W. Brian Arthur on 'Algorithms and the Shift in Modern Science'

“Mathematics is pure language—the language of science,” the mathematician Alfred Adler wrote in the New Yorker in 1972. But over centuries, the language of science may change, says SFI External Professor W. Brian Arthur, and it’s undergoing a shift now.

Throughout the Middle Ages, geometry was the language for investigating the universe. The lexicon shifted in the 17^th century as prominent scientists began to embrace algebra. (Not all of them did: “Thomas Hobbes said algebra was ‘a scab of symbols as if a hen had been scraping there ’,” says Arthur, a complexity scientist, and economist.) As of the late 20^th century, scientists largely continued to combine equations in algebra and calculus with mathematical reasoning to support new theories and ideas.

Today’s shift moves away from equations and toward algorithmic thinking, Arthur argues in an essay published last year by the Beijer Institute of Ecological Economics, part of the Royal Swedish Academy of Sciences. Arthur says he first began noticing signs of the transition when he joined SFI in 1987 and encountered physicists, mathematicians, and computer scientists who built their ideas on algorithms and computational thinking, rather than equation-based reasoning.

Traffic provides a telling example of why this shift is useful and illuminating. Equations can describe the motion of individual cars, but for a real-world scenario, the equations quickly become unwieldy and cumbersome. An algorithm, on the other hand, can better describe the dynamics of the system. With a tool like agent-based modeling, in which every car is represented as an independent element, researchers can let the system run in time and observe emergent phenomena like traffic jams.

“Standard equations lend themselves to static pictures or simple dynamics, where the objects change in predictable ways,” Arthur says. If you know the values of the variables in a differential equation, you know what happens next. “But algorithms are very good at showing where novel things come from, and how they change according to their current pattern they create.”

Algorithmic thinking suggests a way to explore the deep context of systems that unfold dynamically and to better understand complexity in its many scales and forms. Darwin’s theory of evolution essentially describes an algorithmic system for how novel species emerge, Arthur says. Technology, too, evolves according to a kind of evolutionary algorithm.

Algorithms are tools that can help scientists study previously unnoticed phenomena in nature, Arthur says. Thinking algorithmically gives researchers a way to study ideas like unpredictability and emergence; it offers a world view based on action and change, rather than stasis.

“With algorithms,” Arthur says, “something novel might happen, and suddenly the whole story can change.”