This work argues against Frege's theory of arithmetic and points out the logical and mathematical discrepancies of his system according to which arithmetic could be built on purely logical grounds and hence could have foundations devoid of intuitions. It concentrates mainly on the "definite article premise" as one of the faults of Frege's theory of arithmetic where the definite article serves to classify something as an object. Although Frege's formulation had failed, it is so alluring to build arithmetic on logical grounds with the new rules introduced by him and again so tempting to ascribe numbers to concepts, that appearance of these discrepancies did not stop pursuing logicists trying to find better formulations still for similar purposes. This book explains that why such attempts fail and why intuitions are required in mathematics.