In article ,
Shankar Bhattacharyya wrote:
(Paul Schlyter) wrote in
:


And that's the case with all our knowledge: it's based on at least
_some_ assumptions, assumptions which appears to remain valid
through an extremely large number of observations and therefore we
trust them. But we're still dependent on these assumptions.

Absolute proofs exist only in mathematics and in formal logic, but
not in the natural sciences.

Would you actually claim that that is really true of mathematics and
formal logic?

Mathematics depends on a set of axioms. Axioms are "obvious
assumptions". Formal logic provides the tools of inference by which we
extend axioms into theorems and evaluate information for consistency,
implication, so on so forth.

The absolute proofs of mathematics depend on the assumptions.

In some sense, in possibly some situations, maybe in many, the proofs
may not depend on the assumptions, in a mechanistic sense. However, what
the proofs prove presumably does depend on the assumptions. Otherwise we
would not need axioms.

You're right here, of course: the axioms of mathematics defines the
"universe" which mathematics "lives" in. You can change the axioms
and get another "math universe" which is equally valid and sometimes
even can be useful. One good example is Euclids parallell axiom:
"Through a point beside a line, one can draw exactly N lines parallell
to the first line" where N=1 for classical plane geometry. By
changing N to 0 you instead get e.g. spherical geometry, and by
setting N to "larger than 1" you get the open, "saddle-shaped" geometry.
These alternate geometries were mathematical curiosities when first
invented, but later turned out to be useful for e.g. cosmology.

Anyway, my real point wasn't about the existence or non-existence of
axioms, but about the existence or non-existence of absolute proofs:
such proofs exist only in math and formal logic (with suitable axioms
of course). They don't exist in sciences attempting to deal with the
real world, such as physics, chemistry, biology, psychology, sociology
which attempt to deal with, in turn, more and more complex systems.

--
----------------------------------------------------------------
Paul Schlyter, Grev Turegatan 40, SE-114 38 Stockholm, SWEDEN
e-mail: pausch at stockholm dot bostream dot se
WWW: http://www.stjarnhimlen.se/
http://home.tiscali.se/pausch/