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Mathematicians who forget the mistakes of history

Dr. Richard S. Wallace

a review of
Engines of Logic
by Martin Davis
Norton, 2001 (Paperback, 257 pages)

Dr. Richard S. Wallace
Engines of Logic book cover

Martin Davis, one of the greatest living mathematicians and computer scientists, and one of the few academics supportive of my disability claim against NYU, has written a remarkable book. Taking a page from Turing's biographer, Andrew Hodges, another professional mathematician, Davis plunges into the most nonmathematical of subjects: history.

When I last saw Martin Davis at NYU in 1995, he told me was working on a book about the history of logic. Around the same time, I attended a talk Davis gave at the New York Academy of Sciences. The subject was "Leibniz's Dream", also the title of this book's first chapter. Engines of Logic, originally published in hardcover under the title The Universal Machine, is the culmination of those efforts.

Only one degree of separation takes us from Martin Davis to the generation of Einstein, Gödel, Turing, Mauchly, Eckert, Atanasoff, von Neumann, Church, Post and Ulam, many of whom the author met or knew personally. Martin Davis is both living history as well as writing it. He is the link between that generation and those of us working today on logic, artificial intelligence and "thinking machines."

If you want to know why predicates are called "predicates" in AIML, and not "properties" or "variables", read this book. The story begins with Leibniz, who, along with Newton, invented the calculus. Gottfried Wilhelm Leibniz also had a dream, as Davis puts it, "He dreamt of an encyclopedic compilation, of a universal artificial mathematical language in which each facet of knowledge could be expressed, of calculational rules which would reveal all the logical interrelationships among these propositions. Finally, he dreamed of machines capable of carrying out calculations, freeing the mind for creative thought". This was around 1680 in Germany.

Engines of Logic is one of those few books to have affected me profoundly. Readers of the AIML mailing lists know that I often rail against computer scientists reinventing mathematical wheels. "Object oriented programming (OOP)" provides a good example. What does an OOP system contain that was not already modeled a century ago as sets of objects and maps between them? At some level "OOP" is just marketing hype, window dressing for concepts well known long before computer programmers appeared on the scene. Lisp, on the other hand, or Prolog or SETL, are more grounded in the mathematical tradition that forms the topic of Davis's book.

Even more significantly, Davis unveils the personal lives of the great logicians. This is the kind of book that makes me wish I had read these stories years ago, because they would have helped me understand my own life and emotional problems. The contrast between the presentation of mathematicians and their work in mathematics classes, and the reality of their lives in some cases, is striking.

Men Of Mathematics

Reading through Davis's book, you would almost have the impression that mental health problems were the rule rather than the exception among mathematical geniuses. Of the seven scientists named in chapter titles [Leibniz, Boole, Frege, Cantor, Hilbert, Gödel, Turing], only two, Leibniz and Boole, escape the experience of "madness" in themselves or their immediate family. Even in the case of Leibniz we cannot be so sure. Davis says, "We have little idea what he was like as a person." His misfortune with the Dukes of Hanover and his dispute with Newton over the invention of the calculus, remain part of the mythology of Leibniz' life.

Leibniz faced a particular form of torment all too familiar to frustrated scientists today. The Duke of Hanover, his employer, felt it would be preferable for Leibniz to work on the Duke's family genealogy, rather than waste his time on unprofitable pursuits such as logic, philosophy and mathematics. Leibniz must have bristled knowing that his rival, Newton, had secured an academic appointment at Cambridge.

Boole's contribution to computer science is so significant that his name now usually appears in lower case, as in "a boolean expression". Davis tells the incredible story of this hardscrabble English schoolmaster who transformed logic into algebra. Formal logic began with Aristotle, but little progress was made until the 19th century. Boole systematized Aristotelian logic, putting it on the same footing as algebra and calculus by applying mathematical symbols.

Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz

I first heard the story of Frege and Bertrand Russell when I was an undergraduate studying logic in the philosophy department. Frege's life work was a magnum opus called Grundlagen der Mathematik, an attempt to reduce mathematics to precise statements in symbolic logic. The first volume was published, and the second was at the printer, when Frege, a German, received a letter from the English mathematician Bertrand Russell. In one page, Russell had demolished the very foundation of Frege's entire Grundlagen. Although Frege was thoroughly discouraged by what is known today as "Russell's Paradox", his work remains significant for the introduction of logic concepts still in use today.

Frege is not described as clinically depressed, but he did cease work on logic altogether after Russell's letter, and died a bitter man. In order to practice his profession as a scientist, Gottlob Frege accepted a nonpaying appointment as a lecturer at the University of Jena. Davis says, "Because his colleagues didn't really value his work, he was never appointed to a full professorship."

