NATS 1700 6.0 COMPUTERS,  INFORMATION  AND  SOCIETY Lecture 9: From Hollerith to Turing | Previous | Next | Syllabus | Selected References | Home | | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 |   Introduction The Harvard 'Computers'. "The first mass data crunchers were people, not machines." In an article that appeared in Nature (vol 455, p37, 4 September 2008) Sue Nelson looks at the discoveries and legacy of the remarkable women of Harvard’s Observatory. Go and read the article in the Library. As we approach the twentieth century and the advent of the modern computer, we need to make another detour and examine a mathematical thread that eventually led to the foundations of computer science. For this and the next lecture a good reference is  A Very Brief History of Computer Science written by Jeffrey Shallit at University of Waterloo. Before doing so, however, I want to mention an important figure in our history of computing,  Herman Hollerith,  who in 1884 invented the modern punched card system, which received its first real-world test in the 1890 US census. The system included punches (similar to a typewriters), counting devices, and readers. The test was so successful--saving not only a substantial amount of dollars to the US Census Bureau, but allowing the processing of the census data to be completed in a fraction of the time taken in the previous occasions--that Hollerith's system was adopted in the 1891 census in Canada and other countries. Hollerith founded the Tabulating Company which, following mergers with other computing companies, eventually became IBM.   Topics Our detour starts with  David Hilbert (1862-1943),  one of the greatest mathematicians of all times, who in 1900 delivered a famous speech, The Problems of Mathematics, at the Second International Congress of Mathematicians in Paris. In this speech Hilbert posed many questions--23, to be precise. In particular he asked: Is mathematics complete?  In plain words, can every mathematical statement be either proved or disproved? Is mathematics consistent?  Is it true that statements such as 0 = 1 cannot be proved by valid methods? Is mathematics decidable?  Is there a mechanical method that can be applied to any mathematical assertion so that (at least in principle) we can know whether that assertion is true or not? This last question became known as the Entscheidungsproblem (the decision problem). To appreciate the context of this speech and of these questions, consider Hilbert's words: "History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of today sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. "The deep significance of certain problems for the advance of mathematical science in general and the important role which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, it is alive. A lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems... "It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the gain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: 'A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.'" Although Hilbert didn't have computers in mind, the three questions above are clearly crucial, at least in principle, to any attempt to automate mathematics and mathematical reasoning. In 1931,  Kurt Gödel  (1906-1978) answered two of Hilbert's questions. He showed that every 'sufficiently powerful formal system' is either inconsistent or incomplete. Furthermore, if an axiom system is consistent, this consistency cannot be proved within the system itself. This was a remarkable, also because unexpected, result. Examples of inconsistent and incomplete, sufficiently powerful formal system are arithmetic and Euclidean geometry. In 1936,  Alan Turing  (1912-1954), and independently Alonzo Church  (1903-1995), provided a solution to Hilbert's Entscheidungsproblem, the third problem, by constructing a formal model of a computer, now known as the Turing Machine, and showing that there were problems such a machine can not solve. The Turing Machine Essentially, a Turing machine can be imagined as a device that can read, (scan) a tape of potentially infinite length, divided into squares, each carrying one symbol. These symbols could be 1's and 0's, or anything else. The machine can read one symbol at a time, erase and replace it if necessary, and move in either direction along the tape, always one square at a time. The machine can be in any one of a number of internal states, and it can change from one state to another, depending both on the symbol read off the tape and on certain rules stored in the machine. If you think of the overall effect of this process, a Turing machine's input is a series (a string) of symbols and its output is another series or string of symbols. As an aside, a physical implementation of the Turing machine has been carried out recently by Ehud Shapiro of the Computer Science and Applied Mathematics Department at the Weizmann Institute of Science (Israel). "Shapiro's mechanical computer has been built to resemble the biomolecular machines of the living cell, such as ribosomes. Ultimately, this computer may serve as a model in constructing a programmable computer of subcellular size, that may be able to operate in the human body and interact with the body's biochemical environment, thus having far-reaching biological and pharmaceutical applications." [ from ScienceDaily ] Going back now to the Church-Turing result, imagine that the symbols on the tape represent a suitable coding of the basic axioms of, say, arithmetic, and that the internal states and rules of the machine represent the rules for logically manipulating such axioms. After the Turing machine has 'processed' the tape, what do we have, really, on the tape? Well, arithmetical theorems are precisely what a logical manipulation of the axioms of arithmetic is all about. Therefore what Church and Turing had proved means that there are arithmetical theorems that the Turing machine can not prove (nor disprove). This is a sobering thought: computers--at least as envisaged by Church and Turing--do have limitations. Turing went farther. He was able to prove that, under certain fairly general assumptions, certain Turing machines are universal, in the sense that they can simulate any other Turing machine. Modern computers are actual implementations of universal Turing machines.. Of course these were theoretical results. The physical construction of a computer was another story. Alan Turing became in fact involved in the construction of the Automatic Computing Engine (ACE) at the National Physical Laboratory, London. Visit  The Turing Archive for the History of Computing.   Questions and Exercises A good site dedicated to the life and work of Turing is  The Alan Turing Home Page.  Another brief, but excellent account of Turing's work is an essay by John M Kowalik. You may also want to download and play with  Visual Turing,  a tool for simulating Turing machines on your computer. Especially if you need to relax, read  As We May Think : a seminal article (1945) by Vannevar Bush about the ways he imagined the new emerging information technology would affect society. Bush was no science fictions writer. He was a reputed electrical engineer who invented the differential analyzer, a large analog computer, one of the last calculating devices to precede the digital computer. In 1996, The Brown/MIT Vannevar Bush Symposium was held to celebrate "50 Years After 'As We May Think'." You may want to read an extended abstract of the event.   Picture Credit: The University of the Virgin Islands Last Modification Date: 04 September 2008