In this highly provocative book, Scoular presents a new paradigm for mathematics. Mathematics is practiced today as if it is the "science of quantity." This forces mathematicians to hide the qualitative elements of mathematics and accept infinite quantities. However, by practicing mathematics as if it is the "science of the qualitative infinite," it is possible to formulate a new foundation for mathematics that both represents the irrational numbers and is philosophically consistent. This book is for philosophers, mathematicians and, most importantly, the general reader who is matheamatically, philosophically or scientifically inclined. It represents a new way of looking at the infinite and, therefore, the philosophy of mathematics.