How the biologists were able to determine the functions of body organs? Using a specific example organ for an answer would be ideal.
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A very common way of getting information on the function of an organ, part of an organ (such as a region of the brain) or a protein is to look at the outcome (symptoms or phenotype) in cases where it is not functional. E.g., a lot has been understood about brain function for speech production by studying aphasia in stroke patients (see wikipedia) and linking the symptoms with the brain region that is damaged.
This is still the way we mostly study living cells or organisms: mutants defective for such or such protein, or in which external disruption (such as ablation) is used and the resulting phenotype or symptoms are studied.
This is indeed very different from what can be done for research in physics e.g., where reduced systems with simpler functions can be studied. The approach in biology is much closer to reverse engineering, where a given system has to be studied as a whole but a rational approach on some subpart allows to build an understanding of that subpart.