Originally published at System Toolbox. Reprinted with permission.

Last time, we took a look at the life and some of the achievements, and near achievements, of Charles Babbage, the Godfather of Computing. Babbage made great leaps in our understanding of what would become the field of computer science by considering, and then demonstrating, that mathematical processes could be carried out quickly, repeatedly and without error through mechanical means. This was such a simple idea, but it was ground breaking in its implications. Babbage had been frustrated by the errors that crept into the lookup tables that serious mathematicians used for their calculations. His drive to create calculating machines grew out of the desire to remove these errors from the process of creating those tables. Babbage was ahead of his time. He was a pioneer of the 19th century. If his work hadn't been rediscovered, his achievements would have been almost entirely forgotten by the time the idea of *automatic calculations* through machines began to take hold in the 20th century.

One of the proponents of such *automatic, mechanical, calculations* was a mathematician in King's College, Cambridge; a young Alan Turing. It's almost a natural progression for this series to move from the *cog wheel brains* of Mr. Babbage to the *theoretical thought machines* of Alan Turing. Out of the necessity to answer one of the most critical mathematical questions of his time, Turing started down the road of what would become the fields of modern computer science and cryptography. As one of the single men whose achievements helped turn the tide of World War II, he is a hero. As developer of some of the original ideas about digital computers and for helping solve Hilbert's final question of Mathematics, he is a genius. Being human, his life is ultimately marked by complexity and, unfortunately... tragedy.

This article will focus on Alan Turing's life leading up to, and including, his invention of the "Turing Machine." Next month, we will tackle his achievements in cryptography during World War II, his ideas on the digital computer, and the controversial events that led to this hero's, one of my heros, tragic death.

Alan Mathison Turing was born to Julius Mathison Turing, an Indian Civil Service officer, and Ethel Stoney on June 23, 1912 in Paddington, England. Alan's father was still under active commission in India and feared the risks of raising family in the remote provinces over which he held jurisdiction. After Alan's birth, his father decided to leave his family in England instead of risking those uncertainties, choosing instead to make the trip back and forth between India and England while leaving his family with friends in England.

Like Babbage (and many others in this field), Turing showed early signs of, what I like to call, the "personality disorder" that leads to a such vocations as engineering and mathematics. Alan's natural inquisitiveness was often confused with mischief, where "planting" broken toys in hopes of resurrecting them was probably interpreted as "getting rid of the evidence." At a very early age, he is said to have taught himself to read in only three weeks and his discovery of numbers brought about the distracting habit of stopping at every street light in order to find its serial number. At the age of seven, while on a picnic in Ullapool, Scotland, Alan had the idea of gathering wild honey for the afternoon's tea. By plotting the flight paths of the bees among the heather, he was able to find the intersection point that marked their hive and provide an unexpected treat for the family.

There's another anecdote that made an appearance in Neal Stephenson's spectacular work of fiction, *The Cryptonomicon*, in which Turing plays a supporting role. It seems that Alan had a bicycle that had a problem with its chain. He discovered that the chain would dislodge itself from the gears after a regular, repeatable, number of revolutions. At first, the young Alan would count the revolutions of the gears throughout his ride until it was time for the chain to be forced to derail. He would then get off his bike and re-adjust the chain. As this got to be cumbersome over longer treks, he finally rigged a mechanical device that would maintain the count and readjust the chain itself. Supposedly, it never occurred to him to just buy a new chain to solve the problem. I believe that it is more likely that the chain's issues presented a unique problem set for Turing's mind to solve. It challenged him to think in a different way. It was challenging and fun; buying a chain was not.

At the age of six, Alan's mother enrolled him in a private day school, St. Michael's, in order for him to learn Latin. Thus began Alan's introduction into the system that would shape his intellectual and personal development for the next fourteen odd years. The English educational system would prove to be both a conflict and a collaboration with Turing's sensibilities. The collaboration is epitomized by his early respect for rules and their relationship to his concept of fairness. These ideas are probably best illustrated by an anecdote of his mother skipping part of *The Pilgrim's Progress*. Judging one section to be too theologically weighty for the youngster, she had skipped it while reading aloud in order to spare him. Alan objected and felt that the story was ruined; skipping parts, in his sensibility, was against the rules of reading.

The conflict, in his relationship with the English school system, was partially rooted in Alan's resolve that he was nearly always right. Personal opinions were held as closely as fact. He was one of those people that *knows* something and doesn't *think*, *feel* or have an *opinion* on them. This type of mind set was definitely at odds with an education system built on tradition and firm in the belief that it *knew* what was best for its charges.

Early on, Alan was marked with the label of "genius" by the Headmistress of St. Michael's, a proclamation that would be echoed a few years later by a gypsy fortune teller. Despite such proclamations, Alan was required to follow the natural order of the English school system and, upon finishing his studies at St. Michael's, followed his brother's path to his next school, Hazelhurst and then to his first *public school*, Marlborough. Public school showed the ugly side of the English school system and Alan had his first troubles with bullies, proclaiming that he learned to run fast in order to "avoid the ball."

