  What is the dimension of spacetime? We address this question in the context
of the AdS/CFT Correspondence. We give a prescription for computing the number
of large bulk dimensions, $D$, from strongly-coupled CFT$_d$ data, where
"large" means parametrically of order the AdS scale. The idea is that unitarity
of 1-loop AdS amplitudes, dual to non-planar CFT correlators, fixes $D$ in
terms of tree-level data. We make this observation rigorous by deriving a
positive-definite sum rule for the 1-loop double-discontinuity in the flat
space/bulk-point limit. This enables us to prove an array of AdS/CFT folklore,
and to infer new properties of large $N$ CFTs at strong coupling that ensure
consistency of emergent large extra dimensions with string/M-theory. We
discover an OPE universality at the string scale: to leading order in large
$N$, heavy-heavy-light three-point functions, with heavy operators that are
parametrically lighter than a power of $N$, are linear in the heavy conformal
dimension. We explore its consequences for supersymmetric CFTs and explain how
emergent large extra dimensions relate to a Sublattice Weak Gravity Conjecture
for CFTs. Lastly, we conjecture, building on a claim of arXiv:0908.0756, that
any CFT with large higher-spin gap and no global symmetries has a holographic
hierarchy: $D=d+1$