On top of all that, Frege was a closet anti-Semite. Although one could perhaps dismiss his anti-Semitic views to the general political atmosphere of the times, such outrageous political views would surely be seen as a "sickness" today, perhaps especially so if they were held by a university professor.

Davis's discussion of Cantor's psychological problems brought me back to my student days, struggling to learn an advanced mathematical concept known as a "Cantor Set". The style of both the instructor and the textbook was a kind of mental perfection, as if understanding the Cantor Set required a superhuman cognitive ability, an ultimate state of rationality that would seem to be the opposite of irrational, mental illness. It seems odd now that neither the book nor the instructor mentioned that Cantor struggled with deep emotional pain.

Davis writes, "Cantor suffered the first in a series of nervous breakdowns in 1884, an intense depression that lasted about two months." Cantor suffered from manic-depressive illness, about which Davis says, "It is now generally understood that the disorder's fundamental cause is rooted in defective brain chemistry."

Hilbert is one of the titans of mathematical history. Years ago I had a historical book called Men of Mathematics. Perhaps it is well that I cannot recall the author's name, because we can not be sure that today he would choose such a gender specific title. Hilbert leapt from history with a sort of mathematical workaholism. His famous "23 Unsolved Problems" became the basis for a century of mathematical research that followed.

Hilbert himself was not afflicted with any mental illness, but his son certainly was. Martin Davis writes, "Franz (Hilbert) was a badly disturbed young man, and it finally became necessary to institutionalize him. The father's reaction to this tragedy was that he no longer had a son; the mother felt otherwise." Does the image of Hilbert as an abandoning parent reduce the size of his towering figure in mathematics?

Goedel and Eisntein

Kurt Gödel and Albert Einstein

Engines of Logic includes a number of black and white photographs, but the most memorable is one of Kurt Gödel and Albert Einstein. The two were close friends at Princeton. This is the Gödel of Gödel, Escher, Bach who, as Davis says, "upset the applecart". The applecart was one of Hilbert's unsolved problems, which now would never be solved. Gödel's Incompleteness Theorem, along with the Church-Turing thesis, set the limits of what can be accomplished with mathematics and computers alone. There will always be true statements which cannot be proved, computer programs we can't tell will terminate.

Remarks similar to those about Cantor and his Set apply to Gödel and his Theorem as well. Only the select few students who make it all the way to the most advanced classes will study the Incompleteness Theorem. One pictures something like the Olympics, in which a series of qualifying rounds eliminates all but the greatest mental athletes. Gödel's theorem is taught as another work of cognitive perfection. His struggle is hidden from view. Surely no one with a

"defective" brain could find his way to the Advanced Symbolic Logic course!

Gödel spent much time in sanatoria, recovering from depression. Davis writes, "The boundary between Gödel's unorthodox view and outright clinical paranoia was not always clearcut. Morgenstern records his surprise that Gödel took ghosts quite seriously. More important, Gödel was convinced that the refrigerator and radiators in his various apartments in Princeton were giving off noxious gases." Sadly, "In a paranoid state over the safety of food available to him...he literally starved himself to death" in 1978.

During my years as an academic scholar, many hours were spent studying the theoretical foundations of computer science using an abstract device known as a "Turing Machine". In another class, the "Turing Test" was an important subject in artificial intelligence. And it goes without saying that Turing was undeniably the father of modern cryptography and codebreaking. Only in hushed tones was it whispered that Turing had met a tragic fate owing to his homosexuality, but this was the 1980's, and gay was becoming mainstream. Today Turing would not be considered mentally ill for being gay, but his suicide indicates depression.

The subtext, if any, was that "this could never happen now. We are far too enlightened to permit this kind of discrimination today." Of Turing's fate, Davis writes "Sex in England had become dangerous, perhaps too dangerous to attempt.... After his conviction, he lost his security clearance.... Alan Turing was hounded to death by the governing authorities of a nation he had---unsung---done much to save." It is the year 2001. I am reading these words on a train crossing the English countryside, to catch the Hydrofoil to Holland, where my vice, marijuana smoking, is not too dangerous, as it remains in England. I am wearing a medal, the Loebner Prize, which bears the likeness of Alan M. Turing.

Davis also mentions the Dutch mathematician L.E.J. Brouwer, who propounded somewhat unorthodox views, largely discredited, on the foundations of logic. "Although Brouwer never recanted his views," Davis writes, "he felt more and more isolated, and spent his last years under the spell of totally unfounded financial worries and a paranoid fear of bankruptcy, persecution, and illness."

