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Mechanizing Calculation: Thinking Machines The Most Radical and Most Revolutionary of All Current Telecommunication Technologies Is the Computer. The commonly known history of the computer selectively downplays the lateness of its development and the comparative slowness of its diffusion. The conceptualization of the computer was slow to arrive because it had to work against Western philosophy established in the 17th century by Descartes, that intelligence and thinking can only be a human characteristic. Therefore the very concept of a thinking machine was unthinkable. Yet it was this same school of philosophers that were intent on enshrining mathematics as the ‘queen’ of all sciences and the clearest evidence available of the glory of God’s creation. Alan Turing studied at King’s College, Cambridge under the mathematician Max Newman. Following a brilliant undergraduate career, Turing was elected fellow, at the age of 22 in 1935. In 1936 he published a paper On Computable Numbers, that dealt elegantly with the Cartesian obstruction. The agenda Turing addressed was at the heart of advanced pure mathematics. By the late 19th century, in the wake of the creation of non-Euclidean geometries (among other developments), mathematics were becoming, for the first time since the Greeks, increasingly concerned about the consistency of the axiomatic systems they used, that is the system of given or absolute truth: If you all recall your high school math classes and having to learn the algebraic axioms, you’ll recall that mathematical systems are defined by a set of axioms, from which theorems are deduced. A mathematical theory might be defined as the set of all propositions that are true under the given set of axioms (theorems). For example, a theory of addition would contain propositions like "1 + 2 = 3", "2 + 5 = 3 + 4", and axioms like "a + b = b + a", "a + 0 = a", etc. As historians recognize today: The creation of non-Euclidean geometry had forced the realization that mathematics is man-made and describes only approximately what happens in the world. The description is remarkably successful, but it is not the truth in the sense of representing the inherent structure of the universe and therefore not necessarily consistent…Every axiom system contains undefined terms whose properties are specified only by the axioms. The meaning of these terms is not fixed, even though intuitively we have numbers or points or lines in mind. It was against this background that Bertrand Russell coined his famous epigram: Pure mathematics is the subject in which we do not know what we are talking about or whether what we are saying is true. The sort of people that slowed computing were led by David Hilbert, the greatest mathematician of his generation, who in the early decades of this century insisted on the primacy of axiomatic method. Well into the 1920’s, Hilbert continued to assert: “that every mathematical problem can be solved. We are all convinced of that…” To back this up, there existed the liar paradox that can be classically expressed in the sentence, “This sentence is false”. Anytime a mathematical problem arose that could not be solved, the liar paradox could be asserted. In 1931, Kurt Godel attacked this approach to mathematics in a paper titled On Formally Undecidable Propositions of Prinicpia Mathematica and other systems. In this Godel demonstrated that it was impossible to give proof of the consistency of a mathematical system by demonstrating a fundamental limitation in the power of the axiomatic method since the existing set of axioms is simply incomplete. Therefore there are true arithmetical statements that can not be derived from an axiom. Godel’s incompleteness theorem highlighted a number of other problems, primarily the question of decidability. If there were now mathematical assertions that could be neither proved nor disproved, how can one determine effective procedures? Godel muted Hilbert’s declaration that mathematical systems had to be consistent and complete and his insistence upon the discovery of effective procedure as a necessary part of mathematics. It was Turing, five years later, who dealt with this problem. Turing had been struck by a phrase in a lecture of Newman’s where Hilbert’s suggestion that any mathematical problem must be solvable by a fixed and definitive process was glossed by Newman as “a purely mechanical process”. Turing, in his paper, found a problem that could not be so decided or in Turing’s language computed. It involved an artificial construct known as the Cantor diagonal argument whereby ‘irrational numbers’ could be created. (Cantor was one of the 19th century mathematicians whose work set the stage for the crisis in axiomatic methods.) To dispose of the decidability problem, Turing constructed a mental picture of a machine and demonstrated that it could not compute certain numbers. Therefore there were mathematical problems which