  The appearance of linear spaces, describing physical quantities by vectors
and tensors, is ubiquitous in all of physics, from classical mechanics to the
modern notion of local Lorentz invariance. However, as natural as this seems to
the physicist, most computer scientists would argue that something like a
"local linear tangent space" is not very typical and in fact a quite surprising
property of any conceivable world or algorithm. In this paper, we take the
perspective of the computer scientist seriously, and ask whether there could be
any inherently information-theoretic reason to expect this notion of linearity
to appear in physics. We give a series of simple arguments, spanning quantum
information theory, group representation theory, and renormalization in quantum
gravity, that supports a surprising thesis: namely, that the local linearity of
space-time might ultimately be a consequence of the linearity of probabilities.
While our arguments involve a fair amount of speculation, they have the virtue
of being independent of any detailed assumptions on quantum gravity, and they
are in harmony with several independent recent ideas on emergent space-time in
high-energy physics.
