  We prove that a finite dimensional algebra is $\tau$-tilting finite if and
only if it does not admit large silting modules. Moreover, we show that for a
$\tau$-tilting finite algebra $A$ there is a bijection between isomorphism
classes of basic support $\tau$-tilting (that is, finite dimensional silting)
modules and equivalence classes of ring epimorphisms $A\longrightarrow B$ with
${\rm Tor}_1^A(B,B)=0$. It follows that a finite dimensional algebra is
$\tau$-tilting finite if and only if there are only finitely many equivalence
classes of such ring epimorphisms.
