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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, e…