In the case of Mathematica, 1-based indexing also makes sense in the context of it being a lisp-like language where everything is an expression that can be repr

In the case of Mathematica, 1-based indexing also makes sense in the context of it being a lisp-like language where everything is an expression that can be represented as a tree. So in the case of exp = x + y + z this is syntactic sugar for exp = Plus[x, y, z] As expected with 1-based indexing exp[[1]] returns x, but the zero index can also be used to access the head of the expression, so exp[[0]] returns Plus. Another nice property of 1-based indexing in Mathematica's case is that by using negative indices you can access elements in the reverse direction, so exp[[-1]] would return z, and exp[[-3]] would return x.