Algebra and Computing

From the end of the 2nd World War in 1945 the world was set for an exponential growth in the use of computers. At this time, Leslie Fox (1918-1992) [9] moved from the Admiralty Computing Service to the National Physical Laboratory (NPL) joining a section including Alan Turing [7] and led by E. T. Goodwin. He was interested in numerical linear algebra and whilst at the NPL he started a line of investigation into using Gaussian elimination to estimate the accuracy with solving linear equations. Prof. Fox went on to become the first director of the Oxford University Computing Laboratory in 1957, becoming Professor of Numerical Analysis in 1963. He stayed there until his retirement in 1983.

J. R. Womersley was the Superintendent of the Mathematics Division at the NPL. In 1946 he noted in a report on the proposed ACE computer that:

... this device is not a calculating machine in the ordinary sense of the word. One does not need to limit its functions to arithmetic. It is just as much at home in algebra ...

In 1951, Christopher Strachey, then a teacher at Harrow School, made contact with Mike Woodger at the NPL via a mutual friend. He started to write a draughts program for the Pilot ACE, and soon progressed to the machine being developed at Manchester University. He obtained a copy of the Programmer's Handbook by Alan Turing and wrote a long letter to Turing on his plans:

... It would be a great convenience to say the least if the notation chosen were intelligible as mathematics when printed by the output ... once the suitable notation is decided, all that would be necessary would be to type more or less ordinary mathematics and a special routine called, say, `Programme' would convert this into the necessary instructions to make the machine carry out the operations indicated. This may sound rather Utopian, but I think it, or something like it, should be possible ...

These ideas were very much in line with those of Turing, who ensured that Strachey was offered an attractive job to make use of his programming skills which tempted him away from teaching. Later Strachey went on to set up and lead the Programming Research Group as part of the Computing Laboratory in Oxford until his untimely death in 1975.

Programs specifically designed to read, manipulate and output mathematical algebraic formulae in a manner similar to that envisaged by Strachey are now readily available. Perhaps the market leader is Mathematica, developed by Wolfram Research Inc., founded by Stephen Wolfram (who attended the Dragon School followed by Eton, and was briefly a student in Oxford during the 1960s and 70s). This tool has been enthusiastically used by researchers such as Dana Scott - previously at Oxford and now at Carnegie-Mellon University in the US - to help speed up mathematical investigations in both teaching and research. As well as symbolic representations of algebraic formulae, graphical output is also possible. The speed and capacity of personal computers and workstations has now enabled such programs to be accessible to many people even though the implementation consists of over 1 million lines of C code.