🪾 If a tree falls in the forest, & no one is around to hear it: does it make a sound? 🧮 If a mathematical problem is solved with certainty, & no one gains a deeper understanding: was it truly solved? For fifty years, mathematics was focused on one thing: proving basic arithmetic. Then in 1930, at age 24, in a discussion session at a conference Gödel announced the First Incompleteness Theorem almost trolling every mathematician that what they were after was itself unprovable. Sitting in the audience was John von Neumann, who independently derived the Second Incompleteness Theorem within weeks. When he wrote Gödel about it, Gödel responded he had already derived it himself & published that same week. That mutual derivation & exchange is what set up Alan Turing just a few years later to think creatively about "Turing Machines" & what they could or could not decide. In the spring of 1935 von Neumann was lecturing at Cambridge & Turing took his class where von Neumann later said he "knew Mr. Turing very well". Reading the history of how these seminal minds influenced each other is why I could not agree more with the 27 Fields medalists who agreed that open problems are "landmarks and lighthouses," because we have seen time & time again that merely solving the foundational mathematics problem is only the beginning. It is the shared understanding that empowered Turing's breakthrough. To remove that human aspect of the outcome is indeed a form of strip-mining as Tao suggests. This piece from two applied mathematicians shows many more examples of the value derived from cross disciplinary shared understanding between pure & applied mathematicians. It is worth reading it instead of summarizing it nytimes.com/2026/09/22/scien…