|
Given a square with one of its diagonals drawn, it is the same as a pair of right triangles with two equal sides. Calling the sides "x" and the hypotenuse "y", it can be analyzed in terms of the Pythagorean theorem, as mathematicians have done. The application of "number" to geometry by use of the Pythagorean theorem can represent both the proper application of number to geometry, as well as a very subtle misapplication of number to geometry that involves the most fundamental concepts of both "number" and "measure".
The Pythagorean theorem states that the squares drawn on the sides of a right
triangle, taken together, are equal in area to the square drawn on the
hypotenuse.
Numerically, that is represented as:
a² + b² = c²
In the case indicated above of one half a square whose side and diagonal are
x, and y, respectively, the representation of the Pythagorean theorem becomes:
2x² = y²
By numerical manipulation that equation can be expressed in the forms:
y = x(\²/2)
or
x = y / (\²/2)
where (\²/2) represents the square root of two.
How is that to be interpreted?
Notice that the Pythagorean equation refers to squares and areas, while the numerically manipulated equations refer to lengths. Notice too, that the linear equations retain some of the properties of the square equation from which they were derived. The numerical representation of lengths expressed by these equations retains the roots of their derivation. The use of radicals to represent such numbers, which are then used to represent such lengths, is a reflection of the radical difference in the implied process of measurement as compared to the simple process of measuring a straight line.
It apparently was not, and still is not, recognized that any problem represented by numbers other than the counting numbers, or integers, requires unit conversions to take place. I hope the work in Appendix 1 with ratios, fractions, decimals, and percentages has helped to clarify the situation of shifting references, which I will review here from a slightly different perspective.
Even using integers alone, unit conversions become necessary as soon as any process of division is required. A simple fraction necessarily implies such a unit conversion. Such unit conversions take place "automatically" in the processes of manipulating numbers. The laws that govern the manipulation of numbers reverse the unit conversions necessary to state the problem in numeric terms when the result of the manipulations is reapplied to the problem being solved. Thus they "drop out" of the equations and can usually be ignored.
The law that mediates or controls such unit conversions is given by the
equation:
1 = 1/N × N.
It could also be expressed as :
x/x = 1
The second form of the equation implies that the "N" in the first equation need
not be expressed as an integer, but that N could equal some function of x. Thus,
the first equation could also be expressed as:
1 = 1/f(x) × f(x).
Such equations necessarily imply a common unit, or sub-unit, of the quantity expressed as "one" with all other numbers used in the conversion of a problem to a numerical problem. If the numbers applied to different quantities in a problem were not consistent with a single unit, and with each other, how useful could they be in representing or solving the problem?
In view of the preceding paragraph, it is paradoxical that a numeric analysis of a problem such as the measurement of two lengths, could lead to the conclusion that there can be no common measure between them. What's wrong? Are the numbers used compatible or incompatible with each other? Are the measuring processes used to derive the numbers applied to the problem actually the same?
For example:
To measure the length or width of a room, a measuring rod is used as a reference
in a very specific and controlled manner. The measuring rod is placed end on end
with the iterations of moving the rod being counted as the measure of length.
Notice that the iterations of placing the measuring rod end on end is not
random. It must be done in a straight line. To do otherwise would not yield a
true measure. Thus the measurement of a straight line implies a reference angle
of measure equal to a straight angle.
It appears that such a process of measurement cannot produce a number such as the square root of two. Without using a reference angle of one half a straight angle, how could a number like the square root of two be derived? Given any two lengths whose origins are unknown or unspecified, how could it be proven that they have no common measure? They could of course actually be the side and diagonal of a square, but specifying one of them in terms of a square root implies a unit of measure that is different from the unit of measure of a straight line, unless it could be shown that a square root is derivable using a straight line and angle of reference without a shift in reference implied in ratios of reciprocals.
It seems intuitive that given any two lengths there is a common unit of measure that fits both of them exactly. Since each line is divisible in an infinite number of ways into an infinite number of equal parts, there must be one or more of those subdivisions that are equal to each other. Even in the case of lengths represented as transcendental numbers there must be at least one common measure, even if that common measure is infinitesimal, i.e., the smallest conceivable unit that could still represent length to be used as a measuring rod.
Even a blind point could see that.
I had originally intended to end this section with the preceding sentence, but the present context lends itself so well to drawing some relations between some very divergent views that I thought that it would be worthwhile to comment on some of them. Some of the comments taken as "anticipated conclusions" might be helpful in putting the steps taken so far in a more meaningful overall context. At least they should help to show the focus of attention with the implied bias and limitations that are the result.
The use of the idea of "taking steps" in the preceding paragraph and in my initial analysis of "equal step transformations", is a case in point. Coupling the idea of "steps" with that of dimensional, though infinitesimal, points could culminate in an idea of measurement in terms of equal steps that is common to linear, curvilinear, and angular quantities, as has already been suggested. Since the steps could be infinitesimal, such a system of measure could tie together many ideas of trigonometry, geometry, and calculus. Is that absurd? Would the calculation of pi as the limit of taking ever smaller steps be so different from other methods that have been employed? Is the idea of a perfect circle distorted any less than is the idea of a pure line as a continuum by considering both of them to be measurable - whether in terms of equal steps or some other notion of quantification?
Yes, dimensional points would introduce many complications and problems. Is it worth the effort? The idea of a point "stretching" itself from where it is to where it wants to be, does stretch the point of applying anthropomorphic qualities as an aid to understanding. Yet all understanding of anything must be reduced to terms that are dependent on the nature of the conscious being considering them. Just as "flexible dimensions" could be used to define topological transformations, systematic topological transformations could be used to define a shift from a linear dimension to a curvi-linear one, or from a Euclidean plane to other non-Euclidean planes with their respective geometries. Of what practical value could this be?
One of the purposes of mathematics is to create a model of the "real", "physical" world. Is there any question about that statement? Is it notable that in it mathematics - not mathematicians - was imbued with the anthropomorphic qualities of "creating" and having a purpose?
Suppose I pose a practical problem of dividing a circle, or any angular space less than a circle into any number of equal parts. Suppose further that I could construct physical models that could accomplish that in such a way that the divisions would be - theoretically - exact. Suppose even further that the physical models I suggest could be occurring in nature as processes not yet understood because the observation of the results can only be ascribed to processes that are "known".
Would the inability to define, or to "measure" the results "exactly" necessarily imply that they could not be the result of such processes? Would the inability to write exact equations that describe the processes necessarily imply that the processes could not occur? Or would such inabilities merely imply that the current state of mathematical development is not up to the task? Could an equation be written for the results of flexible dimensions that tend toward an equilibrium of the "forces" that might be at work in a given process?
Given a circle to be divided into "n" equal parts, draw "n" radii and connect them with "n" "equilibrating dimensions".
To obtain a division of any angular space less than a full circle, proceed as above, leaving out the one "equilibrating dimension" that would be "outside" the angular space defined by the "fixed", "outside" radii.
To construct a wheel that has seven spokes, attach seven springs that have equal tension at an equal distance from the hub. Connect the spokes with a ridged material, and "round out" the resulting perimeter.
Theoretically, it doesn't matter whether the "springs" are of the "extension" or "compression" variety, so long as their "elasticity" is "exactly" the same, the division will be just as theoretically "exact".
Given "points" that divide themselves by "mitosis" until they are "besides themselves" ........
|
|
|
