Mathematics is often based on intuition, on the deep feeling of what should be true. However, intuition can be at a dead end. Another example of this is a recent refutation of one of the hypotheses in the theory of probabilities, known as a hypothesis. This result forced to revise approaches to such tasks and raised questions about how evidence in mathematics works.
The hypothesis proposed in the 1980s is associated with graphs, which are a set of points (peaks) connected by lines (ribs). If you build a copy of such a graph above the original and connect them with vertical ribs, the structure will resemble a bunk bed. The hypothesis claimed that the probability of finding the path between two points at the lower level is always greater or is equal to the probability of a path with a transition to the upper level.
The idea looked logical. It was intuitive that the additional transitions between levels complicate the search for the path. The hypothesis also had applications in physics, in particular, to study the properties of liquids in porous materials. However, it was not possible to prove it. Skeptics indicated that the statement is too wide to be universal.
A team of three mathematicians was able to to find a counter example, which refutes the hypothesis. To do this, they used a combination of computational methods and theoretical approaches. At the initial stages, a computer search for graphs was carried out that could break the hypothesis. However, the scale of the task quickly exceeded the possibilities of analysis. Even using machine learning, proof remained unattainable due to the probabilistic nature of the approach, which could not guarantee complete confidence.
The breakthrough occurred when a Cambridge mathematician, who worked with hypergraphs (expanded versions of graphs), found a counter example for a more general formulation of the task. This inspired researchers to reconsider the approach. They adapted the methods of hypergraphs to classic graphs and built a complex structure containing thousands of peaks and ribs. Their proof with complete confidence showed that the probability of a path at the upper level could be greater than at the lower one, which refuted the hypothesis.
This result emphasizes the importance of a critical attitude to mathematical assumptions. Although the hypothesis has been considered almost obvious for decades, its refutation recalls that intuition can mislead. The discussion of the role of computational methods in mathematics remains important. Although in this case they did without the final use of machine learning, the development of such technologies raises questions about the future of evidence based on probabil