You wrote
"In my opinion the word "ontology" is not appropriate here. By the phrase
"ontology of reals numbers" you seem to understand a formal definition of the
concept. However, if by "ontology" we understand a study concerned with the
*nature* (and existence) of real numbers, what IS a real number, really?..."
By "ontology" of the real numbers (perhaps the word should not be used here)
I mean, for instance, the "construction" of reals from the empty set
according to Zermelo Fraenkel set theory in the version of Halmos. There is
one set, which is empty, then we stipulate rules to form other sets; then
there is an axiom of infinity that grants the existence of natural numbers
(as sets) etc., until we reach a point like this: there is a set whose
elements are the pairs of half lines of rationals. This set is there, it
exists. The "ontology" of its elements is granted (from our agreement about
the axioms). Then comes the definition: let as call each element of this
set a Real number". This is what Tall and Winner call "concept definition".
I would not use the expression "definition of the concept" in any other
sense. Then our concept images have to abide by the concept definition as a
consequence of our agreement about the axioms. This was not the point of
view of Dedekind about definitions. This point of view (according to Boyer)
was first expressed by Bertrand Russell in 1924.
Yours truly
Baldino