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"Linux Gazette...<I>making Linux just a little more fun!</I>"
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<H1><font color="maroon">A Pioneer for a New Century -- Alan Turing, part 1</font></H1>
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<H4>By <a href="mailto:jones@systemtoolbox.com">G James Jones</a>
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<BR>Originally published at <A HREF="http://www.systemtoolbox.com/">System Toolbox</A>. Reprinted with permission.</H4>
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<P>
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Last time, we took a look at the life and some of the achievements,
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and near achievements, of <A
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HREF="http://www.systemtoolbox.com/article.php?articles_id=43">Charles
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Babbage</A>, the Godfather of Computing. Babbage made great leaps in
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our understanding of what would become the field of computer science
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by considering, and then demonstrating, that mathematical processes
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could be carried out quickly, repeatedly and without error through
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mechanical means. This was such a simple idea, but it was ground
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breaking in its implications. Babbage had been frustrated by the
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errors that crept into the lookup tables that serious mathematicians
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used for their calculations. His drive to create calculating machines
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grew out of the desire to remove these errors from the process of
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creating those tables. Babbage was ahead of his time. He was a
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pioneer of the 19th century. If his work hadn't been rediscovered, his
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achievements would have been almost entirely forgotten by the time the idea
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of <I>automatic calculations</I> through machines began to take hold
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in the 20th century.
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<P>
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One of the proponents of such <I>automatic, mechanical,
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calculations</I> was a mathematician in King's College, Cambridge; a
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young Alan Turing. It's almost a natural progression for this series
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to move from the <I>cog wheel brains</I> of Mr. Babbage to the
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<I>theoretical thought machines </I> of Alan Turing. Out of the
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necessity to answer one of the most critical mathematical questions of
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his time, Turing started down the road of what would become the fields
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of modern computer science and cryptography. As one of the single men
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whose achievements helped turn the tide of World War II, he is a hero.
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As developer of some of the original ideas about digital computers and
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for helping solve Hilbert's final question of Mathematics, he is a
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genius. Being human, his life is ultimately marked by complexity and,
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unfortunately... tragedy.
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<P>
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This article will focus on Alan Turing's life leading up to, and
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including, his invention of the "Turing Machine." Next month, we will
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tackle his achievements in cryptography during World War II, his ideas
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on the digital computer, and the controversial events that led to this
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hero's, one of my heros, tragic death.
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<P><H3>Early Signs of a Remarkable Mind</H3>
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<P>
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Alan Mathison Turing was born to Julius Mathison Turing, an Indian
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Civil Service officer, and Ethel Stoney on June 23, 1912 in
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Paddington, England. Alan's father was still under active commission
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in India and feared the risks of raising family in the remote
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provinces over which he held jurisdiction. After Alan's birth, his
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father decided to leave his family in England instead of risking those
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uncertainties, choosing instead to make the trip back and forth
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between India and England while leaving his family with friends in
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England.
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<P>
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Like Babbage (and many others in this field), Turing showed early
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signs of, what I like to call, the "personality disorder" that leads
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to a such vocations as engineering and mathematics. Alan's natural
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inquisitiveness was often confused with mischief, where "planting"
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broken toys in hopes of resurrecting them was probably interpreted as
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"getting rid of the evidence." At a very early age, he is said to
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have taught himself to read in only three weeks and his discovery of
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numbers brought about the distracting habit of stopping at every
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street light in order to find its serial number. At the age of seven,
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while on a picnic in Ullapool, Scotland, Alan had the idea of
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gathering wild honey for the afternoon's tea. By plotting the flight
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paths of the bees among the heather, he was able to find the
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intersection point that marked their hive and provide an unexpected
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treat for the family.
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<P>
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There's another anecdote that made an appearance in Neal Stephenson's
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spectacular work of fiction, <I>The Cryptonomicon</I>, in which Turing
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plays a supporting role. It seems that Alan had a bicycle that had a
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problem with its chain. He discovered that the chain would dislodge
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itself from the gears after a regular, repeatable, number of
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revolutions. At first, the young Alan would count the revolutions of
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the gears throughout his ride until it was time for the chain to be
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forced to derail. He would then get off his bike and re-adjust the
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chain. As this got to be cumbersome over longer treks, he finally
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rigged a mechanical device that would maintain the count and readjust
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the chain itself. Supposedly, it never occurred to him to just buy a
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new chain to solve the problem. I believe that it is more likely that
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the chain's issues presented a unique problem set for Turing's mind to
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solve. It challenged him to think in a different way. It was
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challenging and fun; buying a chain was not.
