![]() ![]() Maybe the Four-Color Theorem or Hales’s Theorem, for which every known proof requires a massive computer enumeration of cases, belong to this class. No one has any idea how to prove any of the above statements-and yet, just on statistical grounds, it seems clear that it would require a ludicrous conspiracy to make any of them false.Ĭonversely, one could argue that there are statements for which we do have a proof, even though we lack a “convincing explanation” for the statements’ truth. π is a normal number (or even just: the digits 0-9 all occur with equal limiting frequencies in the decimal expansion of π).Īnd so on.The Collatz Conjecture (the iterative process that maps each positive integer n to n/2 if n is even, or to 3n+1 if n is odd, eventually reaches 1 regardless of which n you start at).The Twin Primes Conjecture (there are infinitely many primes p for which p+2 is also prime).You might make a more general point: there are many, many mathematical statements for which we currently lack a proof, but we do seem to have a fully convincing heuristic explanation: one that “proves the statement to physics standards of rigor.” For example: In fact, the argument above basically already did it!” Many have objected: “but despite Gödel’s Theorem, it’s still easy to explain why G(F) is true. Therefore F is either incomplete or unsound. If, on the other hand, G(F) is false, then it’s provable, which means that F isn’t arithmetically sound. If G(F) is true, then it’s an example of a true arithmetical sentence that’s unprovable in F. G(F) = “This sentence is not provable in F.” In kindergarten, we all learned Gödel’s First Incompleteness Theorem, which given a formal system F, constructs an arithmetical encoding of ![]() ![]() Here’s an observation that’s mathematically trivial but might not be widely appreciated. ![]()
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