Re: Calculus XOR Probability

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

<snip>

When you say there's a "problem" with my limit, what do you mean?

I mean that it doesn't take into account anything but location, and you're
using it to measure distance.


Distance and length are real numbers.

Is lim n->oo {C_n} a real number, or is it a set of points?

It's a sequential set of points, that is a line of some sort, with a real
measure called length.


So why do you claim that I am using the limit to measure distance or
length? I am using the limit to take the limit of a sequence of curves,
nothing more.

To measure distance or length of a set of points in R^2 with the usual
metric, I use the usual approach, just like everyone else.

That's good, because this approach doesn't work. When you prove the length in
the limit is 2 instead of sqrt(2), that's an indication that the object in the
limit is NOT exactly the same as the diagonal line, and the reason for the
discrepancy is easily seen to be due to the angle at which each of the
approximating segments intersects the diagonal it's supposedly measuring. Even
though the staircase in the limit is infinitesimally close to the diagonal
line, it doesn't actually contain the same points, because the derivative in
the limit remains discontinuous rather than constant. The slope is not 1 at any
given point on the staircase, without some kind of smoothing technique, like
Han was talking about.



* Do you mean that my definition of limit is not well-defined...

No, the locations of the staircase in the limit are indistinguishable from the
locations of the points on the diagonal.


Ok; so you (for the moment) understand what I mean by limit of {C_n} as
a statement about curves in R^2 using the usual metric.

Do you understand what I'm saying about the elements of the staircase needing
to be parallel to the diagonal in order for the "usual metric" to work?



* Do you mean that it fails to have some property which is
mathematically required before one is allowed to use the word "limit"?
If so, what property is that?

No, it fails to have a property that is required to use that limit to measure
the diagonal.

Then you are misunderstanding me.

No, you are misunderstanding me. I reject your example as demonstrating that
the concept of an inductive proof of equality holding in the infinite case is
invalid, because the error in your staircase example is easily explainable in
the way I've been describing.


I use the limit to find the limit of a sequence of curves. I don't
"measure" anything with it; nor have I claimed that you /should/ be
able to "measure" anything with it.

Fine so far.


I use the /usual method/ to "measure" the lengh of any curve, both
before and after taking the limit, based on sets of points in R^2 using
the usual metric.

What is the usual method for measuring the staircase in the limit? As far as
I'm concerned, you proved it had length 2.


Finally, I use the principle of "infinite induction" to /deduce/ that
"length(limit)=limit(length)"; i.e.:

Yes, that's very nice, and I have no problem with that. You're measuring the
length of an infinite number of infinitesimal steps, not a straight line.


THE PRINCIPLE OF INFINITE INDUCTION is what I (erroneously) "use to
measure" the length of the limit of the curves;

Don't be so hard on yourself. You got the right answer. You did very well.


because the principle of infinite induction claims that I /can always/
"use" the limit of the lengths of the curves "to measure" the length of
the limit of the curves.

Yeah, and that worked out pretty good, didn't it?


It is this /third/ assertion (premise B, "infinite induction") which
causes a "problem", not the first two.

No, you're just confused because you think that location is all there is to a
set of points. When you're talking about a curve, you're talking about an
uncountable sequence of points, and there is a connectedness between successive
points that has direction. One can just as easily view a curve as a set of
infinitesimal segments between points, as a set of points separated by
infinitesimal segments.


The third assertion is simply false: limit(length of curve) is not
equal to length(limit of curves), where "limit", "length", and "curve"
refer to objects in the domain of R^2 with the usual metric. Therefore,
the principle of "infinite induction" /does not apply/ in the case
where we are working in R^2 with the usual metric; and I think you'll
find that it doesn't apply in most situations.

What do you mean by "the usual metric". Maybe this is the problem here. Is the
metric usually wrong?


There is no reason to tack on additional ill-defined (or even
well-defined) concepts like "therefore there are points with no, some,
and all directions, at infinitesimally different locations" that are
not part of my argument, and are certainly not a part of R^2 with the
usual metric, which is, after all, the usual domain of curves,
including all the familar the results of calculus.

Yeah, well, as I understand it, the usual metric is generally used parallel to
whatever one is measuring.


To add these other "features" of points in is an amusing diversion; but
it is a diversion: it is like saying "the natural number satisfying
'2*x = 3' is x = 3/2". Fine, but 3/2 is not a natural number.

No, it's nothing like that.


Although it may /look/ like a simple commutation of functions f(g) =
g(f), in fact limit(length) is a function from a sequence of reals to a
real, and limit of curves is an /entirely different/ function from a
sequence of curves to a curve. So it is unsurprising that given
functions f, limC, and limR, we often find that f(limC) != limR(f).

I wouldn't call it unsurprising. I would call it an indication of a problem in
our metric. As far as I'm concerned, the error of sqrt(2) is the inverse of the
cosine of the 45 degree angle each riser and tread makes with the diagonal. The
accurate measure of curves depends directly on parallel metrics.


This is not true only in R^2 with the usual metric; in fact, you have
to work very carefully for your "infinite induction" to hold in a
paricular domain.

I don't think you (or Han) are being that careful, in general, when you
invoke it; and that is the point I have been trying to make.

Well, I think you're wrong, and I don't think you're example illuminated
anything other than basic calculus ideas of approximation. It has nothing to do
with probability over an infinite set, and didn't show why infinite induction
of equalities ever fails. It's just your assumption of "usual metric" that
fails.


