06.10.2006           [Previous Postings]

Computers & Mathematics – A Project for a slightly alternative History 1930-1960

1 But very few human inventions have had an overall impact on society as has had and still has the digital computer. The computer may thus very well be called a secular phaenomenon, a society-changing if not defining device that only arises every some hundred years. It is therefore only logical that many books and articles have been devoted to the historiography of the computer, and has by now reached a quantity scarcely to be overlooked by a single person.

2.1 Histories about and around the computer can be had in many varieties. Of course, a large part of the literature consists of personal recollections, and biographies of some of the main actors (e.g. Turing, Mauchly/Eckert) have been written. Books and/or collections written by historians and trying to give a picture of the global development, or a part of it, however, tend to fall in two categories, that again diversify in two classes. (Remark: In the following we will focus on books on the origins of the computer, neglecting everything happening after, say, 1960)

2.2 A first category mainly focuses upon the descriptive, fact-based side of the origin and development of computers, anecdotes only occurring marginally. Since these descriptions involve a lot of technical and specialised knowledge, this category of books either concentrate on the hardware side (1), or on the software side. (2)

2.3 The other category is somewhat different, narrating the advance of the digital computer but in the light of broader, cultural perspective. Not seldomly, these authors connect the history of the modern computer to older forms of computing or processing. Indeed, even before 1940, mechanic or electric machines were used to handle specific data. Babbage’s Difference Engine, devised for correct table making, L. Comrie’s re-wiring of accounting machines for mathematical computations, and of course calculating machines may serve as illustrations for these proto-computers. Hence, the history of computers has been linked to table-making (3), information processing (4), and in particular, number systems, abaci and the like. (5)

2.4 These works already show the links existing between history of mathematics and history of the computer, since, e.g., the history of number systems has always been part of the introductory chapters in a history of mathematics. Another link, however, also exists. The books of 2.3 may described as putting the computer into a larger, cultural perspective, but taking it from the hardware side, starting from concrete devices. Looking at it from the software side though, is also possible. The theoretical framework in which computer languages are studied owes much to logic, in particular to the theory of recursion, combination, Turing machines &c. (6) Some books then try to integrate the history of the computer into the history of logic. (7)

2.5 Concludingly, the computer has many histories, and an integral history will, I guess, never be written, neither should it be written, because the partial histories answer partial questions, try to untangle specific contingencies.

3.1 The studies on the computer have also been important in reviving some often forgotten chapters in the history of mathematics, and more generally spoken, the invention of the computer has had and is having a not unimportant impact on modern historiography of mathematics. Of course, as for every other discipline, the computer changes the processing methods and tools available and is rapidly creating new ones. But, the history of the computer being linked up with parts of the history of mathematics has been an eye-opener, e.g., for the history of mathematical tables, and, far more importantly, mathematics itself is changing due to the computer. Other disciplines have changed and are changing as well, but because mathematics was present at the birth of the computer, it has a longer history along these lines. Thus, one might hope to be able to view some things already now, the distance and time elapsed being favourable to the historian.

3.2 This text thus is a kind of research proposal, a plan for a much longer text I will hopefully write not all too distant in the future. The outcome would be yet another history of the computer, but also a chapter in the history of mathematics, mainly unwritten yet, but complementing much of the already existing stories.

4.1 The starting point of this to be written history would be the journal Mathematical Tables and other Aids to Computation (1943-1959), later on continued and renamed by Mathematics of Computation (1960-...). This journal was founded by R.C. Archibald and D.H. Lehmer, a most interesting combination. Archibald was a historian of mathematics with a famous and large library, Lehmer was a number theorist, though not of the ”transcendental” kind, but of the kind interested in numbers and mechanical devices to compute these numbers. Armed with their near encyclopedic knowledge of mathematics, tables and machines, the consecutive issues of the journal nearly seamlessly flow one into the other, though in 1946 the ENIAC as the first of its kind appears on its pages. Whereas the computer is often seen as a discontinuity, it here appears as a logical consequence, a part in a continuous history.

