We weren't quite sure what to expect. We were familiar with the 200-seat Tusten Theatre in Narrowsburg, and we were pretty sure that it couldn't really accomodate an opera. The stage is quite small and there isn't even an orchestra pit.

What's more, the tickets for the Delaware Valley Opera's production of Mozart's Don Giovanni — a tragicomical chronicle of the last few days of archetypal sexual compulsive Don Juan — were only $25 a seat, which seemed frighteningly low for a three-hour opera with eight solo roles, chorus, and orchestra.

Yesterday afternoon we discovered how it's done: The small Delaware Valley Chamber Orchestra — two violins, viola, cello, bass, French horn, bassoon, clarinet, oboe, flute, and timpani — occupied the space between the stage and the first row of center seats. DVO Director Jim Blanton conducted from behind a synthesizer set to a harpsichord program for the recitativi. The sets were minimal but otherwise it was complete authentic Don Giovanni — uncut or nearly so and sung in Italian — and great fun. There were even English supertitles projected on a small screen at the top of the stage.

The eight roles were very well sung, particularly Jeremy Moore as a young, virile, and arrogant Don Giovanni; Julie Ziavras as a powerfully voiced Donna Anna; and Jeanne-Marie Lowell as Donna Elvira, never quite sure if she's hunting down the Don or stalking him. They were joined by the dozen or so fine voices of the DVO Choral Ensemble.

The minimal sets — with some action occassionally in front of the closed curtain — helped to keep the pace of the opera going, and only towards the end was I disappointed with the staging. Normally the Commendatore appears on stage as a marble statue. In this production he was merely a disembodied voice coming from the rear of the theater, and Don Giovanni had to reach into space as the Commendatore supposedly took his hand. Almost in compensation for this deficiency were the wonderful writhing demons who crawled across stage to encircle the Don and drag him down to hell.

In those final few moments of the opera, the problem with using a chamber orchestra also became most apparent. A strong orchestra plays a crucial part in introducing the Commendatore with the same chords heard three hours ago in the overture, and in these chords and the subsequent increasingly frenetic music, the chamber orchestra became not quite adequate.

Still, it was a real treat to see Don Giovanni in such an intimate setting; the Delaware Valley Opera has two more performances of Don Giovanni coming up, as well as Donizetti's Don Pasquale sung in English, and two other programs of songs and opera excerpts.

ABBA. QED.

Julie Lerman tagged me to answer a few questions that have been circulating.

The Short Version

How old were you when you first started programming?
18.

How did you get started in programming?
College course (1971).

What was your first language?
FORTRAN IV.

What was the first real program you wrote?
Probably health insurance claims statistics (c. 1976).

What languages have you used since you started programming?
PL/I, APL, 8080 assembly, Z-80 assembly, MBASIC, CBASIC, 8086 assembly, Turbo Pascal, C, C++, C#, VB .Net.

What was your first professional programming gig?
Getting paid strictly for programming apart from writing has been rare for me until very recently, but in 1988 (I believe) I was paid to write a graphical roulette-wheel for a Fall Comdex (Las Vegas) presentation of the OS/2 Presentation Manager.

If you knew then what you know now, would you have started programming?
Absolutely. Owning a computer without programming is like having a kitchen and using only the microwave oven.

If there is one thing you learned along the way that you would tell new developers, what would it be?
Learn as much as you can because everything fits together.

What's the most fun you've ever had ... programming?
Time-critical interrupt-driven Z-80 machine code for a homebrew digital music synthesizer (c. 1980).

So Who's Next?
I shall tag the three founders of Wintellect: Jeff Richter, Jeff Prosise, and John Robbins.

The Long (Essay) Version

(Portions adapted from a talk I gave to the C++/Java Sig in New York City in May 2005.)

Stevens Institute of Technology — a small engineering and science school in Hoboken, New Jersey — was the first college to require all incoming freshman to take a programming course. In 1971 when I entered Stevens Tech as a freshmen, the course was E11 and the language was FORTRAN IV.

Stevens Tech had a DEC PDP-11 in the basement of the library, along with a bunch of IBM keypunch machines. You typed each statement of your program on a card, and then submitted the deck to the snotty guru behind the counter, who would sometimes riffle through the cards and say "I see a few syntax errors in there, but let's run it through anyway." And sometime later you would pick up a stack of printed paper. If your program was so bad it couldn't run, you'd get a tiny stack. But sometimes if your program ran and ran and ran, you'd get a very tall stack of paper. That was the most embarassing.

There was also a room in the library basement with several teletypewriters connected to the PDP-10 where you could do stuff in real time, but you had to belong to the ACM to use the teletypes, and those guys were strange.

I didn't much like programming then. I had gone to college to be a mechanical engineer. I liked making stuff with my hands, and computing seemed much too abstract for me. I took only two other programming courses in college. One was on numerical analysis using FORTRAN, and the other was a course in assembly language.

The course in assembly language was taught by a math professor, Myron E. White, whose was universally known as Myron Epsilon White. In the year I graduated, Professor White was the recipient of the annual Golden Ream Award — a gruesome-looking large gold-plated screw mounted on a board, which was supposed to symbolize what the professor had done to his students.

There was no textbook available for assembly language programming on the PDP-10 so Professor White had written and published his own, a book entitled Meet Macro-10. I'm not sure what technology he used to publish this book, but a typewriter was definitely involved. Each page looked like a typewritten sheet of 8½ × 11-inch paper. This course is assembly language was totally baffling to me. I think I must have understood that assembly language actually translated to the machine code of the computer and hence was very fast and efficient, but when submitting programs on cards and getting back paper, processing speed was not something you witnessed first hand.

I eventually became a math major and when I graduated in 1975 I moved to New York City and got a job at New York Life Insurance Company as an actuarial student.

Some of the younger actuaries programmed, and new actuarial students were expected to program as well. The actuaries coded in FORTRAN using punched cards on an IBM 1130 — a castoff computer from an earlier time. The company's main computer was a big IBM 370, and within a couple years non-IT people like myself also got access to the 370 via a terminal system called Time-Sharing Option or TSO.

New York Life was one of the few major financial institutions that didn't do COBOL. New York Life was a PL/I shop. By this time, I was liking programming a lot better than in college, partially because I was getting pretty good at it, and also because I could make and correct my own syntax and runtime errors in the privacy of my own terminal, thank you. No more snotty gurus behind the counter riffling through card decks.

PL/I was a glorious language. It had a little something for everybody. You could do reports in PL/I but you could also do hard-core science and engineering programming. There was a FIXED DEC data type where you specified the number of decimal places. The math library was very rich. The DO loop was particularly versatile (and I may not have the syntax entirely correct but you'll get the idea):

DO I = 1 TO 5, 6 TO 20 BY 3, 100, 120, 1000 TO 1010 BY 2;

To this day I encounter problems where I think "I'd really like to use the PL/I DO loop for this." You could also put these loops right in your WRITE statements.

Meanwhile, at home, I was working with my hands. I had an interest in electronic music (beginning with the 1968 release of Switched On Bach) and by about 1977 I had started building electronic music instruments out of CMOS chips. With the help of a couple books (Don Lancaster's "cookbooks" were instrumental) I had taught myself digital logic circuitry, and I was wiring up chips with a wire-wrapping gun to make little electronic-music sequenzers. I had seen Philip Glass's Einstein on the Beach at the Metropolitan Opera House in 1976, and an early performance of Steve Reich's Music for 18 Musicians (and other works) at The Kitchen around that time, and I really liked the idea of creating repetitive music. Except I didn't want to play it — I wanted a machine to play it for me.

After building two sequencers, I remember designing a third where I'd have certain codes in memory that would govern how notes stored in other parts of memory would be retrieved and moved around and repeated, and how the individual bits of these codes would trigger other chips, and I suddenly thought "I bet this is how computers work!" It was an epiphany the likes of which I have never ever experienced.

This was sometime in 1978, I believe, and I knew there were chips called microprocessors, but I hadn't really looked at them because they had seemed like overkill for what I was doing. Suddenly they seemed ideal. I realized that having a microprocessor at the center of an electronic music synthesizer would be very beneficial. At that time there were two major 8-bit microprocessors available for hobbyist use: the Intel 8080 and the Motorola 6800.

The Barnes & Noble store on 5^th Avenue and 18^th Street (the very first Barnes & Noble store) had a good selection of technical books, but I think they happened to be sold out of books about the 6800 at the time I checked. Instead I gravitated towards the 8080, the beginning of a life centered around Intel architecture. I sometimes wonder how my life would have been different if the Barnes & Noble had a good book about the 6800 in stock at the time.

After my hands-on experience with digital logic design, 8080 assembly language was a snap. It was obvious to me exactly what every instruction did in terms of gates and registers.

Around 1979 I started building a computer-controlled digital electronic music synthesizer. It took about two years and was based on a block diagram in an article in the Computer Music Journal, Vol. 1, No. 2 (April 1977) entitled "Design of a Digital Oscillator which will Generate up to 256 Low-Distortion Sine Waves in Real Time" by John Snell. I ended up using the Zilog Z-80 microprocessor which was binary compatible with the 8080 but also supported additional instructions such as indexed addressing that I found very handy.

The computer part had 64K of dynamic RAM, with another 64K accessible sequentially through an I/O port. The synthesizer part was entirely TTL (for speed). It generated not the 256 sine waves promised by John Snell but 80 sine waves in real time, with separate envelopes for frequency and amplitude, combined in pairs in simple frequency-modulation (FM) synthesis, so I got 40 simultaneous voices. I built everything myself except for one 4K static RAM board. The machine had a five-octave music keyboard, and a regular keyboard and video display capable of 16 lines of 32 characters, which basically let me type hexadecimal bytes into memory.

The software system controlling the synthesizer was entirely interrupt driven, and the machine code had to be as efficient as possible. As I coded, I literally counted clock cycles, and I soon memorized the length of each instruction. I would code and recode, shaving a cycle here and a cycle there.

