We give a general overview of progress and expectations in three directions. 1. What kind of mathematical results can we prove only by going beyond the usual axioms for mathematics (ZFC)? (Some systematic investigations in the integers). Buzzwords: templates, Boolean Relation Theory, Order Calculus. 2. Is there a relationship between ordinary commonsense naive thinking and abstract mathematics? (They are unexpectedly close, and in some sense even equivalent). Buzzwords: interpretation, Concept Calculus. 3. In what sense and to what extent can we achieve certainty, at least in mathematics? (With the use of existing computer technology, we can, in various senses). Buzzwords: proof assistants, certificates.