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* So, the good news is that I turned in the manuscript for The Damn Book last week. The bad news is that I've been pretty much brain dead ever since after months of feverish writing. So instead of blogging this weekend as originally planned, I napped and indulged in DVD marathons. My bad. I will be back this coming week, refreshed, and ready to tackle one of the many blog post topics in my teeming fodder file. One of those would have been an acknowledgment of Gordon Brown's belated apology to Alan Turing, but intrepid guest blogger Alex Morgan beat me to it with the following personal rumination.*

Only once in my life have I ever heard a mathematician refer to a contemporary as a "genius." The person doing the referring was the well known topologist, Peter Hilton. He was visiting Yale for a few days in the early 1970's, and, since I was a graduate student doing my dissertation research in topology, I was invited to the supper in his honor. I remember a broad-faced man who laughed easily and was brightened by conversation. I told him I was interested in applications, an eccentric impulse by the standards of the mathematicians then at Yale, who were proudly "pure."

Hilton surprised me by saying that he himself had done some applied mathematics. He was British, and during World War II he had been enlisted into the group at Bletchley Park (north of London) working to break the German secret codes and ciphers. He didn't disapprove of my interest in control theory and differential equations, but he urged me to complete my work in topology. He assured me that to be a good applied mathematician, you have to be a good pure mathematician. (I would respectfully debate the point with him today.)

It was in recounting his experiences with the British cryptographers that he let drop his unusual comment about genius. He said something like, "Normally, somebody tells you about a result he's proven, and you think 'I could have done that. I'm just as bright as he is.' Of course you wouldn't say it out loud. But Alan would do something, and I'd be forced to admit to myself: I never would have been able to think of that."

Hilton went on to tell the story of how Alan Turing had taken a set of documents encoded with the German Enigma machine, went away by himself to think, and proved something important about how they had been generated -- proved in the sense of a mathematical proof and important in the sense that it led to breaking the German naval codes, allowing Britain to defeat the German U-boat blockade in 1941 and thereby survive the war.

I thought of my chat with Prof. Hilton when Turing was mentioned recently in the news. The *Baltimore Sun* headline on Sept. 11, 2009, was "British government apologizes for 'inhumane' treatment of gay wartime codebreaker Alan Turing." The article, credited to Jill Lawless of The Associated Press, begins: "British Prime Minister Gordon Brown offered a posthumous apology Friday for the 'inhumane' treatment of Alan Turing, the World War II codebreaker who committed suicide in 1954 after being prosecuted for homosexuality and forcibly treated with female hormones. The mathematician helped crack Nazi Germany's Enigma encryption machine -- a turning point in the war -- and is considered a father of modern computing."

Turing's early academic work centered on the logical foundations of mathematics, an area far removed from the physical world. Yet, if you look at his papers, what jumps out is his range of interests, his "eccentric" insistence that machines, quantum physics, neuroscience, extrasensory perception, biology, and "the soul" were all worthy of careful consideration. If you want to encounter Turing "face to face," there are two papers worth looking at: "On Computable Numbers, with an Application to the Entscheidungsproblem" (1936) and "Computing Machinery and Intelligence" (1950). Turing's writing has a striking clarity, so that a "general reader" can glean the gist of his ideas, even without understanding the details.

The 1936 paper focuses on a "challenge problem" posed in 1928 by the venerable German mathematician, David Hilbert. This problem is referred to in German as the *Entscheidungsproblem*, in English "the decision problem." Its spirit is something like: Can big hunks of modern mathematics be derived from postulates, like Euclidean geometry? The logician Alonzo Church solved this problem (the answer: "usually not") and published his results some months ahead of Turing. Normally, this would be a disaster for a young graduate student, and Turing was disappointed. However, the actual "result" was far less important than Turing's method of proof and his overall perspective (which were different from Church's).

In his paper, Turing defines an abstract "machine" -- now always referred to as a Turing machine -- and he then proceeds by reinterpreting the decision problem in terms of different possible ways his machine might function. Turing wrote his paper on two levels: he establishes the mathematical result that is the "official" point of the paper, but he also describes his vision for a computer as we know it today. Much of his explanatory language is less dauntingly technical than a typical mathematics research paper, as if Turing is hoping for a more general audience. Hilton notes in his memoir, *Reminiscences of a Codebreaker*: "What very few knew then [in 1936], and somewhat more know now [in 1998], is that, even in those early days, his machine was not merely, in his mind, a metaphor but also a blueprint for a machine which could actually be built, that is, a computer."

"Computing Machinery and Intelligence" was written for the philosophy journal, *Mind*. By 1950, Turing had hands-on experience building and working with digital computers, so in this paper he can give an even better conceptual overview of their design. To most people in the 1950's who knew about computers, they were simply "computers," that is, devices for generating tables of numbers using mathematical formulas. Although wonderfully more powerful than the human "computers" that preceded them -- the word was first used in English in 1613 to refer to people who compute -- they were crude and slow by the standards of the Twenty-first Century.

Imagining one of these early computers to be a "thinking machine " is a remarkable extrapolation. Turing proposes a clarification of what it would mean for a computer to think (afterward, called the Turing test), formulates responses to various objections to the idea that a machine might think, estimates how large a memory and how fast a processor a thinking computer would have to have, and finally considers how a machine might be programmed to learn.

In this paper, Turing makes the overly optimistic prediction that human-level machine intelligence would be achieved in fifty years; that is, by the year 2000. The field of artificial intelligence (AI) was founded six years after "Computing Machinery and Intelligence" was published (and two years after his death). Many of his ideas were further developed in the following decades, especially his perspective that thinking was "logic processing," that is, a step-by-step application of rules and deductive reasoning. AI has achieved many fine successes, but the development of human-level thinking machines remains elusive.

In 1952, Turing's house was robbed. He reported the theft to the police, who, in the course of their investigation, uncovered evidence that he was having a relationship with one of the men associated with the robbery. Turing was charged with homosexuality and eventually convicted of the "crime." The hormone treatment he was forced to undergo as an alternative to prison was not his only penalty. He lost his British security clearance and thereby his access to government support for his research. The U.S. authorities refused to grant him a visa, on the grounds that he was a felon, to continue joint work at Princeton. Isolated, lonely, and depressed, he committed suicide on June 7, 1954.

Britain hounded its war-hero genius to death because he was gay. Of course, no citizen should be so persecuted. Still, I can't help but be astonished by yet another example of the self-destruction implicit in a tribe's impulse to repress its outliers. Turing's colleague, Jack Good, summarized this way (as quoted in Hilton's memoir): "Fortunately, the authorities at Bletchley Park had no idea Turing was a homosexual; otherwise, we might have lost the war."

Where did the leggo computer come from and can you point to more detail please

Posted by: Roy Vannini | September 25, 2009 at 02:31 PM

The Leggo Turing machine is from http://www.mapageweb.umontreal.ca/cousined/lego/5-Machines/Turing/Turing.html

Posted by: Alex Morgan | September 25, 2009 at 03:15 PM

I've played around with the Leggo Turing machine. It's lots of fun.

Your story reminds me of when I was in college and one of my friends was talking about a graduate school he really wanted to attend. When somebody asked him "You applied?"

He responded "No, theoretical."

Posted by: The Science Pundit | September 27, 2009 at 08:21 PM

"otherwise, we might have lost the war.." Who, the British? I think that is probably an exaggeration.

Posted by: changcho | November 04, 2009 at 03:27 PM