After it was completed in 1981, I was able to program entire music compositions and play them back, and I spent about a year doing this until it became very, very clear that I had no talent at all for composing music. But it was fun and obviously a valuable learning experience. (Without this experience there obviously would have been no Code: The Hidden Language of Computer Hardware and Software.)

Sometime in the future I'd like to post some photographs and documents about this computer-controlled digital synthesizer, and even some of the "music" I wrote.

About 1982 I bought an Osborne I, which ran CP/M on a Z-80. The simple reason I bought an Osborne rather than an IBM PC was because the Osborne was $2,000 and the IBM PC was $5,000. About the same time, New York Life got some IBM PCs, and I picked up 8086 assembly language fairly quickly as well.

I finally bought a two-diskette $5,000 IBM PC early in 1984. It wiped out two credit cards, so I decided the machine needed to pay for itself. I wrote an article about ANSI.SYS and the PROMPT command, sent it to PC Magazine and they sent me $800. It was the first time I had ever been paid for something I wrote.

In the summer of 1984, PC Magazine was doing the first of what became an annual roundup review of printers, so they called all the people who had written for PC Magazine who lived in the New York City area and asked them to come in to the office to review printers. The PC Magazine offices were at One Park Avenue just a few blocks north of the New York Life offices at 51 Madison Avenue, so it was very convenient for me. I reviewed printers and also started showing people at the magazine some small assembly-language programs I had written. I was soon writing little 300-500 byte .COM file utilities for PC Magazine

By the summer of 1985 I was getting so much freelance work from PC Magazine that I was able to quit my job at New York Life Insurance. I haven't had a real job since.

I think I learned C in 1984 or 1985. I know that I was pretty familiar with C when I started working with the beta Windows SDK in 1985. Another fortuitous connection occurred: Jonathan Lazarus was a VP at Ziff-Davis (which published PC Magazine) at the time, and he was very enthuiastic about Windows. Jon left Ziff and contracted with Microsoft for his consulting firm (H. Roark & Associates!) to publish a Windows programming magazine. But they chickened out and decided that Microsoft Systems Journal would cover both DOS and Windows programming. Jon knew a few PC Magazine authors so he recruited us to write some articles. I wrote the article "A Step-by-Step Guide to Building Your First Windows Application" for MSJ, Vol.1, No. 2 (December 1986) which I believe was the first article about Windows programming to appear in a magazine.

This early connection between PC Magazine and Microsoft Systems Journal continued for several years. Editors and writers from the two magazines would often socialize together, both in New York City and at industry events such as Comdex.

At some Microsoft-related function in 1986 I told Tandy Trower how much fun I was having writing articles about Windows programming for MSJ. When Microsoft Press editor-in-chief Susam Lammers asked Tandy if he knew anybody who could write a book about Windows programming, my name came up, and Susan gave me a call. As I result, I was contracted to write a book and I wrote the first edition of Programming Windows from January through August 1987.

I think from that point my programming career has been somewhat more conventional....

Everybody agrees: Blogging is a crucial tool for marketing books. It is from blogs that people first hear about books; it is through blogging that the potential reader's interest is aroused; and it is through blogging that the reader is ultimately seduced into purchasing the book, resulting in yet another $2.25 of income to the author.

What nobody tells you is this: It's not the author's blog that's important. The author's own blog is pretty much inconsequential. Nobody reads that thing, and really — why should they?

It's actually other people's blogs — and in particular Jeff Atwood's take on The Annotated Turing — that perform the magic feat of causing a book's Amazon.com sales rank to climb from ~20,000 to 368 throughout the course of a single day.

I'm sure the sales rank for The Annotated Turing will soon be back to normal, but meanwhile, thanks Jeff!

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In January, 1952, English mathematician and computer pioneer Alan Turing was arrested for having sex with another man.

The law that Alan Turing broke was the infamous Section 11 of the Criminal Law Amendment Act of 1885. Section 11 prohibited "any act of gross indecency with another male person" and specified that offenders "be imprisoned for any term not exceeding two years, with or without hard labour."

The term "gross indecency" was not specifically defined in the law, but it was understood to mean any sexual contact other than that denoted by the quaint English legal term "buggery." Oral sex or mutual masturbation would fall under the category of "gross indecency."

Section 11 was controversial from its very beginning. The Criminal Law Amendment Act of 1885 described itself as "An Act to make further provision for the protection of women and girls, the suppression of brothels, and other purposes." The law raised the age of consent for girls from 13 to 16, and contained several provisions intended to prevent women from exploitation, such as being drugged in brothels or abducted into prostitution.

Section 11 was added to the Act just a week before it was passed. At the time it was noted that the acts prohibited by Section 11 had never been illegal in England before when performed in private, and that the law could easily be abused through blackmail. The most famous victim of Section 11 was playwright and poet Oscar Wilde, who served two years hard labor from 1895 to 1897. His health severly compromised by the prison term, Oscar Wilde died three years after his release at the age of 46.

By the time of Turing's conviction, alternatives to imprisonment were offered in the form of hormone treatments called "organotherepy" but more commonly known as "chemical castration."

Experiments with treating gay men with hormones had begun in the 1940s. At first it was believed that homosexuality was caused by insufficient "maleness," so the treatment involved testosterone. As we know now, increased levels of testosterone are associated with aggressiveness (sexual and otherwise), so it's not surprising to learn that the experimental testosterone treatments did not have the anticipated effect!

By Turing's time, they had switched to estrogen treatments. The implant they gave Turing for a year rendered him impotent and caused his breats to grow.

The early 1950s were a terrible time to be identified as a gay person. As part of my research in writing The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine, I wanted to get a good understanding of what it was like. Of course I got some great information from the concluding chapters of Andrew Hodges's biography Alan Turing: The Enigma (Simon and Schuster,1983).

I also found a fascinating book by historian David K. Johnson about the situation in the United States entitled The Lavender Scare: The Cold War Persecution of Gays and Lesbians in the Federal Government (University of Chicago Press, 2004), and which I discussed in a blog entry two years ago.. The title plays off the concept of the "Red Scare" initiated by Senator Joseph McCarthy and others to weed out communists in the State Department. There actually weren't very many communists in the State Department, but there were plenty of closeted gay people working in the U.S. government and they were fired by the hundreds. The term "security risk" used during this period was basically a euphemism for "homosexual."

I was hoping to find something similar to The Lavender Scare but about England. I am now convinced that such a book does not exist, but I did benefit a great deal from H. Montgomery Hyde's The Love That Dared Not Speak Its Name: A Candid History of Homosexuality in Britain (Little, Brown and Co., 1970), published in England under the title The Other Love. This book covers a much broader period of time but provided me with background on Section 11 and details regarding the early-1950s persecution of gay people in England.

As a result of his conviction, Turing's job options were certainly diminished. Any type of security clearance would be out of the question. (At the time, his crucial work on breaking codes used by the German military during World War II was still top secret.) He would no longer be able to visit the United States. A 1952 law prohibited admission to the U.S. of "‘aliens afflicted with psychopathic personality," which was interpreted to mean homosexuality.

Alan Turing committed suicide in June, 1954, at the age of 41. He left behind no note, and his suicide has some mysterious aspects, but it is generally assumed that it was related to the humiliation he suffered from his prosecution under Section 11.

That certainly seems like a long time ago! Today we are as appalled by the 1950s persecution of gay people as much as we are appalled by the discrimination against persons of color during the same period. As a society, we have made significant moral progress since that time, and I am encouraged by these trends.

In Europe and North America, discriminatory laws against gay people have largely been repealed. Full marriage equality — certainly an important milestone in the progressive historical advancement of human rights and individual dignity — is becoming a reality. Today five countries (Canada, Belgium, the Netherlands, South Africa, and Spain) offer full marriage equality for gay couples, and Norway will join their ranks next year.

Marriage equality in the United States will probably proceed one state at a time until some kind of tipping point is reached, or the U.S. Supreme Court makes a definitive decision. Today only Massachusetts and California have full marriage equality. Here in New York, Eliot Spitzer was recently elected Governor with a platform that included marriage equality. Although he's gone now, his successor David Paterson recently affirmed that same-sex marriages outside the state will be recognized here (which is, of course, the normal procedure except that some states have specifically gone in the opposite direction). A fascinating article in the New York Times last month described Governor Paterson's long affinity with gay people, and indicated that he is also a supporter of marriage equality.

But are we really so removed from the laws that drove Oscar Wilde and Alan Turing to their early deaths? Not really.

It was as recently as 2003 that the United States Supreme Court decision Lawrence v. Texas struck down state laws prohibiting gay sex. (In 1998, based on a false report of a man with a gun, the Harris County police entered the home of John Lawrence and found him having sex with another man. Rather than apologize profusely and quickly leave as normal people would have done, the police arrested the two men and charged them under Texas's Homosexual Conduct law.) At the time of Lawrence v. Texas some 13 other states had laws prohibiting homosexual sex acts, and even after the decision many of these laws are still on the books.

What scares me is the real agenda of the people behind the movement opposing marriage equality. It seems that for some of these people, the real motivation has nothing to do with "marriage" or "family" or "children" or "home" or "values." It's really all about gay sex, and the Lawrence v. Texas decision plays a major role in their worldview.

For example, one of the best-known and acknowledged nationwide leaders of the movement opposing marriage equality is James C. Dobson, head of the organization Focus on the Family. It is quite clear from reading Dr. Dobson's book Marriage under Fire: Why We Must Win This Battle (Multnomah Publishers, 2004) that the real problem for him is the Lawrence v. Texas decision — in short, that the Supreme Court's decision in giving gay people the legal right to have sex also implies the legal right to marriage.

