Many things in life are more than the sum of their parts. Whether its the behaviour of crowds of people, flocking birds or shoaling fish, the unpredictable patterns of the weather or the complex structure of the Internet, it's often the interaction between things, rather than the things themselves, that generates complexity. It's a challenge to science, whose traditional approach of taking things apart and looking at the individual bits doesn't work when faced with emergent complexity. But there are mathematical techniques to understand this phenomenon.
What would you think if the nice café latte in your cup suddenly separated itself out into one half containing just milk and the other containing just coffee? Probably that you, or the world, have just gone crazy. There is, perhaps, a theoretical chance that after stirring the coffee all the swirling atoms in your cup just happen to find themselves in the right place for this to occur, but this chance is astronomically small.
Results in mathematics come in several flavours — theorems are the big important results, conjectures will be important results one day when they are proved, and lemmas are small results that are just stepping stones on the way to the big stuff. Right? Then why has the Fields medal just been awarded to Ngô Bào Châu for his proof of a lemma?