Church’s answer to ‘what is computable?’ — the same year as Turing’s machine, but with NO machine at all. Just functions: a way to write one (λ), a way to apply one, and a single rule — substitute the argument. Astonishingly, numbers, arithmetic, and loops can all be BUILT from pure functions. A number n is just ‘apply f, n times’. Slide the numeral and watch it count.
The lambda calculus (Church, 1936) has three constructs: variables, abstraction λx.M (define a function), and application (M N). Its one computational rule is beta-reduction: (λx.M) N → M[x:=N] (substitute the argument). It is TURING-COMPLETE — equivalent in power to the Turing machine (the Church–Turing thesis). Data is encoded as functions: the Church numeral n = λf.λx. fⁿ(x) applies f to x exactly n times; successor, addition, and recursion follow. It is the root of every functional language. A fail-loud self-check throws unless the numeral 3 applies f three times and add(1,2)=3. ◆ real theory of computation, node-verified.
Encodings (Church numerals, booleans) are one of several; the beta-reduction rule and Turing-completeness are exact. The ‘count the applications’ view is the standard reading, not a simplification.