Re: Truth Among Mathematikers and Empirics
- From: "T.H. Ray" <thray123@xxxxxxx>
- Date: Sun, 05 Aug 2007 15:24:12 EDT
"T.H. Ray" wrote:
assumption totheorems are true
because the transformation of terms from
result is a closed loop.
Although we might spend weeks discussing the always
popular
question of what constitutes mathematical truth, I
think
that you have captured the essence of what makes a
theorem
a theorem, whether or in what sense it is "true", in
the above
quote. I wonder if you would care to amplify and
clarify what
you mean here.
1) Please follow up to sci.logic as well as sci.math,
as
I am a sci.logic subscriber.
2) Please ignore Lester's gibes. No one takes him
seriously.
--
hz
Thanks, Herb, but I will have to forgo the invitation
at least for the time being. Maybe later. At the
present, I am struggling with a pending job transfer
and other personal matters. Actually, I feel a bit
guilty spending all this time with Zick; it isn't
as taxing or time consuming, though, as a serious
exchange would be.
My reason for making reply to this nonsense at all, is
that the world is largely populated with Zicks, i.e.,
with irrational believers. Mathematics and Logic are
necessarily rational enterprises.
So far as what I write here, you are welcome to quote
me in your other group, and I stand ready to defend any
statement I make, if asked, though I cannot always
promise an immediate reply.
So far as expanding on my statement you quote above, I
tried to do that in the previous post, using that
ring of questions and answers that forms a closed loop
from "the theorem is proved," back to "the theorem is
proved." I reproduce that article in full here, and we
can strive for more depth in due course:
Well, at least Mr. Zick is making a positive statement,
even if it's easily proven logically wrong. His
claim--like that of any believer--is shown to be
irrational.
The fact is, that if existence is self-contradictory,
logic itself doesn't exist. Aristotle's first
principle (identity), and his second (non-contradiction)
necessarily assume existence. The third (excluded middle)
might be jettisoned, but one does not successfully
make claims for existence (even the existence of self-
contradictory existence) without assuming existence.
Think of it in terms of parity--two negative terms
multiplied together, have a positive result.
Were Mr. Zick to get a little education in the subject,
he would very quickly learn that theorems are true
because the transformation of terms from assumption to
result is a closed loop. I know that this has all been
patiently explained by Randy Poe in this thread, but
I will make another attempt for the benefit of lurkers
who might not be convinced that truth can be known
rationally.
Q1: How do we know a theorem is true?
A: The theorem is proved.
Q2: How do we know the proof is true?
A: There are no gaps of deduction between one
statement and another. (All mathematics is the study
of propositions of the form, "A implies B.")
Q3: How do we know that the deductions are true?
A: They are consistent with the axioms from
which the conclusions follow.
Q4: How do we know the axioms are true?
A: The axioms are a self consistent system of
assumptions without proof.
Q5: How do we know the system of axioms is true?
A: We don't.
Q6: How do we know a theorem is true if the system of
axioms may not be true?
A: The theorem is proved.
So if one might be tempted to think that Zick has
brilliantly discovered a flaw in mathematical
logic--reflect on the fact that a theorem is true
_because_ it is tautological. Proofs close the gap
between what we believe is true and what we can prove
is true. Closing the gap with the fewest number of
unproven assumptions is the goal of what is known as
mathematical epistemology.
Much progress has been made in the last 200 years or
so, in demonstrating the limits of how we define
systems, both mathematically and empirically. Another
in this thread has discussed what is probably the
best known example--the demarcation between Euclidean
geometry and non-Euclidean space. Rational truth,
unlike Zick's private notion of truth, is progressive.
Our rational approach to mathematical logic has been
very productive in adding to our knowledge--If Gregory
Chaitin is right in his research into the limits of
computation, even simple arithmetic holds enough
unsolved mysteries to keep us busy for thousands more
years.
Tom
.
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