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Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, wa ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
LambdaCalculus and Functional Programming tions.
"... The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ..."
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The lambdacalculus is a formalism for representing funcBy the second half of the nineteenth century, the concept of function as used in mathematics had reached the point at which the standard notation had become ambiguous. For example, consider the operator P defined on real functions as follows: ⎧f(x) – f(0) for x 0 P[f(x)] = ⎨ x ⎩f ′(0) for x = 0 What is P[f(x + 1)]? To see that this is ambiguous, let f(x) = x 2. Then if g(x) = f(x + 1), P[g(x)] = P[x 2 + 2x + 1] = x + 2. But if h(x) = P[f(x)] = x, then h(x + 1) = x + 1 P[g(x)]. This ambiguity has actually led to an error in the published literature; see the discussion in (Curry and Feys