In forming your answer you may choose to address any or all of the following aspects of the question:
- Descriptive
- What part do arguments from authority actually play in mathematical reasoning?
- Normative
- What part do arguments from authority ideally play in mathematical reasoning?
- Regulative
-
What if any discrepancies exist between the actual and the ideal?
What if anything should be done about the discrepancies that exist?
Recycled from a question I asked on MathOverFlow.
Inquiry Into Inquiry 
I can only offer the perspective of the mathematically naive and I apologize if its not germane:
It depends. At my level of reasoning and scholarship there is no abstract general “authority”. There are authoritIES, such as the Encyclopedia of Mathematics or someone I recognize (rightly or wrongly) as a famous mathematician, such as Newton, Feynman, or Kenneth Arrow. I can only judge the merit of an argument from authority in relation to my belief (justified or not) in the probable degree of merit of a particular argument-authority combination and the probable degree of fitness of that argument-authority for a particular purpose–for example I might quote a perceived authority to back up some half-baked mathematical argument of my own as in “Lookback Homeward, Angel,” an essay about lookback time and Hubble’s Law. (http://almanac2010.wordpress.com/2012/08/15/lookback-homeward-angel/)
This is a pragmatic (if naive) answer but there’s a theory buried in there somewhere. 🙂
PR
I can’t speak directly to what role authority actually holds, but I am inclined to believe that role is much less in mathematics than it is in a field I’ve a little more familiarity with, gravitational cosmology. Gatekeepers like Hawking, Krauss, Greene, and Carroll loom very large over the field, so that only individuals who are themselves very well established (Smolin, Woit, Disney, Steinhardt) can afford to make any serious challenges to the standard model. This is extremely problematic, because that standard model of gravitational cosmology (SMGC) is deeply flawed. (I’ve blogged about it myself, in my discussions of “model centrism,” so I’ll not go into any details here.)
The problem with SMGC is that the evidence is so very thin, while the model is so very complicated. In the case of mathematics, however, it strikes me that the “evidence” is usually quite “dense”, that is, everything is already there on the pages in front of you. So I wonder if authority carries a little less weight because one can more directly challenge that authority by pointing out errors more explicitly? Unlike cosmological data, which is vague enough even when it is interpreted (and must be heavily interpreted before it can even qualify as “data”), the mathematics is not only less ambiguous, but “all there”. Obviously this is a highly idealized characterization of actual mathematical inquiry, as people like Tymoczko, of Davis and Hersh, have shown.
As for what role authority should play, well again ideally, it “should” play little or none. But realistically, if I’m studying an article in the Journal of Symbolic Logic, and the author cites work by Saharon Shelah, well, I’m just going to take it for granted that Shelah’s work is correct.