Summary: While the most important role of mathematics in the physical sciences is that of a deductive tool, outside the physical sciences special methods are developed in which mathematics appears also as an instrument of observation and a tool of induction. In nonphysical sciences problems of measurement arise that have few if any counterparts in the physical sciences. Consequently measurement and scaling theory becomes an important branch of mathematics in its nonphysical applications. Statistical inference is also of far greater importance in the nonphysical sciences as they become more rigorous. Finally, the deductive role of mathematics in the nonphysical sciences is somewhat different from that in the physical sciences. The former rests on very few fundamental laws that have not only an empirical but also a vastly important theoretical significance, reflecting as they do the essential properties of space, time, matter, and energy. All physical theories rest solidly on these laws. No counterparts of such basic laws exist outside of physics. The construction of specific mathematical models of specific aspects of nonphysical reality thus becomes the principal task of mathematics in its deductive role outside the physical sciences.