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:
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" reality.
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.
"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)."
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