Dec 24, 2013

Cambridge Computing

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.

 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.

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.

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).
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.

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 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. _____________________________________________

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.

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.

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