The principle of linear superposition holds for the integers, eg. 3 + 4 = 7. However, when we take the integers out to infinity we now obtain nonlinear behaviour. Infinity plus any amount is still infinity, so linearity has broken down.
It seems paradoxical that the system of integers possess linear behaviour, under addition, and then that linear behaviour fundamentally changes at infinity. How can linearity result in nonlinearity out of seemingly nowhere?
This seems fundamentally paradoxical to me.
A worrying aspect of this is that logical induction breaks down. By induction, we can predict linear superposition for the addition of integers for higher and higher values, and then at infinity the induction process collapses.
Is this not a philosophical worry, in general, for proofs by induction?