 
Summary: ON COMPUTABILITY
Wilfried Sieg
1 INTRODUCTION
Computability is perhaps the most significant and distinctive notion modern logic
has introduced; in the guise of decidability and effective calculability it has a
venerable history within philosophy and mathematics. Now it is also the basic
theoretical concept for computer science, artificial intelligence and cognitive sci
ence. This essay discusses, at its heart, methodological issues that are central
to any mathematical theory that is to reflect parts of our physical or intellec
tual experience. The discussion is grounded in historical developments that are
deeply intertwined with metamathematical work in the foundations of mathemat
ics. How is that possible, the reader might ask, when the essay is concerned solely
with computability? This introduction begins to give an answer by first describ
ing the context of foundational investigations in logic and mathematics and then
sketching the main lines of the systematic presentation.
1.1 Foundational contexts
In the second half of the 19th
century the issues of decidability and effective calcu
lability rose to the fore in discussions concerning the nature of mathematics. The
divisive character of these discussions is reflected in the tensions between Dedekind