were not decidable; but Turing wrote “It is possible to invent a machine which can be used to compute any computable sequence.” Because of this metaphor of a machine, “On Computable Numbers” had, beyond its immediate significance in pure maths, broader implications. Turing’s proof involved imagining a machine which read, wrote, scanned and remembered binary numbers inscribed on a tape. It might not be able to compute the irrational numbers of Cantor’s trick but it could, in theory, deal with a vast range of other computations. Turing had conceived of a tremendously powerful tool as he christened it a universal engine. Of course, he had no intention of building such a machine. When he wrote “computer” he meant, as did all his contemporaries, a person who performs computations as in the following quote from Turing: The behavior of the computer at any moment is determined by the symbols which he is observing, and his ‘state of mind’ at that moment. We may suppose that there is a bound B to the number of symbols or squares which the computer can observe at any one moment. If he wishes to observe more, he must use successive observations…Let us imagine operations performed by the computer to be split up into ‘simple operations’ which are so elementary that it is not easy to imagine them further divided. Every such operation consists of some change in the physical system consisting of the computer and his tape. The human computer and his tape were to become the machine computer and its program. Of course, with mathematicians all over the world attempting to solve decidability or rather as it is popularly referred to by mathematicians Entscheidungsproblem, it was inevitable that Turing would have competitors. In mid-April 1936 he presented his paper to Newman in Cambridge. On April 15th, Alonzo Church of Princeton sent away his demonstration of a different unsolvable proposition for publication. In his Appendix, Turing had to acknowledge Church’s similar conclusions about the Entscheidungsproblem. In October 1936, Emil Post, a mathematician at the City University of New York (now NYU), submitted a paper to Church suggesting a mechanical device -- a ‘worker’ -- for demonstrating Church’s proposition along the lines of Turing’s universal machine. Post acknowledges the power of Turing’s approach by coining the phrase Turing machine. These men stand in a line of mathematical logicians traceable back to the self-taught 19th century English mathematician, George Boole. Boole is by many considered the man who discovered pure mathematics by showing that an exact agreement between two classes of operations, Calculus and Algebra, exists. He did so by reducing certain types of thought to a series of on/off states or 0s and 1s. This is a binary system of notation that dates back in its modern mathematical form to Bacon. Boolean algebra is the means by which a Turning machine can be said to think, make judgements and learn. These men are part of the ideation process that prepare the ground of scientific competence which could be transformed by technology into the computer. In the 1930s, there was much going on in mathematics which would help to translate activities popularly considered as uniquely human into forms that would be ‘machine readable’. In 1938 Claude Shannon, whom Turing was to meet during a wartime visit to the United States, published his MIT master’s thesis, A Symbolic Analysis of Relay and Switching Circuits in which the insights of Boolean algebra were applied to telephone exchange circuit analysis. This produced a mathematization of information which not only had immediate practical applications for his future employers, Bell Labs, but also established another part of modern computer science’s foundation. Information Theory as Shannon’s work is called, defines information as the informational content of signals abstracted from all specific human information. It concerns not the question, “what sort of information?” but rather “how much information?” In a telephone exchange, design requirements dictate that there be less concern about the content of messages than with the accuracy with which the system will relay them. Information becomes reified, quantifiable so that it can be treated as a measure of the probability or uncertainty of a symbol or set of symbols. By how much does the transmitter’s message reduce uncertainty in the receiver? By that much can the informational content of the message be measured and capacity of the channel of communication be determined. A binary code made up of 0 and 1 or off and on, is made up of binary digits. A set of binary digits that carry a message we will call a bit, the more complicated the message is the more bits it will require. Each wave, each transmission of a bit, reduces our uncertainty. The “bound B tot he number of symbols or squares which the computer can observe at any moment,” of which Turing wrote can be expressed as the capacity of the computer, human or