488 | NOTES TO PAGES 56–59
- A. W. Burks, H. H. Goldstine, and J. von Neumann, ‘Preliminary discussion of the logical design of an
electronic computing instrument’, Institute for Advanced Study (28 June 1946), in Collected Works of
John von Neumann, Vol. 5 (ed. A. H. Taub), Pergamon Press (1961), Section 3.1, p. 37. - Notoriously, Turing’s ‘Proposed electronic calculator’ (Turing (1945)) did not mention the universal
machine of 1936, leading some commentators to wonder whether it was a direct ancestor of ACE at
all. However, some fragments of an early draft of Turing’s proposal (Turing (1945a)) cast light on this
issue: they were published for the first time in Copeland et al. (2005). There Turing specifically related
ACE to the universal Turing machine, explaining why the memory arrangement described in Turing
(1936) could not ‘be taken over as it stood to give a practical form of machine’. - Turing (1947), pp. 378 and 383.
CHAPTER 7 HIlBERT AND HIS fAmOUS PROBlEm (COPElAND)
- This chapter includes material from my article ‘What did Turing establish about the limits of comput-
ers and the nature of mathematics’, Big Questions Online, John Templeton Foundation (February
2013) (https://www.bigquestionsonline.com/). I am grateful to the John Templeton Foundation for
permission to publish the material here. - For a more detailed account of this pioneering time, see B. J. Copeland, C. Posy, and O. Shagrir, ‘The
1930s revolution’, in Copeland et al. (2013). - K. Gödel, ‘Uber formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’
[On formally undecidable propositions of Principia Mathematica and related systems], Monatshefte
für Mathematik und Physik, 38 (1931), 173–98; an English translation is in M. Davis (ed.), The
Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable
Functions, Raven (1965). To be exact, in 1931 Gödel proved that the system of arithmetic set out
by Bertrand Russell and Alfred Whitehead in their seminal Principia Mathematica is (if consistent)
incomplete, in the sense that there are true statements of arithmetic that are not provable in the sys-
tem. Using Turing’s discoveries, Gödel was later able to generalize this result considerably. The details
are explained in The Essential Turing, pp. 47–8. - Nowadays this prima facie unpromising escape route has its advocates: see C. Mortensen, Inconsistent
Mathematics, Kluwer (1995), G. Priest, In Contradiction: a Study of the Transconsistent, Oxford
University Press (2006); and R. Sylvan and B. J. Copeland, ‘Computability is logic-relative’, in G. Priest
and D. Hyde (eds), Sociative Logics and their Applications, Ashgate (2000). - M. H. A. Newman, ‘Alan Mathison Turing, 1912–1954’, Biographical Memoirs of Fellows of the Royal
Society, 1 (November 1955), 253–263, p. 256. I say ‘seemed to leave open one of the most fundamental
problems’ because Gödel’s proof of Theorem X of his 1931 paper does entail the unsolvability of the
Entscheidungsproblem (given the Church–Turing thesis: see Chapter 41), but this was not noticed
until 1965, by Martin Davis. It is probably fortunate that Gödel did not settle the decision problem in
1931, for if he had, then the young logician Alan Turing would not have taken it up, and the history
of the computer might have unfolded quite differently—and perhaps much less satisfactorily, in the
absence of Turing’s fundamental concept of universality. - D. Hilbert and W. Ackermann, Grundzüge der Theoretischen Logik [Principles of Mathematical Logic],
Springer (1928). - D. Hilbert, ‘Mathematical problems: lecture delivered before the International Congress of
Mathematicians at Paris in 1900’, Bulletin of the American Mathematical Society, 8 (1902), 437–79. - Turing (1936), p. 84.
- Hilbert and Ackermann (Note 6).
- In Gödel’s notes for his 1939 introductory logic course at the University of Notre Dame, published as
P. Cassou-Nogues, ‘Gödel’s Introduction to Logic in 1939’, History and Philosophy of Logic, 30 (2009),
69–90, on p. 85. - Gödel in Cassou-Nogues (Note 10), p. 85.
- See B. J. Copeland, ‘Turing’s thesis’, in A. Olszewski, J. Wolenski, and R. Janusz (eds), Church’s Thesis
after 70 Years, Ontos Verlag (2006), and B. J. Copeland and O. Shagrir, ‘Turing versus Gödel on com-
putability and the mind’, in Copeland et al. (2013).