How mathematics invented new realities

Although Euclid’s ideas about geometry in 300 BC were rooted in physical reality, the field became ever more abstract throughout the twentieth century.
In her new book, historian Alma Steingart reveals how this push for abstraction was mirrored by, and often triggered, parallel trends in economics, sociology, psychology and political science. (Axiomatics: Mathematical Thought and High ModernismAlma Steingart, Univ. Chicago Press 2022)

“‘Mathematizing’ a problem did not mean to measure and compute, but to reveal a hidden skeleton of conceptual relationships: to formulate the underlying idea in abstract mathematical language,” writes reviewer Davide Castelvecchi.

Euclid’s Elements provided the basis for mathematics.
Credit: Tim Ockenden/PA/Alamy

Taken to excess

Ironically, even as some mathematicians argued that abstract thinking was the key to applying maths to other disciplines — and suggested that even the most abstract maths was worthy of public funding — most mathematicians in academia seemed singularly uninterested in getting involved.
For decades, much of the progress in applied mathematics ended up taking place not in universities, but in think tanks and industry laboratories, or in newly established departments devoted to fields such as computer science or statistics.

Eventually, an excess of abstraction caught up with mathematicians

Meanwhile, Eilenberg and Steenrod’s approach to pedagogy came to be seen as a cautionary tale, with their techniques, although still widely used, affectionately being called ‘abstract nonsense’.

Sometimes I’ve believed as many as six impossible things before breakfast”.

White Queen in “Through the Looking-Glass” (1871)

But in another of the twists of fate often seen in the history of maths and physics, some physicists now consider abstract-nonsense techniques a promising approach to devising a quantum theory of gravity — perhaps delivering another route to the very real, if abstract, goal of a theory of everything.

 “On each decision, the mathematical analysis only got me to the point where my intuition had to take over.”

Robert Jensen

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