Re: Calculus XOR Probability

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

<snip>

A mathematical argument usually proceeds something like this:

1. Premise A; because (supplies proof of A).
2. Premise B; because (supplies proof of B).
3. Therefore C, because (A and B) implies C.

Yes a classical deductive argument starts with a finite set of premises and
derives a conclusion logically from them. Correct.


A mathematical argument is not "C must be true; because it would be
absurd otherwise; or maybe you could make C work if you did some other
thing besides A and B".

To wit:

((A&B)->C)->(~C->~(A&B))

In other words, if the conclusion is incorrect, then one of the premises is
incorrect.


Right now, we're working on Premise A: the limit of the sequence of
stairsteps curves is the set of points D, which is the diagonal line.

Well, what we are doing is debating whether premise A or premise B is at fault
for the faulty conclusion C.

If a point is a line segment in the limit at the length goes to 0, then a point
has direction...

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? 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? If
you mean the normal, self-parallel diagonal line, then this is not a correct
statement. Once the stairs are reduced to infintiesimal size and appear to you
as points, they maintain their directions as infinitesimals segments, and the
line is everywhere discontinuous, and self-orthogonal. One page I found
discussed the problems with counting pixels to determine the perimeter of a
raster region, saying that it created errors or 41.42135%, or in other words,
an error or sqrt(2) (but I can't find it right now :(). Does that sound
familiar?


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.


<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.

- it is insufficient to claim that simply
because the limit of the curves is D, that therefore the length of D is
limit of the length of those curves, 2.

Not if D is defined as the staircase in the limit, which indeed has a length of
2, while having endpoints only sqrt(2) apart. That's not a straight line, and
not the normal diagonal of length sqrt(2). Consider the derivative of each
line. For the normal diagonal defined by y=x, the derivative is everywhere a
constant 1. On the staircase, the risers have infinite slope, while the treads
have zero slope, so we have a slope of 0 with point discontinuities at f'=oo.
When you take this shape to the limit, you have an indeterminate derivative at
every point. They're not the same animal.


It is equally insufficent to claim that simply because the limit of
finite sets of natural numbers is N, that therefore the sum of a
uniform distribution on N is 1.

There is nothing insufficient in claiming that a set of n elements with uniform
probability distribution will always have elements each with a likelihood of
1/n, no matter what n is. To try to form a uniform probability distribution
over the standard naturals is admittedly impossible, but not germane to whether
this statement holds for infinite n.


That is the parallel between these two incorrect arguments.

I don't really see a parallel. Your argument has at its roots a problem with
distinguishing directionless points from segments in the limit, and the second
rests on irrelevant problems with the unboundedness of the finite naturals
which have nothing to do with a set with a well defined maximal element. Of
course, despite the obvious counterexample in the reals in [0,1], you associate
this maximal element with a finite set, but there's not much I can do about
that except to point it out.


<snip>

So, you claim that the set of the points (x,y) satisfying "x + y = 1",
where x >= 0 and y >= 0 does not uniquely identify the diagonal line?
What other thing could "the diagonal line" possibly refer to?

Does that refer to the staircase?

No; it refers to the well-defined /limit/ of the staircases.

I don't see any reference to the treads or risers. Maybe you can point those
out for me.


It doesn't seem to describe a staircase. If
you differentiate the infinite staircase at any point, is the derivative equal
to -1? No, it's either 0 or oo and indeterminate.

If we can't agree on what it means to say "(something) is the limit of
(something else)", then it makes no sense to start talking about
differentiating, which is already defined in terms of limits.

Uh huh. So take the derivative and tell me what you get.




<snip>

I think not. You look at the set of points and see that they are
"indistinguishable" from the points in D. The notion of distinguishability is
crucial here, as it is for the Archimedean principle.

Do you also believe that the set of points (x,y) such that x^2 + y^2 =
1 is somehow "distinguishable" from the circle centered at the origin
with radius 1?

No, that accurately describes a circle.


Or that the graph of "y = mx + b" is somehow different from the line
with slope m and y axis intercept b?

No, that sounds right. Calculus works on that. You're describing curves
formulaically, but none of them have the differential discontinuities of the
staircase.


So you agree that the diagonal really is just a set of points; so when
I show that the set of points I mean by "the limit" is that set of
points D, that formulaic definition is identical to the formulaic
definition "the set of points with x+y = 1, x,y >= 0", which yields the
identical set of points D, the diagonal.

No, there is something inherent in those continuous formulas which sets the
direction of the points, whereas any definition of the staircase which you
could take to the limit would have inherent differential discontinuities.



When things are
infinitesimally close, they are not distinguishable on the finite scale. So,
you cannot detect the difference.

If you cannot detect the difference between two things, then I don't
see how it makes mathematical sense to claim that they are different.

I think you can if the points are to be integrated into lines. I'm not sure how
to show you this. Perhaps if you derived the formula for the staircase in the
infinite case and tried to take the derivative, you'd see what I mean. Hmm...


How can you take the derivative, if you can't decide what it means to
say "X is the limit of Y"?

They are one and the same. Why do you think I brought it up?


<snip>

If you are unable to articulate /why/ this limit is invalid except to
talk about something unrelated, such as the conclusion of some argument
that uses this premise, then you are not addressing this claim; you're
addressing a /different/ claim.

If I claim "6=9", and part of my proof is "3+3=6", you can't claim that
"3+3=6" must be false, simply because by my logic, that would mean
"3+3=9", which is absurd. It's true that "3+3=9" is absurd; but that in
no way implies that therefore "3+3=6" is false.

(sigh) Yes, I know all that, but when we are faced with a proof of something
absurd, we need to think carefully about the cause, and I'm sorry, but "hey,
it's infinite, that's why it doesn't make sense" doesn't work for me as an
explanation of anything.

That's not what I'm claiming.

No? What did you say "RIght" to, above?

Right now, I'm just claiming Premise A:
the limit of the staircases is D, the diagonal line. You don't need to
appeal to anything else yet; just evaluate that claim.

It's been done. There is an inherent difference between the two lines, which is
accurately reflected in their different lengths, as amply shown to result from
the cosine of the angle between the diagonal and the steps.

Cheers - Chas



--
Smiles,

Tony
.

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