Empiricism can't prove the Law of Non-contradiction; it is an axiom, a conclusion without proof. Physics armed with the Law of Non-contradiction has made many wildly inaccurate predictions, as well as failing to predict the 96%of the universe that dark energy and dark matter together comprise.

In math, Godel indicates you can be consistent, or complete. Nature is complete; therefore, nature is inconsistent. Evidence of nature's inconsistency is all around me: in the contradictory lines of mountains, in the juxtaposition of very different rocks in stream beds, in conflicting birthdates for Julius Caesar on Wikipedia.