The Essence of Mathematics

The history of Mathematics evinces a transition in the concept of Number from concrete to abstract, i. e. from tool of enumeration, to Natural Number, to Integer, to Rational Number, etc., to, more recently, Transfinite Number. A central debate in Philosophy of Mathematics is whether the essence of Mathematics is in its concreteness, i. e. its rootedness in, experience, or in its abstractness, i. e. in which its structures are purified of any contingency. Cassirer sees the development itself as most significant, because it symbolizes the growth of Human Knowledge from particularity to universality. Hence, in his System, the essence of Mathematics is that it serves as a transition between empirical and theoretical knowledge. Formaterialism agrees with Cassirer that the entire development is significant, but disputes the need to seek the essence of Mathematics outside of its sphere, i. e. in its relation to other branches of knowledge. For, it regards Number as fundamentally Ordinal, and finds the nature of Number sufficiently entailed in the notion of enumeration, in the establishment of a unit for the basis of further operations. That is, the unit is concrete, and what it anticipates is abstract, so, different theories tend to focus on one aspect or the other. For example, counting is the recursive process of achieving, and then exceeding, finitude, so, all counting can be described as a process of 'transfinition'. Hence the 'first Transfinite Number' is always the next number. So, the Transfinite realm of some theories is the result of a detachment and an hypostasization from the process of counting, and neither a self-subsistent Platonistic world nor a fulfillment of a potential.