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For most of us, mathematics represents the best example of what it means to think rigorously. Starting from a handful of initial definitions and axioms, all the conclusions to which their demonstrations lead us seem to be unquestionable truths, the result of the application of the ironclad logic of deductive reasoning. Or so it seemed to us for more than two thousand years, during which Euclid's book Elements became a paradigm of rational thought. But that placid certainty in which mathematicians lived has been cracking in the last two hundred years. The appearance of non-Euclidean geometries was the first crack, to which other paradoxes were added (such as Russell's famous paradox) that shook the foundations of mathematics, until Gödel dealt the final blow with his incompleteness theorem. Mathematicians suffered a tremendous shock when they discovered that perhaps some of the foundations on which they had so confidently built their discipline might fail. A "crisis of the foundations" whose echoes still resonate today, as it calls into question the project of building a machine capable of reasoning automatically, mechanically or algorithmically.