Condemned To Repeat Them

Academics who have reviewed my disability case against NYU have said, even though they used terms like "psycho" and "lunatic" to describe my condition, "We didn't know you were sick." Or in another variant, even though they documented much irrational conduct and poor judgment, they didn't know the "severity" of my illness. Apart from the obvious objection that not knowing the law doesn't let them off the hook for violating the law, Davis's book makes an even more substantial ethical point: They should have known better. Don't these mathematicians know their own history?

This reviewer is not claiming to be a Leibniz, a Cantor, a Gödel, or even a Martin Davis. I have certainly never saved the world from the Nazis as Turing did. But I have experienced discrimination in an academic, scientific research setting because of my own mental illness.

Many advanced logic and math classes in my academic training included deep discussions of Cantor's Set, Gödel's Incompleteness Theorem, and Turing's Machines. But seldom, if at all, were their personal lives discussed. Thanks to Martin Davis's book, the gap between their lives and their work has been more completely filled.

Once at NYU I met the incoming director of the Courant Institute of Mathematical Sciences. I asked him if he knew that Richard Courant, for whom the Institute is named, had fought on the side of the Germans in World War I. The new director did not know, and indeed seemed most surprised. This apparent historical paradox is easily explained by the fact that many loyal German Jews, who had served and even been decorated in the Kaiser's army, were among those who were driven from or murdered in Germany during the later Nazi regime. The point of the story is that it makes you wonder: how many mathematicians know their own history? I mean, the guy was heading up the Courant Institute!

One can find few faults in Engines of Logic. When the mathematician approaches the subject of history, however, the results can be somewhat curious. Writing about the origins of the First World War, Davis writes, "In the summer of 1914, in response to the assassination of Archduke Ferdinand, and with German encouragement, the Austrians began World War I by attacking Serbia. To stress their determination that Austria not be permitted to destroy their fellow Slavs, Russia began mobilization." And so on. He states the bare facts of August 1914, but one wonders about the intended audience. Surely not historians interested in the history of mathematics, but if it is mathematicians interested in history, one has to wonder why they would be ignorant of such basic facts.

The cause of World War I has often been said to be "miscalculation". It would be interesting to read a mathematician's interpretation of that "calculation" which went so terribly wrong. The few other times Davis delves into political or world history, the words add little to the story and if anything only highlight the ontological gap between mathematical and historical knowledge.

Besides those digressions into nonmathematical history, I objected only to two adjectives in Martin Davis's remarkable book. The first was his use of "defective" to describe the "defective brain chemistry" at the root of manic depression. The label "defective" is a heavy burden for a mental health patient to bear. The simple logic most people follow says "defective brain = defective mind = defective person". No amount of drugs or therapy can treat that. The sentence would have worked just as well without the adjective.

The second offensive adjective was his description of the Courant Institute building in New York as "handsome". That building looks like a Hilton Hotel and has about as much charm as a suburban cinderblock high school. When I worked at the Courant Institute, I always preferred the older building at 715 Broadway, where some of the computer science department was housed. That stone edificed 19th century monument was vastly more "handsome" than the Courant building.

The Best Of All Possible Worlds

Davis's Epilogue reminds me of the IT worker who said, "If it weren't for Turing, I'd either be unemployed, or working for the Nazis."

History has a way of crucifying its most creative children, but at the same time history has a way of settling scores. Those who forget the mistakes of history do indeed repeat them. Yet "mad scientist" is a meme in our society for a reason. Scientists today

Martin Davis

Martin Davis

have no justification for being surprised that members of their profession suffer from the same mental health problems which afflicted scientists in the past.

Scientific progress does not proceed along the same neat, linear paths as it is presented in textbooks. Scientific revolutions are known to be painful for the revolutionaries. Mathematics and computer science texts, on the other hand, are written in a style that makes the evolution of knowledge seem quite orderly, proceeding neatly from proposition, to lemma and theorem.

Martin Davis reveals a far less tidy picture of the mental life of mathematical revolutionaries. More of his colleagues could do worse than study this oral history of their profession, and recite it to the younger generation of students, alongside the axioms and theorems of mathematics.

Davis writes: "The Dukes of Hanover thought they knew what Leibniz should be doing with his time: working on their family history. Too often today, those who provide scientists with the resources for their lives and work try to steer them in directions most likely to provide quick results. This is not only futile in the short run, but more importantly, by discouraging investigations with no obvious immediate payoff, it shortchanges the future."

They should have known better.

Thanks to Martin Davis for pointing out a few minor errors in an earlier draft of this document. Thank you also to Noel Bush for editing the text. -RW