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Topic: Circular reasoning on a curve
Replies: 14   Last Post: Nov 11, 2013 12:40 PM

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Posts: 148
Registered: 4/13/13
Circular reasoning on a curve
Posted: Oct 28, 2013 7:06 PM
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In Thomas's _Calculus and Analytic Geometry_, 2nd Ed., 1953, the author
provides the following

Example 6-3.1: Find the equation of a circle whose center is at C(h,k)
and whose radius is r.

/Solution/. From the definition of a circle as the locus of points in
the plane which are at constant distance from a fixed point, we must have

CP = r

for any point P on the circle and, conversely, if CP = r, then P is on
the circles. This condition is the same as

sqrt((x - h)^2 + (y - k)^2) = r

from Eq. (1) [Pythagorean theorem], or

(x - h)^2 + (y - k)^2 = r^2

which is the equation of the circle."

First notice that he speaks of points in _the_ plane, as if there is
one, and only one, plane in the mathematical universe. For the present,
I grant for discussions of two dimensional plane geometry, that there is
only one plane.

In classical physics, when one speaks of the distance between two
points, he is using the abstraction of a zero-dimensional object whose
location relative to other objects in the system is exactly knowable.
Given two distinct points, assumed to be at relative rest, it is
hypothetically possible to measure the distance between them using an
idealized rigid measuring rod of infinite accuracy.

That is, the geometrical space under consideration has an objective
reality which conforms to our naive experience.

But, in mathematics, we can invent methods of "measuring" distance by
which the diameter of a circular disk is infinite. For example:

So how can one speak of geometric conditions for a curve, separately
from the algebraic formulation? For example, an ellipse can be
specified as the locus of points such that the sum of the distances
between each point and two fixed foci is constant. Placing the foci a
distance c from the origin in either direction along the x-axis we have:

PF1 = sqrt((x + c)^2 + y^2)
PF2 = sqrt((x - c)^2 + y^2)

2a = PF1 + PF2 = sqrt((x + c)^2 + y^2) + sqrt((x - c)^2 + y^2)

To say any point satisfying this condition is on the curve, and any
point on the curve satisfies this condition seems trivially tautological.

In the case of the ellipse, Thomas begins with with the above
formulation calling 2a = PF1 + PF2 the "geometric condition" and derives
this "algebraic equation":

(x/a)^2 + (y/b)^2 = 1, where b = sqrt(a^2 - c^2)

He then shows that we can begin with the /algebraic equation/ and arrive
at the /geometric condition/.

He is apparently treating objects such as PF1, PF2 and CP as geometric
objects, existing prior to any coordinatization. In other words, PF1,
PF2, and CP are things we can "pick up, and move around". They carry
with them, in some abstract sense, an objective, and even "physical"

My problem is that when a physical concept such as a rigid, straight
measuring rod is abstracted into the mathematical universe, it becomes
subordinated to the godlike powers of the mathematical imagination. For
example, the Escher drawing linked above shows a representation of an
infinite plane which has been confined to a finite disk. It is possible
to define rules of measurement such that all the individual tiles have
the same size, and all of their edges are segments of straight lines, in
the sense that I could communicate these measurements to a person using
traditional drafting tools and that person would draw figures with those
characteristics using the values provided.

On page 78 of Wely's _Space-Time-Matter_ we find him protesting as follows.

"Various attempts have been made to set up a standard terminology in
this branch of mathematics involving only the vectors themselves and not
their components, analogous to that of vectors in vector analysis. This
is *highly* *expedient* [emphasis, mine] in the latter, but very
cumbersome for the much more complicated framework of the tensor
calculus. In trying to avoid continual reference to the components we
are obliged to adopt an endless profusion of names and symbols in
addition to an intricate set of rules for carrying out calculations, so
that the balance of advantage is considerably on the negative side. An
emphatic protest must be entered against these orgies of formalism which
are threatening the peace of even the technical scientist."

I, too, find geometric reasoning highly expedient, but is it appropriate
to treat abstract geometric objects e.g., a circle, as existing
independently of our algebraic representations. To say a circle is the
locus of all points equidistant from a given fixed point is only
meaningful to me if I am given a metric with which to determine
distance. In the case of Euclidean geometry, it is the Pythagorean
formula r^2 = x^2 + y^2. To say C is all (x,y) such that r^2 = x^2 +
y^2, and then to add, /if r^2 = x^2 + y^2, then (x,y) is on C, and if
(x,y) is on C then r^2 = x^2 + y^2, seems merely be needlessly redundant.

Now here is the other hand:

"In this book, a central theme will be a Geometric Principle: The laws
of physics must all be expressible as geometric (coordinate-independent
and reference-frame-independent) relationships between geometric
objects, which represent physical entitities(sic)."


sevot yhtils eht dna,gillirb sawT'
ebaw eht ni elbmig dna eryg diD
,sevogorob eht erew ysmim llA
.ebargtuo shtar emom eht dnA

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