With Category Theory, Mathematics Escapes From Equality | Mathematics - Quanta Magazine

Two monumental works have led many mathematicians to avoid the equal sign. Their goal: Rebuild the foundations of the discipline upon the looser relationship of “equivalence.” The process has not always gone smoothly.

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The equal sign is the bedrock of mathematics. It seems to make an entirely fundamental and uncontroversial statement: These things are exactly the same.

Photo: Ana Porta for Quanta Magazine

But there is a growing community of mathematicians who regard the equal sign as math’s original error. They see it as a veneer that hides important complexities in the way quantities are related — complexities that could unlock solutions to an enormous number of problems. They want to reformulate mathematics in the looser language of equivalence.

“We came up with this notion of equality,” said Jonathan Campbell of Duke University. “It should have been equivalence all along.”

Jacob Lurie, a mathematician at the Institute for Advanced Study,
was awarded the $3 million Breakthrough Prize in Mathematics in 2014.
John D. & Catherine T. MacArthur Foundation

The most prominent figure in this community is Jacob Lurie. In July, Lurie, 41, left his tenured post at Harvard University for a faculty position at the Institute for Advanced Study in Princeton, New Jersey, home to many of the most revered mathematicians in the world... 

Lurie published his first book, Higher Topos Theory, in 2009. The 944-page volume serves as a manual for how to interpret established areas of mathematics in the new language of “infinity categories.” In the years since, Lurie’s ideas have moved into an increasingly wide range of mathematical disciplines. Many mathematicians view them as indispensable to the future of the field. “No one goes back once they’ve learned infinity categories,” said John Francis of Northwestern University... 

Lurie’s work was hard to swallow in other ways. The volume of material meant that mathematicians would need to invest years reading his books. That’s an almost impossible requirement for busy mathematicians in midcareer, and it’s a highly risky one for graduate students who have only a few years to produce results that will get them a job.

Lurie’s work was also highly abstract, even in comparison with the highly abstract nature of everything else in advanced mathematics. As a matter of taste, it just wasn’t for everyone. “Many people did view Lurie’s work as abstract nonsense, and many people absolutely loved it and took to it,” Campbell said. “Then there were responses in between, including just full-on not understanding it at all.”

Emily Riehl, a mathematician at Johns Hopkins University,
is helping to lead the development of higher category theory.
Photo: Will Kirk/Johns Hopkins University

Scientific communities absorb new ideas all the time, but usually slowly, and with a sense of everyone moving forward together. When big new ideas arise, they present challenges for the intellectual machinery of the community. “A lot of stuff got introduced at once, so it’s kind of like a boa constrictor trying to ingest a cow,” Campbell said. “There’s this huge mass that’s flowing through the community.”
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Source: Quanta Magazine