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In forming your answer you may choose to address any or all of the following aspects of the question:

  1. Descriptive. What part do arguments from authority actually play in mathematical reasoning?
  2. Normative. What part do arguments from authority ideally play in mathematical reasoning?
  3. Regulative. What if any discrepancies exist between the actual and the ideal and what if anything should be done about them if there are?
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I would say that ideally, arguments from authority should play no part in mathematical reasoning. In practice, people very frequently rely on results whose proof they do not completely understand; in such a context, the confidence in these results typically comes from authority, in some sense; namely, confidence in the mathematicians who produced those results, together with a confidence in the community around those mathematicians, who have vetted the results. – Emerton Jun 14 '10 at 3:59
I think this should be a community wiki. I also question some of the tags. – Charles Staats Jun 14 '10 at 4:41
One common appeal to authority is "This conjecture is worth my time working on it, and your time hearing about it, because it came from someone who is reputed to have good taste." But I guess that's not about mathematical reasoning. – Allen Knutson Jun 14 '10 at 11:26
How about a computer program that checks all the cases, when there are too many for any one human to reasonably do single-handedly. Is that "appeal to authority"? – Gerald Edgar Jun 14 '10 at 14:04
The quotation <…; is pretty relevant :) – Mariano Suárez-Alvarez Jun 14 '10 at 14:57

Appeals-to-authority, of course, are widely regarded in many contexts as logical flaws, as something to be avoided in logical discourse. But I believe that there is a sense, nevertheless, in which appeals-to-authority are a commonplace in mathematics. Namely, we all cite other mathematician's work in our own work, and such citations amount in a sense to an appeal to authority, appealing to the authority of that other mathematician and of the editors and referees of that journal that the cited theorem is correct, even though a proof of it is not immediately given in the citing text. Although it is sometimes stated that a mathematician should only cite work that he or she has personally checked in detail, this suggestion is surely not universally observed. In any case, even when it is observed, the citation itself still does not provide the full logical force of the conclusion that is drawn from it---one must still trust the cited theorem in order to proceed with the logical conclusion at hand. Thus, I find it to be essentially an appeal to authority.

But this kind of appeal to authority seems relatively mild in comparison with the truly objectionable kind one might imagine in other contexts. The reason is the open nature of the appeal; it can in principle be checked. That is, when we cite a published theorem, our readers can in principle look up the proof of that theorem themselves and determine if it is correct.

A more pernicious instance of appeal-to-authority, of course, would be less open, and not easily checkable. Although such appeals sometimes occur in mathematics, I believe that most mathematicians do not regard them as constituting proof. And so I don't think that we actually have a problem with this. Rather, when a mathematicians says that so-and-so famous mathematician asserts $X$, then I think that this by itself is rarely taken as a proof of $X$. Instead, it is taken as an indication that there likely is a proof of $X$, and that perhaps that famous mathematician has such a proof, and that perhaps we might search for such a proof ourselves.

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Anecdotal evidence suggests that an assertion X by a famous so-and-so is often intended precisely to $\textit{discourage}$ others from searching for a proof themselves, whether or not so-and-so possesses it, --- it's a form of staking the territory and warning others to stay away. (Note that the previous sentence also involves an appeal to authority.) – Victor Protsak Jun 14 '10 at 8:16
Victor, I have heard this before, but haven't observed it much, except perhaps a few times in connection with one person. So perhaps the word "often" in your remark is an exaggeration. Also, in the case that the famous person is also pompous, then I think the strategy is a bad one for them, since any such announcement will also attract people attempting to refute the claim, just for the pleasure of showing them wrong. But I suppose famous people who are also pompous seldom realize it... – Joel David Hamkins Jun 15 '10 at 12:31

I can't bear to just post the links as comments, so please forgive an answer that recalls two Sidney Harris cartoons:

alt text alt text

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You missed off the caption on the first! "I think you need to be a bit clearer on this step.", if memory serves me right. – Loop Space Jun 14 '10 at 18:08
"I think you should be more explicit here in step two". – Steve Huntsman Jun 14 '10 at 19:14

You could interpret your question outside of the context of mathematical proof. It seems like your question was phrased broadly-enough to allow this interpretation.

There are what are perhaps the most subjective aspects of mathematical reasoning:

1) What questions do we consider important?

2) What results do we consider worthy of publication?

3) What do we consider worthy of teaching our students?

I suppose this is a type of metamathematical reasoning. But arguments from authority frequently get used in these situations.

Especially in grad school I seemed to encounter far too many people with fantastical images of certain mathematicians, frequently fields-medalists, sometimes with little understanding of what these people have accomplished. The further I get from grad school the less of this I see, thankfully.

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You read me right. I use words like "inquiry" and "reasoning" to invoke the greater genus of which deductive proof is but one species, however favored a place it may occupy in the mathematical bestiary. A line of thinking that extends from Aristotle through Kant to Peirce and beyond holds that the less exact or "non-demonstrative" kinds of inference also serve the progress of science and even mathematics, that they have functions and structures all their own, and that it repays the effort of analysis to clarify their roles and their rules. – Jon Awbrey Jun 16 '10 at 2:08
4) Whom do we consider worth giving a job? – Arend Bayer Jul 24 '10 at 9:02

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