Rediscovery of calculus in 1994: what should have happened to that paper?

Online publishing, at the face of it, should be giving editors some room for maneuver. In this case, they could append the online version of the paper with an editorial note like

EDITORIAL NOTE: Citing this paper as "Tai's method" is strongly discouraged, as the method is actually known under the name of trapezoidal rule. See also Comments for further explanations.

The academic community affected by this rediscovery should have taken actions accordingly, and editors in other journals should have made sure that citations to the paper are only given if something other than the trapezoidal rule is being discussed.

John Paul II apologized for a number of misdoings by the Catholic Church. The editors should've apologized to their research community for committing this peer review failure. It's OK to be wrong from time to time; science moves by rectifying things that come into contradiction with other bits of knowledge. However it's not OK to withhold knowing that you are wrong and insisting on it.

The answer to this question depends on what we should term a peer-review failure. There are tons of journals in the academic market: some journals are happy to publish results as long as they are correct even if they do not push the boundary of innovation. There are some journals that are happy to publish unsound experimental work, and on the other hand, some that encourage theoretical models that are severely impractical and are essentially mathematical mumbo-jumbo.

Publishing something blindingly obvious even to a high-school student is ridiculous, but we could infer two things: one, the editor and the reviewers had not found this so obvious, and two, the journal did not (or does not?) have high standards on innovation anyway. Given it is a field that potentially could have practitioners who know not a thing about calculus, the publication should not come as a big surprise.

The reality is that publication of known facts is not all that uncommon in academia. The hoo-ha over this paper is basically because of the elementariness of the concept. One could expect apologies from journals for wrong results but not necessarily for stale ones.

Personally, if I had anything to do with the acceptance of a paper such as this, I'd want to suppress any knowledge that I had anything to do with this paper at all as thoroughly as possible.

I would certainly hope that in such a case the author in question would be so embarrassed by "rediscovering" something known for hundreds of years that the author would willingly retract the paper. But if that has not happened, there's not much that can be done. The peer-review process failed, but it would be up to either the journal editor or the author to demand retraction. Newton and Simpson are dead, and nobody else could realistically make a claim to be "injured" by this momentous "discovery."

If you want to ask the question must the paper be retracted, then the answer is no, unless there was actual bias or foul play in the peer review process. If the reviewers knew there was a problem, and intentionally allowed the paper to move forward, then a retraction is necessary. Otherwise, though, the paper should not de rigueur be retracted. It should be shunned by the academic community, however, and people should refrain from citing it (except perhaps for articles and works showing how not to do research!).