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Vladimir_Nesov comments on Logic: the science of algorithm evaluating algorithms - Less Wrong
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You are looking for essences at every turn, ask what certain things "really are", but the world doesn't work like that. There are often multiple explanations (interpretations), none of them "more real" than the other. And in math that is more apparent than elsewhere, so morphisms of every kind between all kinds of structures get introduced to describe particular explanations, relations between different things that show in what way do properties of one thing tell something about the other.
Exactly. There are enough connections within mathematics that almost any field can be seen as an abstruse development of almost any other field, and indeed, most mathematicians have the belief that their field is the most truly fundamental one. What we just saw here is a computer scientist looking to suborn mathematical logic as a special case; naturally, the opposite interpretation is just as valid.