I will quote at length (pages 39 to 41) without ellipses so you can get a proper sense of Dobson's not-so-hidden agenda:

And now the nation's courts have wreaked havoc on the institution of marriage. In one decision after another, the judiciary has torn into the fabric of the home. I won't review all those unfortunate cases at this point, but there is one that stands above the rest. On June 26, 2003, the U.S. Supreme Court considered the legality of homosexual behavior and found that, lo and behold, the Constitution guaranteed a right to sodomy. Yep, it was tucked right there (somewhere) in the original document.

With this ruling, our Founding Fathers must have rolled in their graves. Our august justices "made up" this new constitutional right and used it to strike down the Texas law prohibiting sodomy. The case is now known as the infamous Lawrence v. Texas decision. Very few Americans agreed with the decision, but they were never asked. They no longer determine their own destinies. Abraham Lincoln said in the Gettysburg Address that ours is a government "of the people, by the people, and for the people," and yet "the people" have now been co-opted by an unelected and unaccountable judiciary, appointed for life, that determines all the great moral issues of our day. Each time the Supremes meet, it's as through they are holding a "Constitutional Convention," because the foundational document becomes whatever any five of these justices say it is. This is called an oligarchy — a government by the few — and it is taking us ever further down the road to moral relativism.

Writing for the majority in the Lawrence case, Justice Anthony Kennedy — whom I consider to be the most dangerous man in America because of his determination to rewrite the Constitution — stated that, speaking of the prohibition of sodomy, the law's "continuance as precedent demeans the lives of homosexual persons." By ruling that sodomy is a constitutionally protected "right," the highest court in the land declared, in effect, that considerations of morality and decency were irrelevant.

It was this regrettable decision that has created the present turmoil throughout the nation. It has emboldened rogue commissioners, mayors, and legislators to begin overriding laws prohibiting homosexual marriage. They have been passing out marriage licenses like candy. These minor bureaucrats now have things going their way, and they are going to strike while the iron is hot. This is why we are in the state of peril that faces our nation today. Like Adolf Hitler, who overran his European neighbors, those who favor homosexual marriage are determined to make it legal, regardless of the democratic processes that stand in their way.

The logic in these paragraphs is so tortured that I can hardly believe that Dr. Dobson actually believes anything he's writing here. The only "rights" Lawrence v. Texas has eliminated are those of a government to decide what consenting adults can do in the privacy of their bedrooms! Dr. Dobson seems to feel his rights have been restricted, yet the "rights" that he wants is the power to use the government to enforce his own narrow view of proper sexual behavior. That concept should scare heterosexuals and homosexuals alike!

The introduction of Adolf Hitler in Dr. Dobson's argument is astounding. Surely Dr. Dobson knows which side of the gay rights debate the Nazis were on! Surely Dr. Dobson knows that laws prohibiting gay sex in Germany were almost repealed during the Weimar Republic, and that when the Nazis gained power they actually strengthened the laws and made homosexual sex a felony rather than a misdemeanor. Surely Dr. Dobson knows that homosexuals were imprisoned by the Nazis in concentration camps, and that they had their own special triangle to identify them (notice the 5th column):

It could be argued that the extremist views of Dr. Dobson do not represent the mainstream of the movement opposing marriage equality, and that most people really do not want to reinstate laws that prohibit gay sex. I don't know. But I have not heard anyone on the anti-equality side actually denounce Dr. Dobson's position on this issue. I am forced to asume that Dr. Dobson's ugly and immoral views are actually quite prevalent.

Anyone who proposes the reinstatement of laws that prohibit consensual sexual contact between adults must consider the consequences of these laws, and that means to be familiar with the people who have been their victims. Alan Turing might still be alive today — he would have turned 96 earlier this week — had it not been for a law that made him a criminal, and which stripped him of his dignity and freedom.

The best way to prevent these laws from coming back is to give gay men and women the fullest protection of the law, and that means full marriage equality.

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I'm pretty sure I've never been to MTV's web site before, and I definitely won't try to claim that I'm familiar with everyone mentioned in the article Lil Wayne, Kanye West And LL Cool J Fly Girl Talk's Friendly Skies, Unknowingly, In Bigger Than The Sound, but check out what book Peter Fonda was reading on the plane (4th paragraph, 4th line).

Or perhaps the guy just looked like Peter Fonda, like so many other readers of The Annotated Turing.

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Wed, 25 Jun 2008 22:37:00 GMT 2008/06/220824.html Comments (0) June 22, 2008
Roscoe, N.Y.

The Collected Works of Alan Turing are published by Elsevier, an Amsterdam-based publishing house. The four volumes reproduce Turing’s 30-odd papers and articles, as well as other material, mostly in facsimile with valuable introductory essays. The only drawback is the price: the individual volumes range from $165 to $190.

An excellent low-cost alternative is the The Essential Turing (Oxford University Press, 2004), edited by B. Jack Copeland, Professor of Philosophy at the University of Canterbury in New Zealand and Director of the Turing Archive for the History of Computing, www.AlanTuring.net. The Essential Turing includes all of Turing's major papers on mathematical logic (including “On Computable Numbers, with an Application to the Entscheidungsproblem” that is the subject of my recent book) and artificial intelligence, along with essays written by Jack Copeland and others. In particular, I found the essay “Corrections to Turing’s Universal Computing Machine” by Donald W. Davies to be indispensable in analyzing Turing's paper for my book.

Another Turing-related book edited by Jack Copeland was published by Oxford University Press in 2005. Alan Turing’s Automatic Computing Engine: The Master Codebreaker's Struggle to Build the Modern Computer is all about the ACE computer project that Turing worked on after World War II. Prior to the announcement of this book, the title of my book-in-progress was Alan Turing’s Computing Machine, but Copeland's title was so similar I felt obliged to change mine!

Another somewhat-Turing-related book edited by Jack Copeland, Colossus: The Secrets of Bletchley Park's Codebreaking Computers was published by Oxford University Press two years ago, but somehow I missed this one. I spent much of today reading big chunks of it, and found it very illuminating. The book is a collection fo 26 essays, 12 technical appendices, and 16 pages of photographs threaded into a coherent narrative concerning the purpose and design of the 1943 Colossus computer. A good chunk of the book is written by veterans of Bletchley Park.

As is well known now, Alan Turing and about 10,000 other people spent the years during World War II at Bletchley Park, a large estate and mansion purchased by the British government to consolidate code-breaking activity. Bletchley is very conveniently located: From the Bletchley train station you can go south to London, east to Cambridge, or west to Oxford. Turing worked mostly on cracking the German Enigma encryption machine and designed a parallel simulation device called the Turing Bombe to help work out possible combinations of Enigma settings.

One of the primary motivations of Colossus is to correct a common misconception that Turing also designed the Colossus computer at Bletchley . He did not. That honor belongs to Thomas H. Flowers who contributed two essays included in this book before his death in 1998, and who appears in the prominent frontispiece photograph.

Tommy Flowers worked at the Post Office Research Branch at Dollis Hill since 1930, and from his work there in telephone switching networks, he pioneered the use of vacuum tubes (called "valves" in England) to replace slower electromechanical relays. The common wisdom at the time was that tubes were too prone to burnout to be used in large quantities in critical applications. Flowers knew different, and he applied that knowledge in building the Colossus, which is now regarded as the first functional electronic (that is, using tubes rather than relays) digital computer. "Flowers was (as he himself remarked) possibly the only person in Britain who realised that valves could be used reliably on a large scale for high-speed digital computing." (73)

The Colossus was not always awarded its proper place in computing history due to the secrecy that surrounded its implementation. Although ten Colossi were built, all but two were dismantled and destroyed after the end of the war. For some 30 years it remained a state secret that an electronic computer had actually been built at Bletchley Park!

The Colossus was built specifically to help break coded messages transmitted by the Lorenz Schlüsselzusatz 40 or SZ40. The code-breakers at Bletchley Park gave the different German code machines names of fish, and this one was known as Tunny (tuna). Unlike the Enigma (which required that coded messages be manually transcribed from flashing lights and sent by Morse Code), Tunny directly transmitted five-bit Baudot-Murray code by radio, and printed unencrypted messages on a teletypewriter. A series of 12 wheels generated a five-bit key that changed from character to character, and was combined with the text using a bitwise exclusive-or operation. Applying the same key again decrypted the text.

Information about the Tunny was only declassified by the British government in 2000; a 500-page Report on Tunny is the result. The release of that document freed up Bletchley Park veterans to talk about the Tunny, which made Colossus the book possible.

Colossus contains a wealth of information about Tunny and Bletchley Park, but some bonuses as well, including more biographical information about Maxwell Herman Alexander Newman than I've ever seen. Max Newman taught the Foundations of Mathematics course at Cambridge that inspired Turing to write his "On Computable Numbers" paper, and he helped guide the paper to publication after it became obvious that Turing had been scooped by Alonzo Church. Newman was also at Bletchley Park and then at the University of Manchester where he ran the Mark I computer project. Colossus includes a short biography of Max Newman by his son William. (I was very surprised to learn that William Newman coauthored one of the classic books on computer graphics that I've owned for many years, Principles of Interactive Computer Graphics (McGraw-Hill, 1973), known popularly as "Newman and Sproull.")

Catherine Caughey, one of the Wrens (Women's Royal Naval Service) who worked at Bletchley Park, tells this story about Max Newman:

One day, waiting for the train to take me home on leave, I saw him on the platform. He was dressed in a shabby old Burberry raincoat and was carrying a dead hare by the hind legs. He appeared to be searching the platform for something so I went up to ask if I could help. He gave me a distressed look and said that he had lost his ticket. We searched together but were unsuccessful, and as my train came in I tried to cheer him by saying I was sure the guard would believe him. His reply was: 'Oh no, that is not my problem — until I find my ticket I cannot remember whether I am going to Oxford or Cambridge.' (166)

Catherine Caughey later talks about the strong emphasis on secrecy at Bletchley Park and for decades afterwards: "My great sadness is that my beloved husband died in 1975 without knowing what I did in the war." (171) It's a common refrain among the people who worked at Bletchley Park that their triumphs were completely unknown to the general public for decades afterwards.

