Abstract
The work of Comrie and others demonstrated the benefits that could be obtained from even a partial automation of the processes of scientific computation. Human input was still required in order to control operations and to ensure that steps in a calculation were performed in the right order and with the right data, but increasingly all that was required was the ability to perform routine labour which involved little skill or initiative. During the 1930s, the extent to which even this residual human agency could be replaced by machines began to be investigated more systematically, both in theory and in practice. This chapter describes the various accounts of mechanical computation, or effective computability, constructed by mathematical logicians in the mid-1930s, and the associated account of formal languages, which made concrete the idea of a notation that could be processed mechanically, and so by extension could be read and interpreted by actual machines.
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Notes
- 1.
This ‘confluence of ideas’ was analyzed by Robin Gandy (1988), who did not, however, go on to consider contemporary technological innovations.
- 2.
- 3.
Boole (1854).
- 4.
Frege (1879).
- 5.
Whitehead and Russell (1910).
- 6.
Gödel (1931).
- 7.
Hilbert and Ackermann (1928).
- 8.
See Barwise and Etchemendy (1987).
- 9.
Gödel (1931), p. 147, emphasis in original.
- 10.
Gödel (1931), p. 147.
- 11.
Skolem (1923), p. 333, emphasis in original.
- 12.
Hilbert (1926), p. 388.
- 13.
Gödel (1931), p. 159.
- 14.
Ackermann (1928).
- 15.
See Cutland (1980), for example, for further discussion of the function ψ.
- 16.
Gödel (1934).
- 17.
Kleene (1936a).
- 18.
Schönfinkel (1924).
- 19.
Curry (1929).
- 20.
Church (1932).
- 21.
Rosser (1984), p. 345.
- 22.
- 23.
Church (1936), p. 349.
- 24.
Rosser (1984).
- 25.
Church (1936), footnote to p. 346.
- 26.
- 27.
- 28.
Jon Agar (2003) has described how similar ‘mechanical’ processes had been introduced in non-numerical areas, particularly in the British Civil Service, and speculates that awareness of this was an additional factor leading to Turing’s mechanical definition of computability.
- 29.
Post (1936), p. 103, emphasis in original.
- 30.
Post (1936), p. 104, emphasis in original.
- 31.
Post (1936), p. 105.
- 32.
See Chaitin (2001), p. 16, for example.
- 33.
- 34.
Turing (1936), p. 232.
- 35.
Turing (1936), p. 233.
- 36.
Turing (1936), p. 234.
- 37.
Turing (1936), p. 233.
- 38.
Turing (1936), p. 234.
- 39.
Turing (1936), p. 236.
- 40.
Turing (1936), p. 236.
- 41.
- 42.
- 43.
Gödel (1931), p. 159, footnote.
- 44.
Turing (1936), p. 237.
- 45.
Turing (1936), p. 237.
- 46.
Turing (1936), p. 237.
- 47.
Turing (1936), p. 238.
- 48.
Turing (1936), pp. 236, 239.
- 49.
Turing (1936), p. 253.
- 50.
Turing (1936), p. 241.
- 51.
Turing (1946), p. 21.
- 52.
Church (1932), p. 352.
- 53.
- 54.
- 55.
Tarski (1933), p. 167.
- 56.
Carnap (1937), p. 4, emphases in original.
- 57.
Carnap (1937), p. 53.
- 58.
Tarski (1933), p. 172.
- 59.
- 60.
Carnap (1937), p. 4.
- 61.
Tarski (1933), p. 166, italics in original.
- 62.
Carnap (1939), p. 6.
- 63.
Tarski (1933), pp. 165–166.
- 64.
Tarski (1933), p. 193.
- 65.
Morris (1938), p. 3.
- 66.
Morris (1938), pp. 6, 13.
- 67.
Carnap (1939), p. 4.
- 68.
Pickering (1995).
- 69.
Turing (1936), p. 233.
- 70.
Turing (1936), p. 243.
- 71.
Morris (1938), p. 16.
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Priestley, M. (2011). Logic, Computability and Formal Systems. In: A Science of Operations. History of Computing. Springer, London. https://doi.org/10.1007/978-1-84882-555-0_4
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