How can a piece of mathematics be criticized? Concerning its correctness there cannot be anything more than a YES or NO.
Who are the critics? What is the status of self-criticism? Does self-criticism lead to a self-reference or to an infinite regress? Who criticizes the critics? The issue of critique leads to the issue of authority as well as to issues like anonymous vs. non-anonymous and private vs. public critique. ‘Objective’ critique is another idealistic phrase: can critique become automatized and will automatized critique be more than just a local checking of the correctness of a proof?
The (by now) hidden process of refereeing seems to be crucial for detecting specific characteristics of ‘living mathematics’!
What is the difference between an ‘authorized’ referee and an ‘ordinary’ reader? This is a main question!!
If the often proclaimed unity of mathematics was the case then there would be also a certain unity in the criteria upon which a piece of mathematics is criticized. But the problem is that there are actually many sub domains with their own more or less separated communities and thus with their own values. The separation is best seen in what is called school-mathematics versus university-mathematics. There might be a similar but much smaller separation between undergraduate mathematics for non-mathematics students (engineering but also business studies, biology, medicine, etc.) and mathematics for future mathematicians.
This seems to be the result of a wrong belief – mostly met in mathematics and in the sciences as well – that correct statements ‘speak for themselves’. This is an important negative consequence of the split between the sciences and the humanities. Critique serves not just for quality control and for evaluation. It serves also to create a channel of communication between the author (the sender) and the audience (the receiver). It is a message about a message, so there is an issue of self-reference. Not only because of this but also because of the mediating function of critique it is clear that it does not make sense in a closed system. It actually serves to ‘open’ the system. The ‘insiders’ don’t need critique. They agree because they share very similar values and experiences. They only nod and the type of nodding depends upon whether the (mathematical) item in question has been appreciated or not.
It would be interesting to look at cases where the members of a specialized mathematical sub-community disagree. I would suppose that the disagreement would be more about the correctness than about the quality. Here we get back again to the reviewing process. Disagreement in mathematics is not allowed to become public. It is either ruled out by not accepting a specific paper for publication or the issue of disagreement gives birth to a new subfield like nonstandard analysis or like constructive mathematics. Maybe nowadays through the many platforms for informal discussion the issue of disagreement could be studied on a more concrete basis.