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The way mathematicians use the word "result" is quite unfamiliar and unusual in the viewpoint of non-mathematicians. Although I know that "results" usually mean theorems and prepositions that have been proven, I struggle to define the word exactly. Whenever I write something like "This result in complex analysis is very enlightening", somebody will ask me "are you sure this is a correct use of the word 'result'?", and I will struggle to explain what I mean by this word.

What exactly does the word "result" mean in mathematics?

There are few synonyms of the word "theorem" other than "result". Maybe the word "result" plays a role in reducing the repetitiveness of "theorems"?

Ma Joad
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  • Result is quite the right word: we have a "result" on the sum of consecutive numbers $1,2,\ldots ,n$, namely that $1+2+\cdots +n=n(n+1)/2$. What else could we have with these numbers? A knowledge? A finding? You are right, one would also say, a "proposition", or even a "theorem", although this is probably too much for the formula. – Dietrich Burde Sep 08 '19 at 13:30
  • What were the words before "this result"? If I showed you a framed photograph and said, "This result in photography is one of my favorites," you might think it an odd way of telling you I like the photograph, although of course the photograph is a result (of some photographer's artistic vision and of the process of taking and printing the photograph). – David K Sep 08 '19 at 13:32
  • "This result in photography?" This would denote a statement in photography. The photographer's artistic vision is perhaps "This result of Photography is one of my favorites." – Dietrich Burde Sep 08 '19 at 13:35
  • @DietrichBurde True, the word "photography" does not lend itself to being used with "in" in the way "complex analysis" does. Even if I said "of photography," however, "result" would still be a strange way of describing it unless I had previously explained what it was a result of. – David K Sep 08 '19 at 13:38
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    Also, if I add $2$ and $3$ the result is $5,$ but $5$ is not a theorem. A "result" is too broad a term for some contexts. Again, it would depend on the context in which you used the word. Although to be honest I have a hard time imagining a context in which your sentence would not be OK. – David K Sep 08 '19 at 13:41
  • Yes, you are right. I would reformulate your title as "What does the words "result in (some area)" mean in a mathematical context?" Like a result in complex analysis, or a result in elementary number theory. – Dietrich Burde Sep 08 '19 at 13:42
  • @DavidK Exactly! As I point out in a clarification comment under one of the below answers, the word result is so broad as to embrace within its ambit any result of a mathematical investigation, even if one has not formally proven or published it. Indeed, there are such results that are often well-known to members of your field of expertise, which no one has really written up somewhere. Indeed, mathematics is not necessarily born only when it's been formally written down. We carry it within us first. That's its first residence, before the paper or book. – Allawonder Sep 08 '19 at 19:35

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A result is the outcome of some process. As you indicated, mathematicians often call theorems results, since they are the outcomes of investigations. So, it's quite right to say something like There's is a result in complex variables to the effect that, etc.

Allawonder
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A word that has a close meaning to "theorem" is "lemma". It also means "a mathematical statement that has been proven", like a theorem, but usually lemmata are in a sense secondary theorems that are being used in main theorems. I hope this makes some sense...

A "corollary" is also a proven statement, but usually one that almost directly follows by another result.

Perhaps, if a statement has a proof, you can present it as a "fact"? But maybe that use of the word is a bit far-fetched.

"Result" is also a commonly used word, abstract enough to cover all above cases. In a way, "proven statement" can be seen as the outcome of the process of proving that statement (therefore a result of that process) as Allawonder also points out in his answer.

frabala
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    A small clarification (cf. your last paragraph) -- I wouldn't say, though, that the word result refers (only, as implied) to the outcome of the process of proving some statement. That's far too strong. For example, mathematicians also say things like So-and-so proved the result... This shows that a result is so weak and general as to cover outcomes of investigations that one has not formally proven or published yet, especially if it's well-known in your field of expertise. So, to emphasize what I did say in my answer, I'd use result to mean the outcome of a mathematical investigation. – Allawonder Sep 08 '19 at 19:26