A typo, a spelling error etc., in a published article, is definitely not enough for issuing an erratum.
If a mistake destroys a main result, then an erratum is definitely necessary, and the proof should be rewritten, if it's possible to fix.
What about cases in between? What if a small lemma, remark is not correctly stated? What about a mistake in the proof that is actually very easy to fix? What about a wrong number in the calculation? What if … (you name the case) …?
Best Answer
The evidence on arXiv does not seem to support the assertion that mathematicians are significantly more reluctant to issue an erratum than, say, physicists:
errata in mathematics (92 items)
errata in physics (160 items)
The factor of two in the number of errata between physics and mathematics correlates well with the total number of arXiv submissions (about three times as many in physics than in mathematics). Incidentally, only a very small percentage of arXiv submissions are errata (the total number math articles is about 150,000).
I do notice that many of the mathematics errata are limited to the arXiv version, and not incorporated in the journal. The idea being, I would think, that issuing an erratum for a relatively minor issue is a service to the reader, who will likely consult the arXiv version and will therefore be alerted to the error. This would obviously be unacceptable for an error that invalidates the entire work, but those happen more rarely.
In this connection I find it interesting to note that mathematicians withdraw more papers from the arXiv than physicists, 128 versus 90 in the last year. I presume this says something about the different consequences of a small mistake in the two disciplines.