Alan was introduced to science through Edwin Tenney Brewster's *Natural Wonders Every Child Should Know*. Brewster's book sought to introduce topics that help children understand their place in the world and what they had in common and how they differed with and from other living things. This discovery, and that of mathematics, would sustain Turing in a life-long love affair. The rules and discoveries of science and mathematics fit his general sensibilities of the world; it had order and could be explored with reason. Sense could be made of life if observed in the correct way. Brewster's book was probably is the first to link the concept of machine and biology in Alan's mind, explaining that the human body was a complex *machine* with complicated processes that carried out the duties and chores of maintaining life.

While school offered many torments, it also opened up a world of knowledge to the young Turing. He showed an early interest and ability in languages, especially French, and treated it as a code that would allow him to carry on covert communications. Also, having always had a fascination with various process oriented activities, Alan was exposed to chemistry for the first time and fell instantly in love. Turing would go on to dabble in chemistry for the rest of his life, often co-opting family basements and guest rooms as chemistry labs. His habit of concocting various chemical solutions would later play a part in his untimely death as a adult.

At the age of 13, Alan was enrolled to attend the Sherborne boarding school. At the time of the school's summer term of 1926, England had just been brought to a stand still by the first day of the general strike. No buses or trains were running. Turing made something of a stir, being reported in the local newspaper, by bicycling the sixty miles from his home in Southampton to Sherborne, staying overnight in an Inn at a halfway point.

Sherborne and Alan were not the best match. Sherborne, as many English schools of the time, was concerned with creating *citizens* and not *scholars*. The headmaster, at the time of Alan's enrollment, espoused the idea that school was originally created to be a miniature society. Students would learn to navigate the complexities of their later adult lives by learning to survive the power plays of their current public school life. Authority and obedience held more sway than the "free exchange of ideas" and the "opening of the mind." Not long after arriving, the already shy Turing became even more withdrawn.

Alan sought solace in his books and course work. In 1927, he was able to find the infinite series of the "inverse tangent function" from the trigonometric formula for tan1/2x (tan^{-1}x = x - x^{3}/3 + x^{5}/5 - x^{7}/7 ...) without the aid of elementary calculus (Alan had yet to be exposed to it). It was a significant enough achievement to have his mathematics instructor include himself among the roster of people that had proclaimed the boy's genius. Such a proclamation didn't hold much sway with the school. While the accomplishment was extraordinary, Sherborne's headmaster, not a particular fan of science, felt he was wasting his time and was in danger of becoming a scientific specialist and *not* an educated man. This disrespect of science was not uncommon at the school. Alan's autumn form-master, a classicist who was enthralled with Latin, called scientific subjects "low cunning" and felt that the only reasons that the Germans lost World War I was because they placed to much faith in science and engineering and not enough in religious thought and observance.

Alan's dogged persistence to study such *low* subjects, finally earned him some respite. As long as he made a few concessions to the formalities of the school, he was left to his own devices. In 1928, he became enthralled with the theory of relativity and lost himself in the English translation of Einstein's *Relativity: The Special and General Theory*. Probably one of only a few, if any, sixteen year olds who actually grasped Einstein's theories, Turing was able to fully grasp Einstein's doubts of the veracity of Galilei-Newtonian laws. He was even able to deduce Einstein's Law of Motion ("the separation between any two events in the history of a particle shall be a maximum or minimum when measured along its world line") from his readings alone (it wasn't specifically stated in the text). By 1929, Alan had begun to study quantum physics. It was a heady time as Schroedinger and others turned what was considered a "dead" science on its head. Schroedinger's quantum theory of matter was only three years old and Alan and his friend Christopher Morcum immersed themselves in these emerging discoveries. Alan was in his element.

Turing had originally planned on attending Trinity College at Cambridge. As far as he was concerned, it was the center of scientific and mathematical thought in England and he wanted to attend. After a number of failed attempts at passing his final examinations, more out of abstinence in engaging his "classical" work, he finally missed a scholarship to Trinity but was able to obtain one to King's, the college of his second choice.

King's College agreed with Alan. Though he was still somewhat of a social misfit, his studies and the freedom from the petty tortures of public school life allowed him to relax and find his rhythm. King's also turned out to be a good fit due to the caliber of its faculty. Turing's mathematics professor was one of the most distinguished mathematicians of his time, G.H. Hardy, who had recently left Oxford to take up the Sadleirian Chair at Cambridge. He was also among 85 other students engaged in scientific study, as compared to the one or two he had to seek out during his Sherborne days. As happens today with many high school geeks, college offered a chance for Alan to emerge from his protective shell and begin to engage the world on his own terms.

During the 20's, Cambridge had moved to establish itself as second in the world in the field of new maths. It had been able to stake this claim on the developments that its faculty and students were making in the realms of quantum theory and *pure* mathematics. It was widely regarded as second only to Gottingen University in Germany, a place that supported such genius as John Von Nuemann.

