The formalist school of math philosophy thinks that meaningful questions have to be phrased in terms of finite computational processes. But if you try to write a program for determining the truth value of "this statement is false", you'll see it recurses and never terminates:

def f():
return (not f())

See also Kleene-Rosser paradox. This may or may not dissolve the original question for you, but it works for me.

There's more to be said about the paradox because it keeps turning up in many contexts. For example, see Terry Tao's posts about "no self-defeating object". Also note that if we replace "truth" with "provability", the liar's paradox turns into Godel's first incompleteness theorem, and Curry's paradox turns into Löb's theorem.

ETA: see also Abram Demski's explanation of Kripke's fixed point theory here on LW, if that's your cup of tea.