This book has shown nonsense arising from sense, a breaking with form as a consequence of formalism. In mathematics, the pursuit of logic led to the denial of its universal potency, the negation of a tacit assumption that had been an underpinning of scientific enquiry. This spirit found its way into literature in the nineteenth century, and exploded in the twentieth, with an explicit recognition of the arbitrariness of language and the evanescence of meaning. This phenomenon cannot simply be labelled a reaction to the stodgy nineteenth-century fascination with form, for it is an outgrowth of that fascination.
This work has barely scratched the surface of this story. It is nothing but a few representative case studies and a general paradigm. I have found that the deeper I delve into a study, the more links I discover. To explore fully the development of formal reasoning and its consequences, one would have to relive the lives of all the individuals who constituted the social systems in which this development occurred. Indeed, to be utterly complete, one would have to study every detail of human civilisation and physical existence meticulously. This, of course, as Charles Dodgson would delight in pointing out, would take more time than has elapsed in all of human history; it is an undertaking that the Bellman might consider. As for us, as this work has shown, we must be content with approximations.
The developments discussed in this book are not parts of a finished tale. A shrewd observer can recognise their thread in recent history and in current events. It is appropriate to close this book with a not-closing, to discuss some more contemporary events and to leave this story open-ended.
* * *
In the middle of the night on August 24, 1970, a powerful car bomb destroyed an addition to Sterling Hall on the Madison campus of the University of Wisconsin that housed a mathematics research centre funded by the United States Army. One researcher was killed, one student was injured, and the research of twenty-nine people was obliterated. The governor of Wisconsin refused to be reactionary and place the campus under martial law. His comments on this refusal show a recognition of the causes of the shootings that had occurred at Kent State University the previous spring, in which the National Guard of the United States had fired into a crowd, massacring the very citizens for whose defence the armed forces exist. "You have to recognize there is a limitation. You can't make every campus into an armed camp. People don't like that kind of surveillance."
It is ironically appropriate that it was a mathematics research centre that was bombed. In the nineteenth century, the developments of abstract algebra were initially decried by those who thought that mathematics was straying too far from concrete reality. How much closer to reality can mathematics be brought than that ultimate physicality that comes with the pulverising of its edifice?! The people who planted the bomb were radicals of the New Left, which by that time had become factionalised and was rapidly disintegrating and losing power. The army represented the concrete potency of the American social system, the Establishment, and Sterling Hall was a convenient symbol of that abstraction, an edifice that could be acted against physically, as were Reserve Officer Training Corps buildings on many campuses, which had also been targets of destructive attacks. Thus the selection of a mathematics research centre was serendipitous. This makes the attack all the more ironic, and paints it as a result not of the four individuals who placed the bomb in the car knowing that it would explode and destroy the building, but of the social system of which they were a part. This recalls the views of the early social scientists.
The actions of these radicals were motivated by political beliefs. Many of these people expressed only facile, puerile justifications for their actions and were just along for the ride, but the core that supplied the impetus for the movement had reasoned out their positions, often with the aid of information acquired during university educations. Thus the anarchic, irrational act of destruction of a university building was a consequence of reason and of the university!
The tragedy of this event is not to be taken lightly. Yet, at the same time, one can marvel at and even laugh about its eery mathematical irony.
* * *
In chapter 0, I spoke of abstract systems as prisons. Willie Bosket, known as the most violent inmate in the New York prison system, makes this statement concrete. Like the theme of this book, Bosket seems to have two sides that constitute a whole: a rational, intellectual side that looks on and analyses, and a nihilistic side, utterly opposed to systematisation. Bosket has been made an object of the criminal justice system since he was nine years old. When he was fifteen, he was convicted of murdering two subway riders and attempting to murder a third. In prison, he takes every opportunity that he can get to damage or to obstruct the system. He is kept in solitary confinement. He sets fire to his cell, throws his own feces at the guards, assaults the guards, even eats light fixtures. Each day in the life of Willie Bosket is a Day of Rage.
