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From Bulletin of the AMS

Poincaré on the value of reasoning machines

orange Bulletin cover Volume 61 Number 3 July 2024

by Colin McLarty

Abstract

Hilbert's Foundations of Geometry in 1899 made Poincaré think of "reasoning machines" before Hilbert did. Poincaré found the idea "deadly for teaching, and desiccating for researchers" but indispensable for telling when intuitions have been fully expressed. A machine will use stated axioms without the vague intuitions Poincaré considered vital to learning and research. Years of famously intuitive creativity, plus boundless faith in technology, as well as the impact of Hilbert, led Poincaré to see that machines could aid human intuition but not replace it, precisely because machines have no intuition. This relates to recent machine achievements in Lean and HoTT, and to the issues in Akshay Venkatesh's essay.

Over one hundred years ago, Henri Poincaré imagined machine proofs:

One would be able to confide [Hilbert's axioms for geometry] to a reasoning machine, such as Stanley Jevons's "logical piano," and one would see all of geometry come out. There is no need to say how deadly this would be for teaching, [and how] desiccating for researchers whose originality would soon run dry. But it is explained and justified in the hands of Monsieur Hilbert when we recall his goal.⁠1 [27]

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