Von Nuemann and Turing were to cross paths a number of times throughout their lives. In 1932, Turing read Von Nuemann's *Mathematische Grundlagen der Quantemechanik* and was deeply affected by the text. His interest in quantum theory continued into the studying of the works of other luminaries like Schrodinger and Heisenberg. This exposure to the *greats* in an emerging field totally engaged the young Turing and set him to exploring the questions that their discoveries raised. It was this exposure and new found focus that put Turing on an crash course with Hilbert's *Three Questions of Mathematics*.

In 1928, developments in *pure* mathematics seemed to be unraveling the foundations of the field. It seemed that the world was on the cusp of unlocking the vary foundations of mathematics. It wouldn't be long before core axioms were nailed down and mathematics would be just a set of easily applied rules that would lead directly, inevitably to the solution of any problem. No problem would be beyond the reach of mathematics. Appropriately applied, mathematics would make the world a better place (sounds kind of like the commotion surrounding the Internet, doesn't it?).

It was during this period, in 1928, that Hilbert, already famous for his development of Hilbert quantum spaces, posed a number of questions about the core of mathematics, whose unexpected answers would shake the field and push it into new realms of discovery and reason. Hilbert's agenda was to find a general algorithmic procedure for answering all mathematical inquiries, or at least proving that such a procedure existed.

- Three of those questions at the heart of his agenda were:
- Was mathematics
*complete*? Meaning, could every assertion be proven or disproven with the*rules*of math? - Was mathematics
*consistent*? Meaning, could a false statement never be proven true with the*rules*of math? - Was mathematics
*decidable*? Meaning, were there definite steps that would prove or disprove an assertion?

While nobody, including Hilbert, had been able to offer solutions to these questions by proof in 1928, Hilbert was confident that the answer to each was *yes*. In his mind, there had to be a solution for every problem, if only to prove that it was unsolvable. This failed assertion, as bad as it sounds, would actually save mathematicians a lot of effort spent pursuing blind alleys. So, it was still a solution; its a *math* thing.

The issue lay in *proving* that mathematics was *complete*, *consistent*, and *decidable*. At the same gathering, the young mathematician Kurt Godel dealt a serious blow to this line of queries, by showing that math must be *incomplete* because, as he showed, there are assertions that can be stated that can be neither proved nor disproved. An assertion, encoded in the form of mathematics, that said, in effect, "this statement is unprovable" showed this disturbing (if you are into that sort of thing) property. An attempt to prove it true or untrue leads to contradiction. At least in the form of the question phrased by Hilbert, Godel had proved that arithmetic was incomplete. There are nuances to this, of course, but it was still damaging. Godel also showed that mathematics could *not* be proven consistent *and* complete. However, he was not able to shake loose an answer to Hilbert's question as to the *decidability* of arithmetic.

Alan's professor Hardy, for one, was happy that Godel couldn't topple Hilbert's final question. In his view, a mechanical process that could perform a solution to all mathematical problems would put every serious mathematician out of a job. Everything would have been done.

It was time for the student to instruct the teacher, at least in part. After a day of running, an activity that Alan found to nicely clear the mind, he stumbled onto the idea of a machine of simple, though improbable, design that could tackle any sort of problem put to it. The powerful machine would only understand the digits `0`

and `1`

; the first binary computer. It would move a read/write mechanism across an infinite tape of these numbers and, based on their particular arrangement, solve various types of problems. Alan's breakthrough was that he had defined, in specific language, what a *general algorithm* actually was. The Turing Machine, as his construct would be called, was a thought experiment that helped codify the features of algorithms. During his exploration of the wonderful ideas that this *machine* inspired, Turing found that, despite the simple, general, nature of his algorithm, there did exist problems that it could not solve. This discovery *proved* Hilbert's assertions were incorrect, the answer to Hilbert's final question, the *Entscheidungsproblem* was "no, mathematics is not decidable."

The young mathematician from King's College, Cambridge had bested one of the greatest mathematicians of his time at the age of 23. He gained a fair measure of acclaim for his achievement and the word "genius" began to be tossed around again. Had he done only this, he would be remembered in some history books and higher math students would get acquainted with him at some point. At any rate, a small amount of historical immortality, as obscure as it may be, would be granted in his memory. However, it was what he did next that changed the course of human history.

Next month, we will explore the workings of a Turing Machine and follow Alan into the war effort. We will see how a single man's true genius can turn the tide of war, and we will shake our heads in disbelief at a hero's humiliation and eventual death. Stay tuned.

------

*© 2001 G. James Jones is a Microcomputer Network Analyst for a mid-sized public university in the midwest. He writes on topics ranging from Open Source Software to privacy to the history of technology and its social ramifications. This article originally appeared at System Toolbox (http://www.systemtoolbox.com). Please email and let me know where it is being used. This article is dedicated to the memory of Dr. Clinton Fuelling. Verbatim copying and redistribution of this entire article is permitted in any medium if this notice is preserved.*

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