# when does one use the word 'fact' in mathematics

I am having trouble using the word 'fact' when speaking/writing Mathematics. For instance, suppose we have proved a new theorem, I then apply the theorem to deduce some other results.

Can I regard what have been proven, i.e. the ingredient of the theorem as facts?

• As per dictionary definition, they are facts. But in my experience, fact in mathematics is used as a result which is known to be true but the proof is missing from the source which you're reading. Feb 26, 2015 at 13:35
• Definitely, e.g you don't prove pythagorean theorem every time you use it.
– AvZ
Feb 26, 2015 at 13:38

Don't use the "f word" in mathematics.

Use theorem, lemma, axiom or conjecture, whatever applies, and clearly state your assumptions. Like the "fact" that $2+2 = 4$... sure, but not in $\mathbb{F}_3$.

• If you're gonna use $2$ and $+$ that freely, then you should use $4$ as freely as well: $4=1$ and there's nothing wrong. Feb 26, 2015 at 13:46
• OK, take the "fact" that the sum of the interior angles of a triangle is 180° then... in the Euclidean plane.
– user139000
Feb 26, 2015 at 13:50
• I recently used the word "fact" to refer to a very obviously true proposition about a strictly increasing sequence $(y_n)$ occurring in a proof: it was contained in $(-1, 1)$, so I referred to "the fact that $\sum_n (y_{n+1} - y_n) < 2$". Is such a usage not defensible? Feb 26, 2015 at 14:40

First, getting angry about notation or terminology is not a healthy habit. I take the approach to always try to learn when I am reading a piece of mathematics, not to get upset about how the author has expressed him(her)self.

Most authors stay away from the word 'Fact' unless assuming a mature audience. In this context, the word 'fact' is (usually) equivalent to a Theorem, i.e. by stating a fact one will be stating a theorem, but without providing a proof or an introduction to the theorem.

It is perfectly valid, once having proven a theorem, to continue and say 'now, as shown above, it is a fact that ...', as long as it is clear and the audience understands.

• Your answer makes good sense to me! Feb 27, 2015 at 13:14