In a previous blog, I discussed the case of whether platinum, palladium or gold could form oxo compounds. The argument that they could was published, with mountains of data that allegedly supported the case. Eventually, these assignments were found to be wrong, and one critic blamed the referees for permitting the original publication. Throughout the discussion, however, nobody seemed to come to grips with the problem: what comprises proof? One question that fascinates me is why do undergraduate science courses inevitably omit to address this question? Why do no such courses include the prerequisite for some, even brief, course in logic?
We sometimes see the argument, often attributed to Popper, that you cannot prove a statement in science; all you can do is falsify it. I think that is unnecessarily restrictive. One answer was given by Conan Doyle: when all possible explanations but one for an observation are eliminated, then that one, however unlikely it might seem, must be the truth. The reason why Popper's argument fails in such cases is that there was an observed effect, and therefore something must have caused it.
Set theory provides a formal means of answering the title question. Suppose I carry out some operation that addresses a scientific question and I obtain an observation, and to simplify the discussion, assume I am trying to obtain a structure of a molecule. There will be a set of structures consistent with that observation. Suppose I do another; there will be a further set of structures consistent with it. Suppose we keep making observations. If so, we generate a number of such sets, and because the molecule remains constant, the desired structure must be a member of the intersection of all such sets. The structure is proved when such an intersection contains only one element. Of course, this raises the question of the suitability of data. Simply reproducing the same sort of observation many times merely produces much the same set many times.
The problem, of course, is to ensure that the sets of structures that might give rise to the observation is complete because the logic fails when the truth was not considered, and proof fails when the sets are incomplete. (The truth not being considered may show up if the intersection of all sets is the empty set.) We may now guess at a problem with the oxo compounds: there was an awful lot of data consistent with the argued structure, but it was not definitive. This is one place where it is possible that the referees failed, but then the question arises, is it reasonable to expect the referees to pick it? Referees have general expertise, but only the author really knows what was observed. Should authors have to outline their logic? I think so, but I know I am in a minority.
I should also declare an interest in that last comment. I published a series of papers devoted to determining the substitution patterns on red algal polysaccharides, and I followed that logic. Accordingly, the papers tended to have a large number of set relationships, and a number of matrices. Rather interestingly, eventually the editor told me to desist and write papers that looked like everybody else's. Since I am not paid to write papers, and they made no difference whatsoever to my well-being, I simply desisted.
