For the past three decades, peer review practices have received much attention in the literature. But although this literature covers many research fields, only one previous systematic study has been devoted to the practice of peer review in mathematics, namely a study by Geist, Löwe, and Van Kerkhove from 2010. This lack of attention may be due to a view that peer review in mathematics is more reliable, and therefore less interesting as an object of study, than peer review in other fields. In fact, Geist, Löwe, and Van Kerkhove argue that peer review in mathematics is relatively reliable. At the same time, peer review in mathematics differs from peer review in most, if not all, other fields in that papers submitted to mathematical journals are usually only reviewed by a single referee. Furthermore, recent empirical studies indicate that the referees do not check the papers line by line. I argue that, in spite of this, mathematical practice in general and refereeing practices in particular are such that the common practice of mathematical journals of using just one referee is justified from the point of view of proof validity assessment. The argument is based on interviews I conducted with seven mathematicians.

We present a taxonomy of errors in the scientific literature and an account of how the errors are distributed over the categories. We have developed the taxonomy by studying substantial errors in the scientific literature as described in retraction notices published in the journal Science over the past 35 years. We then examine how the sorts of errors that lead to retracted papers can be prevented and detected, considering the perspective of collaborating scientists, journal editors and referees, and readers of the published articles.

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Retractions are rare in science, but there is growing concern about the impact retracted papers have. We present data on the retracted paper in the journal Science, between 1983 and 2017. Each year, approximately 2.6 papers are retracted; that is about 0.34% of the papers published in the journal. 30 % of the retracted papers are retracted within one year of publication. Some papers are retracted almost 12 years after publication. 51 % of the retracted papers are retracted due to honest mistakes. Smaller research teams of 2 to 4 scientists are responsible for a disproportionately larger share of the retracted papers especially when it comes to retractions due to honest mistakes. In 60 % of cases all authors sign the retraction notice.

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