Exploring the mathematical universe–connections, contradictions, and kale

Science and maths skills are widely celebrated as keys to economic and technological progress, but abstract mathematics may seem bafflingly far from industrial optimisation or medical imaging. So, to extend geometry to surfaces that aren’t flat, an open-minded mathematician might propose a new definition of a triangle: pick three points and connect each pair by the shortest path between them. The parallel postulate problem Around 300 BC, the Greek mathematician Euclid wrote a treatise on planar geometry called The Elements. This is exactly what you’ll measure in every flat triangle, but later mathematicians debated whether the parallel postulate should be a foundational principle or just a consequence of the other fundamental assumptions. Salad Days People sometimes debate whether mathematics is discovered or invented, but both points of view feel real to those of us who study mathematics for a living.