Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning.
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From 'Systems of Logic Based on Ordinals', published in Proceedings of the London Mathematical Society.
Date: 1938-1939
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Mathematical thinking involves two distinct mental operations: intuition—those sudden, unexplained insights that arrive without deliberate reasoning—and ingenuity, the creative application of method. Turing argues that intuition produces valid mathematical judgments spontaneously, outside conscious logical steps. This challenges the assumption that all valid thought must be mechanically traceable. Human mathematical understanding involves flashes of recognition that precede and exceed any formal proof chain, making the mind irreducible to pure step-by-step procedure.
This quote comes from Turing's 1939 Princeton PhD thesis on ordinal logic, written as he was formalizing computation itself. It reveals his lifelong preoccupation: can machines replicate human thought? By identifying intuition as something beyond mechanical procedure, Turing implicitly defined the boundary his own Turing machines could not cross. This tension directly seeded the Turing Test—his attempt to ask whether machines could produce outputs indistinguishable from minds that possess genuine intuition.
Written in 1939, this quote emerged amid a crisis in mathematical foundations. Gödel's 1931 incompleteness theorems had shattered Hilbert's dream of fully formalizing all mathematical truth, proving some truths lie beyond any formal system. Turing himself had just published his groundbreaking 1936 paper defining computable functions and the universal machine. Against this backdrop, what human mathematical intuition adds beyond any mechanical process was the defining intellectual problem of the age.
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