Imagine that a Greek philosopher promised to his queen that he would determine the greatest prime number. He failed. Do you think that the mathematical fact that primes are infinite was a cause of his failure? I'm asking this because I guess most philosophers think that mathematical facts have no causal effects.

You've asked an interesting question, one related to what's often called the "Benacerraf problem" in the philosophy of mathematics (see section 3.4 of this SEP entry). I'm not sure that the problem is peculiar to mathematics. Imagine that the philosopher tried to impress his queen by creating a colorless red object. Was his failure caused by the fact that colorless red objects are impossible? If facts about color and facts about redness in particular can have causal power, can the fact that colorless red objects are impossible have causal power?

Part of the problem may be that these questions assume that we have a better philosophical grasp of the concept of fact and the concept of cause than we actually do. Given our currently poor grasp of those concepts, I don't think we should be confident that mathematical explanations or mathematical knowledge must depend on the causal power of mathematical facts.

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