Alan Turing is not a major player in the story of the Colossus but he makes frequent appearances in the pages of this book. Few people can resist telling their Alan Turing anecdotes. Here's Max Newman's son William writing about Turing during the University of Manchester days:

When he later bought a house in Wilmslow he would sometimes run the dozen or so miles from there to our house in Bowdon. Once I heard a noise in the early hours of the morning and went to the front door to find Alan dressed in running gear. He wanted to invite us to dinner and, thinking us all asleep but having nothing on which to write, was posting through our letter box an invitation scratched on a rhododendron leaf with a stick. (187)

I was familiar with some of the Turing anecdotes, but here's one told by Peter Hilton that was new to me:

At BP he developed a real delight in playing tennis, and especially enjoyed playing doubles. He was very good at the net, where his speed and good eye enabled him to make many effective interceptions. However, he was dissatisfied with his success rate: too often he intercepted a return from an opponent, but set the ball into the net. Applying his remarkable thinking processes to a mundane problem, he reasoned as follows: 'The problem is that, when intercepting, one has very little time to plan one's stroke. The time available is a function of the tautness of the strings of my racquet. Therefore I must loosen the strings.' And being Alan Turing, he then carried out the necessary alterations to his racquet himself. At this point my recollection may be coloured by the great distance in time, but I seem to recall Turing turning up for his next game with a racquet somewhat resembling a fishing net. He was absolutely devastating, catching the ball in his racquet and delivering it wherever he chose — but plainly in two distinct operations and, therefore, illegally. He was soon persuaded to revert to a more orthodox racquet! (198-199)

Copeland’s biography on the dust jacket of Colossus indicates that his next book is entitled Turing’s Machines. I'm looking forward to it.

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Chapter 9 of Turing-biographer Andrew Hodges' new book One to Nine (which I discussed Monday) takes on mathematics education, which probably everyone agrees is pretty much a disaster. One of the aspects of math education that Hodges seems to be against is memorization:

Whether it is even essential to memorize the base-10 multiplication table, I doubt. It is more important to have a good general sense of what people call ballpark estimates; to know motes from beams, gnats from elephants, sledgehammers from nuts. (pg. 286)

Yet he also seems to think that students should be able to perform rudimentary calculations without a calculator:

You must know what numbers are all about. If you don't, you end up writing down answers a hundred times too big, or upside down. For this reason it is essential to learn about adding and multiplying without a calculator. (pg. 289)

For quite awhile in my teenage years, I knew there was something wrong with the way I multiplied certain numbers from the basic multiplication table. For most of the single-digit multiplications, the product popped immediately into my head. "What's 7 times 4?" would be followed by an unthinking "28."

But not always. There was a little section of the multiplication table where something different went on in my head, and as I began to analyze my thought processes, I understood what this difference was. Apparently at some point in early childhood before the entire multiplication grid was permanently etched upon my neurons, I had stopped memorizing when I realized I could instead mentally calculate the product. This "defect" applied to just three cells involving the numbers 6, 7, and 8. But not the squares: The squares of 6, 7, and 8 popped instantly into my head as 36, 49, and 64. It was the other combinations that I had to calculate "manually."

This is how I performed these three calculations in my head: For 7 times 8, I'd think "7 plus 7 is 14, times 2 is 28, times 2 is 56." Similarly, for 6 times 8 I'd think "6 and 6 are 12, times 2 is 24, times 2 is 48." The final problematic product was 6 times 7, and for that I'd think "7 times 3 is 21, times 2 is 42."

As I just now write down my mental process for these calculations, I realize that for 7 times 8, I could have started with 7 times 4, and similarly with 6 times 8 I could have started with 6 times 4, but for some reason I didn't. I don't know why, but maybe my consistency in how I did these multiplications is a key to the mental process: I had memorized the algorithm for these multiplications, but not the actual result! Moreover, despite the hundreds of times I must have performed these convoluted calculations, the products themselves never took root.

What had happened in my early education to truncate the memorization process before the entire multiplication table had been fully assimilated? Did I just find it easier at some point to double numbers a couple times in my head rather than to memorize these three products? I don't know. Nor do I know which approach is mentally healthier. But if you're going to multiply without a calculator (as Andrew Hodges evidently believes you should be able to do), then at some point you need to come up with the product of 7 and 8, and either you're actually figuring it out in your head the way I did, or you can skip that step and simply spit out the memorized value.

I think memorization is important. The quicker the simple sums and products come into your head, the easier you'll manage more complex calculations because you don't have to keep track of so many intermediate results. I think memorization should actually be emphasized much more than it is. I think most people would benefit from having memorized two-digit tens-complement calculations — for any two-digit number, to know what 100 minus that number is. In real life this is making change from a dollar in your head.

I went through my childhood and almost all my teenage years multiplying 6 times 7, 6 times 8, and 7 times 8 in the way that I described. But in the summer before I went to college, I figured I should finally do something about my multiplication problem. It seemed illicit to be attending an engineering and science school without having first completely memorizing the multiplication tables. I deliberately set out to memorize these three products. It didn't take long, of course, and from that time until today, I can retrieve them instantly. Now when I hear the numbers 7 and 8, I immediately think 56 even if I have no need to be actually multiplying those two digits.

Several years ago, while reading Andrew Hodges’ Alan Turing: The Enigma in the course of researching my book The Annotated Turing, I learned that Turing spent some time before beginning his studies at Cambridge in 1931 reading G. H. Hardy's classic A Course of Pure Mathematics (Hodges, pg. 58), first published in 1908 and still in print today in the 10th edition.

And there we have one of the many differences between Turing and Petzold: Turing prepared for college by reading G. H. Hardy's math textbook. Petzold prepared for college by learning his multiplication tables.

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Anyone writing a book that has anything to do with Alan Turing (such as my book The Annotated Turing) benefits from having at the outset a valuable and ultimately indispensable resource: Andrew Hodges' extraordinary biography Alan Turing: The Enigma published in Great Britain and the United States in 1983. This biography is a model of its kind: very well researched, beautifully written, accurately informative about the mathematics, and quite moving.

My book The Annotated Turing is certainly not another biography of Turing, but rather an extended discussion of his 1936 paper that introduced the concepts of computability and the Turing Machine. However, in Chapters 4 through 11, I attempt to interweave biographical information about Turing with my presentation and discussion of the first 22 pages of Turing's paper. (It was one of the very fun challenges of writing these chapters to make chronological connections between Turing's post-1936 life and various sections of the paper without overstrained seques.) The number of footnotes in The Annotated Turing that reference Alan Turing: The Enigma is just one indication how much I relied upon Hodges' biography.

Andrew Hodges is an English mathematician whose "interest in Alan Turing developed partly through mathematics and partly through his participation in the 'gay liberation' movement" (to quote the biography in the original Simon and Schuster edition). Hodges also runs a website devoted to Turing. His mathematical work has involved working with Roger Penrose and Penrose's theory of twistors.

Andrew Hodges has recently written a new book entitled One to Nine: The Inner Life of Numbers published last year in Great Britain by W. W. Norton, and recently in the United States. The book's overall structure is simple: It has nine chapters, each of which is devoted to one of the first nine natural numbers. (Zero gets involved in the first and last chapters.) This might sound like a dreary over-determined exercise, but Hodges makes it a completely entertaining and even rollicking ride.

To Hodges, certain numbers seem engrained in the fabric of the universe, and reveal themselves in the structure of electromagnetism, spacetime, relativity, quarks, and quantum mechanics. At times there seems as much physics in this book as mathematics. (The Penrose influence?) Alan Turing shows up frequently, of course, and takes center stage in Chapter 8, which focuses on digital computing, but many other mathematicians, physicists, and ideas make appearances as well. This book darts among numerous topics with eager enthusiasm, and manages to be interactive as well, with little challenging number puzzles punctuating the text.

One to Nine is also interactive in the sense that I often found myself arguing with the author. I was very surprised to see Hodges acknowledge the univesality of 10 as a number base (or partial number base in the case of the ancient Babylonians) but question where it comes from:

The universality of ten is not fully explained by the form of the human hands. It would surely have been just as natural to use the eight fingers, not including thumbs. Eight-based counting would have been better suited to the repeated splitting of differences which is so natural for sharing and trading, as indeed for music. (p. 15)

I'm not sure Hodges really believes this himself — to me the cultural dominance of 10 is older and deeper than any explanation other than the digits at the end of our arms — but he waits for the last page of Chapter 9 (p. 320) to reveal why he believes 10 to have an additional existence independently of our fingers.

This is very much a book of the moment: Hodges uses the logic puzzle Sudoku as a motif of sorts, with its use of the digits 1 to 9 in their manifestation as unique symbols unburdened by culture or the physical world. There is also much pop culture in One to Nine. I counted three references to Desperate Housewives, and a great deal of music. (Hodges is also a composer.) Bach and Mahler show up, but also the Pet Shop Boys, who get so many mentions that they merit an entry in the index.

Hodges is also reeling from the continuing horrors of the Bush/Blair years, and a quiet rage about global warming and the Iraq War seems to simmer behind the pages. One of Hodges' examples involves statistics of the number of Iraqis killed as a result of the war (211-212), and a combinatorics example offers sardonic advice for interrogators of the "American-led coalition" (180). Yet with a brief shudder, Hodges is able to move on to yet another connecting topic. This book is nothing if not freewheeling!

At the very end Hodges presents a Sudoku puzzle containing 26 initial squares marked A through Z, each of which is a one-digit solution to a series of 26 additional little number puzzles. ("C: Find the penultimate digit of the trillionth power of 2.") Information about the book and solutions to the puzzles appears on Hodges' website Cool Math.

Throughout One to Nine, Hodges also compares and contrasts his text with Constance Reid's first book From Zero to Infinity: What Makes Numbers Interesting published in 1955 and based on a similar premise. And there we have another connection between this book and mine: Constance Reid (who turned 90 years old in January) shows up in the The Annotated Turing in two chapters. She is the author of the only major biography of David Hilbert in English, which I used a lot in Chapter 3, and she is the older sister of the late mathematician Julia Robinson, who is featured in my last chapter on the solving of Hilbert's Tenth Problem.