mechanical to address a discrete number of bits. The quantification of information in Information Theory parallels and perhaps determines the reification of information which is so crucial a part of the “Information Revolution”; that is to say the rhetorical thrust which has the production of information in so- called “post-industrial societies” substituting for the production of physical goods depends upon such reification. It allows people to be comfortable with the somewhat curious notion what we can survive by making information instead of producing things. The implication of all this work in the 1930s at the outer edges of advanced mathematics was not immediately apparent even to the mathematical community. Pure mathematical logic was so pure that few human activities could be considered. But once they understood Turing, many became rich. The importance of Turing’s On Computable Numbers was that he moved the computer from number-cruncher to symbol manipulator. Turing threw the first plank across the Cartesian chasm between human being and machine. John von Neumann, a mentor of Turing’s, was one of the fathers of the American computer and a mathematician of enormous range -- from game theory to nuclear devices -- who dominated the field in the first decade after the war. Von Neumann wrote, in First Draft of a Report on the EDVAC, the document that contains the original master-plan for the modern computer: “Every digital computing device contains certain relay-like elements, with discrete equilibria… It is worth mentioning, that the neurons of the higher animals are definitely elements in the above sense. They have all-or-none character, that is two states: Quiescent [or inactive] and excited”. The seductiveness of the analogy between human neural activity and digital symbol manipulators has proved irresistible. Drawing such parallels is not new. It has been a characteristic of Western thought throughout the modern period, beginning with Lamettrie’s L’Homme Machine in 1750. Seeing humanity in the image of which ever machine most dominates contemporary life is what might be called mechanemorphism. With Lamettrie it was the clock. The combustion engine followed. Freud thought electromagnets were a good metaphor for the brain. Today this tendency finds its most extreme expression with the computer, especially among the proponents of ‘strong’ artificial intelligence. Mechanemorphism has conditioned not only our overall attitude to computers but also the very terminology which has arisen around them. For example what crucially distinguishes the computer from the calculator that precedes it is its capacity to store data and instructions -- ‘memory’. And the codes that control its overall operation ‘language’ facilitates contemporary mechanemorphism. There was also an element in the ground of scientific competence which has to do with the basic architecture of what a Turing machine might look like. When Turing thought to call his metaphor a universal engine he was honoring Charles Babbage. By 1833, the English mathematician Charles Babbage was abandoning work on a complex calculator which had occupied him for the previous decade. Instead he now envisaged a device which could tackle, mechanically, any mathematical problem. This universal analytic engine was never built, but the design embodied a series of concepts which were to be realized a century or more later. The machine was to be programmed by the use of two sets of punched cards, one set containing instructions and the other the data to be processed. The input data and nay intermediate results were to be held in a ‘store’ and the actual computation was to be achieved in a part of conditional branching operations, basic logical steps, by hopping backwards and forwards between the mill and the store. It was to print out its results automatically. This is so close to the modern computer - the mill as the CPU, the operational cards as the ROM and the store as the RAM, etc. that Babbage has been hailed as its father. However the possibility that the analytic engine could alter its program during its computations alluded Babbage, his thought remained one crucial step away from the computer proper. Babbage’s proposed device was a calculator of a most advanced type, a number cruncher not a symbol manipulator. Babbage’s interest in automatic calculation sprang from the common root -- boredom. Like Leibniz, Pascal and Napier before him and Mauchly, Zuse and Aitken, three major computer pioneers a century after him, Babbage disliked the computational aspect of mathematical work. In one story he reportedly muttered to a colleague, the astronomer Herschel, “I wish to God these calculations had been executed by steam.” Herschel is said to have replied: “It is quite possible.” Unfortunately Babbage died before he completed work on his advanced calculator. It was left to George Scheutz, Swedish lawyer, and newspaper publisher to build Babbage’s difference engine. Scheutz’s