Bloch states the principle in Latin as: Nihil est in infinito quod non prius fuerit in finito, and explains this as follows: Every proposition in whose statement the actual infinity occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in finite terms.
Bloch mainly applied this principle to the theory of functions of a complex variable. Thus, for example, according to this principle, Picard's theorem corresponds to Schottky's theorem, and Valiron's theorem corresponds to Bloch's theorem.
Based on his Principle, Bloch was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes, Hayman's result that an exceptional set of radii is unavoidable in Nevanlinna theory.
In the more recent times several general theorems were proved which can be regarded as rigorous statements in the spirit of the Bloch Principle:
A family of functions meromorphic [analytic] on the unit disc is not normal if and only if there exist (a) a number (b) points (c) functions (d) numbers such that spherically uniformly [uniformly] on compact subsets of where is a nonconstant meromorphic [entire] function on 
Zalcman's lemma has the following generalization to several complex variables. The first thing to do is to make a precise definitions.
A family of holomorphic functions on a domain is normal in if every sequence of functions contains either a subsequence which converges to a limit function uniformly on each compact subset of or a subsequence which converges uniformly to on each compact subset.
For every function of class we define at each point a Hermitian form and call it the Levi form of the function at
If function is holomorphic on set This quantity is well defined since the Levi form is nonnegative for all In particular, for the above formula takes the form and coincides with the spherical metric on
Now, we have is the following important characterization of normality, based on Marty's theorem.
Suppose that the family of functions holomorphic on is not normal at some point Then there exist sequences such that the sequence converges locally uniformly in to a non-constant entire function satisfying .
Let X be a compact complex analytic manifold, such that every holomorphic map from the complex plane to X is constant. Then there exists a metric on X such that every holomorphic map from the unit disc with the Poincaré metric to X does not increase distances.