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I am writing a master's thesis (in Number Theory) and there are multiple places where I need to claim that something is not yet known. Since I am not an expert (and even if I were) I would like to reference some more authoritative source than myself or, say, Wikipedia.

For example, consider the following statement:

In 1980 Schmidt proved that [some property holds]. No other necessary or sufficient conditions are currently known, though.

For the first sentence I can cite Schmidt's original paper, but how can I substantiate the second claim?

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    Negative proofs are hard... – jubobs May 21 '15 at 16:55
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    Is your master's thesis supposed to be original research? If so, then are you not asserting that the thesis itself is (believed to be be) unknown but knowable. – emory May 22 '15 at 21:25
  • @emory Yes, my master's thesis is supposed to be original research, but I don't understand your point. In the case of the example statement, my result has some nice consequences for numbers with some property P, and Schmidt proved that P holds for a certain class of numbers. Some sufficient conditions for P, say, could have allowed me to generalise those consequences, but alas... – A.P. May 22 '15 at 22:05
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    I think you are overthinking this. You are the expert. "No other necessary or sufficient conditions are known." is sufficient. If you prefer: "We do not know of any other necessary or sufficient conditions." – emory May 22 '15 at 22:36
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One way, especially since you are not an expert of the field (and even if you were), is to not make such an absolute claim:

To the best of our knowledge, no other necessary or sufficient conditions are currently known.

At least in my field (robotics), this is quite common and I think an appreciated amount of humility.

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    +1. You'll be surprised at how often reviewers (who, after all, should be experts in the field) do know such things and point you to sources you and your advisor never found. If you were too certain in your formulations, this can be rather humiliating. – Stephan Kolassa May 21 '15 at 9:58
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    In addition to not making absolute claims, it is also helpful when you can cite a recent survey or tutorial paper where it is mentioned as open problem. – CrepusculeWithNellie May 21 '15 at 10:03
  • It's not uncommon in Number Theory either. – Kimball May 21 '15 at 17:15
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    I agree fully with this. I am -- like the commenter directly above me -- a professional number theorist and write lines like the highlighted one in my papers rather often. If it works for me, it should be appropriate for a master's thesis as well. (I might take out the word "currently": what else could you mean: that you have a proof that it is impossible to know? If so, say so!) – Pete L. Clark May 21 '15 at 17:56
  • when would it be appropriate to write: "to the best of our knowledge, some other necessary or sufficient conditions are currently known."? can we reduce the word count? – emory May 23 '15 at 22:22
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Lifesayko's answer is good, but it should not be the first way to address the problem.

Scientists tend to be aware of the gaps in their knowledge. So they write about them. You will find such sources:

  • In review articles and metastudies. They say things like "We reviewed 8 cross sectional studies on the link between zugs and wugs. We found that the presence of zugs is a strong predictor of the subsequent appearance of wugs, but the exact mechanism behind this is not yet known".
  • In the "future work" sections of articles in the area. For example, "In this paper, we found that, after 3 months of delay, we can measure a correlation of 0.8 between the presence of zugs and wugs. A next step would be to establish the mechanism by which zugs contribute to the appearance of wugs".
  • In other articles which work are based on the same theory as your work. "We are exploring the link between zugs and wods. In the past, zugs have been linked to wugs, although the exact mechanism is unknown".
  • In articles which outline the challenges in a certain discipline for the coming years. Usually published by established professors in major journals, they explain what is missing in their area. "One of the major unresolved problems in zug research is how they cause the appearance of wugs".

Of course, for all three of them, it is important that your source is fairly recent, because somebody can well have found out the missing information in the meantime. A defensive formulation is never out of place. But it does not absolve you from looking for sources first.

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    Yes, one should search the literature, but the amount of things not known is much greater than the number of things we have recorded we don't know. In this case, what I would recommend is using MathSciNet and Google Scholar to check the papers that cite Schmidt's paper, and see if they prove anything related. – Kimball May 21 '15 at 17:20
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    You write about "Scientists" but the question concerns mathematics. There are cultural similarities and cultural differences between mathematics and the sciences (in particular, with apologies to the NSF, mathematics is not a science). In my experience, "calling attention to what is not known" is something that scientists do significantly better than mathematicians. At the beginning of my career, most of my papers ended with a section on open questions and/or further work. Frequent negative comments by referees clued me in to the fact that this is actually rarely done.... – Pete L. Clark May 21 '15 at 19:35
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    ....In general the discipline of mathematics tends to view such text as "wasted space". It is much more common to answer any question about extensions to your work by a vague reference to "work in progress". If you are lucky enough to find a "This is not known pronouncement" in the literature, then you should cite it to justify your case but there is a bit of a paradox here: my saying "This is not known" in year X+Y has to be a stronger statement than "This is not known" appearing in the literature in year X, unless further evidence is presented (which you could also present).... – Pete L. Clark May 21 '15 at 19:40
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    ...Nevertheless I agree that "To the best of my knowledge" can cover up a situation in which you made no effort at all to check what is known, and I certainly agree that one should make such an effort. This is a tough one! – Pete L. Clark May 21 '15 at 19:41
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    @ChrisWhite: I see two reasons you caught more flak than Pete did: 1) your comment was in the middle of an already-contentious debate, and 2) your comment claimed to speak for others and Pete's did not. That said, I agree with you both. – Mark Meckes May 22 '15 at 7:26
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What I usually do when I'm not sure, despite having put in a lot of effort trying to get to the bottom of the issue by doing extensive literature research, is to contact one or more experts in the field. You can just send an email to one or more leading experts in the field, likely they will be able to clear up the issue. Also they may point to some sources that you may have missed. You can then cite those sources, also you can cite the information in the communication as a "private communication" and, of course, in the acknowledgements, you should thank them for assistance.

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The answers that say there's no call to make absolute statements like that are very good, but its still embarrassing to make a temporizing statement like "to the best of my knowledge", and then have a referee show you that your knowledge isn't good enough.

Your ability to assert that something is not known and be believed is related to your credibility. Master the literature, or work with a co-author or advisor that has, and cite it wisely and thoroughly. Make referees feel like you've done your due diligence.

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I can't speak specifically to the academic angle, as others already have, but language might help here - namely, E-Prime (simply implemented as the removal of all forms of the verb "to be").

Restating your problem sentence:

No other necessary or sufficient conditions are currently known, though.

...without the "are" requires an actor; I presume you would use yourself and perhaps other contributors. Thus you could say:

The authors know of no other necessary or sufficient conditions.

This seems literally true and equivalent (less the implied but non-existent surety) to the original, but it thus has lost any force. To provide some content to the statement, perhaps then reference the resources used, date, and search terms, as in:

The authors found no other necessary or sufficient conditions. (JSTOR, 23 May, 2015: 'search term'; EBSCO discovery service, 23 May, 2015: 'search term')

  • If you think that a master's thesis could have more than one author, then I think you are too far away from "the academic angle" to be answering questions like this one. – Pete L. Clark May 24 '15 at 8:57

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