Basically, I suppose the justification for studying these lattice models is very simple: they are relevant and they can be solved, so why not do so and see what they tell us?
R.J. Baxter, Exactly Solved Models in Statistical Mechanics (Academic Press, 1982)

I got really fascinated by these (1+1)-dimensional models that are solved by the Bethe Ansatz and how mysteriously they jump out at you and work and you don't know why. I am trying to understand all this better.
Richard Feynman (1988)

Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field.
P.A.M. Dirac, The principles of Quantum Mechanics (Oxford, 1930)