Today in Edworking News we want to talk about How Failure Has Made Mathematics Stronger May 22, 2024 Kristen Norman for Quanta Magazine
Introduction
Reading a math paper is a bit like having dinner at a nice restaurant. The entrée might taste delicious, but it doesn’t tell the full story of how it was made. Clever recipes that end up tasting funky don’t make the menu; undercooked dishes are usually not served to customers. But missteps in both conception and execution are important parts of the process. Danny Calegari, a topologist at the University of Chicago, wants mathematicians to be more transparent about what goes on in the kitchen.
A Mathematical Journey
Calegari grew up in Melbourne in a mathematical household. His father taught calculus and his mother statistics at a local technical college. Their bookshelves were filled with textbooks that he and his brother Frank would browse through for fun. Initially, he considered becoming a writer or a cognitive scientist. However, one topology class at the University of Melbourne changed his life: “It was just electrifying,” he said. This encounter ignited his passion for math, leading him to major in the subject and later earn a doctorate from the University of California, Berkeley.
Throughout his journey, Calegari has remained an avid reader of literature, a passion that helps him see the world through different lenses. However, mathematics and its complex problems, especially in low-dimensional topology and geometric group theory, keep him up at night. His fascination with topology was like getting "run over by a train," he described.
Kristen Norman for Quanta Magazine - Calegari explaining a mathematical theory.
The Importance of Failure
In a recent essay for the Notices of the American Mathematical Society, Calegari discussed the importance of failure in mathematics. He believes that disappointment is both a crisis and an opportunity. His personal experiences with failure began early, including not making the International Math Olympiad team. However, he saw these early failures as opportunities for deeper learning.
Resistant but Resilient
His major professional disappointment was with "stable commutator length." Despite proving some unexpected theorems and uncovering fundamental relationships, his work received little recognition. Few mathematicians built on it, leaving him feeling isolated. This experience taught him to care less about external validation and more about his own curiosity. For anyone in mathematics, the act of doing math itself must be the reward. Recently, Calegari has returned to this area of study to formulate more precise conjectures.
Calegari explains that failure and collaboration make the field more dynamic. For example, Bill Thurston's early failed attempts at proving the geometrization conjecture led to his deeper understanding and subsequent groundbreaking work. These experiences underscore the value of resilience and the necessity of collaboration in mathematics.
Collaborative Efforts and Teaching
Mathematicians increasingly collaborate, resulting in multi-author papers that often contain fewer errors. Collaborative projects like the Polymath project allow failed proof attempts to become part of the public record, enriching the field. By learning from these experiences, both seasoned and new mathematicians can better understand the cutting edge of research.
Teaching also plays a vital role. By presenting mathematical concepts as a series of explorations, including wrong turns, educators can more effectively communicate the complexities and intrigue of the subject. This approach aligns with Calegari's appreciation for literature, which he believes allows us to see inside other people's heads.
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Remember these 3 key ideas for your startup:
- Embrace Failure as an Opportunity:
- Just like Calegari, view failures as opportunities to delve deeper into your subject. Utilize setbacks to refine and improve your strategies rather than being discouraged by them. Consider these risk mitigation strategies for better preparation. - Value Collaboration:
- Collaboration is crucial. Multi-author projects and shared efforts can yield more accurate results and inspire innovative solutions. Look into large-scale public collaborations similar to the Polymath project to continually move forward as a community. - Focus on Intrinsic Rewards:
- Let curiosity and passion drive your work. External validation is not always guaranteed, so the intrinsic satisfaction of achieving your goals should be the primary reward.
By embracing these principles, your startup can navigate challenges more effectively and harness the power of collaboration and intrinsic motivation for sustained success.
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