The place of mathematics in hypotheticodeductive processes and in biological research is discussed. (Natural) Selection is defined and described as differential elimination of performed sets at any level. Sets and acting sets are groups of units (themselves sets of smaller units) at any level that may or do interact. A pseudomathematical equation describes directional change (evolution) in sets at any level. Selection is the ram of evolution; it cannot generate, but can only direct, evolutionary energy. The energy of evolution is derived from molecular or chemical levels, is transmitted upwards through the increasingly complex sets of sets that form living systems, and is turned in directions determined by the sum of selective processes, at different levels, which may either supplement or oppose each other. All evolutionary processes conform to the pseudomathematical equation referred to above, use energy as described above, and have a P/OE (ratio of programming to open-endedness) that cannot be measured, but can be related to other P/OE values. Phylogeny and ontogeny are compared as processes af directional change with set selection. Stages in the evolution of multi-cellular individuals are suggested, and are essentially the same as stages in the evolution of some multi-individual insect societies. Thinking is considered as a part of ontogeny involving an irreversible, nonrepetitive process of set selection in the brain.