  My purpose is to examine some concepts of mathematical logic, which have been
studied by Carlo Cellucci. Today the aim of classical mathematical logic is not
to guarantee the certainty of mathematics, but I will argue that logic can help
us to explain mathematical activity; the point is to discuss what and in which
sense logic can "explain". For example, let's consider the basic concept of an
axiomatic system: an axiomatic system can be very useful to organize, to
present, and to clarify mathematical knowledge. And, more importantly, logic is
a science with its own results: so, axiomatic systems are interesting also
because we know several revealing theorems about them. Similarly, I will
discuss other topics such as mathematical definitions, and some relationships
between mathematical logic and computer science. I will also consider these
subjects from an educational point of view: can logical concepts be useful in
teaching and learning elementary mathematics?
