A point of clarification. Your statement that a "...collection of
observations... can *never* contain a contradiction" needs to be
qualified. I agree with you that the world itself is consistent.
Therefore, if we observe the world accurately then our observations
will be consistent. (02)
However, neglecting observational error is one common way in which
contradictory statements arise. It is notoriously easy to generate
a contradiction by incorrectly stating observations as facts about
the entities being observed. Here is a simple example. Suppose there
is a 7 cm bar being observed by two different sensors, each of which
makes a noisy observation about its length. Specifically, Sensor A
observes the bar length as 7.1 and Sensor B observes the bar length
as 6.8 cm. (03)
* The statement "The bar has length 7.1 cm AND the bar has
length 6.8 cm" is a contradiction. (04)
* The statement "Sensor A observed the bar to have length 7.1 cm
AND Sensor B observed the bar to have length 6.8 cm" is not
It is essential that observations be identified as observations and
not as statements about the entities being observed. It is also
essential to say something about who or what is doing the observing
and about the conditions under which the observation occurs (e.g.,
accuracy of sensors A and B under various lighting conditions;
lighting conditions at time of observation; etc.). Of course, this
information may be unavailable. In this case, it is all too common
to say nothing about potential observational error, and for the
consumer to simply assume the observation is perfectly accurate.
This often generates contradictions. (06)
At 5:54 PM -0500 12/15/07, John F. Sowa wrote:
>That statement has no connection with either mathematics
>or the universe:
>DT> It seems to me that as valid as mathematics may be for the
> > purposes of engineering, architecture, other worldly purposes,
> > and as a tool for analytical thinking, that it's requirement
> > for self-consistency automatically negates its capacity to
> > faithfully model the natural universe and its ever changing
> > landscape.
>First of all, I don't want to defend all uses of logic and
>mathematics because people often apply the wrong kind of
>math to the wrong problems for the wrong reasons.
>However, there are two very important points to make:
> 1. The fact that something x at time t1 is changing to
> something y at time t2 does not in any way create any
> kind of contradiction. Certain kinds of mathematics,
> such as differential equations, for example, describe
> change and rate of change very nicely.
> 2. Any collection of observations about the universe or any
> part of it either at one instant of time or at different
> times can *never* create any contradiction of any kind,
> whether stated in a natural language or translated to logic.
>The explanation of point #2 is very simple:
> a) Any observation statement in logic or English or any other
> language requires only two logical operators: conjunction
> (i.e., 'and') and the existential quantifier (i.e., 'there
> is something'). No negations ('not'), disjunctions ('or'),
> implications ('if-then'), or universal quantifiers ('all'
> or 'every') are required to state an observation.
> b) Therefore, any collection of observations about the universe
> will contain only positive statements whose only logical
> operators are existence and conjunction.
> c) Any collection of statements as in (b) can *never* be
> contradictory with one another. Therefore, all observations
> of the universe or any part of it must always be consistent.
>Contradictions can only arise from combinations of observation
>statements with other statements that contain an explicit or
>implicit negation. Such implicit negations are typical in the
>scientific theories and in so-called "common sense" ideas about
>how the world (or the universe) works. But those statements
>are *never* reports of observations. They are always somebody's
>interpretation or hypothesis about the observations.
>If a contradiction arises between an observation statement and
>some theory, hypothesis, or commonsense idea, then there are
> 1. The observation might have been erroneous or perhaps slightly
> 2. If the observation was correct, it is possible that the person
> who was applying the theory or idea made some mistake in
> applying it. For example, the theory was about subsonic air flow,
> and the person applied it to an airplane in supersonic flight.
> 3. If the observation was correct and the person applied it correctly
> and the contradiction still remained, then the original theory or
> idea is incorrect and must be revised or rejected.
>Both science and common sense have followed these procedures for
>all recorded history (several thousand years). And people have
>frequently found that many theories and ideas about the universe
>But *nobody* has ever detected a single example where the universe
>or any part of it was self-contradictory.
>If you can ever discover such a case, it would truly be a miracle
>that is even more miraculous than anything reported in any religious
>text of any kind. The miracles reported in any religious text
>might contradict theories or ideas about the universe. But as
>observation statements, they cannot be self-contradictory.
>If you ever observe something that is self contradictory, you
>have witnessed a miracle that is far more miraculous than
>anything ever reported in any religion.
>If you do, don't bother to tell us about it, because we would
>simply dismiss you a nut case.
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