In Math, Rigor Is Vital. But Are Digitized Proofs Taking It Too Far?

Summary

The real legacy of these books, however, is not their content but their style. Bourbaki prioritized abstraction over all else, eschewing concrete examples and calculations in favor of only the most general statements. They presented each proof as a series of logical deductions, usually without reference to any underlying intuition or rationale. One of Bourbaki’s textbooks. “It’s a style that is very formal,” said Leo Corry, a historian at Tel Aviv University. “Very austere.” Bourbaki’s philosophy quickly spread, influencing mathematics almost everywhere. “It had an enormous influence,” said Patrick Massot of Paris-Saclay University. “The most successful parts have become so much part of the common mathematical knowledge and style that it’s hard to think of what it was before.” The subject became far more airtight, if increasingly abstract and difficult. “This was not a brilliant decision from a pedagogical point of view,” wrote Maurice Mashaal, the longtime editor of Pour la Science and the author of a book on Bourbaki. But the group’s emphasis on abstraction molded research mathematics in ways that are harder to assess. Some mathematicians revere Bourbaki’s approach. Massot argues that through abstraction and elegance, “you are forced to understand what really makes things work, and what is just noise.” Formalization, in this view, has brought clarity. But Bourbaki’s project had unforeseen consequences as well. Their vocabulary and style weren’t a natural fit for every type of mathematics. For example, the fields of combinatorics (often described as the science of counting) and graph theory (the science of networks) tend to be highly concrete and visual. Perhaps, then, it’s no surprise that for decades, they were sidelined at most prestigious institutions in the United States and parts of Europe; they were only able to thrive in places where Bourbaki’s influence was limited, like Hungary. “Graph theory [was] the slum of topology,” said Béla Bollobás of the Universi...