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Yesterday I received a few “hot off the presses” copies of The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine, kindly forwarded to me by Wiley, the book's publisher. Even after writing 15 books over the past 20 years, I still get a thrill when I see the first copies.

Just to get a sense how the book is shipping to people other than the author, a couple weeks ago Deirdre and I submitted test orders to Amazon and Barnes & Noble. This morning B&N sent me an email saying that the book is "now packed and ready to leave our warehouse." I guess they're just waiting for the UPS guy to show up. Amazon hasn't been quite as prompt; let's just hope they don't mess up royally like they did with 3D Programming for Windows.

I wish I could say that The Annotated Turing was "10 years in the making" but it's really only been 9 years. I created the first Word file for the book on May 12, 1999, and I sent my final fixes to the production editor on May 8, 2008. From its conception to its completion, the idea behind the book remained constant: to put Alan Turing's 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" in an historical and intellectual context, and then to annotate the hell out of it.

Whether this was a great idea for a book or a really stupid idea for a book, I never really decided, but undecidability is what Turing's paper is all about, so I wasn't very concerned.

Now the book is yours rather than mine, and I'm interested in hearing your reactions to it.

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Last Wednesday, May 28, 2008, was the hundredth anniversary of the birth of Ian Fleming, the creator of James Bond. A front page article in today's Arts & Leisure section of the Sunday New York Times by Charles McGrath entitled "That License to Kill Is Still Unexpired" includes a photo of the famous Enigma machine with the caption "The first Bond novelist, Ian Fleming, worked for British naval intelligence during World War II and was concerned with devices like the German Enigma, an encryption device. (The online version of the story includes the caption but not the photo.)

The Enigma machine was also Alan Turing's big concern while working at Bletchley Park, the center of Great Britain's code-breaking activities during the war. As discussed by David Kahn in Seizing the Enigma: The Race to Break the German U-Boat Codes, 1939-1943 (Houghton-Mifflin, 1991), 124-126 and Stephen Budiansky in Battle of Wits: The Complete Story of Codebreaking in World War II (Free Press, 2000), 158-159, Turing and his people were eager to get their hands on a month's worth of Enigma settings to help them understand the Enigma's mathematical characteristics.

Lieutenant Commander Ian Fleming of the Naval Intelligence Division came up with a plan to get those settings, and in a memo dated September 12, 1940, he described how it would work. (I'm quoting the version in Kahn including the material in brackets)

I suggest we obtain the loot by the following means:

1. Obtain from Air Ministry an air-worthy German bomber.

2. Pick a tough crew of five, including a pilot, W/T [wireless telegraph] operator and word-perfect German speaker. Dress them in German Air Force uniform, add blood and bandages to suit.

3. Crash plane in the Channel after making an S.O.S to rescue service in P/L [plain language].

4. Once aboard rescue boat, shoot German crew, dump overboard, bring rescue boat to English port.

In order to increase the chances of capturing an R. or M. [Räumboot, a small minesweeper; Minensuchboot, a large minesweeper] with its richer booty, the crash might be staged in mid-Channel. The Germans would presumably employ one of this type for the longer and more hazardous journey.

The plan was given the codename RUTHLESS, and was approved as high up as Churchill. It wasn't quite clear that a crash landing could be faked without serious damage to the crew, but the biggest problem was the lack of an opportune moment when a German ship was operating nearby. Kahn quotes one of the other codebreakers after the cancellation of the operation:

Turing and [Peter] Twinn came to me like undertakers cheated of a nice corpse two days ago, all in a stew about the cancellation of Operaton Ruthless. The burden of their song was the importance of a pinch. Did the authorities realise that ... there was very little hope, if any, of their deciphering current, or even approximately current, Enigma ... at all.

Turing and Fleming never met, and quite likely never even knew about each other. Ian Fleming wrote thirteen James Bond novels between 1953 and his death in 1964. Only Casino Royale and Live and Let Die were published before Turing's suicide in June 1954.

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I didn't know it had been publicly announced until I saw Tim Sneath's very kind blog posting this morning, but it's true: I'll be at this year's Professional Developers Conference in October with a full-day pre-conference tutorial on programming for the Windows Presentation Foundation entitled WPF Code and Concepts. How exciting!

The idea of converting my WPF book into a presentation did not appeal to me at all. (Once I've written something, I'm pretty much done with it. Isn't that what's writing for?) So that's not what I'm going to do here. I've been entirely re-thinking the material, the approach, the emphases, and the demo programs.

You've probably seen talks before where the speaker builds an application just by dragging and dropping controls inside Visual Studio. That's not the type of thing I'm interested in doing here. Instead, I'd like to focus much more on how to write your WPF classes so they're conducive to building those apps — in effect, how to make your classes "XAML ready."

I don't do fluff. This is going to be hard-core stuff. We're going to be looking at a lot of code, and discussing it, and analyzing it, and getting a good feel for what makes WPF such a powerful framework for Windows programming. I hope to see you there.

(I'm going to be travelling today and tomorrow, and possibly staying at a hotel without WiFi, so if you post a comment, it might not show up until late Saturday.)

I recently discussed how I cut some pages from my 1999 book Code: The Hidden Language of Computer Hardware and Software in a futile attempt to get the page count down. Those pages focused on some logic puzzles of the Victorian author beloved by geeks worldwide: the Rev. Charles Dodgson, a.k.a. Lewis Carroll.

Yesterday evening I rummaged around in my attic — actually I did the New York City equivalent, which involved visiting the Manhattan Mini Storage building on Spring and Varick — and found the box in which in the summer of 1999 I dumped all the existing printed and hand-written pages of Code and some magnetic media. The magnetic media turned out to be four 100-megabyte Zip disks, and fortunately we still own a Zip drive with a USB cable, and fortunately the disks were still readable. (It's only been nine years. I suspect the magnetic-media-anxiety-factor is much worse at the 20-year time period.)

Here's the original beginning of Chapter 10:

10. Babies Are Illogical

By profession, Charles Lutwidge Dodgson was a mathematician. Born in 1832 to a well-to-do family in Cheshire, England, he earned a degree in mathematics at Christ Church, Oxford, and became Mathematical Lecturer there in 1855, a post he held for over 25 years. But Dodgson’s talents went far beyond mathematics. Ever since childhood he enjoyed drawing pictures, writing stories and poems, and creating games and puzzles for the amusement of his family and friends. In 1856 he purchased a camera and photographic chemicals, and pursued his new hobby with near-total abandon. As a result, Charles Dodgson is today regarded as one of the best portrait photographers of Victorian England, and his photographs of children are unrivaled. Dodgson had a special rapport with children that was also revealed when he wrote and published under the pen name Lewis Carroll two enchanting books about an adventurous girl named Alice.

Although most of Charles Dodgson’s books on mathematics were written for his students and fellow mathematicians and published under his real name, a significant exception is the book Symbolic Logic, published in 1896 just two years before his death. Despite its forbidding title, Symbolic Logic was intended for a general audience—perhaps not for every reader of the Alice books but maybe some of them. It is a peculiar mix of difficult concepts in the field of mathematical logic enlivened with examples that are pure Lewis Carroll. The book culminates with a collection of 60 whimsical logic puzzles known as sorites (se-RYE-teez). The first one goes like this:

(1) Babies are illogical;
(2) Nobody is despised who can manage a crocodile;
(3) Illogical persons are despised.

Not all of the statements in a sorites need necessary be true. These certainly aren’t. (Most of us know likeable illogical persons and can easily imagine a despicable crocodile wrangler.) But the idea here is to assume that all the statements are true, and then to arrive at an overall conclusion. This a fairly easy sorites, and after mulling it over for awhile, most people are able to come to the correct solution:

Babies cannot manage crocodiles.

The 60 sorites in Symbolic Logic get progressively more difficult and more involved, and the last one in the book is a monster:

(1) The only animals in this house are cats;
(2) Every animal is suitable for a pet, that loves to gaze at the moon;
(3) When I detest an animal, I avoid it;
(4) No animals are carnivorous, unless they prowl at night;
(5) No cat fails to kill mice;
(6) No animals ever take to me, except what are in this house;
(7) Kangaroos are not suitable for pets;
(8) None but carnivora kill mice;
(9) I detest animals that do not take to me;
(10) Animals, that prowl at night, always love to gaze at the moon.

Clearly this is beyond the limits of the juggling power of most human brains. Some kind of clever technique would be useful—perhaps mathematical in nature.

The earliest extensive writings on the subject of logic date from the 4th century B.C.E. with the collection of Aristotle’s teachings known as the Organon....

I replaced this all with the sentences "What is truth? Aristotle thought that logic had something to do with it. The collection of his teachings known as the Organon (which dates from the fourth century B.C.E.) is the earlier extensive writing on the subject of logic....

Later in the chapter (equivalent to page 92 of the printed book) I deleted the following after discussing the "All men are mortal..." syllogism:

Using Boolean algebra may seem like overkill for proving the obvious fact (particularly considering that Socrates proved himself mortal 2,400 years ago), but let’s see how it helps with the first of Lewis Carroll’s sorites from Symbolic Logic. Of course, Lewis Carroll was familiar with the work of George Boole. Indeed, without George Boole few mathematicians would have been engaged in the study of logic in the late 19th century. Lewis Carroll used his own notation for solving syllogisms and sorites, but we can use Boole’s. Here’s the first of Lewis Carroll’s sorites again:

(1) Babies are illogical;
(2) Nobody is despised who can manage a crocodile;
(3) Illogical persons are despised.

In Symbolic Logic Lewis Carroll also helps out by defining the universe (that is, the things we’re talking about here) and suggested letters for the qualities of various classes that comprise this universe. Here’s his suggestion:

Univ. “persons”; a = able to manage a crocodile; b = babies; c = despised ; d = logical.

