It is often claimed that 2 is special because it is the only even prime. This is circular because "even" is defined in reference to 2. But the reasons that mod 2 is more interesting than mod n for n≠2 are reasons that 2 is special.
Groups arise naturally as automorphism groups. I tend to think of "group" as meaning "thing that is isomorphic to an automorphism group", and take the group axioms as a convenient way to characterize which things are groups.
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Yes, but what is an automorphism group? A collection of endomorphism tuples (m_1, m_2) such that m_1 * m_2 = m_2 * m_1 = 1, composing under the rule (m_1, m_2) * (n_1, n_2) = (m_1 * n_1, n_2 * m_2). But perhaps a 2 has been snuck in here implicitly…
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Given a set with some sort of structure on it (which necessarily includes =), its automorphism group is the set of all functions from the set to itself preserving that structure, endowed with every operation that can be naturally assigned to such sets in a constructive manner.
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By preserving that structure, I mean in the iff sense, not just that structure must be carried over in one direction.
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Admittedly what I said doesn't apply to automorphism groups of objects in non-concrete categories. I think of these as "things that work in much the same way that sets with structure do".
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Alright, now the 2 comes from your focus on "functions"; that is, on particular 2-ary relations. But we might also look at 3-ary correspondence relations. E.g., R(x, y, z), such that for any choice of x, y, or z, there are designated corresponding values for the other arguments.
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Once you focus on the relations that are like functions in that they satisfy conditions like "for any value of _ there is unqiue _", there is something 2-ish going on, because the places in the relation are being partitioned into 2 pieces, the domain piece and the codomain piece.
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Even if you try to generalize bijections by using a condition that the relation must be function-like for multiple domain/codomain partitions (e.g. for any values of any k places, there are designated values for other n-k), each partition is into 2 parts.
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A ternary correspondence is a subset of X x Y x Z such that each element of X, Y, or Z appears in precisely one element of the subset.
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