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Reference: alt.philosophy"Constants"
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From: Anthony G. Rubino
Date: Fri, 26 Sep 2003 18:54:10 -0400 (EDT)
Subject: Constants
I am impressed by the many posts that indicate an understanding of the sophisticated mathematics, topologies, and geometries, that are the foundations of contemporary physics and quantum theories. Yet no one here made a judgement of the validity of a simple theorem in plane Euclidean geometry that I presented. Too elementary I suppose. Still I will again present some elementary ideas.
Much of conntemporary physics is based on "c", (the speed of light), considered as a constant, which in turn is based on a number of assumptions and presuppositions. In mathematics there are a number of "constants" that are much simpler than "c" and which are generally considered to rest on the strongest foundation possible: i.e., they are considered to be beyond question. One of those constants is "pi", the ratio of the circumference of a circle to its diameter.
My analysis of "pi", however, raises some interesting caveats. To begin with, it is based on measurement, and therefore, cannot avoid some general problems inherent in measurement itself. Also, any evaluation of "pi" is based on an approximation, or idealization, of the method used to measure its two components.
One of the simplest, and most primitive, notions of measuring length is to apply some unit of length, taken as a standard, to some other length to be measured, in equal steps, and to count the steps: the "count" is then taken as the "measure" relative to that standard. The specification of a measured length necessarily implies such a concept by definition. Attempting to apply that concept directly to the measurement of a circle, however, is not simple. The result is that "pi" is not a constant. The actual value of "pi" will vary depending on the relative size of the circle in relation to the unit that is taken as a standard. Think about it. Try it.
For example:
If there is such a thing as the smallest unit, then it would not be possible to construct a circle with that diameter. To construct the smallest circle possible, using the smallest possible unit as radius, would produce a circle with a diameter of 2. It would take exactly 6 equal steps (using a compass, or divider) to "walk" around such a circular path, and "pi" would be equal to 3, exactly. That first "perfect" circle is the only one that yields such a "perfect" result.
Then, if the radius is increased by increments of that smallest possible unit, each circle will have a unique value for "pi", the larger the circle, the closer the approximation to the generally accepted value.
For most intents and purposes, there is no problem treating "pi", "c", or any other value, as a constant. It should, however, be understood that their constancy is contingent upon the intent and purpose in a given frame of reference. There could be (probably are) other alternative interpretations.
For example:
At the quantum level where an orbit of an electron is traversed in quantum distances (smallest posible equal steps), would any equations using "pi" to describe that motion benefit from some "correction"?
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From: Edgar Svendsen
Date: Sat, 27 Sep 2003 02:42:17 GMT
Subject: Re: Constants
I'm not sure of this but it seems to me that you are mixing apples and oranges. The circle whose radius to circumference ratio is defined by pi is a construct of a particular geometry. Of course, there are no real circles, only imperfect approximations to the geometrical definition. Any constructed 'circle' will be only an approximation. In that geometry there are no smallest units; distance is infinitely divisible. If you define another geometry in which distance is quantized, then the relationship between the radius and circumference will be quite different.
Ed
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From: Anthony G. Rubino
Date: Tue, 30 Sep 2003 12:58:06 -0400 (EDT)
Subject: Re: Constants
To Edgar Svendsen:
Some minor points first:
1. The radius to circumference ratio is not generally considered to be defined by pi. It is the definition of pi. (Technically, the definition of pi is given in terms of the diameter, but there is no material conceptual difference in the present context.) The only way pi could be considered to define a particular ratio of a circle would be if, and only if, a unique value of pi was associated with each unique circle, which could be the case for the way I have measured pi. To avoid confusion with the usual notion of pi, I will refer to it as pii to make
the statement:
Pii = 3, exactly, in one, and only one circle which has a radius of 1.
2. I, of course, disagree with the assertion that there are no real circles: circles are real enough to be recognized by almost everyone.
3. Any circle, whether it is perfect, or not, has an exact measure: The difficulty is the ability to determine that measure.
4. I realize that some would question the possibility of 'a smallest possible unit'. That's why I presented it in a conditional statement.
5. Defining a distance which is "quantized" seems like a reasonable thing to do since I cannot imagine distance without dimension, which brings me to the more important issue:
"Distance is infinitely divisible."
'Infinite divisibility' is an interesting concept that is not limited to just distance or length, but it is incompatible with measurement. It may be useful in theoretic discussions that do not involve measurement, but as soon as any measure is specified, divisibility is limited by the implied unit of measure, and infinite divisibility is no longer applicable. Without infinite divisibility, a 'smallest possible unit' relative to that measure is necessarily implied: the inability to specify such a unit exactly notwithstanding.
Measurement, in its intuitive form, does not require units of measure or divisibility.