What’s implied here is that the universe is divided into the class of persons who can manage crocodiles and the class of those who cannot. The universe is also divided into people who are babies and people who are not, and into people who are despised and people who are not. And finally, the world is divided into logical people and illogical people.

“Babies are illogical.” This means that the intersection of the class of babies (B) and the class of persons who are illogical (1 – D) is the class of babies:

B × (1 – D) = B

Like many of the statements here, this can also be written in other ways, for example, (B × D) = 0. This means that the intersection of babies and logical persons is nothing.

“Nobody is despised who can manage a crocodile.” This means that the intersection of the class of despised persons (C) and the class of persons who cannot manage crocodiles (1 – A) is the class of despised persons:

C × (1 – A) = C

“Illogical persons are despised.” The intersection of the class of illogical persons (1 – D) and despised persons (C) is the class of illogical persons:

(1 – D) × C = (1 – D)

To solve this, we can begin by substituting the third equation in the first:

B × ((1 – D) × C) = B

and then use the associative law to regroup:

(B × (1 – D)) × C = B

But from the first equation we know that B × (1 – D) equals B, so:

B × C = B

Now substitute the second equation into this:

B × (C × (1 – A)) = B

Regroup again:

(B × C ) × (1 – A) = B

But we already know that (B × C) equals B, so:

B × (1 – A) = B

which means that the intersection of the class of babies and the class of people who cannot manage crocodiles is the class of babies. At this point, the sorites is basically solved, but let’s go a little further using the distributive law:

(B × 1) – (B × A) = B

and then simplifying

B – (B × A) = B

The only way that this can be true is if (B × A) equals 0:

B × A = 0

Thus, the intersection of the class of babies and the class of people who can manage crocodiles is 0. In other words: Babies cannot manage crocodiles. (But you knew that.)

Boolean algebra can also be used to determine if something satisfies a certain set of criteria. Perhaps one day you walk into a pet stop ...

The words sorites is Greek for "heap" and refers here to a heap of syllogisms. There is also a "sorites paradox" that involves a real heap of grain. Take one grain out and you still have a heap. By induction, you can continue removing grains and it still remains a heap even when no grains are left.

Boolean logic is a logic of classes. You can also solve sorites using predicates and implication. While working on Chapter 12 ("Logic and Computability") of my forthcoming book The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine I tried to get Lewis Carroll involved once again. This is a rather sketchy passage from an early draft of Chapter 12:

Analyzing Aristotelian syllogisms (“All men are mortal; Socrates is a man; hence…”) or the straightforward sorites devised by Lewis Carroll. Here’s a typical Lewis Carroll sorites of moderate difficulty:

(1) No kitten, that loves fish, is unteachable;
(2) No kitten without a tail can play with a gorilla;
(3) Kittens with whiskers always love fish;
(4) No teachable kitten has green eyes;
(5) No kittens have tails unless they have whiskers.
[Lewis Carroll, Symbolic Logic: Part I. Elementary (Macmillan, 1896), pg. 118.]

We could assign five letters or little words to these five sentences, but we wouldn’t know what to do with them because the letters do not represent anything about the structure of the sentence.

We can slightly modify our thinking by assigning letters (or words) not to sentences but to predicates.

A universe of discourse. We’re talking about kittens.

Monadic predicates: LovesFish, Teachable, Tailed, Whiskered, GreenEyed, GorillaFriendly.

(1) LovesFish → Teachable
(2) GorillaFriendly → Tailed
(3) Whiskered → LovesFish
(4) Teachable → – GreenEyed
(5) Tailed → Whiskered

By taking them in the order (2), (5), (3), (1), (4) we have:

GorillaFriendly → – GreenEyed

The conclusion, of course, is “No kitten with green eyes will play with a gorilla.” [Ibid, 160] But you knew that.

(Professor Dodgson uses very different notation and techniques.)

Google Book Search has a copy of Symbolic Logic, and the book was also republished by Dover. The sample sorites begin on page 112, but be forewarned that some of the puzzles in this book are anti-Semitic and rascist, which comes as a shock even if you're accustomed to reading Victorian literature.

Although Symbolic Logic says "Part I. Elementary" on its title page, Carroll did not live to publish subsequent parts. The book Symbolic Logic, edited with annotations and an introduction by William Warren Bartley, III (Harvester Press, 1977) has a recreation of the uncompleted and long-lost Part II.

The second chapter of my forthcoming book The Annotated Turing is entitled "The Irrational and the Transcendental" and innocently begins:

We begin counting 1, 2, 3, and we can go on as long as we want.

That's not true, of course. "We" simply cannot continue counting "as long as we want" because "We" (meaning "I" the author and "you" the reader) will someday die — probably in the middle of reciting a very long (but undoubtedly finite) number.

What the sentence really means is that some abstract ideal "somebody" can continue counting, but that's not true either: Counting is a temporal process, and at some point everybody will be gone in a heat-dead universe. There will be no one left to count. Even long before that time, counting will be limited by the resources of the universe, which contains only a finite number of elementary particles and a finite amount of energy to increment from one integer to the next.

We tend to accept simple statements such as the one that begins Chapter 2 of my book because our culture is hopelessly entangled in a Platonistic view of the natural numbers. We feel that the natural numbers exist somewhere "out there" independent of the people who use them to count. This assumption is not often closely examined, but it actually implies the opposite of what we think. We like to pretend that mathematics is the most "objective" and least human-bound intellectual endeavor, but our view of the natural numbers reveals mathematics to be founded on a very human metaphysical conceit. The natural numbers are not, in fact, "natural" — that is, intrinsically part of nature — but arise out of human discourse.

Is this a problem? And if so, can mathematics ever be truly liberated from metaphysics?

This is the job that the brave mathematician and philosopher Brian Rotman has taken on in his book Ad Infinitum... (Stanford University Press, 1993). I discovered this book in 1999 while searching out resources for writing the book that eventually became The Annotated Turing, and I was amused by the book's subtitle. Actually, Ad Infinitum... has three subtitles, and it's the first subtitle that made the book come up in an Amazon search. The complete title seems to be Ad Infinitum… The Ghost in Turing’s Machine: Taking God Out of Mathematics and Putting the Body Back In: An Essay in Corporeal Semiotics.

The western world has mostly been comfortable with the dual concepts of infinity described by Aristotle in the Physics, Book III, Chapters 4 through 8. Aristotle differentiated between an "actual" or "completed" infinity, of which he denied the existence, and a "potential" infinity, such as that of the natural numbers, which he found to be acceptable:

Infinity turns out to be the opposite of what people say it is. It is not ‘that which has nothing beyond itself’ that is infinite, but ‘that which always has something beyond itself ’. [Aristotle, Physics, translated by Robin Waterfield (Oxford World’s Classics, 1996), Book III, Chapter 4, page 73.]

Although Aristotle's rejection of the completed infinity has been often ignored over the centuries — most notably in the use of "infinitesimals" in the creation of the differential calculus, and more recently, in the work of Georg Cantor and his mathematics of the transfinite — it remains the gold standard for our allowance of some forms of infinity into mathematics but not others.

Brian Rotman, however, finds even Aristotle's potential infinity to be hopeless metaphysical and seeks to eliminate even that. This is not just an issue between the two philosophies of mathematics roughly divided as Platonists (a.k.a., Realists) and Constructivists, because many of the historical Constructivists have accepted the pre-existence of the natural numbers. Leopold Kronecker famously said, "God made the integers, all the rest is the work of Man" (quoted by Rotman on page 38) and to L. E. J. Brouwer, the natural numbers were a shared Kantian intuition among all mathematicians.

Rotman instead asks:

But what if no deity created the numbers, what if they are not the work of a pure angelic being outside history, but rather the production of a secular, time-bound, and empirically tainted culture? What would follow from a conception of the whole numbers as humanly constructed? Would we not at the beginning have to repudiate their closure in the past — God's creation of them in their entirety — and replace it with an openness framed in terms of the discursively manufactured present-future: numbers as things which exist or can be imagined to exist, which are humanly constructed and constructible, through some ongoing symbolic process? [ pg. 49 ]

What Rotman attempts to introduce as an alternative is a "non-Euclidean arithmetic." Geometry as formulated by Euclid sought to derive everything from self-evident postulates; yet, the fifth of these postulates was forced to bring infinity into play by including the phrase "if produced indefinitely" (in Sir Thomas Heath's translation) in describing the conditions under which lines are not parallel. Just as nineteenth-century mathematicians formulated non-Euclidean geometries by making assumptions contrary to the fifth postulate, Rotman attempts to formulate an arithmetic where the natural numbers cannot be realized indefinitely because of the limitations of resources and human cognizance.

Ad Infinitum... is not an easy book. It requires a familiarity and sympathy with constructivist mathematics and post-modern theory. (The longest entry in the bibliography belongs to Jacques Derrida.) And although I didn't follow all of Rotman's arguments entirely — and even found myself often lost in the middle of single sentences — I found Ad Infinitum... to be a fascinating and challenging assault on conventional notions of infinity.

Although Alan Turing figures in the first of the three subtitles of Ad Infinitum..., he doesn't play a big role in this book and his imaginary computing machines are only discussed in any detail on page 99. Rotman indicates that Turing's paper "was conceived entirely within classical, infinitistic terms," which is unsurprising given how much the work of Cantor and Hilbert hangs over it.

The business of carrying out the computation, of articulating what or who might, in principle, be involved or called upon to "physically" run or be able to run his ideal machine — the ideal typist as it were — formed no part of Turing's picture. Or rather, no problematic part, since the classical context in which he operated already internalized an infinity of numbers immediately accessible to a — necessarily disembodied, dephysicalized — counting agent; hence, for example, the lack of any qualms involved in Turing's assumption of an infinite memory tape as part of his machine's specification.

