Re: Galileo's Paradox


Tonico, let me revisit my comments below to present the same analysis
in slightly different terms.

If we take some conclusion, C, and want to know if it's true there are
two different ways to proceed.Aristotelian syllogistic inference bases
the truth of C on the truth of its constituent premises such as B. In
other words it says "if B then C" which is a truism because C would
certainly be true if B were true.

However then we're just faced with exactly the same problem with B. So
we further regress analysis of the truth of C to the truth of B to
find the truth of B relies on the truth of some constituent premise of
B such as A with the result that we wind up with "if A then B then C".

This is exactly how classical syllogistic inference has always worked
in the context of science and mathematics. To support the truth of
some conclusion such as C there is an indefinite regression of
problematic premises and this regression is what I call empiricism. In
ordinary science this regression stops at what would appear a logical
boundary of sensory and perceptual experience whereas in mathematics
it stops with axioms and axiomatic assumptions of truth.

Now this doesn't mean that truisms like "if A then B then C" cannot be
true only that their truth can never be known in exhaustive terms. The
most we can hope for is to stumble on some syllogistic regression that
turns out to be true and employ it to ground further speculations. In
effect syllogistic regressions such as "if A then B then C" become a
line of reasoning or in the parlance of modern math a "model" of truth
because the truth can never be known absolutely with such a method.

Now I analyze the same problem from exactly the opposite perspective.
Instead of asserting the truth of C relies on the truth of constituent
premises I maintain the truth of any conclusion such as C relies on
the falsity of alternatives to C, in other words what is "not C'.

Thus we form a tautological regression of "C, not C" instead of the
syllogistic regression "if A then B then C" and find that C can and
must be true only if "not C" must be false and "not C" must be false
only if it is self contradictory.

In any event I hope this clears up what I mean by empiricism and truth
in the context of mathematics and science.

On Fri, 01 Dec 2006 12:04:42 -0700, Lester Zick
<dontbother@xxxxxxxxxxx> wrote:

On 30 Nov 2006 23:55:39 -0800, "Tonico" <Tonicopm@xxxxxxxxx> wrote:

[. . .]

Perhaps it is that we don't really understand what each other means by
"empirical". For example, what empirical evidence (of what, where,
when...?) does the axiom stating the existence of a unit element in
group theory have? Or the axiom in Topology that states that the empty
set is part of the set of open sets?

I agree we don't understand the term "empirical" the same way. Not my
fault since I've discussed the subject at length over the past couple
years here and elsewhere. Basically any tautologically undemonstrated
judgment is empirical. Doesn't matter whether the judgment is sensory,
perceptual, cognitive, or whatever. If you assume an axiom such as "a
straight line is the shortest distance between points" the assumption
is empirical until and unless demonstrated true analytically. The same
applies to definitions.

Most people completely misunderstand the meaning of an empirical
judgment. Most think it means getting out the tape measure, scales,
and so forth. The problem originated with Aristotle and his concept of
syllogistic inference. Aristotle was history's first empiricist in
formal terms. He found he could not establish the truth of any
conclusion syllogistically except by regression to further premises
whose truth he could not establish either except by further regression
ad infinitum. Which meant he could establish no truth syllogistically
at all without some kind of true basic premises which he set out to
find in unreducible perceptual terms. Which left us epistemologically
exactly where we are today in terms of all kinds of mathematical and
scientific methodologies.

In point of fact however empirical judgment is nothing more than input
to a process of tautological regression whose ultimate goal is
reduction to self contradictory alternatives. That's how the mind and
brain work, tautological rather than syllogistic inference because it
can produce reductions to truth in exhaustive mechanical terms.

~v~~

~v~~
.

Relevant Pages

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