Re: Calculus XOR Probability

Matt Gutting said:
Tony Orlow wrote:
Matt Gutting said:
Tony Orlow wrote:
Virgil said:
<snip>

The length of a broken or polygonal line depends only on the location
of of the "joints" and not at all on the direction of the segments
between joints.
What is the slope of the staircase at x=0. It's infinite. Does that ever change
as the staircase approaches the limit? No. That point at x=0 always has slope
infinite, as opposed to the slope of the diagonal. The direction between 0,0
and 0,1/n never changes.
But you're taking into account multiple points (2 points). The sequence of
staircases converges pointwise; you don't *have* a staircase as the limit.


Uh, yeah, you do have an infinite sequence of infinitesimal steps, and the
slope at x=0 remains infinite.

I hate to repeat myself, but in the absence of a precise, non-circular
definition of "infinite," the truth value of the above statement can't be
evaluated.

Oh come on. Infinite slope isn't even quaestionable. The stair is always
vertical at the point x=0.



Virgil:
That may be a TO requirement, but it is not a mathematical one. The only
mathematical requirement for a curve to have a length is the the lengths
of its polygonal approximations have a finite upper bound, and then
that upper bound is then defined to be the length.

Tony:
If the endpoints are on the curve.
Whether the endpoints are on the curve or not, as long as the series of
lengths is convergent.

Gee, the series of your lengths on the staircase "converges" to the same value
it has in all finite cases. Some point on the segments must be parallel to the
curve at a point on the curve perpendicular to a point in that segment, or your
measure is off.


How can a point be parallel, or perpendicular, to anything?

The segment is parallel to the curve at a point.


Matt

*** Matt orders all his beanie babies from http://www.teranews.com ***


--
Smiles,

Tony
.

Relevant Pages

  • Re: Calculus XOR Probability
    ... and the object is no longer a staircase. ... segments, rather than a set of locations, the difference is quite detectable, ... but 1, and n/0 is still infinite, even when n is infinitesimal. ... The limit of a curve is curve satisfying the following criteria: ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... According to TO's infinite induction principle it would necessarily be ... The length of the staircase in the limit IS 2, ... The segments have directions, which alternate between vertical and horizontal, ... We're not defining the curve by the endpoints any more. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... curve, which is precisely what causes the problem here. ... infinite case, and that is why the proof fails. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... your concept of the diagonal line being the staircase in the limit is ... segments which constitute the staircase. ... infinite case, and that is why the proof fails. ... The "length" of smooth plane or space curve is only defined on intervals ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... So, the diagonal line as a whole is the limit of the staircase as a whole, as ... are where the risers meet the treads, but then there are infinitesimal segments ... Besides - points don't have directions. ... You postulate something magical happening in the infinite case ...
    (sci.math)