machine was based on an account he had read of Babbages work in 1834. When finished, the Scheutz machine, which had four differences and fourteen places of figures, punched results on to sheet lead or paper mache from which printing stereotypes could be made. The machine was operable by 1844 and refined to the point where duplicates were possible by 1855. The Scheutz engine was built in advance of any real supervening necessity. But by the 1860s and 1870s the need had arisen through the train, the modern corporation, the modern office and all the things that helped it run -- the phone, mechanical calculator (1875), modern shift-key typewriter... Commercial desktop calculators were in immediate production and the office equipment industry was born. After the first key-driven calculator was demonstrated, an elegant roll-paper printing mechanism was added. Other machines were more desk-like than desktop. The Millionaire, for example, had built-in multiplication tables and was manufactured continuously from 1899 until 1935. The new motive power, electricity, was also used. In 1876, at the very same Centennial Exposition in Philadelphia where Bell made such a stir, an engineer, George Grant, showed an electrically driven piano-size difference engine. For the 1890 American census, Herman Hollerith designed a device of even greater significance both in its use of electricity and for the fortune it made his company, eventually to become IBM. The decennial census was proving ever more difficult to complete. By 1887, the Census Office (it became a Bureau in 1920) realized that it would still be processing the 1880 data even as the 1890 returns were being collected. In a public competition held to find a solution to this problem, Hollerith proposed an electro-mechanical apparatus. He resurrected the punched cards that Babbage had intended as the input for the analytic engine. Hollerith’s cards contained the census data as a series of punched holes. The operator placed the card into a large reading device and pulled a lever. This moved a series of pins against the card. Where the pins encountered a hole, they passed through the card and into a bowl of mercury, thereby making an electrical circuit which activated the hands of a dial. In the test which won him the contract, Hollerith’s machine was about ten times faster than its rival. Six weeks after the census, the Office announced that the population stood at 62,622,250. Hollerith declared himself to be “the world’s first statistical engineer”. The census was completed in a quarter of the time its predecessor had taken and the Tabulating Machine Company was founded. Hollerith’s enterprise became part of the Computing Tabulating and Recording Company (C-T-R) which in turn became IBM in 1924. By the 1930 and into the 40s business’s requirements for automated machinery were completely satisfied. It can be argued that these requirements were so satisfied that there was no need for more advanced calculators, calculators that could alter their programs and thus be classed as computers. The possibility of building an electro- mechanical digital calculator along the general lines proposed by Babbage had been first outlined by the Spanish scientist Leonardo Torres y Quevedo in a work published in 1915: An analytic machine, such as I am alluding to here, should execute any calculation, no matter how complicated, without help from anyone. It will be given a formula and one or more specific values for the independent variables, and it should then calculate and record all the values of the corresponding functions defined explicitly or implicitly by the formula… Torres y Quevedo suggested the way to do this was with switches “a brush which moves over a line of contacts and makes contact with each of them successively”. In 1920, Torres y Quevedo built a prototype to illustrate the feasibility of his suggestions, the first machine to have a typewriter as the input/output device. It wasn’t till a few years later that Bell Labs began to think along the Torres y Quevedo line of enquiry and it wasn’t until the late 1930’s, over 15 years later that devices appeared in the metal. The Model K (for kitchen table) Bell ‘computer’ was made of old bits of telephone exchange mounted on a breadboard, one weekend in 1937, by George Stibitz. Stibitz, a staff mathematician at the Labs, was convinced he could wire a simple logic circuit to produce binary additions because the ordinary telephone relay switch was a binary - an on/off - device, and over the weekend that is exactly what he did. There was little immediate enthusiasm for the breadboard calculator but a pilot project was funded and the first complex calculator was built. The internal supervening necessity was the endless calculations necessary in the developing theory of filters and transmission lines. The complex calculator (Model 1) was finished by 1940 and did the job the Labs required. Apart from building the first binary machine, at a meeting