Nikolaos Aletras, Dimitrios Tsarapatsanis, Daniel Preoţiuc-Pietro and Vasileios Lampos have performed just such an analysis, described in their paper, Predicting Judicial Decisions of the European Court of Human Rights: a Natural Language Processing Perspective. Specifically, based on an analysis of the textual description of the case, they manage to predict with an accuracy of about 80%, the binary outcome of the case; i.e. whether or not a violation of human rights has been judged to have occured.
P-adic numbers are a way of "filling in the gaps" in the rational numbers to complete the number line. However, a new metric is introduced so that the distance between two points is not the intuitive distance as measured by a ruler. These p-adic metrics allow statements such as "the infinite sum of the powers of 2 equals -1" to make sense.
The Banach Tarski Paradox shows that it is possible to cut a 3D object such as a sphere into 6 pieces and re-arrange them into 2 spheres identical to the original in both size and density. It can be done with as few as 5 pieces and although only translations and rotations are required for re-aligning the pieces, the pieces are dense scatterings of uncountably infinitely many points. If this feels counterintuitive to you, realise that it relies on the Axiom of Choice being true and this is where you should investigate your gut feelings.
Lesson 1 part 1: Euthyphro and the Bad Dad [08:19]