In 'Thoughts on the Principles
of Descartes', Leibniz -- in the context
of discussing the demonstration of axioms --
observes that had Euclid given a "good
definition" of straight lines, he could
have proven that two straight lines can
have only one point in common.
I remember Leibniz saying somewhere
that had Euclid adequately defined the
geometric figures, then he could have
dispensed with the Postulates (but
not the mereological axioms, of course),
as for example in Book V Euclid was able
to give adequate definitions and dispense
with postulates (but not of course the
mereological axioms).
I have just madly searched Leibniz,
but have not been able to find this more
general claim.
I have a series of questions.
(1) Is this a false memory?
(2) Even if it is a false memory, would
Leibniz have agreed to it?
(3) Would Euclid have agreed to it, that
is, would Euclid have taken it for granted
that he needed ths postulates only because
either his definitions in Book I were
inadequate or his powers of deduction
"from" the definitions were inadequate?
(4) Historically, rigor in mathematics
seems to be firmly associated with giving
good definitions. We can begin our
deductions of a mathematical subject matter
by framing definitions in, e.g., the
language of set theory.
(4a) Perhaps a framework
for making good definitions
would have been available to Euclid had he
expanded the mereological structure implicit
in his axioms?
(4b) Why did Hilbert in FoG not go
the route of definitions in a more
general frame rather than "axioms" not
dependent on definitions?
(4c) When did -- and
why -- this break with the Euclidean style
of rigor (good definitions first) occur?
Clearly (?) Hilbert's axiomatic thinking"
is not Euclidean, but, rather, sets the
issue of definition aside in favor
of special axioms.
robert tragesser