of the American Mathematical Society, Stibitz also performed the first remote control calculation by hooking up the teletype input keyboard in a lecture hall at Dartmouth and communicating via the telephone wire with the Model 1 in New York. This is the earliest application of telegraph technology to computing. This would eventually lead to the Internet. Stibitz now looked to make more complex devices, but the cost, $20,000, kept the Lab administration from making further computers for several years. When the USA joined the Second World War in 1941, Stibitz expertise did find a proper outlet, in building a series of specialized electro-mechanical complex calculators. The Model 2 had a changeable paper-tape program and was self-checking. It was designed specifically to test anti-aircraft detectors. The Model 3, designed for anti-aircraft ballistic calculations, made, like its predecessor, for unattended operation, rang a bell in the staff sergeant’s quarters if for any reason the computation was halted. Stibitz’s experience during these years parallel that of Konrad Zuse, who was also drawn to automata because of boredom: In 1934 I was a student in civil engineering in Berlin. Berlin is a nice town and there were many opportunities for a student to spend his time in an agreeable manner, for instance with the nice girls. But instead we had to perform big and awful calculations. In 1936, Zuse began building electro-mechanical binary calculators out of relays, machines which were to occupy most of the living room of his parents’ apartment. The Z1 and Z2 were test models. He had failed to interest the German office machine industry in his project except for some partial support from one manufacturer. In 1939, work on the Z2 was halted as Zuse was inducted into the Wehrmacht. As Zuse explained to his American interrogator in 1946, the German Aerodynamics Research Institute was interested in his Z2 and so he was relieved to continue this work. By 1943, still finishing the Z3 he began building the special purpose S1 to increase missile production by speeding calculation. Zuse 3 Zuse 4 The S1 worked so well that in the later part of 1943 Zuse, supported by the Air Ministry established his own small firm. The Z3 was, like the contemporary models in America, programmable, but by holes punched in 35mm film rather than tape. The Z3 was a floating- point binary machine with a sixty-four-digit store and was the first of its general purpose class to work, by December 1941. At the war’s end Zuse was working on a Z4. As the Allies closed in on Berlin he was given a truck on which he loaded the Z4 and headed south. By the surrender he was holed up in the village of Hinterstein in the Bavarian Alps close to the Austrian border, the Z4 hidden in a cellar. Howard Aitken, the builder of the third major line of electromagnetic machines, came to computers, like the others, while slogging through calculations he needed for his thesis. By 1937 he had a proposal ready for mechanizing the process. He showed it to the military but he also took it to a calculating machine company which decided the device was impractical, against the opinion of its chief engineer. The engineer then directed Aitken, to IBM. In 1939, Aitken was contracted to build his electromagnetic machine at IBM’s Endicott Lab, the money to come from the US Navy and a million-dollar gift from Thomas Watson Sr, the then president of IBM. Aitken was given the reserve rank of Naval Commander and his staff was navy. IBM furnished the space and equipment, mainly as with Stibitz and Zuse, relay switches, but it also provided a team of engineers. Within four years the machine, the Mark 1, was working. It was transferred to Harvard as a present from IBM. Watson insisted it be clothed in gleaming, aerodynamically molded steel, an IBM version of 1940s high tech. Aitken thought it should be left naked in the interest of science. IBM called the machine the ASCC (Automatic Sequence Controlled Calculator); Aitken called it the Harvard Mark I. IBM ignored the Mark I as a prototype for a business machine yet nevertheless set about building a bigger and better one and the world got its first glimpse of a “robot brain”. And yet Stibitz, Zuse and Aitken invented nothing. Their machines were built out of readily available parts and this was to be the case with the all- electrical calculators which came next. These accepted prototypes speak directly to the lack of a real supervening necessity. Occasional tasks, like Bell Labs’ needs might produce a machine. Inspired amateurs, like Zuse, might do so too. And in certain corners of the military needs were perceived and met, but basically the world worked well enough with these various prototypes. A universal analytic engine was simply not needed. It was to take two wars, one hot and one cold, to change that perception.

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posted: | 3/10/2012 |

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