nt.number theory - Is there a high-concept explanation for why characteristic 2 is special?

I think 2 is not special, we just see the weirdness at 2 earlier than the weirdness at odd primes.

For example, consider Ext_E(x)(F_p , F_p) where E(x) denotes an exterior algebra over F_p. If p=2 this is a polynomial algebra on a class x₁ in degree 1 and if p is odd this is an exterior algebra on a class x₁ tensor a polynomial algebra on x₂. I say these are the same, generated by x₁ and x₂ in both cases and with a p-fold Massey product < x₁,...,x₁>=x₂. The only difference is that a 2-fold Massey product is simply a product.

In what sense are the p-adic integers Z_p the same? One way to say it is that if you study the algebraic K-theory of Z_p you find that the first torsion is in degree 2p-3. If p=2 this is degree 1, and K₁(A) measures the units of A (for a reasonable ring A). If p is odd it measures something something more complicated. Another way to say it is that Z_p is the first Morava stabilizer algebra and there is something special about the n'th Morava stabilizer algebra at p if p-1 divides n. If you study something like topological modular forms, this means the primes 2 and 3 are special.

The dual Steenrod algebra is generated by xi_i at p=2 and by xi_i and tau_i at odd primes. But really it is generated by tau_i with a p-fold Massey product < tau_i,...,tau_i>=xi_i+1 at all primes, after renaming the generators at p=2. (Again a 2-fold Massey product is just a product.)

I could go on, but maybe this is enough for now.