Re: Calculus XOR Probability

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?

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.


* 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 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.

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.

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.

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

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

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.

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

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.

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.

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.

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).

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.

<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?

<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.

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".

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"?).

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.

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.

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.

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"})\{}".

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.

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.

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.

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!

Cheers - Chas

.

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