Bosket is quite arguably one of the most barbaric, nihilistic, and repulsive human beings in the world. And it is not difficult to conceive of him as a hero. Bosket does every day what almost all of us would like to do, at times. He is a continual explosion of hatred. He refuses to acknowledge the dominion of the administrative systems that the rest of us must acknowledge in our everyday existences of jobs, ambitions, and governments.
While Bosket was being interviewed by Matthew Worth, a journalist helping him write his autobiography, he pulled a home-made knife and stabbed his guard. "It was so random, so senseless and stupid", said Worth. "He didn't even know the guard." Worth seems not to recognise the meaning of his own comments. He regards Bosket's actions from a rational frame of reference. But Bosket vehemently rejects rationalism! His entire life has been an expression of that rejection. In stabbing the guard and thus destroying the chance for completion of his autobiography, Bosket was the perfect anarchist. Forget the autobiography! It has importance only within a system that values information.
Bosket's rational side is a shrewd observer. It speaks of the system's having created the monster in Bosket. Bosket knows what he is. And he seems not to care; his rational part stands back and lets the monster do what it will. "I laugh at this system because there ain't a damn thing that it can do to me except deal with the monster it has created", he has said. Although Bosket's formal education ended with the third grade, he is as much aware of the power of abstract social systems as was Quetelet. A psychological test given at a reform school found Bosket "precocious, warm, and empathetic". But these qualities were not nurtured. Instead, the system confined Bosket to maximum-security institution, where his hatred festered. Thus Bosket seems quite correct in observing that it is the system that has created him as a monster.
Bosket's life bears a fascinating resemblance to that of his father, whom he never knew as he was growing up. Like his son, William James Bosket Senior received little formal education and was placed in a reformatory. While Bosket's mother was pregnant with him, his father was arrested for a double murder. He escaped and robbed a bank before being captured and imprisoned. But at this juncture the life of Bosket's father diverges from that of Bosket. While in prison, the father seems to have arrived at an understanding with the system, a feeling that he could still maintain his identity and individuality internally while satisfying the system externally. While he was a prisoner, he attended college and graduated with honours. A few years later he was released, but then arrested again, on a charge of molestation. And at that point the façade of civilisation seems to have crumbled. He escaped from prison, and when finally cornered by police in a shootout and running out of ammunition, he shot himself dead, rather than face another prison term. This final act was characteristic. It was a sanity of insanity. The seemingly inevitable similarities between William Bosket Junior and his father form a case that would have fascinated Galton and the eugenicists.
The following excerpt from one of the letters of William James Bosket Senior to his son summarises the understanding that evolved in him, but was never given the chance to evolve in his son:
In your letter you seem taken with the ideas and writings of George Jackson[47] most, and the need for some form of "revolutionary suicide". Frankly, that's a bit too much excitement for me, and it has been my observation that the energies from such a thought basis tend to dissipate unfruitfully before the onrush of hard pragmatic realities. But then, I'm just old folks.
* * *
It's crazy to be writing about this. Anarchy has no place in a library. The very concept of a library promotes organisation. Mikhail Bakunin, the founder of the international anarchist movement, started writing many works on anarchy, and finished not a single one. His collected works are a collection of fragments, broken rubble. Bakunin spent his time not creating information, but destroying, helping to rip down old systems. He travelled throughout Europe, and wherever discontent was producing revolutionary feelings, Bakunin was there, helping it along, mischievously tipping the carefully balanced social systems of Quetelet off their precipices and sending them crashing down under their own weight. "The urge to destroy is also a creative urge", Bakunin wrote, and this has become one of his most famous sayings. It takes skilled engineers to design and erect edifices, and it takes demolition experts who are just as skilled to raze them. There is a primal, almost orgasmic joy in blowing something up, in playing with fire.
While contemplating this concluding chapter, I got an urge to read Abbie Hoffman's book Steal This Book!. It is out of print, and I could not find it in any used book store (though, undoubtedly, there exists a copy that someone wouldn't mind selling, out there somewhere in that approximation to the Universal Library). I checked in the library at Cornell, and found that both copies owned by Cornell had been stolen. How appropriate! How mathematically beautiful! I no longer felt the need to read the book.
