„Keep an open mind” is often used as an argument against science, or to justify ignoring evidence more broadly. Let’s distinguish two cases of keeping an open mind: before vs after the evidence comes in. It is good to keep an open mind before data is obtained – no hypothesis is ruled out. In reality, all possibilities have positive probability, no matter how great the amount and quality of information, so one should not dogmatically rule out anything even given the best evidence. However, for practical purposes a small enough probability is the same as zero. Decisions have to be made constantly (choosing not to decide is also a decision), so after enough scientific information is available, it is optimal to make up one’s mind, instead of keeping it open.
Intellectually honest people who want to keep an open mind after obtaining evidence would commit to it from the start: publicly say that no matter what the data shows in the future, they will ignore it and keep an open mind. Similarly, the intellectually honest who plan to make up their mind would also commit, in this case to a policy along the lines of „if the evidence says A, then do this, but if the evidence says B, then that”. The latter policy resembles (parts of) the scientific method.
The anti-science or just intellectually dishonest way of “keeping an open mind” is to do this if and only if the evidence disagrees with one’s prior views. In other words, favourable data is accepted, but unfavourable ignored, justifying the ignoring with the open mind excuse. In debates, the side that runs out of arguments and is about to lose is usually the one who recommends an open mind, and only at that late stage of the debate and conditional on own weak position. Similarly, “agreeing to disagree” is mostly recommended intellectually dishonestly by the losing side of an argument, to attempt to leave the outcome uncertain. This is an almost logically contradictory use of “agreeing to disagree”, because it is mathematically proven that rational agents putting positive probability on the same events cannot agree to disagree – if their posterior beliefs are common knowledge, then these must coincide.