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A computational no-coincidence principle — Alignment Research Center

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In a recent paper in Annals of Mathematics and Philosophy, Fields medalist Timothy Gowers asks why mathematicians sometimes believe that unproved statements are likely to be true. For example, it is unknown whether 𝜋 π is a normal number (which, roughly speaking, means that every digit appears in 𝜋 π with equal frequency), yet this is widely believed. Gowers proposes that there is no sign of any reason for 𝜋 π to be non-normal -- especially not one that would fail to reveal itself in the first million digits -- and in the absence of any such reason, any deviation from normality would be an outrageous coincidence. Thus, the likely normality of 𝜋 π is inferred from the following general principle: No-coincidence principle (Gowers): If an apparently outrageous coincidence happens in mathematics, then there is a reason for it. In other words: suppose that the digit 3 accounts for not 10% but 11% of digits in the decimal expansion of 𝜋 π . Intuitively, there ought to be an explana

In a recent paper in Annals of Mathematics and Philosophy, Fields medalist Timothy Gowers asks why mathematicians sometimes believe that unproved statements are likely to be true. For example, it is unknown whether \(\pi\) is a normal number (which, roughly speaking, means that every digit appears in \(\pi\) with equal frequency), yet this is widely believed. Gowers proposes that there is no sign of any reason for \(\pi\) to be non-normal -- especially not one that would fail to reveal itself in the first million digits -- and in the absence of any such reason, any deviation from normality wou

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