In his book on how mathematicians think, which I have discussed here, Hadamard devotes a chapter to inquiring how mathematicians choose the topic to work on. He starts with Claparède’s view that there are two kinds of invention: one is given a goal, and looks for a means to reach it; the other discovers a fact and then imagines what it might be useful for.

He is of the opinion that many scientists, especially mathematicians, use the second method. Some of his examples are very relevant to the vexed question of impact. Here are a few.

About 2400 years ago, the Greek mathematicians investigated the properties of ellipses; “they did not think and could not think of any possible use” for their discoveries. But, if they had not done so, Kepler could not have formulated the laws of planetary motion, and Newton could not have formulated the law of universal gravitation.

Balloons, in the early days, tended to be filled with light but flammable gas such as hydrogen. Now we have light non-flammable gas. “This progress has been possible for two reasons: in the first place, because one has succeeded in knowing which substances exist in the atmosphere of the Sun and which do not; secondly, because research was started, by Lord Rayleigh and Ramsay among others, to determine the density of nitrogen” to four significant figures rather than three. (An unknown element was identified in the Sun’s atmosphere by spectroscopy, and was named helium. Meanwhile, chemists doing accurate measurements found a discrepancy between the density of nitrogen produced by chemical reaction and nitrogen obtained by removing the other known constituents of air; they discovered by careful analysis the inert gases, the lightest of which was identified with the element found in the Sun’s atmosphere.)

Elie Cartan’s work on group theory in 1913 was applied fifteen years later to the theory of the electron.

Bernoulli invented “calculus of variations”, but Volterra and others did the job more wholeheartedly, producing a calculus in which functions took the place of numbers and “functionals” operated on them. This seemed an aesthetic but impractial construction, until it turned out to play a fundamental role in quantum mechanics.

One of Hadamard’s own results (on determinants) was later fundamental in Fredholm’s work.

He also mentions in passing the descovery of imaginary numbers, which he attributes to Cardan, the Renaissance doctor, mathematician, and generally extraordinary character who gave the world the solution to the cubic.