On Geometrical DemonstrationBlaise Pascal
About On Geometrical Demonstration
Pascal's De l'esprit géométrique (On the Geometrical Spirit) was written around 1658 but published posthumously. It is a treatise on method: what makes a demonstration rigorous, and what lies beyond the reach of demonstration altogether.
Pascal distinguishes the geometrical method (defining every term, proving every proposition from axioms) from the perfect method (defining all terms and proving all propositions absolutely). The perfect method is impossible, because every definition uses prior terms and every proof rests on prior truths. Geometry succeeds not because it achieves absolute foundations but because it begins with terms so clear that no definition could make them clearer, and with truths so evident that no proof could make them more certain.
The essay also contains Pascal's reflections on infinity and the infinitely small. The human mind, he argues, is situated between two infinities: the infinitely large and the infinitely small. It can approach neither limit but is stretched between them, and this position defines the peculiar character of mathematical thought. The work anticipates themes that will recur in the : the disproportion of man, the limits of reason, and the strange power of the mind to recognize what it cannot fully comprehend.