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<p>
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<H3>Getting an Education</H3>
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<P>
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At the age of six, Alan's mother enrolled him in a private day school,
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St. Michael's, in order for him to learn Latin. Thus began Alan's
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introduction into the system that would shape his intellectual and
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personal development for the next fourteen odd years. The English
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educational system would prove to be both a conflict and a
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collaboration with Turing's sensibilities. The collaboration is
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epitomized by his early respect for rules and their relationship to
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his concept of fairness. These ideas are probably best illustrated by
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an anecdote of his mother skipping part of <I>The Pilgrim's
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Progress</I>. Judging one section to be too theologically weighty for
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the youngster, she had skipped it while reading aloud in order to
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spare him. Alan objected and felt that the story was ruined; skipping
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parts, in his sensibility, was against the rules of reading.
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<P>
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The conflict, in his relationship with the English school system, was
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partially rooted in Alan's resolve that he was nearly always right.
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Personal opinions were held as closely as fact. He was one of those
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people that <I>knows</I> something and doesn't <I>think</I>,
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<I>feel</I> or have an <I>opinion</I> on them. This type of mind set
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was definitely at odds with an education system built on tradition and
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firm in the belief that it <I>knew</I> what was best for its charges.
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<P>
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Early on, Alan was marked with the label of "genius" by the
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Headmistress of St. Michael's, a proclamation that would be echoed a
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few years later by a gypsy fortune teller. Despite such
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proclamations, Alan was required to follow the natural order of the
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English school system and, upon finishing his studies at
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St. Michael's, followed his brother's path to his next school,
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Hazelhurst and then to his first <I>public school</I>,
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Marlborough. Public school showed the ugly side of the English school
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system and Alan had his first troubles with bullies, proclaiming that
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he learned to run fast in order to "avoid the ball."
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<p><H3>Brushes with Science</H3>
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<P>
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Alan was introduced to science through Edwin Tenney Brewster's
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<I>Natural Wonders Every Child Should Know</I>. Brewster's book
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sought to introduce topics that help children understand their place
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in the world and what they had in common and how they differed with
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and from other living things. This discovery, and that of
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mathematics, would sustain Turing in a life-long love affair. The
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rules and discoveries of science and mathematics fit his general
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sensibilities of the world; it had order and could be explored with
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reason. Sense could be made of life if observed in the correct
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way. Brewster's book was probably is the first to link the concept of
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machine and biology in Alan's mind, explaining that the human body was
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a complex <I>machine</I> with complicated processes that carried out
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the duties and chores of maintaining life.
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<P>
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While school offered many torments, it also opened up a world of
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knowledge to the young Turing. He showed an early interest and
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ability in languages, especially French, and treated it as a code that
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would allow him to carry on covert communications. Also, having
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always had a fascination with various process oriented activities,
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Alan was exposed to chemistry for the first time and fell instantly in
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love. Turing would go on to dabble in chemistry for the rest of his
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life, often co-opting family basements and guest rooms as chemistry
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labs. His habit of concocting various chemical solutions would later
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play a part in his untimely death as a adult.
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<p><H3>Sherborne</H3>
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<P>
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At the age of 13, Alan was enrolled to attend the Sherborne boarding
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school. At the time of the school's summer term of 1926, England had
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just been brought to a stand still by the first day of the general
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strike. No buses or trains were running. Turing made something of a
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stir, being reported in the local newspaper, by bicycling the sixty
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miles from his home in Southampton to Sherborne, staying overnight in
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an Inn at a halfway point.
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<P>
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Sherborne and Alan were not the best match. Sherborne, as many
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English schools of the time, was concerned with creating
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<I>citizens</I> and not <I>scholars</I>. The headmaster, at the time
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of Alan's enrollment, espoused the idea that school was originally
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created to be a miniature society. Students would learn to navigate
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the complexities of their later adult lives by learning to survive the
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power plays of their current public school life. Authority and
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obedience held more sway than the "free exchange of ideas" and the
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"opening of the mind." Not long after arriving, the already shy
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Turing became even more withdrawn.
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<P>
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Alan sought solace in his books and course work. In 1927, he was able
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to find the infinite series of the "inverse tangent function" from the
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trigonometric formula for tan1/2x (tan<SUP>-1</SUP>x = x -
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x<SUP>3</SUP>/3 + x<SUP>5</SUP>/5 - x<SUP>7</SUP>/7 ...) without the
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aid of elementary calculus (Alan had yet to be exposed to it). It was
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a significant enough achievement to have his mathematics instructor
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include himself among the roster of people that had proclaimed the
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boy's genius. Such a proclamation didn't hold much sway with the
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school. While the accomplishment was extraordinary, Sherborne's
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headmaster, not a particular fan of science, felt he was wasting his
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time and was in danger of becoming a scientific specialist and
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<I>not</I> an educated man. This disrespect of science was not
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uncommon at the school. Alan's autumn form-master, a classicist who
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was enthralled with Latin, called scientific subjects "low cunning"
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and felt that the only reasons that the Germans lost World War I was
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because they placed to much faith in science and engineering and not
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enough in religious thought and observance.
