Re: Courage?
- From: Matt Gutting <tchrmatt@xxxxxxxxx>
- Date: Tue, 03 May 2005 08:23:39 -0400
Albert Wagner wrote:
Matt Gutting wrote:
Allan C Cybulskie wrote:
"Giuseppe Bilotta" <bilotta78@xxxxxxxxxx> wrote in message news:lzhzadynydfd$.88ivp0qk2ycj$.dlg@xxxxxxxxxxxxx
On Thu, 28 Apr 2005 08:54:14 -0500, Albert Wagner wrote:
A line is *not* constituted of points. A line is *not* composed of points.
You keep saying that. Is there anything you can do to you your statement is true and Robert's statement is false?
His reasoning was:
If points are dimensionless, then they cannot add up to the length of a line, and so a line would have no length.
If they have a dimension, then there cannot be an infinite number of them
in a line of any fixed length.
You are assuming that the length of a line consisting of points is
defined as the sum of the lengths of its points.
He is assuming nothing. He is showing that logical contradiction results from ignoring the question of point dimension. It is a logical absurdity to claim that points simultaneously do and do not have dimension.
I don't see that "logical contradiction results from ignoring the question of point dimension". Allan is saying that either points have dimension, or they do not. This is certainly true. He then reasons that if points were dimensionless, then "they [could] not add up to the length of a line, and so a line would have no length". He goes on to say that if points were not dimensionless, then there could be only a finite number in a line segment of finite length. Finally, he concludes that either lines have no length, or they consist of a finite number of points.
The problem with this is that the axioms don't say anything about how we are allowed to use the word "length". In particular, they don't say anything about how the length (if any) of a point might be related to the length (if any) of a line. It is thus unwarranted to conclude that the length of a line *must be* the sum of the length of its points. Both of Allan's conclusions depend on this statement; and since it is not logically warranted by the axioms, it cannot be used to derive statements necessarily consistent with the axioms.
Since "length" is not mentioned in the axioms, we are not free to use it in reasoning about the concepts described by the axioms.
Which axioms? Hilbert's? Or Cantor's? Or some mix of the two?
Hilbert's.
The length (or as Stephen would say 'diameter') of points is easily inferred from Hilbert's axioms given the usage of points by Hilbert in the axioms. Unless you argue that there is some hidden super axiom that forbids logical inference from axioms.
There is nothing forbidding logical inference from the axioms. But the point of constructing an axiomatic system is to produce a framework which can lead to conclusions about the words whose usage is described in the axioms, *and only about those words*. This is not a "hidden super axiom" but simply a limitation of the way in which axiomatic systems are used.
It seems to me that amateur mathematicians misunderstand Hilbert's intent in specifying point, lines, etc as *initially* undefined. It is roughly analogous to starting a computer program with labels initially unattached to values, and then assigning value to those labels by specifying in the code how variables are assigned. Likewise, Hilbert wished to avoid any pre-existing definitions of points, lines, etc. in order to specify them new in his own explanations, the axioms.
Exactly.
Hilbert himself defines points, lines, etc. when he explicitly describes their use. The axioms are themselves a backdoor definition.
No one has yet been willing to give me the axiomatic basis for statements that describe a line segment as consisting of an infinite set of points, and how to avoid the inherent contradiction in such statements of points being simultaneously dimensioned and not dimensioned.
Okay, two points here. I already dealt with the second point above, noting that while a point either has or does not have dimension, Hllbert's axioms do not by themselves allow us to draw conclusions about dimension.
As far as the first point, I could step through a quick proof that line segments consist of an infinite number of points, but first I'll have to go back and check all of Hilbert's axioms to see exactly how he describes the correct usage of the term "line segment" in his system.
Matt .
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