Re: Calculus XOR Probability

cbrown@xxxxxxxxxxxxxxxxx said:
Tony Orlow wrote:
cbrown@xxxxxxxxxxxxxxxxx said:

<snip>

Let's return to the original argument false argument:

A1: The limit of the staircases is the diagonal line.
A2: The length of the nth staircase is 2.
B: Therefore the length of the diagonal line is the limit of (2,2,2...)
= 2.

Which of the above statements is the one that makes the argument false?

The conclusion falsely assumes that your definition of limit affords an
accurate measure of the object.

Precisely!!

In other words, because your definition of
limit does not distinguish between parallel and non-parallel elements, but
deals only with directionless points, the conclusion that the length of the
diagonal is the same as the length of the staircase "in the limit" does not
follow.

Bingo! And that assumption (that the limit is always, perforce, an
"accurate measure" in every sense) is assumption B. It simply doesn't
follow from saying "the limit of {x_n} is X". Whether or not it is an
"accurate measure" of some particular property depends on what
definition of "limit" is being used.

As you have noted, the limit I gave /is/ an "accurate measure" of the
limit "as location": in other words, with respect to closeness measured
by distance of points. But that's all it's good for. Which sometimes,
is exactly what we want, for example in the 1 dimensional case of the
limit of a sequence of points on the real line.

<snip>

When Han says "let N be the limit of {1..n} as n -> oo", I expect N to
be /something/ - a set? a number? Something that we can actually talk
about that allows us to say "N has such-and-such a property P".

Han's argument has this form:

A1: The limit of the {1..n} as n-> oo is N.
A2: The sum of n*1/n = 1 for all n.
B: Therefore it makes sense to say that there is a uniform distribution
on N.

Which if the above statements is the one that makes Han's argument
false?

The conclusion attributes to the unbounded set a property associated with
bounded initial segments of the set.

Do you see any justification for that assumption (i.e., B)? I don't;
except by some vague appeal to the limit as "always" having this
property (i.e., that the limit is an "accurate measure" in every
sense). But that is exactly what premise B was in the above argument as
well; and we see that it is not always the case.

Because one really cannot pin down the
number of finite naturals, one can't define what the probability of each is.

More basically, since Han has not specified what it means to say "the
limit of {1..n} is N", and why it isn't, say "R" or "Q" or "the empty
set" instead; we can't draw /any/ particular conclusions at all.

He might as well have started his argument by saying "let the piirjoon
of {1..n} as n ->oo be N".

Fine - but what's a "piirjoon"? (Besides being the Estonian word for
"limit").

But, IF one has such an n as maximal element, one CAN define a uniform
probability distribution, even if n is infinite.


I'll let others disabuse you of notions like these. My interest has
been explaining that "the limit" is not a magic wand - it is the result
of a definition. Whether or not it preserves some property of the
elements of the sequence as "an accurate measure" depends entirely on
the definition It doesn't always follow automatically.

Whenever you see the word "limit", you need to ask: what does it mean
exactly in the given context?

Cheers - Chas



Well yes, Chas, that's true. I'm gald you seem to finally getting what I'm
saying and agreeing (I think) that the definition of the limiting curve you're
using doesn't gurarantee that it is really the same object as the curve in the
limit in every respect. The problem with your example, as I see it, is exactly
that, the difference between considering or not considering the derivative of
position, as well as the position, as one travels along the curve. It's a
matter of ignoring the direction of the curve in trying to measure it, not a
matter of taking an equality to be true in the infinite case.

Does this problem really apply to Han's limit? I don't see that. The problem
with Han's limit is that the finite naturals don't go to oo, and there is no
well defined endpoint to them. If Han offered the argument you recount above,
then it suffers from that problem, as the notion of expected value does for the
same reason. But, if we leave N out of the discussion, and simply say for a set
with n elements equally likely the probability of each is 1/n, there is nothing
wrong with that statement. It doesn't even involve a limit when so stated. The
limit comes in as n->oo, and the prime question here is whether the individual
probabilities are truly 0, or some infinitesimal value equal to 1/n. The former
violates additive probability, so I go with the latter, especially as it meshes
perfectly with the rest of what I've been arguing for a year. It makes me feel
good, like I'm not totally misuderstood by the entire planet, and like we may
be getting somewhere. :)
--
Smiles,

Tony
.

Relevant Pages

  • 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
    ... What is the slope of the staircase at x=0. ... infinite, as opposed to the slope of the diagonal. ... As long as those locations are ON the curve, ... The mathematics involved is quite straightforward and pedestrian. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... the locations of the staircase in the limit are indistinguishable ... mathematical requirement for a curve to have a length is the the lengths ... it cannot be used as a measure of a curve. ... The mathematics involved is quite straightforward and pedestrian. ...
    (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. ... According to the definitions of arc length in calculus, ... parallel to the curve they're approximating. ...
    (sci.math)