Why care about the interpretation of mathematical diagrams

Judges use  photo finish to determine who won a race, a driver stops at a corner to ask a passer by for directions, a radiologist examines a patient’s x-ray before giving diagnosis, a traveller checks the screen at the airport to get information about her flight, a scientist checks the reading on her nanometer to determine the length of her samples, a mathematician looks at a diagram to gain insight into a new conjecture, etc. What all these cases have in common is that in all of them a person tries to get information about the world not by direct observation but by the use of representations of different sorts. Just as the radiologist need not have any direct contact with the patient, we usually do not need direct contact with whatever aspect of the world we want to know about. In every case, the information might be more or less accurate, the method we use more or less reliable, but in all of them the information is mediated by a representation: a photograph, some words, an x-ray, etc. Consequently, in every case, the person jet go through two different cognitive processes in order to get the information she wants: She must determine both the content of the representation, and also whether she ought to trust the representation and, therefore, incorporate the content of the representation to her system of beliefs about the world or not. The lab scientist must know both how to use her instrument in order to get a reliable reading,  but she must also know how to read it to extract this information from it. An error in either process could result in a bad belief, either false or unjustified.
In other words, we must distinguish between the hermeneutic and the properly epistemic aspect of the use of representations. We need to go through a hermeneutic process of interpretation to determine how the world is represented in the representation, while we need to go through a process of epistemic assessment to determine whether the world actually is as represented in the representation. Information and abilities relevant to one process might not be relevant to the other. There will be cases where the difference will be big, and others where it might be subtle and difficult to draw. When we as for directions, for example, our linguistic skills will be essential for the hermeneutic process, but they will likely be of little use when deterring whether to trust the person giving us directions or not. When a scientist looks at a sample on the microscope, both her interpretation of the image and her trust on the information therein contained are permeated by her knowledge of the causal process.
That is probably why this distinction is well known and assumed in literature on testimony; unfortunately its importance has not been as widely recognised in the rest of the cases. This has given rise to unnecessary confessions in the literature, specially in the philosophy of science, where the hermeneutic and the epistemic questions are both closely dependant on the causal processes taking place in the instrument. Another area of inquiry where the distinction has not been properly addressed is in mathematics, where the epistemic question has been given the lion share of the attention, while very little work has been done on the interpretation of any other kind of representation not in a formal language. My work aims to partially fill this void. In particular, I want to argue for the claim that Geometrical diagrams are interpreted more like pictures than like formulas. This means that general symbolic conventions play but a very small part in determining their content; instead, the appearance of the diagram guides us in our interpretation process so that the default interpretation of a figure is that it stands for a similar figure, and contextual information serves to defeat this presupposition, so that whatever conventions are needed to fix on an intended interpretation are ad hoc and circumscribed to the particular use of the diagram.