We assume that a one-dimensional object is linear, because by definition a line is an extension along one dimension. But there can be no linearity without measurement and there can be no measurement in a first dimension without a second dimension, because there is no way to ascertain units of length within the confines of one dimension. You simply cannot compare one unit with another. A measuring stick cannot be created without the second dimension.
So because measurement is impossible in the first dimension, no finite linear object can exist in it. To propose that there may be only one object in the first dimension, an infinite line, is tantamount to saying there is no object at all, only dimensionality itself. To speculate that there could be a finite line whose existence is not invalidated by its lack of a property of measurability from within its own dimension, is to ignore that the quality of finitude is dependent on measurability; it is not an a priori property but one that has to be derived by systematic observation. Therefore, the first dimension is nonlinear—on its own terms.
However, here in the world of the third dimension we are very linear indeed. Even though we don't have to think in one-dimensional terms, we do anyhow. There is something about the crushing gravity of materiality that compresses thought. Perhaps Hamlet could be bounded in a nutshell and count himself a king of infinite space. But there are too few Hamlets in the world, who have discovered the existence of depth.