The Patterns of Counting: From One to Zero and to Infinity

    Numbers arise from counting. The addition and product rules are twoprinciples to follow when facing counting finite number of objects. However, when facing counting infinitely many objects, cardinals arise in set theory by applying Cantor's one-to-one correspondence, but cardinals seem to produce no rich mathematics so far. In this talk we demonstrate a few examples of counting discrete and continuous objects from combinatorial viewpoint of finitely additive measures. These examples are selected from the topics of subspace arrangements, chromatic polynomials, group arrangements, Grassmannians, and counting points of algebraic varieties. The conclusion is that various polynomials and power series arise from the counting patterns of infinitely many objects with structures.