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Theorem can be proved. But massive accretions of evidence support it so strongly that to deny it the status of ‘fact’ seems ridiculous to all but pedants. The same is true of evolution. Evolution is a fact in the same sense as it is a fact that Paris is in the Northern Hemisphere. Though logic-choppers rule the town,* some theories are beyond sensible doubt, and we call them facts. The more energetically and thoroughly you try to disprove a theory, if it survives the assault, the more closely it approaches what common sense happily calls a fact.
    
    
    I could carry on using ‘Theory Sense 1’ and ‘Theory Sense 2’ but numbers are unmemorable. I need substitute words. We already have a good word for ‘Theory Sense 2’. It is ‘hypothesis’. Everybody understands that a hypothesis is a tentative idea awaiting confirmation (or falsification), and it is precisely this tentativeness that evolution has now shed, although it was still burdened with it in Darwin’s time. ‘Theory Sense 1’ is harder. It would be nice simply to go on using ‘theory’, as though ‘Sense 2’ didn’t exist. Indeed, a good case could be made that Sense 2 shouldn’t exist, because it is confusing and unnecessary, given that we have ‘hypothesis’. Unfortunately Sense 2 of ‘theory’ is in common use and we can’t by fiat ban it. I am therefore going to take the considerable, but just forgivable, liberty of borrowing from mathematics the word ‘theorem’ for Sense 1. It is actually a mis-borrowing, as we shall see, but I think the risk of confusion is outweighed by the benefits. As a gesture of appeasement towards affronted mathematicians, I am going to change my spelling to ‘theorum’.* First, let me explain the strict mathematical usage of theorem, while at the same time clarifying my earlier statement that, strictly speaking, only mathematicians are licensed to prove anything (lawyers aren’t, despite well-remunerated pretensions).
    
    
    To a mathematician, a proof is a logical demonstration that a conclusion necessarily follows from axioms that are assumed. Pythagoras’ Theorem is necessarily true, provided only that we assume Euclidean axioms, such as the axiom that parallel straight lines never meet. You are wasting your time measuring thousands of right-angled triangles, trying to find one that falsifies Pythagoras’ Theorem. The Pythagoreans proved it, anybody can work through the proof, it’s just true and that’s that. Mathematicians use the idea of proof to make a distinction between a ‘conjecture’ and a ‘theorem’, which bears a superficial resemblance to the OED’s distinction between the two senses of ‘theory’. A conjecture is a proposition that looks true but has never been proved. It will become a theorem when it has been proved. A famous example is the Goldbach Conjecture, which states that any even integer can be expressed as the sum of two primes. Mathematicians have failed to disprove it for all even numbers up to 300 thousand million million million, and common sense would happily call it Goldbach’s Fact. Nevertheless it has never been proved, despite lucrative prizes being offered for the achievement, and mathematicians rightly refuse to place it on the pedestal reserved for theorems. If anybody ever finds a proof, it will be promoted from Goldbach’s Conjecture to Goldbach’s Theorem, or maybe X’s Theorem where X is the clever mathematician who finds the proof.
    Carl Sagan made sarcastic use of the Goldbach Conjecture in his riposte to people who claim to have been abducted by aliens.Occasionally, I get a letter from someone who is in ‘contact’ with extraterrestrials. I am invited to ‘ask them anything’. And so over the years I’ve prepared a little list of questions. The extraterrestrials are very advanced, remember. So I ask things like, ‘Please provide a short proof of Fermat’s Last Theorem’. Or the Goldbach Conjecture . . . I never get an answer. On the other hand,