Math and Logic

'Counting' can be, semi-formally, defined as 'n+1', i. e. the addition of 1 to any given Number. Likewise, 'is less than' can be defined as a relation between any n and n+1. Now, 'Inference' can be, semi-formally, defined as 'If n is less than n+1, and n+1 is less than (n+1)+1, then n is less than (n+1)+1. Now Counting is the fundamental operation of Mathematics, and Inference is the fundamental operation of Logic. Hence, as Wittgenstein has argued, Mathematics and Logic are two different 'language games', and as these examples demonstrate, Mathematics is irreducible to Logic, despite the eminent convoluted efforts to prove otherwise, and, likewise Logic is irreducible to Mathematics. In Formaterial terms, Mathematics is fundamentally a Material operation, i. e. a perpetual extension of a series, whereas is Logic is fundamentally Formal, i. e. is based on the Middle term of an Inference homogenizing a series of terms. This is not to assert that they are each purely one or the other, since they are obviously both elaborate combinations of both, like all Systems. The point, rather, is to isolate the essential feature of each, in order to highlight their essential difference.