There is a famous quote by the mathematician V. I. Arnold that goes like this:
Every mathematician knows it is impossible to understand an elementary course in thermodynamics.
The source is Contact Geometry: the Geometrical Method to Gibbs' Thermodynamics, and It goes on like this (bold is mine):
The reason is that thermodynamics is based —as Gibbs has explicitly proclaimed— on a rather complicated mathematical theory, on contact geometry.
Then the author starts to explain, I imagine, how thermodynamics can be rigorously formulated using the formalism of contact geometry. I say "I imagine" because I have to admit that such a formalism is a bit too obscure for me, and I have no familiarity at all with the concept of "contact geometry". As a matter of fact, it is the first time that I hear about it, and the definition Wikipedia gives of it is completely unintelligible to me...
What I would like to know is, in terms accessible to someone with a "basic" mathematical background like me (mostly calculus): how exactly is the formalism of thermodynamics based on contact geometry?