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| Alan Turing |
Alternate History
Alan Turing was born in 1912 and showed an early interest in the natural world. He studied mathematics at Cambridge University and established himself as a mathematician with a unique perspective and original approach to certain fundamental issues in mathematics.
By the age of 24, Turing had devised the idea of a "Turing Machine", a simple logical device capable of computing all mathematical formulae and numbers that can be computed by a finite algorithm. Turing had also contributed to the solution of the famous Entscheidungsproblem (decision problem), showing that no computational system (such as a Turing Machine) can decide as being either true or false every possible theorem of arithmetic.
Soon after recognizing the computational power of a Turing Machine, Turing became interested in the pragmatic task of learning how to use electronic circuits to build computing devices. As an inititial example and test case for mechanical computation, Turing selected one of the most famous computational problems of mathematics, the approximation of the values of the zeros of the Reimann zeta function.
During World War II, Turing applied his knowledge of mathematics and computing devices to British efforts in military code breaking. In 1946, Turing returned to Cambridge and joined an interdisciplinary program to explore the mechanization of mathematical computations, theorem discovery and proof, and the relationship between formal systems and human thought processes. This program became the influential School of Cambridge Computing.
Introduction
Alan Turing was one of the pioneers of discovering how to mechanize thought. He
lived most of his life in England, but did spend a few years in the
United States of America. In the alternative history
Cambridge Computing, Turing stays
longer
at Princeton but has the chance to start collaborating with
Ludwig Wittgenstein during World War II.
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| Wittgenstein |
Wittgenstein was a relentless critic of conventional thinking in
mathematics and psychology. Before World War II he was Professor of
Philosophy at Cambridge and
lectured
on the foundations of mathematics. Like Turing, Wittgenstein was
concerned with the algorithms by which human thoughts are produced and
communicated.
In this alternative history, Wittgenstein and Turing develop a
constructive relationship and begin a collaboration on algorithms for
machine learning. After the war, they establish an interdisciplinary program in
Mind, Machine and Mathematics at Cambridge and also a private company,
AD Computing (Analog/Digital Computing).
The personal relationship that develops between Turing and
Wittgenstein is critical for their mental and physical health, allowing
them both to have long and productive lives. When they become sources of
support and strength for each other, they are able to begin
participation in the growing movement for revision of British laws
related to homosexuality. By the year 1963 they help Great Britain
decriminalize homosexuality.
Wittgenstein dies in 1968, but Turing continues their work into
the age of integrated circuits and increasing computational capacity. AD
Computing markets its first commercial autonomous learning robot in
1975. Turing dies in 1989, the same year that a model AD-417 becomes
world chess champion. Turing's 1950 prediction that the dawn of the new
millennium will be witnessed by thinking machines is realized at the
turn of the century.
Contents
Moritz Schlick, 1936 .... Kurt Gödel .... meets Turing, Summer 1937 coversations with Wittgenstein deflects to
zeta function for Turing's Ph.D. project.
Princeton The building of Alan Turing's first computer (1938-1940).
1939 lectures on philosophy of mathematical logic by Ludwig Wittgenstin.
Summer 1937, Cambridge England.
Wittgenstein and
Turing have made plans to go see a flick (
Secret Agent).
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| The calculus of contradiction |
Cambridge Computing
Turing arrives in the hallway outside Wittgenstein's rooms just
as the door opens and a flock of philosophy students depart from their
session with Wittgenstein. Turing stands aside and listens to the very
loud voice of Wittgenstein from inside, "You cannot doubt just this and
that. It is the number of things that you
cannot doubt that you can count on one hand."
The voice of the student is hard to hear, something like, "Well, I doubt that."
Wittgenstein laughs and looks out the door, sees Turing. "Here's
Turing! He's a mathematician, he believes an infinite number of
things....he cannot even write a proof that does not include infinity."
Turing nods to the student and does an little dance as they both
try to go through the door at the same time. Wittgenstein is striding
around the room pushing furniture back into place and standing folding chairs in
corners. Turing knows from experience that Wittgenstein will be in no
mood to talk until he quiets the bubbling thoughts that arose during the
session with the students. Finally, all the furniture has been pushed
multiple times and Turing realizes that Wittgenstein is looking for
something. Turing picks up a piece of paper from Wittgenstein's writing
table. On it is a sketch.
