FOM: Uniqueness of axioms ? for Tennant, Detlefsen, et al.
Robert S Tragesser
RTragesser at compuserve.com
Mon Dec 22 06:41:26 EST 1997
I've been following the thread on independent
axiomatization from the point of view of the
question of whether or not arithmetic is a
Science in the Aristotelian sense [and under-
standing this question in the light of Mancosu's
history of the question in the late Renaissance,
and the role of the question in the emergence of
"modern" mathematics -- and this in turn was
inspired by Evert Beth's very hard won observation
that we go desparately if we try to understand
FOM against the background of Kant rather than
Aristotle).
Detlefsen had observed that in an Aristotelian
science the axioms should be atomic, and one way
of understanding this is in terms of their mutual
independence. Hence the foundational importance
of the question of independent axioms.
One learns from Mancosu that it is permitted
in an Aristotelian science that there are valid
deductions which are not causal or explanatory.
(From a point of view, RAA proofs have this
character.) Thus one must also say which of the
possible deductions count as explanatory. One
then needs to show that any deduction from the
axioms can be transformed into or be replaced by
an explanatory/direct proof. So it must be possible
to choose (mutually indepedent) axioms and logic
so that the theorems of arithmetic can be given
such direct proofs?
There is an interesting complication already
suggested by Harvey Friedman in another context:
that not all the truths of arithmetic are
essential truths. That is to say, some truths
of arithmetic may be accidental truths -- they
cannot be understood on the basis of WHAT-IS alone.
(We could then have the prospect of a system of
arithmetic being complete with respect to essential
truths or elementary/elemental truths, but not
with respect to all truths?
QUESTION OF THE UNIQUENESS OF THE AXIOMS:
First, it does seem that we do not need strict
logical independence of the axioms, but only that
no axioms be deducible from other axioms BY
EXPLANATORY DEDUCTIONS (e.g., by a normal proof).
The important thing is that the axioms be atoms of
expanation, not logical atoms.
Seocond, it is important that one be able to
detect the axioms qua atoms of explanation. Is
there then some sense in which the axioms of
PA may be regarded as forced, unique?? Here
is what I have in mind: given axioms which are
independent at least up to explanatory deductions,
is there some strong sense in which any other such
set of axioms are inter-explanatorily deducible?
WHAT'S THE RIGHT LOGIC FOR PURSUING THESE QUESTIONS?
Recall that Aristotelian axioms must have the
character of being definitions. Thus the number-
theoretic books rather than the geometric books of
Euclid are paradigmatic here. [ Seidenberg
argued that Euclid wanted Book I to be based on
Definitions alone, but couldn't resolve all
the Postulates into Definitions.]
That is, the axiom-atoms are to be atomic
elements of species/concepts/ideas. This
suggests that something more like a combinatorial
or lambda calculus might be more appropriate for
resolving arithmetic into its specific/conceptual
elements, rather than a predicate calculus.
I see the fundamental or foundational issue here
is of course not wether we can modernize Aristotle,
but whether or not mathematics can be resolved into
self-sustaining, self-standing spheres, walled
cities, rather than megaopolis sprawl (of the sort
Lakatos envisions)?
robert tragesser
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