Re: Calculus XOR Probability

Virgil said:
In article <MPG.1eaedaa5ce5b549e98ac3b@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

cbrown@xxxxxxxxxxxxxxxxx said:

Look at the definition of "limit" I gave. A point is not the limit
of a line segment. The /set/ of points, the diagonal, is the limit
of the sequence of curves "staircases".

So, the diagonal line as a whole is the limit of the staircase as a
whole, as the number of steps increases and the size of the steps
decreases accordingly, but the steps as they reach infinitesimal size
in the limit are not the points which make up the diagonal line?

We have a sequence of sets of points in the plane whose "limit set"
appears to be a point set having different properties from any member of
the sequence. But such inheritance of properties are nowhere required by
any standard limit processes.



Perhaps they should be required, in the case of a constant relationship. The
staepwise diagonal has length 2 even in the limit.


Perhaps you feel that the points of the line are where the risers
meet the treads, but then there are infinitesimal segments between
them, at right angles to each other. In any case, the problem with
your concept of the diagonal line being the staircase in the limit is
that in the limit you lose the concept of the mutually orthogonal
segments which constitute the staircase. If you maintain the
linearity so that the points are indeed the limit of the segments
then the answer is clear.


Besides - points don't have directions. /Curves/ can be said to
have directions /at/ points. I haven't claimed anything about
maintaining "directions at" points, anymore than I have said
anything about maintaining "lengths between" points. This is still
just Premise A.

Premise A is that the diagonal line is the staircase in the limit,
right?

What is the precise definition of "limit" that TO insists on using here?

That's the question. If you are only looking at the locations of the points, it
would seem the diagonal is the limit of the staircase, but taking into account
the directions of the elements in addition to their locations, we can easily
derive the source of the error from this difference.





These issues are not the domain of Premise A: these conclusions are
drawn later.

No, this issue has direct impact on Premise A. If the staircase in
the limit is distinguishable from the diagonal line, they are not
equal.

But by what definition of "limit" does one determine whether the limit
is still staircased or whether it is now diagonal?

For one thing, one notes that they get the wrong answer of 2 instead of sqrt
(2). More specifically, one makes sure their approximating elements are
parallel to the curve they're approximating. Take the formula for the
staircase, and forulaically find the limit. You won't get f(x)=x.


Note that most definitions of arc length of plane or space curves are
very cautious about points at which there are no unique tangents to the
curve, which is precisely what causes the problem here. In the "limit"
there are no points at which the staircase can have a tangent line.

Exactly. That's why i brought up the derivative, and pointed out that it's 0
with discontinuities where it's infinite, as opposed to the diagonal line with
a derivative of 1 everywhere.





<snip>

Now, you blame premise B, that the inductively proven constant
equality of L=2 is only true in the finite case, and not in the
infinite case, and that is why the proof fails. You postulate
something magical happening in the infinite case with no more
justification than, "what do you expect from infinity?"

RIght. So premise B is false

(momentarily dumbfounded - goes out for a smoke and some sun...)

RIght. Ummm....is that a valid justification? I haven't seen any
reason why a formulaic equality proven for all n>m doesn't hold in
the infinite case, and the postulating of magic doesn't fit the bill
as far as being a mathematical theory. Premise B is not at fault
here. Like I say, one should be as careful as possible when assigning
blame.

The "length" of smooth plane or space curve is only defined on intervals
for which there are "piecewise" unique tangents to the curve. In the
limit required, that only holds if one takes the diagonal version as the
curve.

Yes, I agree. Because the line is not continuous in the derivative and not
smooth, we don't get the answer we might expect, even if it looks like the
diagonal, because our elements cannot be parallel to a line which isn't even
parallel to itself.


So that for the infinitesimal staircase version, no length is definable,
at least in standard mathematics.


Well, I disagree that no length is definable. I think Chas proved quite
convincingly that the length of this fractal diagonal is exactly 2. I am
satisfied with that proof.

It's kind of strange to disagree with someone, because you think their proof is
correct, don't you think? Hmmmmm..... Have a nice day! :)

--
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. ... 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)
  • Re: Calculus XOR Probability
    ... infinite number of infinitesimal stairs, the length IS 2, because that's the nature of the staircase. ... By defining the curve as a sequence of segments, rather than a set of locations, the difference is quite detectable, because the segment definition preserves the notion of direction IN THE LIMIT. ...
    (sci.math)
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