4.2 Another aspect of this journal is also appealing: it contains much information about the first computations (not programmes!) done on computers. As such, Mathematical Tables and other Aids to Computation (known as MTAC) discusses and invents much of the themes that indeed, at some later time, belong to the Mathematics of Computation. Comparing, e.g., the issues from 1950-1960 of MTAC with those of the Journal of the ACM, one sees that in the JACM the more general issues of programming, compiling &c. tend to be studied, whereas MTAC more or less disregards these issues and concentrates on specific algorithms – the dividing line being nowhere all too clear, one must admit. One might compare it to Volumes I and II of Knuth’s Art of Programming, the JACM corresponds to Vol. I, Fundamental Algorithms, MTAC to Vol. II, Semi-Numerical Algorithms (the grey zone being Vol. III Searching and Sorting), both making up much of the knowledge that goes into knowing your computer and programming it.

4.3 A last aspect: Leafing through the consecutive issues, one gets a feel of how mathematics is changing, i.e., one sees not a new subdiscipline getting born, but one sees an old discipline getting a new drive. If one wants to start writing about how the face of mathematics changed through the computer, MTAC might very well serve as a diary of its partial transformations. The irony of the story is, by the way, that this transformation comes as no big surprise, Archibald, Lehmer and the contributors being too aware of their history, and too much with both feet in the ground to be really surprised, but mostly eager to use this new device.

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(1) Some examples of books focussing on the hardware, the architecture of computers: J. Howlett, N. Metropolis and G.-C. Rota, eds. (1980): A History of Computing in the Twentieth Century. New York: Academia Press. M. R. Williams, ed. (1997): A History of Computing Technology. Los Alamitos: IEEE Computer Society Press. R. Rojas and U. Hashagen (2002): The First Computers – History and Architectures. Cambridge (MA): MIT Press.
(2)The software side includes of course programming languages and operating systems, though most of these only were developed after 1955-60. Some earlier references: Donald Knuth (1962): "A history of writing compilers", Computers and Automation, December 1962, 8-18. Saul Rosen, ed. (1967): Programming Systems and Languages. New York: McGraw-Hill Book Company. Jean E. Sammet (1969): Programming Languages: History and Fundamentals. Eaglewood Cliffs, NJ: Prentice Hall. D.E. Knuth and L.T. Pardo (1977): ”The early development of programming languages”, in: J. Belzer, A. G. Holzman and A. Kent, eds. (1979): Encyclopedia of Computer Science and Technology. New York: Marcel Dekker, 419–96.
(3) E.g., M. Campbell-Kelly and M. Croarken and R. Flood and E. R., eds. (2003): The History of Mathematical Tables: From Sumer to Spreadsheets. Oxford: OUP.
(4) E.g., Campbell-Kelly, Martin and William Aspray (1996): Computer: A History of the Information Machine. New York: Basic Books.
(5) E.g., William Aspray (1990): Computing Before Computers. Ames: Iowa State University Press (available on the web as pdf's). Arno Borst (1993): The Ordering of Time: From the Ancient Computus to the Modern Computer. Chicago, IL: University of Chicago Press. Georges Ifrah (2002): The Universal History of Computing: From the Abacus to the Quantum Computer. New York: Wiley.
(6) An overview of the complex entanglement leading to programming theory, M. Mahoney (2002): ”The Structures of Computation,” in: Rojas/Hashagen (2002). See also my own paper that treats of information theory and generative grammars.
(7) Sybille Krämer (1988): Symbolische Maschinen. Die Idee der Formalisierung in geschichtlichem Abriß. Darmstadt, Wissenschaftliche Buchgesellschaft. Quite similar, Martin Davis (2001): Engines of Logic: Mathematicians and the Origin of the Computer. New York, Norton.