* * *
So what does all this have to do with computer science? It is the opportunities for creativity within a system of encoding that lure so many young people to computing. The availability and low cost of microcomputers in the 1980's has created many amateur mathematicians. For programmers are a species of mathematician. In a 1947 lecture to the London Mathematical Society, Turing spoke of a need for "mathematicians of ability" to "do the preliminary research on the problems, putting them into a form suitable for computation."[48] Those adolescents of the 1980's who dabble in computers and become enthused with them, of whom I have been one, are carrying on a tradition of the amateur scientist that recalls the curious dabblings of English gentlemen from the time of Newton on into the nineteenth century. It is the current love affair with technology, especially in the United States, that causes these explorers to gravitate toward computer science and computer technology. Many of the same people, had they lived in the nineteenth century, before much of the technological glitz, would have interested themselves in literature. For, as has been illustrated throughout this work, the two fascinations share a common root.
The Information Age is upon us. We are presented with a choice.
We can forget about reason, slowly but surely. We can abandon the problem of organising this backbreaking load. Specialisation will continue to creep forward, and the body of knowledge in general will fragment and disintegrate.
We can derive anarchy from reason, and not merely abandon reason, but forcibly tear it down. At times this seems childish. At times it seems heroic.
We can slip between the horns of the dilemma. We can choose not to choose. Yes, fine, reason implies unreason; we admit it. But this does not mean that a death warrant for reason has been issued. Chaos and contradiction are present, just around the corner, but they need not be struggled with; they can be accepted. One may say that this book cannot argue such a point, for the writing and reading of books, and even the process of argument, are acts of reason, and this constitutes a prejudice. "Prejudice"! Even that word itself implies reason! The more I say, the more I move in circles. I cannot presume to judge reason, or to judge anarchism. I respect them both, and I spit on them both. They have their places, and it seems best to accept both and to take neither to extremes. In this respect the people who bombed Sterling Hall can learn from the comments of Robert Fassnacht's father after his son died:
He would have been a good teacher. I can't help but feel this is the work of an S.D.S.-type of person.[49] I would like to wipe up the street with them. But I am trying to be a Christian, and if I'm a Christian, I shouldn't want to do that.
* * *
Many people could have written this book. Only the particular style is mine, not the subject. Among computer scientists I am not particularly insightful or capable, and I wish not to convey any false impression that I am. In 1830, George Peacock became the first author of a published text on abstract algebra, not because he was the only leader in the field, but because he had the time to devote to writing a book and the determination to do it. The ideas in that book were "in the air", and so it is with this book. One has only to examine my bibliography to see that this is so. Discovering relationships for oneself is often more attractive than communicating them to others, and if this project had not offered me the enticement of doing both simultaneously I could not have completed it.
There is an increasing clamour in the United States for better science education, and "scientific literacy". It is said that the United States is becoming a nation of scientific illiterates. But this is only because the U.S. has become a nation of illiterates in general. I know; I was one of them. I am shocked at the amount of time that was wasted during the early years of my education, in going over old topics and rote exercises again and again. I spent six years of elementary school "learning" arithmetic before the structure of the educational system would allow me to proceed to beginning algebra. By the beginning of junior high school, I had come to detest mathematics, because all the mathematics that I had seen in six years was the same monotonous, elementary arithmetic.
Hopefully the selection of topics and the presentation of this book will alleviate this situation, by introducing some stimulating topics to secondary school students. I have been afforded the privilege of climbing out of that pit of ignorance into which the American educational system placed me and in which it tried to keep me, and during the past few years I have enjoyed a sense of being awake as I never have been before. My recognition of the fabric that is formed by the threads of different fields of study and sensing the harmonies that play through all of them has been the cause of this. If this book can help to develop that sense in any of its readers, then it has been worth writing.
Perhaps someday the types of relationships explored in this book will become obvious to all humans in the normal course of their education. If and when that happens, such books will no longer need to be written, and this little volume will be able to take its place contentedly among the ranks of the superfluous.