly, but from the few times I met him, and from various of his writings, I should call him a no-nonsense kind of guy. Sharp-witted, often acerb, direct, forceful, holding strong opinions that often coincided and occasionally conflicted with my own, he was a technological realist. His books are as free as one can imagine from unnecessary mathematical maneuvering and jargon, from the vain generalizations---largely in the service of "I'll show you how smart I really am"---that can make mathematical writing tremendously obscure.
I found his philosophy of mathematics quite agreeable, but, as it happens, I diverged on many points. Though it may be a disservice to reduce to a few lines what a person has concluded over a lifetime, I would sum up his view this way: If a piece of mathematics works in the real world, why bother inquiring why (in the philosophical sense). And if a piece of mathematics isn't useful in the real world, you have the option of ignoring it completely.
There is no reason to rely on my words, however, when Hamming himself was a great phrase-maker, as in the following well-known instance:
"The purpose of computation is insight, not numbers."
Placed as an epigraph for his book Numerical Methods for Scientists and Engineers (1962), this short sentence has been repeated over and again by hundreds of authors. Discussions of it can be found on the Web. It is so well known by now that if there were an Academy of Computation in Athens, it would surely be carved on the lintel.
But what does it mean? Like all good slogans-cogito ergo sum, for example-it requires explication. I have the impression that when Hamming himself was asked to explain what he meant, he wiggled around and came up with changing interpretations and constant modifications.
"For more than forty years I have claimed that if whether an airplane would fly or not would depend on whether some function that arose in its design was Lebesgue but not Riemann integrable, then I would not fly in it. Would you? Does Nature recognize the difference? I doubt it!"
"Hermite said 'We are not the master of Mathematics, we are the servant.' I have often said the opposite: 'We are the master of Mathematics not the servant; it shall do as we want it to do.' In truth, I seem to believe in a blend of the two remarks; at times we are driven and at times we are in control of mathematics."
"I know that the great Hilbert said 'We will not be driven out of the paradise Cantor has created for us,' and I reply 'I see no reason for walking in!'"
"Do not dismiss the outsider too abruptly as is generally done by the insiders."
"Mathematics is not merely an idle art form, it is an essential part of our society."
"In science and mathematics we do not appeal to authority, but rather you are responsible for what you believe."
These quotations all call for discussion, and some have indeed been discussed ever since the Greeks invented philosophy. The last two, for example, when properly interpreted, may come as close to expressing social concern as Hamming was able to bring himself to do publicly.
For the remainder of this article, I would like to comment on yet another quotation:
"With the enormous growth of results at well over 100,000 (new?) theorems every year . . . the chance of a new piece of pure mathematics being spotted by you and also being at hand when you need it, and not have to be recreated when needed, is increasingly small. . . . Regeneration is increasingly easier than retrieval."
Now here is a thought that diffuses rapidly into the related but separate questions of research strategies, of originality, plagiarism, rights of intellectual property, royalties.
When I was a graduate student I used to hear horror stories of PhD candidates who, in the final throes of their candidacy, were informed that others had anticipated their major results: Your result, Candidate Doe, can be found in a 1923 paper in the ABC Journal of Mathematics; so go back to the drawing board, and this time come up with something original.
Were these stories based on true occurrences, or were they fairy tales put forward to frighten aspirants into proper standards of accomplishment?
Not so many years after my PhD was awarded, I was working alongside old Professor PDQ, a mathematician of great international reputation. He chanced to read one of my published papers and afterward pulled me by my ear (so to speak) over to the library, hauled down a certain journal dated 1933, blew off the dust, and pointed out an article of his that contained the same result. "You should have known about it," he cautioned me.
As a graduate student, I had relied on my thesis adviser to certify the originality of my results. But now I was on my own, and for a few days I trembled in my boots. On study, it turned out that while my theorem and Professor PDQ's were located in the same specific area of mathematics, my hypotheses as well as the methods I employed were quite different. Saved from professional ignominy!
Much less traumatic was the following experience of several months ago. I gave a talk in which I mentioned theorem T, commonly known as Y's theorem. After my talk, Professor Z came up to me and said, "You know, T is referred to as Y's theorem, but R got it a generation before Y."
I was inclined to reply to Professor Z, "Are you absolutely sure that there wasn't an M who got it before R?" But I refrained because I didn't care to lose Z's friendship.
Fairly recently I worked up a theorem in a very popular area. I wanted to check its originality. I looked in a few appropriate books: I couldn't find it. I e-mailed a few authorities: They'd never heard of it. I went on the computerized mathematical databases and inquired. I was confronted with the "zero or infinity" phenomenon. When I searched in the specific area, the computer came up with more papers than I could deal with in ten lifetimes, half of which may have contained errors. When I added a few qualifiers, I came up with zero hits. Maybe I should have posted a notice on the Web: "I am claiming priority on such and such a theorem. If you have any reason to dispute this, come forward or forever hold your peace."
Back to Hamming's quote. We seem to be in a period of transition, in which claims of priority and originality, on the one hand, make a difference and, on the other hand, being in some cases impossible to validate but easy to invalidate, ought not to make a difference.
One final point. A prospective reader, seeing the title "Mathematics on a Distant Planet," may be led to think that Hamming has written a piece of science fiction. Not at all. He has set up hypothetical mathematicians on a distant planet in order to speculate on which portions of the assumptions, definitions, modes of reasoning, and assignment of values that characterize our mathematical world are universal and which might not be.
In this spirit, I wonder how the scientists of the distant planet are handling the dilemmas of information overload and intellectual property.
Philip J. Davis, professor emeritus of applied mathematics at Brown University, is an independent writer, scholar, and lecturer. He lives in Providence, Rhode Island.