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| duckrabbit |
Wittgenstein notices what Turing is holding and he asks, "What is the most wonderful thing for a mathematician?"
Turing replies, "Finding a simple solution to a problem that has been thought to be difficult."
"And how can anything first appear difficult and then be found to be easy?"
"You just find the right way to look at."
Turing lets the image of the duckrabbit flip between being a duck
and a rabbit in his mind a few times then puts the paper on the desk.
Wittgenstein is still looking around the room. "If searching your room
fails to reveal your coat, what's another way of looking at THAT
problem?"
"Turing, good man, I was just waiting for you to confirm the
EXISTENCE of my coat. Now we can attack the problem of how to find it."
Wittgenstein still can do nothing but look around the room.
Turing goes to the door and holds it open for Wittgenstein. "It's
almost worth testing you to discover if you would stand there an
infinite amount of time waiting for the coat to rematerialize, but I saw
that your coat is downstairs in the common room when I came in."
They leave Wittgenstein's rooms and collect his coat then head
out into the warm evening. Wittgenstein asks, "So have you agreed to
dump your ridiculous Ph.D. project?"
A week before, Turing had described his meeting at Princeton with
Gödel and Gödel's ideas for how to (or not) formalize the process of
finding an infinite set of axioms that would be an infinite basis for
mathematics. Wittgenstein had been outraged and had sputtered, "A
complete waste of effort, Turing! An infinite set of axioms is
oxymoronic. Let Gödel find some other lackey to do his dirty work. If
you want to waste your time you should waste it on something that you
want to do."
Turing had decided to formulate a proposal for a new Ph.D. topic.
"What I need is a mathematics topic that can be solved by a computing
machine."
Wittgenstein guffawed. "Applied mathematics at Princeton? Is that allowed?"
They were walking along
River Cam towards the
Botanic Garden
and would end up at the newly opened Regal theater for the flick. "The
Ph.D. project would be demonstrating an improved approximation method
for the
zeta function. Of course, my real goal is to do the actual calculations by machine."
They left the shore of the Cam and turned up towards the Gardens.
Wittgenstein asked, "Is there really a different approach that you
would take to the zeta function if you were going to have one of your
machines do the calculations?"
Turing replied. "I would be happy to use an existing formula for
zeta, but producing electronic circuits to perform the calculations and
hold all of the calculated values during the calculation is a serious
technical problem. An available trick is to use a
Fourier transform
and to put the problem into a form in which approximations could be
made by analog computing components....sets of interconnected wheels.
This would greatly simplify the amount of electronic components to be
constructed."
They walked through the Garden in silence until Wittgenstein
said, "I'm surprised that nobody has ever solved the Fourier analysis of
the zeta function."
Turing shrugged, "It is not widely known that you can explore the
distribution of prime numbers in terms of harmonic frequencies. The
zeros of the zeta function constrain those frequencies."
Wittgenstein turned to Turing and put his fists on his hips.
"Turing, you are becoming a scientist! Mathematicians cannot prove the
Riemann hypothesis
so you have to build a piece of equipment and collect experimental
data? Do the first 100 zeta zeros fit your prediction? Do the first
1000? No matter how long your calculator runs, you will never know ALL
of the zeros. Its not even science. Its something between science and
mathematics."
They started hurrying out of the Garden so as not to be late for
the flick. Wittgenstein said, "What I want to know is if your computing
machines will ever be able to perform a mental transformation. If you
made a machine that could recognize a duckrabbit as a duck, would it
ever be able to also see it as a rabbit? What changes in your mind when
the image remains the same but your perception shifts? If your machines
cannot see both aspects, they will never do mathematics, they will just
be calculating slaves. What is the formal system that defines
creativity?"
Turing had been asking himself such questions with increasing
frequency. He could only dimly conceptualize a 50 year process in the
future during which computing machines would be built and tested,
revealing the details of their mechanics and testing if they could
replicate all of human thought.
They approached the theater and Wittgenstein elbowed Turing, "Creativity must be a secret agent. A secret agent in the brain."
