In an extensive anthropological joint research project concerning the necessities in the development of life and civilisation my group is concerned with mathematics. This forum seems to be extremely well-suited to answer the following question:

Given the case that a civilisation independent of mankind is or will be existing. How would their mathematics look like and

why?

Some teasers: Integers from simple counting to the Binomial Theorem, Geometry including Pythagoras' Theorem, Probability Theory, and Analysis seem to be dictated by practical requirements. Research on Prime Numbers may be a necessary by-product of calculating with integers. What about more advanced concepts like Grothendieck universes or Category Theory? Can even somewhat be said about branches that we don't yet have exploited ourselves?

(This question should be made wiki but I could not find the button. Seems that something has changed during the recent year.)