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Can I quantify the word "proof", so is it possible to write the phrase many proofs and if yes in which situation is this more appropriate than writing much proof?

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Yes, proof can be quantifiable. One obvious context is the realm of mathematics, where there may be more than one proof of a theorem. So, for example, one could say:

There aren't many proofs for the four-colored map problem.
Many early proofs of Fermat's last theorem were found to contain errors.

On the other hand, proof is sometimes used in a way similar to evidence. (One might argue that evidence would be a better word to use in such situations, but the fact remains that proof is nevertheless used.) For example, in this science editorial, the author wrote:

There is much proof that Mars was once, if not a green planet, a planet that once held flowing water and possibly life.

Both can be used; it really depends on if you're talking about a body of evidence or individual proofs.

As a footnote, other usages of the word proof are clearly quantifiable, such as:

The photographer gave us nine proofs to choose from.

  • Good photo proof example -- I hadn't recalled the use "an advanced copy for checking." – JeremyDouglass Dec 13 '16 at 9:40
  • The author of the second citation should not have written "once" twice. Other trivium: a proof with an error is not a proof, so the "many early proofs" in the first citation were in fact none. – Marc van Leeuwen Dec 13 '16 at 14:31
  • @Marc - I considered putting "proofs" in scare quotes for that reason. (I debated it for a couple minutes, actually.) In the end, though, I opted for simplicity, as have many authors before me. – J.R. Dec 13 '16 at 15:28
  • I believe it is the four color theorem. The map problem is "how many colors?" The theorem is "four." Someone proves a theorem, not a problem: you could prove "P=NP", but cannot prove "the P vs. NP problem." Funny that we chose the same example. – JeremyDouglass Dec 13 '16 at 17:20
  • @JeremyD - I think it can be called either one. Perhaps some of us old geezers inadvertantly fall back on problem because we remember when it was still unproven :-) – J.R. Dec 13 '16 at 18:13

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