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<P>
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Alan's dogged persistence to study such <I>low</I> subjects, finally
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earned him some respite. As long as he made a few concessions to the
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formalities of the school, he was left to his own devices. In 1928,
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he became enthralled with the theory of relativity and lost himself in
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the English translation of Einstein's <I>Relativity: The Special and
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General Theory</I>. Probably one of only a few, if any, sixteen year
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olds who actually grasped Einstein's theories, Turing was able to
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fully grasp Einstein's doubts of the veracity of Galilei-Newtonian
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laws. He was even able to deduce Einstein's Law of Motion ("the
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separation between any two events in the history of a particle shall
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be a maximum or minimum when measured along its world line") from his
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readings alone (it wasn't specifically stated in the text). By 1929,
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Alan had begun to study quantum physics. It was a heady time as
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Schroedinger and others turned what was considered a "dead" science on
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its head. Schroedinger's quantum theory of matter was only three years
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old and Alan and his friend Christopher Morcum immersed themselves in
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these emerging discoveries. Alan was in his element.
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<p><H3>King's College</H3>
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<P>
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Turing had originally planned on attending Trinity College at
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Cambridge. As far as he was concerned, it was the center of
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scientific and mathematical thought in England and he wanted to
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attend. After a number of failed attempts at passing his final
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examinations, more out of abstinence in engaging his "classical" work,
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he finally missed a scholarship to Trinity but was able to obtain one
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to King's, the college of his second choice.
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<P>
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King's College agreed with Alan. Though he was still somewhat of a
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social misfit, his studies and the freedom from the petty tortures of
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public school life allowed him to relax and find his rhythm. King's
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also turned out to be a good fit due to the caliber of its faculty.
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Turing's mathematics professor was one of the most distinguished
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mathematicians of his time, G.H. Hardy, who had recently left Oxford
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to take up the Sadleirian Chair at Cambridge. He was also among 85
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other students engaged in scientific study, as compared to the one or
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two he had to seek out during his Sherborne days. As happens today
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with many high school geeks, college offered a chance for Alan to
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emerge from his protective shell and begin to engage the world on his
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own terms.
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<P>
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During the 20's, Cambridge had moved to establish itself as second in
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the world in the field of new maths. It had been able to stake this
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claim on the developments that its faculty and students were making in
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the realms of quantum theory and <I>pure</I> mathematics. It was
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widely regarded as second only to Gottingen University in Germany, a
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place that supported such genius as John Von Nuemann.
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<P>
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Von Nuemann and Turing were to cross paths a number of times
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throughout their lives. In 1932, Turing read Von Nuemann's
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<I>Mathematische Grundlagen der Quantemechanik</I> and was deeply
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affected by the text. His interest in quantum theory continued into
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the studying of the works of other luminaries like Schrodinger and
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Heisenberg. This exposure to the <I>greats</I> in an emerging field
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totally engaged the young Turing and set him to exploring the
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questions that their discoveries raised. It was this exposure and new
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found focus that put Turing on an crash course with Hilbert's <I>Three
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Questions of Mathematics</I>.
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<p><H3>A Question of Mathematics and Turing Machines</H3>
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<P>
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In 1928, developments in <I>pure</I> mathematics seemed to be
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unraveling the foundations of the field. It seemed that the world was
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on the cusp of unlocking the vary foundations of mathematics. It
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wouldn't be long before core axioms were nailed down and mathematics
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would be just a set of easily applied rules that would lead directly,
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inevitably to the solution of any problem. No problem would be beyond
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the reach of mathematics. Appropriately applied, mathematics would
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make the world a better place (sounds kind of like the commotion
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surrounding the Internet, doesn't it?).
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<P>
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It was during this period, in 1928, that Hilbert, already famous for
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his development of Hilbert quantum spaces, posed a number of questions
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about the core of mathematics, whose unexpected answers would shake
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the field and push it into new realms of discovery and reason.
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Hilbert's agenda was to find a general algorithmic procedure for
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answering all mathematical inquiries, or at least proving that such a
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procedure existed.
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<OL>Three of those questions at the heart of his agenda were:
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<LI>Was mathematics <I>complete</I>? Meaning, could every assertion be proven or disproven with the <I>rules</I> of math?
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<LI>Was mathematics <I>consistent</I>? Meaning, could a false statement never be proven true with the <I>rules</I> of math?
