Irony of mathematics
Although many of Hawking's theories are heavily reliant on mathematical equations, he says that the questions behind these equations are more important: "Even if there is only one possible unified theory, it is just a set of rules and equations. What is it that breathes fire into the equations and makes a universe for them to describe? The usual approach of science of constructing a mathematical model cannot answer the questions of why there should be a universe for the model to describe. Why does the universe go to all the bother of existing?"
Irony of black holes
In the first chapter, Hawking tells us that nobody has ever observed a black hole. Ironically, black holes were discovered to exist before they were ever observed.
Irony of fame
When Hawking first published A Brief History of Time, he was not famous and did not think it would sell that many copies. However, it became a best seller and sold over 10 million copies, perhaps due to the inclusive nature of its form and simplistic language. It is now considered a historical scientific text.