Alan Turing: The Enigma

Part One: The Logical

1. Esprit de Corps (to 13 February 1930)

• Alan was slow to learn that indistinct line that separated initiative from disobedience and resisted the duties of childhood.
• He had identified the crucial point, that Einstein doubted the axioms. Not for Alan the 'obvious duties', for nothing was obvious to him.

2. The Spirit of Truth (to 14 April 1936)

• For a person with a mathematical mind, an ability to deal with very abstract relations and symbols as thought with tangible everyday objects...
• He had thought deeply about Einstein and had broken the rules to do so.
• For science, to Alan Turing, was thinking for himself, and seeing for himself, and not a collection of facts. Science was doubting the axioms. He had the pure mathematical approach to the subject, allowing a free rein to thought, and seeing afterwards whether or not it had application to the physical world.
• The distinction between science and mathematics had only been clarified in the late 19th C. Until then it might be supposed that mathematics necessarily represented the relations of numbers and quantities appearing in the physical world, although this point of view had really been doomed as soon as such concepts as the 'negative numbers" were developed. The 19th C however, had seen developments in many branches of mathematics towards an abstract point of view. Mathematical symbols became less and less obliged to correspond directly with physical entities.
• One remarkable thing about Gödel's special assertion was that since it was not provable, it was, in a sense, true. But to say it was true required an observer who could, as it were, look at the system from outside. It could not be shown by working within the axiomatic system.

3. New Men (to 3 September 1939)

• Curiously, the failure of the Hilbert programme had also meant the end of the point of view advanced by Wittgenstein in his first phase, in the Tractatus Logico-Philosophicus, that every well-posed problem could be solved.
• ... overcoming the line drawn between mathematics and engineering, the logical and the physical.
• It was not science, not 'applied mathematics', but a sort of applied logic, something that had no name.

4. The Relay Race (to 10 November 1942)

• Indeed, discerning a pattern in the apparently patternless was essentially the work of the cryptanalyst, as of the scientist.
• The English mathematician Thomas Bayes had seen how to formalise the concept of "inverse probability" - this being the technical term for the likely cause of an effect, rather than the probable effect of a cause.
• English "eccentricity" served as a safety valve for those who doubted the general rules of society.
• He was fascinated that people could be taking part in something clever, in a quite mindless way.

Part Two: The Physical

5. Running Up (to 2 September 1945)

• He wanted "to build a brain".
• In his view, the physics and chemistry were relevant only in as much as they sustained the medium for the embodiment of discrete "states", "reading", and "writing". Only the logical pattern of these "states" could really matter. The claim was that whatever a brain did, it did by virtue of its structure as a logical system, and not because it was inside a person's head, or because it was a spongy tissue made up of a particular kind of biological cell formation.
• The Turing model did not seek to explain one kind of phenomenon, that of mind, in terms of another. It did not expect to "reduce" psychology to anything. The thesis was that "mind" or psychology could propertly be described in terms of Turing machines because they both lay on the same level of description of the world, that of descrete logical systems. It was not a reduction, but an attempt at transference, when he imagined embodying such systems in an artificial "brain".
• He had rejected the idea of a "we" behind the brain that somehow "did" this signaling and organizing of the memory. The signalling and the organization had to be all that there was.
• The logical transformation of symbols was what mattered, not physical power .
• His "weight of evidence" theory had shown how to transfer certain kinds of human recognition, judgment and decision into an "instruction note" form. His chess-playing methods did the same thing - as did the games on the Colossi - and posed the question as to where a line could be drawn between the "intelligent" and the "mechanical". His view, expressed in terms of the imitation principle, was that there was no such line, and neither did he ever draw a sharp distinction between the "states of mind" approach and the "instruction note" approach to the problem of reconciling the appearances of freedom and of determinism.
• "We do not need to have an infinity of different machines doing different jobs. A single one will suffice. The engineering problem of producing various machines for various jobs is replaced by the office work of "programming" the universal machine to do these jobs."
• He had learned how to build a brain - not an electric brain, as he might possibly have imagined before the war - but an electronic brain.
• An English homosexual atheist mathematician had conceived of the computer (ie, an automatic electronic digital computer with internal program storage).
• Analogue machines measured physical quantities, while digital machines were for organizing symbols. For every increase in accuracy a new analogue machine is needed, while digital machines just need to be reprogrammed.
• After working on a Difference Engine, to mechanize the particular numerical method used in the construction of mathematical tables, Babbage had conceived (by 1937) of an Analytical Engine, whose essential property was that of mechanising any mathematical operation.
• It was the mechanisation of the organizing or logical control of the arithmetic that mattered.
• Without conditional branching, the ability to mechanize the word IF, the grandest calculator would be no more than a glorified adding machine.
• The ENIAC - the Electronic Numerical Integrator and Calculator (started April 1943).
• Every tradition of common sense and clear thinking would tend to suggest that "numbers" were entirely different in kind from instructions. The obvious thing was to keep them apart: the data in one place and the stock of instructions to operate on the date, in another place. It was obvious - but wrong.

6. Mercury Delayed (to 2 October 1948)

• "When we have decided what machine we wish to imitate, we punch a description of it on the tape of the universal machine. This description explains what the machine would do in every configuration in which it might find itself. The universal machine has only to keep looking at this description in order to find out what it should do at each stage. Thus, the complexity of the machine to be imitated is concentrated in the tape and does not appear in the universal machine proper in any way."
• His idea was that anything in the way of refinement or convenience for the user could be performed by thought and not be machinery, by instructions and not by hardware.
• To Alan Turing, the multiplier was a rather tiresome technicality; the heart lay in the logical control, which took the instructions from the memory, and put them in operation.
• He had invented the art of computer programming.
• His mind still straddled mathematics, engineering and philosophy in a way that the academic structure could not accommodate.
• "We believe then that there are large parts of the brain, chiefly in the cortex, whose function is largely indeterminate. In the infant these parts do not have much effect: the effect they have is uncoordinated. In the adult they have great and purposive effect: the form of this effect depends on the training in childhood. A large remnant of the random behavior of infancy remains in the adult. All this suggests that the cortex of the infant is an unorganized machines, which can be organized by suitable interfering training. The organizing might result in the modification of the machine into a universal machine or something like it."
• They had (in Manchester), on 21 June 1948, successfully run the first program on the first working stored program electronic digital computer in the world.
• "To a large extent I agree with you about 'thinking in analogies', but I do not think of the brain as 'searching for analogies' so much as having analogies forced upon it by its own limitations..."