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Consider following proof:

Prove that if $x^2 + y = 13$ and $y ≠ 4$ then $x ≠ 3$.

Proof. Suppose $x^2 + y = 13$ and $y ≠ 4$. Suppose $x = 3$. Substituting this into the equation $x^2 + y = 13$, we get $9 + y = 13$, so $y = 4$. But this contradicts the fact that $y ≠ 4$. Therefore $x ≠ 3.$ Thus, if $x^2 + y = 13$ and $y ≠ 4$ then $x ≠ 3$.

Note the statement "this contradicts the fact that $𝑦≠4$".

Definitions from the internet:

Fact: "a thing that is known or proved to be true"

Assumption: "a thing that is accepted as true or as certain to happen, without proof."

Take a look at the first sentence:

Suppose $x^2 + y = 13$ and $y ≠ 4$.

My question is, is it correct that author used word "fact" when referring to $y ≠ 4$? In my opinion, saying "but this contradicts our assumption that $y ≠ 4$" would be more appropriate.

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    $\begingroup$ No, it's not correct. Should've been "contradicts the assumption" just like you said. At least formally. But you can treat it as a figure of speech, the context is clear. So it's actually ok. $\endgroup$ – freakish Jul 19 at 19:02
  • $\begingroup$ I believe assumption is the correct word to use. We are given a proposition, i.e. "If p then q" and we assume (true or false) p (or q) and say something about q (or p). $\endgroup$ – Hendrix Jul 19 at 19:08
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    $\begingroup$ I disagree with @freakish---there is nothing wrong with the use of language here. The goal is to show that if the hypotheses are satisfied (i.e. if $x^2+y=13$ and $y\ne 4$), then the conclusion follows (i.e. $x\ne 3$). Within the scope of the proof, it is a fact that $y\ne 4$ (and also that $x^2 + y = 13$). At least, that is how I would read it. $\endgroup$ – Xander Henderson Jul 19 at 19:09
  • $\begingroup$ @freakish I understand that it can be considered as nitpicking, but suppose you are writing PhD thesis and used the word "fact" like he did, would it be considered mistake? $\endgroup$ – Nelver Jul 19 at 19:11
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    $\begingroup$ That being said, neither "fact" nor "assumption" is, really, a mathematically rigorous term. The goal here is to communicate, and the language used should be as clear as possible. If I were writing this proof, I would have written "But this contradicts the hypothesis that $y\ne 4$." $\endgroup$ – Xander Henderson Jul 19 at 19:11
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tl;dr It doesn't matter, the meaning is clear and so the usage of the word "fact" is fine.

Long answer: So this a semantic problem. Unlike (most) computer languages natural languages are very heavily context sensitive. And even more: people just love to juggle with words and sometimes they deliberately use words in a non-standard meaning. It makes reading more interesting. The side effect is that sometimes words are misused. And I do agree that in this situation the word "assumption" or "hypothesis" suits the context better than "fact". It's just my preference. But would I point it out as a mistake? Never.

So my point of view is: as long as the meaning is clear and unambiguous then you can use language however you want. For example look at StackExchange. Do we always use words in perfectly correct way? Heck, I'm not a native english speaker, I make mistakes like all the time. But is it really an issue?

Now you've asked in comments "I'd just thought that it might be a problem if you were writing an article or a paper that was targeted at more sophisticated audience". This is a bit different question. Because it asks about possible consequences in a concrete environment. And that is a rather psychological/sociological question. From my experience? It is unlikely that anyone will ever have an issue with that. But is it 100% safe? Well, unfortunately grammar police does exist. But do not overestimate their influence. Also before any paper is published it goes through a review process, so in worst case scenario you just change the word and you're fine. But again: I would be very surprised if that actually happened.

Fact: "a thing that is known or proved to be true"

Reminds me of the famous question: what is truth?

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In a given "mathematical structure" or theory, you start by stating your "assumptions": axioms or postulates. Everything that follows from those are "facts" in that theory.

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I think it's okay to treat an assumption as a fact, because when something is assumed you build up on it. It's like this "assumed something" is really "something" you can argue with or you can imagine that it's 100% correct, e.g. a fact. From the assumption we find conclusions, that are built on the assumptions. That's why "Assumption is the mother of all screw ups". In this proof of contradiction one should learn, for the given equation $x^2+y=13$, that showing $A \Rightarrow B$ or $ (y \neq 4) \Rightarrow (x \neq 3)$ is equivalent to showing that $\neg B \Rightarrow \neg A$ or $(x = 3) \Rightarrow (y = 4) $.That's because $ (A \Rightarrow B) \Leftrightarrow (\neg B \Rightarrow \neg A)$ is a tautology.

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  • $\begingroup$ @user247327 sure, every conclusion obtainable from the assumptions can be again used as an assumption for finding new conclusions! $\endgroup$ – Ahmed Hossam Jul 19 at 19:56

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