The Pilates Method was derived from what Joseph Pilates originally called Contrology. In 1945 Joseph Pilates published his first book Return to Life Through Contrology. Jospeh Pilates recognised that the motor functions of the brain controlled the mobility of the body.
It wasn't until the early 1980's that two students of Joseph's protege Romana Kryzanowska (Philip Friedman and Gail Eisen) published The Pilates Method of Physical and Mental Conditioning. It is this work that defined the six key Pilates principles.
Concentration: Pilates requires focus. Concentration combined with physical exercise can improve brain functions, memory as well as build new neural pathways to assist with ongoing mental development.
Control: Practiced well each Pilates movement requires control of the control of the entire body. There are no superfluous actions or breaths.
Centring: All movements in Pilates start from the Powerhouse/Core, which is the centre of the body.
Flow: Pilates sequences are designed to efficiently use energy and smoothly transition from one to the next.
Precision: It is considered better to perform one movement perfectly than to perform multiple repetitions of an action poorly.
Breathing: Oxygen intake and blood circulation have the ability to improve almost all physical conditions. There is a separate FAQ on breathing where you can find out more.