Gandy did an analysis of Babbage's Analytical Engine (see 'Gandy - The Confluence of Ideas in 1936' quoted in 'Herken, Rolf - The Universal Turing Machine—A Half-Century Survey. Springer Verlag') - and said it did (cf. p. 52–53):
- The arithmetic functions +, −, ×, where − indicates "proper" subtraction x − y = 0 if y ≥ x.
- Any sequence of operations is an operation.
- Iteration of an operation (repeating n times an operation P).
- Conditional iteration (repeating n times an operation P conditional on the "success" of test T).
- Conditional transfer (i.e., conditional "goto").
Then he states
the functions which can be calculated by (1), (2), and (4) are precisely those which are Turing computable.
Then he states:
… the emphasis is on programming a fixed iterable sequence of arithmetical operations. The fundamental importance of conditional iteration and conditional transfer for a general theory of calculating machines is not recognized…
Gandy p. 55
I'm assessing the scope of Gandy's claim here. (Whether it is right or wrong). He seems to be stating that although Babbage seems to have stumbled onto a notion of Turing Completeness (can express any program using (1), (2) and (4) - he didn't have a notion of a Computable Function. (Perhaps Gandy was saying that since the work of Babbage was prior to the work of Hilbert and Godel, he didn't have the mathematical tools to tie down the definition of a universal computing machine.)
My question is: Did Alan Turing's student Robin Gandy assert that Charles Babbage had no notion of a universal computing machine?