Another important part of human thinking is the emotional aspect.
In mathematics, what is intriguing, puzzling, interesting, surprising, boring, tedious, exciting is crucial; they are not incidental, they shape how we think.
However, in formalized expositions, this psychological difference vanishes.
In the same way, any idea in mathematics can be thought about in many different ways, with competing advantages.
In exploring this phenomenon, Hadamard produced one of the most famous and cogent cases for the existence of unconscious mental processes in mathematical invention and other forms of creativity.
Written before the explosion of research in computers and cognitive science, his book, originally titled The Psychology of Invention in the Mathematical Field , remains an important tool for exploring the increasingly complex problem of mental life.
Fifty years ago when Jacques Hadamard set out to explore how mathematicians invent new ideas, he considered the creative experiences of some of the greatest thinkers of his generation, such as George Polya, Claude Le;vi-Strauss, and Albert Einstein.
It appeared that inspiration could strike anytime, particularly after an individual had worked hard on a problem for days and then turned attention to another activity.
The content is almost the same (for correlation, you first project to a hyperplane before measuring the cosine of the angle), but the human psychology is very different.
Each mode of thinking has its own power, and ideally, people harness both modes of thought to work together.