Also, "integrable models" in mathematical physics -- with much the same algebraic structures -- are well understood in dimension 2, not well in dimension 3, and virtually nothing above that! Well... here a question must be... so, what do you think? :D
The statements that follow the question are meaningless. Integrable models are those where the conservation laws are sufficient to constrain the dynamics so that only the initial conditions matter. Non-integrable models are those for which the conservation laws aren't sufficient to fully constrain the dynamics.
The non-trivial question is how to deduce the existence-or not-of non-trivial conservation laws, given some representation of the dynamics.
Most models aren't integrable, which means they display deterministic chaos. So the problem there is to deduce the probability measure on the space of configurations.
While the role of the dimensionality of the space of independent variables does play a role, its role is much more subtle than just knowing how to count to three.
Stam Nicolis: thank you for explaining your views. But have you ever heard of Yang--Baxter equation (YBE)? Which governs 1+1-dimensional exactly solvable QFT's, as well as 2-dimensional integrable models in statistical physics?
At the same time, YBE plays a prominent role in topology (do you know where exactly?).
And there are very far-reaching generalizations: the hierarchy of `n-simplex relations' and also the hierarchy of `(n+2)-gon relations'.
Sure, but, so what? The fact that in 1+1 dimensions something special occurs, that's described by the YBE, doesn't contradict what I stated. And, while the YBE has applications in other parts of mathematics, that, by itself, isn't relevant for deducing the specific properties of integrable models and even less so for how to compute the relevant quantities in each case.
The methods used to solve a problem aren't that significant; what is significant is the solution itself.
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