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In composing a proof that is reliant on proven theorems, does one simply assume the reader's familiarity with said theorems, or does one explicitly include their logic in the new logic?

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There's no need to prove known theorems. If it is not something well known, you can give a reference to the paper where it's proved. –  Karolis Juodelė Dec 31 '12 at 18:37
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Do whatever enhances the clarity of your writing. –  Austin Mohr Dec 31 '12 at 18:39
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But do include a reference to a published proof of the known theorems that you use. Preferably a page-number reference, rather than a whole book that has the result in it somewhere. –  GEdgar Dec 31 '12 at 21:29
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3 Answers 3

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It depends on the context. My masters thesis, for example, began with saying that I expect the reader to be familiar with forcing. I am not expecting the reader to be familiar with other topics which are relatively common, though. These topics are fully explained in my thesis.

When writing something one can usually foresee who is going to read the text, and what the readers are expecting to see written.

Over-detailed writing is very hard to read; but under-detailed writing is very hard to understand. It takes time and experience to find the balance. Consult an advisor or referee regarding your actual text.

This may also be affected by where you are sending the text. When writing a very short article (say 3-4 pages) it's fine to just quote results, but when writing a book it's expected to prove them as well.

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It's a judgment call, depending, in part, on the audience you are addressing, and your purpose for writing a proof, or a paper with proofs, etc. And so, as Asaf answers, the answer to your question: "depends on the context."

If you are writing proofs for classes (assigned or recommended), for example, it is best to error on the side of caution and be more explicit rather than less, and include more rather than less. One rarely loses marks for including more information than needed, whereas it is common to lose marks for not including enough. So when justifying a statement in such a proof: you can refer to proofs/theorems in your class text provided they have been covered in class. You can often do so by referencing the number/letter used in the text or in class, or by referring to it by using a commonly-used name of a particular theorem. E.g., "By the Fundamental Theorem of Arithmetic, it follows that...".

It is often helpful, when writing proofs for classes, to also justify an assertion by referring to definition(s) that your assertion relies upon: e.g. "by the definition of congruence modulo n, we know that...", without needing to restate the entire definition, unless you are introducing an unfamiliar term that you plan to use in your proof.

These suggestions are just as much for your (present and future) benefit as they are for your audience, and/or for demonstrating mastery of the material relevant to the statement you are asked to prove. Of course, for proofs required in coursework or in an exam, I'd highly recommend that you consult your instructor on this matter. If preparing for preliminary examinations, you can access and work out some problems from past prelims, and discuss with your advisor whether your proofs/solutions are adequate: (What should I have included? What could I have excluded? etc).

Attachment I:

You might find the following exposition written by Dr. John M. Lee helpful:

Internet Bibliography:

If interested, you might want to explore the links available at

See also the previous post: What can a writer assume in a proof?.


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I just realized that you have a habit of referencing me (to my answers/comments) and later removing the reference. Strange. :-) –  Asaf Karagila Jan 1 '13 at 9:44
    
@Asaf - I re-referenced you: I had edited my first sentence, and ended up omitting the reference, not intending to have done so...so I just re-edited the first sentence to re-insert the reference. Besides, I'm not altogether clear when others "want" for me to reference their answers, (as they may or may not "endorse" any content I add) - And as for my comments, I sometimes delete them later when it seems they no longer apply or if they add only noise, and such. :-) –  amWhy Jan 1 '13 at 14:00
    
Oh, I don't mind you not referencing my answers/comments/whatever. I just find it peculiar as it wasn't the first time I saw that you began by referring to something I'd said and then removed that reference. –  Asaf Karagila Jan 1 '13 at 14:38
    
@Asaf I like to credit where credit is due (trying not to "hijack" answers)...But I'm never really certain that references are always welcome by the referenced, so I'm a bit ambivalent about when to or not to reference, that's all. Any way, I didn't take you to be complaining, or as being critical - not at all. –  amWhy Jan 1 '13 at 14:54
    
I don't mind being cited! :-) –  Asaf Karagila Jan 1 '13 at 15:04
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With all proofs (in whatever format), you must first identify your intended audience. That will allow you to decide what you can / cannot assume, and how to include in the right amount of detail.

Are they students in your field that are interested in understanding the buildup of the theory? Are they 'professionals' in related fields who believe in the truth of your statements and only want to understand implications of your results? Are they high schoolers who want to know why quintics can't be 'solved' and only know the basics of group theory?

Having additional proofs is preferable. You can helps others along by labeling them as Theorem 1, Proof (or even placed in the Appendix), so if readers are familiar, they can skip it.

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