The aim of this investigation is a consideration, in a historical context, of the first phase of Gottlob Frege's Project of reducing arithmetic to logic, that is, the development of an ideography to represent number in the ordinal sense. In this article, to provide a context for interpreting this phase of Frege's project, two separate investigations are carried out: firstly, Immanuel Kant's view of pure intuition and his project of constructing mathematical objects in pure intuition are evaluated; secondly, the project of a number of mathematicians and philosophers of developing formal systems as a sequel to Frege's concept-writing to represent mathematical proofs in a rigorous manner is presented. A positioning of Frege's project in view of its predecessors and pursuers provides us grounds to make a critical assessment. The initial phase of Frege's project of reducing arithmetic to logic comprises the construction of a formal system within which the concept of ordering (in a sequence) can be rigorously represented. Frege desires in this way to constitute a medium into which nothing intuitive penetrates in an unnoticed manner. Nevertheless, since the constitution of such a medium requires the treatment of units successively, the existence of a relation of succession or subordination should be assumed at the outset. That is to say, a formal system relies on subordination of its elements for its proper constitution. Therefore, Kant seems to be right when he claims that the number (the schema of number) should be presupposed in each and every activity of quantitative construction. In a way Frege's project of reducing arithmetic to logic seems to work at its outward appearance involves some circularity. A formal system constructed to represent all the true propositions of arithmetic rests on number itself and its order in an ontological sense. In other words Frege's system is subject to an ontological circularity.