<snip>

I would like your answer to allow me to say "in example (the example)
the set of points p with (x,y)=(0,0) and direction {insert your value}
uniquely defines (the example) because {insert justification here}".

* x = 0, y = 0

As a point in itself it is without direction or measure. It can be thought of,
in the context of the Cartesian plane, as an infinitesimal square parallel to
the x and y axes.


So it is simultaneously directionless, and yet parallel to two
different lines, which are at right angles to each other?

Use your mind's eye. Each side is parallel to one of the axes.


<snip other "interpretations" of the point (0,0) in R^2>

Directionless point.

Pointless direction. What does this have to do with anything?


According to the definitions I gave, each of the above formulas
identically yields the single point (0,0) in R^2 with the usual metric.
In other words, the figure in the x/y plane described by "{(x,y) :
f(x,y)=c}" for all the various functions f is identical: they are not
merely "indistinguishable", they are equal.

Single points don't have any metric anyway.


You claim that, essentially, they are different: but you are using
logic equivalent to "but there /is/ a natural number such that 2*x =3;
it is x = 3/2".

No, that is about as related to what I am saying as your example is to the
original topic. Like, not.


I wanted to see what space you think you are talking about (as opposed
to the one that my proof is concerned with). I.e. naturals:rationals ::
R^2:...?

When pressed to describe what you mean by "points with direction" you
respond with answers like "all and none", so it appears that points not
only have directions, they have sets of directions; and not just sets
of directions, they have uncountable sets of directions; which also
contain a special direction "none". You don't appear to allow this set
of directions to be empty; i.e., every point has a direction, even if
it is "none" (otherwise, how could it be "all /and/ none"?).

I said that in response to the circle in the limit. What direction does a
circle have? (groan)


In addition, you simply cannot resist moving even further beyond the
definitions given: you claim that the limit as I defined it is not the
point (0,0) in R^2, but instead some "infinitesimally different" point
in X^2, where X is something that isn't R; in order to justify that the
circles centered on the origin has a different limit of "points with
directions" than the circles tangent to the axes.

Look, in the standard sense, what you say is correct about the
distinguishability of those points. So what? You're "usual metric" didn't work
for precisely the reasons I've explained, and you can't blame it on infinite
induction. If you have another example of how infinite induction fails, please
present it, but this one didn't fly.


Doubtless I could come up with an example where you would claim that
the direction is "almost" 0, but "infinitesimally" different from it;
so I assume "directions" are also elements of X.

Directions as tangent lines are elements of curves that exist conceptually at
each point on the curve.


So when /you/ talk about figures in the plane, it appears to be figures
consisting of points in (X^2 x P(X union {"none"})\{}, where X is some
currently undefined algebraic object containing the field R as a
sub-algebra; and where P(X union {"none"})\{} is the power set of X
union{"none"}, excluding the empty set.

No, that's far more complicated than it has to be. A point in space need not
have any direction. A point on a curve has direction as long as it doesn't have
a derivative discontinuity, that is, a corner. A curve is a sequence of points,
or of segments between points with direction.


That's fine; although I have no idea what "X" is, so in general, your
assertions are meaningless to me, mathematically speaking. But that has
/never/ been the domain of /my/ argument: it has always been R^2, not
X^2 x P(X union {none})\{}.

Thus

"is there a natural number such that 2*x =3?"

is to

"yes, 3/2"

as

"is there a difference in R^2 with the usual metric between the
figures x^2 + y^2 = 0 and lim n->oo {y=x,-1/n< x< 1/n}?"

is to

"yes; they have different directions because they
are different elements of X^2 x P(X union {"none"})\{}".

Straw man. That's not my argument.

Can you see that while you have been talking about "a figure in the x/y
plane" you have not actually been addressing my argument at all? You
have been responding to some /other/ argument that is not, and never
was, my argument; which is solely about figures in R^2 with the usual
metric.

The metric that usually fails?


I maintain that the figure in the x/y plane satisfied by each and every
one of the examples given is, identically, the origin of the plane: the
point (0,0) in R^2. Period. At least when we work in R^2, with the
usual metric.

Right, they're all points with zero metric.


I don't claim that my argument neccesarily holds for X^3 x Q x P(X x X)
or any other thing you might propose; and never have.

I made this argument to prove that your "infinite induction" fails in
this case, which should give you pause when invoking it in the future:
since it doesn't hold in R^2, it doesn''t hold in every situation; so
you cannot merely assert without proof that it holds when you are
making an argument regarding, for example, some set of elements in X^2
x P(X union {"none"})\{}.

At least, not if you're making a valid mathematical argument; which is
after all the theme of this newsgroup.

I think we're just going to have to agree to disagree. The problem with your
example is obvious. I'm am not convinced.


Then you're being clueless. You should be able to parse that statement. Or, did
you read "probabilities" as "possibilities" again? Remedial English is down the
hall to the left.

This is Remedial Mathematics. The current topic is "what is a valid
mathematical argument?". Stay in your seat until the bell rings.

If that's your attitude, the discussion's over. Have a nice day.


Ah well. Sorry you feel that way. See ya round then!

I'm not going to tolerate that kind of condescending nonsense. I believe I have
a valid point, and that your refutation by counterexample was invalid. We've
strayed far from the topic, and now you want to make this a remedial class?
Sorry, the remedy isn't needed, thanks anyway.


Cheers - Chas



--
Smiles,

Tony
.

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