The ideal and reduced typist — the computing agency, unmentioned by Turing, who would run a Turing machine — is of course the transcendentally given, infinitistic agent of classical mathematics.

It is unfortunate that Turing displays no interest in the more philosophical matters of mathematics in his paper on computable numbers (although he touches on them in the correction to his paper), but it is Turing's modeling of algorithms as computing machines that reveals the strictly physical and temporal nature of mathematics. The tape is "infinite" only in the sense that whenever an additional length is required, it is available. In use, the tape is always finite. Turing doesn't address what happens when the universe (or just the paper factory) runs out of resources and the tape cannot be extended, but the reader certainly ponders the possibility, simply because the tape is a physical object and not an abstract procedure.

Turing's paper proves that the vast majority of real numbers can't even be computed, thus calling into question even the reality of the "real" numbers. Even for those real numbers that can be computed — the algebraic numbers and a tiny subset of the transcendentals — the computation is always bound to a temporal process and obviously can never be completed. It is Turing's imaginary machine that most cogently reveals these limitations as an intrinsic part of any algorithmic process.

Moreover, there is no general finite algorithm that can analyze a Turing Machine and determine whether a particular digit or pattern of digits will ever be computed, thus implying that these digits do not exist until they are actually computed by the machine. If the computation of these digits requires more time or resources than are available in the universe, they will simply never be known.

When I set out to study Turing's paper in detail, I hardly expected it to have implications for the philosophy of mathematics. Yet to me, Turing's conclusions cast real doubts on a Platonistic interpretation of mathematics and imply instead an extreme Constructivist philosophy where mathematics is limited by time, resources, and energy. I don't think this has been widely appreciated because the Turing Machine is most commonly studied in a reformulation that computes finite integral functions rather than real numbers. It's yet one more reason why it's useful to go back to Turing's original paper.

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For quite awhile I assumed that my forthcoming book The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine would not even mention Dutch mathematician Luitzen Egbertus Jan Brouwer (1881 – 1966). I was seriously mistaken.

Brouwer played a major role in the debates in the early decades of the last century involving the foundations of mathematics. Three distinct approaches emerged, called logicism, formalism, and intuitionism.

Logicism is mostly closely associated with Alfred North Whitehead and Bertrand Russell's three-volume Principia Mathematica, which carried on the work of Gottlob Frege in attempting to derive all of mathematics from basic principles of logic. Formalism is mostly closely associated with David Hilbert, who tried to treat mathematics in a strictly formal manner as the manipulation of symbols. Most important to Hilbert were the establishment of certain metamathematical characteristics of axiomatic systems, such as consistency, completeness, and decidability. Although logicism and formalism took different approaches to mathematics, it might have been possible for Russell and Hilbert to have combined them into a single coherent mathematical strategy had not the Great War intervened at precisely the crucial time.

In opposition to both logicism and formalism — and much less likely to be reconciled — was the philosophy of mathematics called intuitionism by L.E.J. Brouwer to describe his idea of how mathematical entities are formulated by the mind. Using symbols on paper to communicate mathematics is to the formalist the whole point, but to the intuitionist a necessary evil.

Both logicism and formalism relied a great deal on Georg Cantor's set theory and, to a certain extent, his theory of transfinite numbers, and this is one area where Brouwer parted company. Brouwer certainly accepted the difference between the enumerable integers and the non-enumerable real numbers, but because of that very difference he refused to accept sets of non-enumerable objects. The set of all real numbers must be prohibited specifically because there is no way to enumerate or generate the members of that set. Much of Cantor’s theory of transfinite numbers is therefore simply ‘‘without meaning to the intuitionist.’’ (‘‘Intuitionism and Formalism,’’ Bulletin of the American Mathematical Society, Vol. 20 (1913))

Brouwer also felt that certain concepts that had validity in the realm of finite sets had been foolishly and dangerously extended to infinite sets. One concept is particular is the law of the excluded middle — either something does or does not have a particular property. This is not so for infinite sets, and Brouwer used as an example a series whose convergence depends on a certain unknown pattern appearing in the infinite digits of π.

Although I certainly knew about Brouwer when working on the early chapters of my book, he didn't seem quite relevant. The early chapters provide the background necessary for reading Turing's paper "On Computable Numbers, with an Application to the Entscheidungsproblem." This background included a major discussion of Georg Cantor's work in Chapter 2, appearances of Gottlob Frege and Bertrand Russell in Chapter 3, and of course David Hilbert, also in Chapter 3.

Turing obviously knew about Cantor's two proofs of the non-enumerability of real numbers, for he alludes to those in Section 8 of his paper. (I don't think Turing had access to Cantor's papers because he references E. W. Hobson's book The Theory of Functions of a Real Variable and the Theory of Fourier's Series for the first of the two proofs.) Turing also refers to David Hilbert and Wilhelm Ackermann's Grundzüge der Theoretischen Logik — this is the book that introduced the Entschiedungsproblem for first-order logic and which also posed the completeness problem that Gödel tackled — and the first volume of David Hilbert and Paul Bernays' Grundlagen der Mathematik.

Despite Brouwer's important historical role in the mathematical movements of the early twentieth century, he didn't seem to play a significant part in the background to Turing's paper. I felt that a discussion of Brouwer in Chapter 3 would have been more of a distraction than anything else. Besides, other popular histories of the period — for example Martin Davis's wonderful The Universal Computer: The Road from Leibniz to Turing (W. W. Norton, 2000), published in paperback under the title Engines of Logic: Mathematicians and the Origin of the Computer — discussed Brouwer and his role in the mathematical controversies of the 1920's.

I suspect I was also partly influenced by the rather dismissive tone towards Brouwer taken by I. Gratten-Guinness in The Search for Mathematical Roots, 1870-1940: Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Gödel (Princeton University Press, 2000), page 480: "The origins of Brouwer's philosophy lie partly in poor understanding of certain mystical texts, and partly on a naive reading of Kant's views on the place of intuition."

I deliberately excluded Brouwer from my book, and in the draft that I circulated to about a dozen friends in the summer of 2005, the Introduction read "I have also been selective in the historical background I’ve provided. I discuss Cantor but not Kronecker, Hilbert but not Brouwer. Although I’ve streamlined the history, I have tried my best not to distort it." I was freely admitting that I was presenting in the book what Herbert Butterfield famously called a "Whig interpretation of history" — a description of the past that always seems to be leading inexorably towards the glorious present.

I felt entirely justified in doing this because in Chapters 2 and 3 I tried to present the mathematical background that would have been familiar to Turing when he wrote his paper. I believed then — and I still believe — that Turing had at most a negligible knowledge of Brouwer's work at the time he wrote the draft of his paper that he submitted to the London Mathematical Society in May 1936.

However, Turing's paper was not actually quite finished at that time.

Shortly after Turing submitted his paper, he found out that he had been scooped by Alonzo Church at Princeton, who had just published a paper that proved the Entscheidungsproblem had no solution. Normally that would have meant that Turing's paper would not be published, but everyone agreed that the publication of Turing's paper was justified by his unique approach involving the imaginary computing machine soon called (by Church himself) the "Turing machine."

Because Church's paper had already been published, Turing needed to add an Appendix to his own paper showing how his approach and Church's approach were equivalent. Church had used a mathematical tool he had developed called the lambda calculus, which subsequently had an important influence on computing, particularly functional programming languages and lambda expressions. I went back to Church's original papers, and those of his students Stephen Kleene and J. Barkley Rosser — the same papers that Turing must have studied during the summer of 1936 to write his Appendix. Turing's Appendix (with my background on the lambda calculus and helpful explanations) appears in Chapter 15 of my book.

About a year after Turing's original paper was published in 1936, Turing published a three-page paper entitled "On Computable Numbers, with an Application to the Entscheidungsproblem. A Correction," hereafter referred to as Correction. For at least the sake of completeness, I wanted the Correction to be part of my book, but I didn't see a big role for it. Most of the stuff in the first half of the Correction paper indicated formal errors — including a shockingly inadequate formalization of the natural numbers in first-order logic — that I had already incorporated into my earlier chapters.

I thought I'd be able to deal with the Correction paper in the same chapter as the Appendix. Somehow I had neglected dealing with the full implications of the second half of the Correction paper, or perhaps it didn't seem all that significant at the time, or maybe I was in serious denial. When I finally focused on this material, I was quite surprised. The section begins by indicating a problem with the way Turing originally described computable numbers way back in the beginning of the paper. Under Turing's definitions, a "computable sequence" is a string of 0's and 1's generated by a machine. A "computable number" is the binary number you get when you put a period in front of this computable sequence, or when you add or subtract any integer from this binary number.

That sounds very simple and straightforward, and yet in the Correction paper Turing indicates this definition is problematic, for one reason because a machine cannot be defined to compute the Euler constant. I quickly figured out that the particular Euler constant that Turing is referring to is not the famous e, the base of natural logarithms, but the slightly less famous γ, equal to approximately 0.57721..., often called the Euler-Mascheroni constant, and the subject of recent book by Julian Havil, Gamma: Exploring Euler's Constant (Princeton University Press, 2003).

But why was Turing asserting that one of his machines can't calculate γ? I knew people had written computer programs to calculate the digits of γ, so why couldn't a Turing Machine also calculate it? This question bounced around in my head for weeks — fortunately I was able to do other work at the same time! — before I grasped the special problem that γ presented to Turing Machines.

The remainder of the second half of Turing's Correction paper was equally mysterious, but is clarified by a tiny footnote on the last page:

This use of overlapping intervals for the definition real numbers is due originally to Brouwer.

There it is: The explicit Brouwer connection. I mentioned earlier that I didn't think Turing had much familiarity with Brouwer's work at the time he wrote the draft of his paper that he submitted to the London Mathematical Society in May 1936, so what changed in the year that passed before Turing wrote the three-page Correction?

