As confirmed by a series of experimental data, there are two different cognitive systems relating to mathematical skills. The first system is not based on symbols, and it is approximative; it is based on the estimation of quantities; and it involves both a simple process of comparison and a series of basic arithmetical operations like addition and subtraction. The second system is based on symbols, and it is language- and culture-dependent; it is typical of adults; and it is founded on the ability of counting, therefore on a numerical system and on all arithmetical operations. . The aim of this book is to answer these difficult questions by turning to the dual process theories. This theoretical approach is widely used by theorists focusing on reasoning, decision making, social cognition, consciousness, etc. In this book, for the first time this theoretical approach is applied to the studies on mathematical knowledge with the aim of detailing the results brought about by psychological and neuroscientific studies conducted on numerical cognition by a few neuroscientists and laying the foundations of a new potential philosophical explanation on mathematical knowledge.