mathematics & dialectics (fwd)
Alex Trotter
uburoi at panix.com
Fri Feb 24 21:26:40 MST 1995
[I was intrigued by some of the recent discussion about dialectics,
specifically, that which concerned mathematics. But since my knowledge of
mathematics is rather limited, I forwarded
a couple of those posts to a friend of mine who is a mathemetician.
Following is his response. Would anyone care to comment?--AT]
---------- Forwarded message ----------
Date: Fri, 24 Feb 1995 16:37:37 EST
To: uburoi at panix.com
Subject: RE: mathematics & dialectics
Alex, I am always skeptical of assertions by non-mathematicians concerning
the purported existence of contradictions in mathematics or the necessity
of the existence of such contradictions. The citing of the work of Godel
and Turing is common in these assertions and is inevitably a result of not
understanding what their work shows and doesn't show. Suffice it to say
that Godel proved (among other things) that in virtually any formal system
it is possible to come up with statements in the language of that system
which can be neither proven false nor true *within* the given formal system.
This is not at all the same as saying that there exist contradictions in
the formal system. Discussing this further requires a careful examination
of the notions of decidability, provability, interpretations,models and a
few other concepts familiar to anyone who has taken a course in Model Theory
which is a branch of mathematics or mathematical logic. Have you ever done
any reading in this area? If not, I will try to find something written for
intelligent lay people and refer it to you.
Here is a little paradox that results from Godel's work: Given a formal
system (ie a language, rules for manipulating the language and rules of
deduction and inference) and given a statement which is undecidable in
the formal system - i.e. one which cannot be proven either true or false
within the given formal system - one may create two new consistent formal
systems by merely appending the undecidable statement to the given formal
system and asserting that the statement is True in one of the systems, and
is False in the other one. Each new system is consistent and they differ
only in assigning opposite truth values to a particular statement. But
there is no contradiction in any of this. Each formal system has a different
model, they are not "pointing" to the same thing. This happened a number
of times in the history of mathematics and created much worry and gnashing
of teeth on many occasions. The most famous incident being the parallel
postulate in Euclidean geometry. Well, enough of this.... thanks for the
mail. If any of the aforementioned is either new or interesting to you and
if you'd like to have a reference to it, let me know.
Cheers,
Ray
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