Take the word "vector", for example: some picture two points A and B joined by an arrow; a physicist might think of a force acting on an object, while a mathematician will tend to see an abstract element of a set called a vector space. No one now thinks of a vortex or a radius vector—terms introduced by astronomers in the 18th century to describe the invisible force that keeps a planet moving around the Sun—still less of the concept of a vector that Hamilton extracted from the quaternions he had just invented.
Similarly, to a physicist, the word "gradient" has a very tangible physical meaning: it denotes the direction and magnitude of the variation in a quantity near a point in space. A mathematician, by contrast, sees it as the differential of a scalar function, which can be identified with a vector in a Euclidean space.
The charm of quantum mechanics
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The pairing of the words "mechanics" and "quantum" is striking: a word familiar to the general public is coupled with an adjective derived from a Greek root to name a physical theory of the very small, far removed from our intuition. In fact, the first term also comes from Greek, via Latin: mekhane meant any kind of device and, by a tortuous route, gave rise to the French word "machine".
During the Renaissance, scientists used the term "mechanics" to refer to the study of the laws of motion. Hence Lagrange's analytical mechanics and Laplace's celestial mechanics.