What changed is that Turing went to Princeton and got his doctorate under Alonzo Church, and there he met people who were familiar with Brouwer's work. Church himself had visited Brouwer in Amsterdam several years earlier, and was sympathetic to intuitionist mathematics. So too was Stephen Kleene, who later co-authored a book about intuitionism and whose article ‘‘Origins of Recursive Function Theory,’’ in the Annals of the History of Computing, Vol. 3, No. 1 (Jan. 1981) contains a photograph of Brouwer visiting Kleene in Madison, Wisconsin. So too was Hermann Weyl who was at the Institute for Advanced Studies at the time Turing was in Princeton. These were some of the possible influences that caused Turing to question and rethink this particular aspect of his paper.

I was curious to read what others had written about the connections between Brower's mathematics and Turing's computable numbers, but I couldn't find much of anything.

The more I read about Brouwer — an overview by Walter P. van Stigt included in Paolo Mancosu's From Brouwer to Hilbert: The Debate on the Foundations of Mathematics in the 1920s (Oxford University Press, 1998), Mark van Atten's short but illuminating text On Brouwer (Wadsworth, 2004), the paper ‘‘Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum’’ by Mark van Atten, Dirk van Dalen, and Richard Tieszen in Philosophia Mathematica, Vol. 10, No. 2 (2002) — and the more I read of Brouwer's own papers — about ten in From Brouwer to Hilbert, and a few more in those indispensable anthologies Jean van Heijenoort's From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (Harvard University Press, 1967) and William Ewald's From Kant to Hilbert: A Source Book in the Foundations of Mathematics (Oxford University Press, 1996) — the more interested I got. I found Brouwer to be a fascinating personality, and I began wishing I had some excuse to plunge into the two-volume biography by Dirk van Dalen and explore the Collected Works. (My own book would have been much delayed as a result!)

Although I'm sure that Brouwer's concepts of real numbers didn't influence Turing's description of machines that generate real numbers, the connections are quite startling.

To Brouwer, the digits of π do not exist until they are computed, and a real number is not something that is ever completely calculated. It is always an uncompleted process that occurs over time. (In constrast, to a mathematical Platonist, π is an actual complete number.) This is precisely the way Turing Machines compute real numbers. The machines never finish, and each step requires a finite period of time in that process.

Brouwer also developed a completely original approach to the definition of the real-number continuum, always problematic because of its schizophrenic nature: seemingly smooth and continuous, but at the same time a collection of discrete (but non-enumerable) points. Brouwer's solution involved something he called "choice sequences" that never converge but always maintain a type of finite "halo" around a particular real. The choice sequences seemed to me quite similar to the progressive digits computed by a Turing Machine.

The primary purpose of Turing's paper — to show that can be no general decision procedure for first-order logic, i.e., that there is no general way to determine whether a specific statement in logic is provable or not — was, along with Gödel Incompleteness Theorem, one of the death knells for Hilbert's formalist program. Yet, Turing's paper also strengthened Brouwer's arguments about the real numbers.

When we set a Turing Machine going to compute the digits of some real number, it is not possible to predict what will happen, or to determine if the machine ever prints a particular digit. By extension, it is also impossible to determine if a the machine will ever print a particular pattern of digits.

Suppose the opposite of what Turing proved turned out to be true. Suppose we really could develop finite algorithms that analyze Turing Machines and determine what they might or might not do. Suppose we had a Turing Machine that computed the digits of π, and we really were able to analyze this machine with finite algorithms to determine if it will ever print particular patterns of digits. We would then be justified in saying that all the digits of π really do exist in a Platonic sense, because we'd be able to extract information about those infinite digits in a finite period of time.

But no such algorithms exist. We still need to calculate the digits to know what they are. Because of the finite resources of the universe, only a finite number of digits of π will ever be computed.

This last section of Turing's three-page Correction was for me initially the most mysterious passage of his entire paper. As I battled my way through my initial confusion (always a fun process because that's how I learn new stuff) I was forced to confront my own ideas about infinity and the nature of numbers. I soon saw that this exploration into the connections between Turing Machines and Brouwer's philosophy of mathematics had to become its own chapter that I called "Conceiving the Continuum." It was a tough chapter to write, but very rewarding as well.

I like all the chapters in my book, of course, but I have a very special fondness for this one.

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If I have one hope for my book, The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine, it is to help readers understand the difference between Turing's original Turing Machine, and the Turing Machine as it's commonly encountered in college courses and textbooks.

Two decades after Turing's paper that introduced his computing machines, the Turing Machine was reformulated by Stephen Kleene in Introduction to Metamathematics (1952) and Martin Davis in Computability and Unsolvability (1958). These reformulated machines — which generally compute functions — dominate the current literature on computability. The "halting problem" (the term is Martin Davis's) involves the existence of a general algorithm to determine whether an arbitrary machine finishes properly and halts, or whether it goes bad and runs forever.

As I've mentioned before ( 2006-08-11, 2007-11-26, 2007-12-02, 2008-05-12), it is not correct to associate the halting problem with Turing's original work. Turing's Turing Machines are supposed to run forever.

Turing's paper describes some machines that compute functions, but these are only special cases. The original conception is a machine that computes the infinite digits of a real number. Of course, this is not typical computer activity (unless you're one of those people who set computers going calculating the infinite digits of π) but it's certainly a good example of an algorithm at work.

The advantage of Turing's approach is that certain theoretical results are available almost immediately. Once you see how a Turing Machine (and hence, any finite algorithm) can be described by a single integer, it becomes obvious that Turing Machines are enumerable. Since real numbers are not enumerable, it follows that the vast majority of real numbers cannot be computed algorithmically. (If you're hazy on the concept of enumerability, that's the focus of Chapter 2 of The Annotated Turing.) Digital computers are thus intrinsically limited in what they can do.

If only a subset of the real numbers can be computed, what are all those other real numbers that can't be computed? The vast majority of real numbers are basically strings of random digits without any pattern whatsoever. You simply can't generate random digits algorithmically.

Georg Cantor proved the non-enumerability of real numbers in two very different ways (although both were reductio ad absurdum proofs). The second of Cantor's proof involves the famous diagonalization process. (Again, see Chapter 2 of The Annotated Turing for a full discussion.) If you can list the real numbers, you can derive a new number based on the diagonal of the numbers in this list, but which differs by 1 in each digit. This must be a real number but it's not in the list. The conclusion is that you really can't list the reals.

If you subject the computable numbers to Cantor's diagonalization process, what do you get? You're defining an algorithm to compute the diagonal from all the other computable numbers, so the diagonal must be a computable number. However, if it is a computable number, then it's not in the list, which means that computable numbers are not enumerable, and we already know that's not so.

The only possible way out of this seeming paradox is the inescapable conclusion that you actually can't construct a diagonal of the computable numbers. You can't construct it because you can't determine which Turing Machines compute numbers, and which get "jammed up" or stuck in undesirable loops. It also follows that there is no general process to determine whether a particular machine ever prints a particular digit, or a particular pattern of digits.

This is yet another limitation of digital computers. To determine what a Turing Machine (or algorithm) ultimately does, you essentially need to trace through the steps — in effect, to simulate the machine.

The indeterminacy of Turing Machines is a combination of bad news and good news: It means we can't write a generalized "debugging" program that can analyze code and reveal all the bugs that hide inside. Extensive testing of software is still required, and even then we can't ever be fully confident that all the bugs have been eliminated.

But imagine if there did exist a finite algorithm that could determine the ultimate fate of Turing Machines. What would that imply about models of the human mind that are based on Turing Machines? Or cosmologies that visualize the universe as a giant Turing Machine? The nasty hobgoblin of determinism is not exactly eliminated by the indeterminacy of Turing Machines, but it's almost rendered impotent.

There is no algorithmic process to determine the future — whether it's the future of a computer program, a thought process of the human mind, or the universe as a whole.

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A friend mentioned John L. Casti's The Cambridge Quintet: A Work of Scientific Speculation (Perseus Books, 1998) to me recently because Alan Turing plays a major role in this fictional dialogue concerning machine intelligence.

The premise is this: In 1949, C.P. Snow wants to advise the British government on the possibilities and potentials of machine intelligence, so he hosts a dinner party at his home in Cambridge with four guests to discuss the issues: Turing, philosopher Ludwig Wittgenstein, geneticist J. B. S. Haldane, and physicist Edwin Schrödinger. Over seven chapters corresponding to the courses of the dinner (sherry, soup, fish, meat, salad, dessert, and cigars and brandy), Casti narrates this imagined conversation. The result is an entertaining introduction to some of the philosophical concepts associated with artificial intelligence and human consciousness.

The philosophical dialogue is a time-honored genre, although traditionally it's populated by character types (such as in the dialogues of Cicero, Galileo, and Hume) rather than actual historical personages. The personalities definitely make The Cambridge Quintet more compelling than it would have been otherwise. We get to see the rumpled Turing often excitedly stuttering when describing his research and ideas, and Wittgenstein already suffering from the cancer that will kill him but taking the stage to deliver his profound pronouncements.

The debate is primarily between Turing and Wittgenstein, with Snow playing the moderator. It's not always clear why Casti specifically brought Schrödinger and Haldane into the fray — although Schrödinger sometimes introduces a Eastern religious perspective, and Haldane bemoans the Lysenko travesty in the U.S.S.R.

Casti warns us in the Author's Note that he incorporates ideas that arose much later than 1949, and it is indeed startling to hear Wittgenstein adopt a lightly altered version of John Searle's famous Chinese Room argument, and for Turing to describe ideas normally attributed to Noam Chomsky: "Suppose," Casti's Turing says "there is a structure in our brains given to us by evolution that is specialized just for language, a kind of language 'organ", if you like." (pg. 121) I know of no analysis of language by Turing at all, let alone one like this.

The anachronism that bothered me the most, however, was Alan Turing describing his 1936 computing machines in terms of the "halting problem" (pg. 42). If there is any hope at all I have for my forthcoming book The Annotated Turing it is for readers to appreciate the difference between Turing's original machines and the reformulated machines in Stephen