In case representing pairs as procedures wasn't mind-boggling enough, consider that, in a language that can manipulate procedures, we can get by without numbers (at least insofar as nonnegative integers are concerned) by implementing 0 and the operation of adding 1 as
(define zero (lambda (f) (lambda (x) x))) (define (add-1 n) (lambda (f) (lambda (x) (f ((n f) x)))))
This representation is known as Church numerals, after its inventor, Alonzo Church, the logician who invented the λ calculus.
Define one and two directly (not in terms of zero and add-1). (Hint: Use substitution to evaluate (add-1 zero)). Give a direct definition of the addition procedure + (not in terms of repeated application of add-1).
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