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<LI>Was mathematics <I>decidable</I>? Meaning, were there definite steps that would prove or disprove an assertion?
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</OL>
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<P>
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While nobody, including Hilbert, had been able to offer solutions to
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these questions by proof in 1928, Hilbert was confident that the
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answer to each was <I>yes</I>. In his mind, there had to be a
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solution for every problem, if only to prove that it was
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unsolvable. This failed assertion, as bad as it sounds, would actually
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save mathematicians a lot of effort spent pursuing blind alleys. So,
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it was still a solution; its a <I>math</I> thing.
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<P>
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The issue lay in <I>proving</I> that mathematics was <I>complete</I>,
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<I>consistent</I>, and <I>decidable</I>. At the same gathering, the
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young mathematician Kurt Godel dealt a serious blow to this line of
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queries, by showing that math must be <I>incomplete</I> because, as he
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showed, there are assertions that can be stated that can be neither
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proved nor disproved. An assertion, encoded in the form of
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mathematics, that said, in effect, "this statement is unprovable"
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showed this disturbing (if you are into that sort of thing)
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property. An attempt to prove it true or untrue leads to
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contradiction. At least in the form of the question phrased by
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Hilbert, Godel had proved that arithmetic was incomplete. There are
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nuances to this, of course, but it was still damaging. Godel also
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showed that mathematics could <I>not</I> be proven consistent
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<I>and</I> complete. However, he was not able to shake loose an
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answer to Hilbert's question as to the <I>decidability</I> of
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arithmetic.
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<P>
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Alan's professor Hardy, for one, was happy that Godel couldn't topple
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Hilbert's final question. In his view, a mechanical process that
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could perform a solution to all mathematical problems would put every
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serious mathematician out of a job. Everything would have been done.
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<P>
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It was time for the student to instruct the teacher, at least in part.
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After a day of running, an activity that Alan found to nicely clear
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the mind, he stumbled onto the idea of a machine of simple, though
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improbable, design that could tackle any sort of problem put to it.
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The powerful machine would only understand the digits <code>0</code>
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and <code>1</code>; the first binary computer. It would move a
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read/write mechanism across an infinite tape of these numbers and,
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based on their particular arrangement, solve various types of
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problems. Alan's breakthrough was that he had defined, in specific
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language, what a <I>general algorithm</I> actually was. The Turing
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Machine, as his construct would be called, was a thought experiment
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that helped codify the features of algorithms. During his exploration
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of the wonderful ideas that this <I>machine</I> inspired, Turing found
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that, despite the simple, general, nature of his algorithm, there did
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exist problems that it could not solve. This discovery <I>proved</I>
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Hilbert's assertions were incorrect, the answer to Hilbert's final
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question, the <I>Entscheidungsproblem</I> was "no, mathematics is not
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decidable."
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<P>
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The young mathematician from King's College, Cambridge had bested one
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of the greatest mathematicians of his time at the age of 23. He
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gained a fair measure of acclaim for his achievement and the word
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"genius" began to be tossed around again. Had he done only this, he
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would be remembered in some history books and higher math students
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would get acquainted with him at some point. At any rate, a small
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amount of historical immortality, as obscure as it may be, would be
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granted in his memory. However, it was what he did next that changed
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the course of human history.
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<P>
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Next month, we will explore the workings of a Turing Machine and
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follow Alan into the war effort. We will see how a single man's true
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genius can turn the tide of war, and we will shake our heads in
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disbelief at a hero's humiliation and eventual death. Stay tuned.
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<P>------
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<P>
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<I>© 2001 G. James Jones is a Microcomputer Network Analyst for a
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mid-sized public university in the midwest. He writes on topics
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ranging from Open Source Software to privacy to the history of
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technology and its social ramifications. This article originally
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appeared at System Toolbox (<a
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href="http://www.systemtoolbox.com">http://www.systemtoolbox.com</a>).
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Please email <a href="mailto:jones@systemtoolbox.com">me</a> and let
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me know where it is being used. This article is dedicated to the
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memory of Dr. Clinton Fuelling. Verbatim copying and redistribution
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of this entire article is permitted in any medium if this notice is
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preserved.</font>
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<!-- *** BEGIN bio *** -->
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<!-- *** END bio *** -->
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<!-- *** BEGIN copyright *** -->
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<P> <hr> <!-- P -->
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<H5 ALIGN=center>
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Copyright © 2002, G James Jones.<BR>
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Copying license <A HREF="../copying.html">http://www.linuxgazette.com/copying.html</A><BR>
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Published in Issue 75 of <i>Linux Gazette</i>, February 2002</H5>
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<!-- *** END copyright *** -->
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