Math’s ‘Bunkbed Conjecture’ Has Been Debunked
Arthur T Knackerbracket has processed the following story:
Much of mathematics is driven by intuition, by a deep-rooted sense of what should be true. But sometimes instinct can lead a mathematician astray. Early evidence might not represent the bigger picture; a statement might seem obvious, only for some hidden subtlety to reveal itself.
Unexpectedly, three mathematicians have now shown that a well-known hypothesis in probability theory called the bunkbed conjecture falls into this category. The conjecture - which is about the different ways you can navigate the mathematical mazes called graphs when they're stacked on top of each other like bunk beds - seemed natural, even self-evident. Anything our brain tells us suggests the conjecture should be true," said Maria Chudnovsky, a graph theorist at Princeton University who was not involved in the new work.
[...] In the mid-1980s, a Dutch physicist named Pieter Kasteleyn wanted to mathematically prove an assertion about how liquids flow throughout porous solids. His work led him to pose the bunkbed conjecture.
[...] The bunkbed conjecture says that the probability of finding the path on the bottom bunk is always greater than or equal to the probability of finding the path that jumps to the top bunk. It doesn't matter what graph you start with, or how many vertical posts you draw between the bunks, or which starting and ending vertices you choose.
For decades, mathematicians thought this had to be true. Their intuition told them that moving around on just one bunk should be easier than moving between two - that the extra vertical jump required to get from the lower to the upper bunk should significantly limit the number of available paths.
[...] In June, Lawrence Hollom of the University of Cambridge disproved a version of the bunkbed problem in a different context. Instead of dealing with graphs, this formulation of the conjecture asked about objects called hypergraphs. In a hypergraph, an edge is no longer defined as the connection between a pair of vertices, but rather as the connection between any number of vertices.
[...] In the meantime, Pak says, it's clear that mathematicians need to engage in a more active discussion about the nature of mathematical proof. He and his colleagues ultimately didn't have to rely on controversial computational methods; they were able to disprove the conjecture with total certainty. But as computer- and AI-based lines of attack become more common in mathematics research, some mathematicians are debating whether the field's norms will eventually have to change. It's a philosophical question," Alon said. How do we view proofs that are only true with high probability?"
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