Princeton
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| Princeton University, 1938 |
When Alan Turing is nearing completion of his
Ph.D. dissertation in 1938 he is offered a job by
John von Neumann.
If Turing returns to England, he can continue on his research
fellowship at Cambridge. While trying to decide if he should accept
von Neumann's offer, Turing's research adviser (
Alonzo Church) shows Turing a copy of
Claude E. Shannon's journal article describing the key results from his Masters degree thesis. By this time, Turing has begun working in
Princeton's
physics department workshop on key electronic components for a
prototype computer circuit. Shannon's work on how to do Boolean algebra
using electronic circuits is clearly an example of how to put into
hardware the logical subsystems of a Universal
Turing Machine.
Turing arranges for Shannon to visit Princeton and give a lecture
on his thesis work. Turing shows Shannon the partially completed
electronic computer components that he has been building and explains
the concept of a Universal Turing Machine. Shannon decides to do his
Ph.D. research in Princeton working with Turing. Shannon first completes
the digital calculating circuit that Turing has already started to
construct then builds the
special purpose computer to calculate the
Riemann zeta-function,
a calculating device that Turing has long dreamed of constructing. In
order to continue working with Shannon on this computer, Turing takes
the job offer from von Neumann, delaying his return to England until
after England enters the
war.
The ability of Shannon and Turing to quickly produce a
functioning electronic computer depended on support from Shannon's
masters thesis adviser
Vannevar Bush
and Turing's new boss, von Neumann. It was only after the defeat of
Nazi Germany when Turing learned the fact that Shannon already had a
U.S. government security clearance as Bush's student. While at
Princeton, the electronic components Shannon explained as coming from
"the electronic shop at
M.I.T."
were actually from U.S. corporations working under government contract
on military electronics research. For most of the electronic components
used by Shannon and Turing, a request from Shannon to the office of
either Bush or von Neumann would result in delivery of the items within a
week, often by military carrier. The physical nature of the silicon
carbide circuit components was a military top secret. It should be noted
that Turing was able to recognize the silicon carbide components
because of his familiarity with European crystal radio technology. John
von Neumann arranged that the physics department workshop was also
quickly provided with six new research assistants who worked under
Shannon assembling and testing the analog circuits of Turing's zeta
function machine.
______________________________________________
The Shannon-Turing ZZ-1 special purpose computer that was built by Shannon and
Turing in 1938 used electric circuits to generate the harmonics
required in the Turing formula for the zeta function.
The diagram to the right shows the basic analog oscillator element of the Shannon-Turing ZZ-1. 3600 of these oscillators were constructed at Princeton from April to November 1938.
_____________________________________________
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| Cambridge England, 1939 |
In 1939 at
University of Cambridge
Wittgenstein lectures on the nature of the foundations of mathematics
and logic. Wittgenstein's work on the philosophy of mathematics was of
interest to several mathematically trained students who decide to attend
Wittgenstein's lectures. In particular, Donald C. MacPhail establishes a
relationship with Wittgenstein, becoming the "star pupil" in
Wittgenstein's 1939 lectures.
MacPhail, stimulated by Wittgenstein's iconoclastic views of
logic, devises a "calculus of contradiction". This new branch of
mathematical logic is basically a method for working with a large system
of theorems, some of which are contradictory. MacPhail and Wittgenstein
realize that MacPhail's methods for dealing with mathematical
contradiction are applicable to the semantic networks that form inside
human brains when children learn. MacPhail applies
Markov chains
in the "calculus of contradiction" and publishes a paper on the subject
with Wittgenstein: "Population Dynamics of Logical Atom Fields
Containing Contradictions". This article attracts the attention of
Turing, eventually leading to constructive interactions between Turing
and Wittgenstein during World War II. Ultimately, this leads to profound
changes in the development of machine learning and autonomous robots
with significant progress towards human-like artificial intelligence.
Neurodiversity
I never pushed
Cambridge Computing past the events described above for the year 1939. Still, it is fun to imagine how the history of our world would have been different if people like Turing and Wittgenstein had been free to be themselves, if they had not been tormented by forced confinement in a culture where being different and thinking unusual thoughts was treated as a crime.
Related reading:
Alan Turing's pardon.
Next:
Dr. Jeppson and Thomas Iwedon