It is a comparative process: a refinement of the intuitive sense of equality and inequality. When things are equal, it almost doesn't matter what standard of measure is used to determine that they are equal. When they are unequal, however, it becomes necessary to express that inequality in terms of some reference. After all, what does unequal mean? Is it bigger, longer, fatter, prettier, . . . ? If 'X' is larger than 'x', any fool could see it: just put them together. If they are not exactly equal, then they are unequal. The problem of measurement comes about when the question: How much? is asked. Then a third element is required that specifies a relationship by which both can be measured in the same way.
Removing the 'infinite' from divisibility still leaves a problem of continuity. Here, too, as soon as any measure is specified, continuity is corrupted, distorted, or pushed into the background. Yet, continuities, such as time and motion, are measured, in effect, by treating them as contiguous rathen than continuous.
Although it may not be immediately apparent, perception and measurement are closely related. What I have said here with regard to measurement is also applicable to perception. To understand that relationship, it is only necessary to realize that a measure is just one kind of observation, perception, or value. Perception can then be considered as the first measure of reality which undergoes continual refinements.
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From: Edgar Svendsen
Date: Wed, 01 Oct 2003 13:35:51 GMT
Subject: Re: Constants
It's an assertion about definitions. One, geometrical, definition of circle is; 'a line everywhere equidistant from a center point.' Another definition is 'any approximately round closed line .' Everyone does recognize objects that correspond to the second definition. I still say there are no real examples of the first definition. Circles corresponding to the first definiton have properties that ones corresponding to the second definition don't have.
I propose the term 'infinite calculability' for the characteristics of conceptual geometric object like circles. I can calculate the circumference of a geometric circle for any length of diameter. The conceptual distance is calcuable to any infinitesimal distance. 'Real' circles must be measured, and so are quite different objects. 'Real' circles probably never have equal distance from the center to the circle line so pi is different for each segment of the circumference. I completely agree that if you draw a very tiny circle (second definition), when you measure it and calculate pi for the segment you measured the value of pi will vary wildly.
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From: Kamerynn
Date: Sun, 05 Oct 2003 12:13:19 -0600
Subject: Re: Constants
I'm not sure that an analysis of pi should be based on measurement. We can speak of pi in theoretical terms without actually measuring anything, as you've done (below). Above.
Also, any evaluation of "pi" is based on an approximation, or idealization, of the method used to measure its two components.
I assure you that I've thought about it once again, prompted by your asking me to do so, and I cannot understand why pi should ever vary. This must be because I'm thinking in terms of how pi is derived (in modern calculus), and it is always the same irrational number.
Eudoxus used the "method of exhaustion" (in the fifth century B.C.E.) to prove that A=pi*r*r. The method goes something like this: inscribe a triangle within any circle so that its angles all rest on the edges of the circle, and then derive the area of the triangle. Next, do the same thing with a square. Continue this method with polygons, increasing the number of angles and sides by one each time. As the number of angles/sides approaches infinity, the inscibed polygon approaches circularity (approaches the shape of the circle it is inscribed within), and its area approaches pi times the square of the radius.
Here's a picture of what I'm talking about:
http://www.ugcs.caltech.edu/~shulman/pub/writings/core1/node2.html
This sort of reasoning is the basis for (some) calculus. Using n as the number of sides/angles, we say, in modern times, that A (the area of any circle) equals the limit of An (not A times n) as n approaches infinity.
When, above, you assert that "it would take exactly 6 equal steps to "walk" around such a circular path," you are right - we can divide the circle into 6 equal steps. But, we will not find that pi equals 3. Each of these six steps takes place around a curve - each step would be 2pi divided by six, as pi is understood today. Because each step takes place along a curve, it is slighly longer than 1 of the aforementioned fundamental units.
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To Kamerynn:
I appreciated your thoughtful reply to my post. Basically what you say is correct. The difficulty is that your still thinking 'inside the box'. I'm not.
A minor point of departure in our thinking is your doubt about 'pi' being based on measurement. Although the intuitive sense of 'difference' does not necessarily require a consideration of 'measure', as soon as the question: How much? is asked, or answered, the implication of 'measure' cannot be dismissed completely. If X is bigger than x, how much bigger? A little. A lot? Immeasurably? If X is more satisfying than x, how much more? (That reminds me of an old cigarette commercial: I'd walk a mile for a Camel. How far would you walk for X rather than to settle for x?)
Thus, if anything is specified with a specific number, such as 'constants' are, then there should be no doubt that measurement is involved. That's one of the points I was trying to make in starting this thread: All mathematical constants are measurements, pi being just one example. Finding a different way to measure whatever it is that's being measured could lead to a new understanding, new insights, simplifications, and new areas of inquiry or development that would otherwise not be feasible.
Another more important point of departure in our thinking is captured in your assertion:
". . . because each step takes place along a curve, it is slightly longer than any one of the aforementioned fundamental units."
One way to see the problem is to consider the rhetorical questions:
When you walk around a circular path, do you take straight steps, or curved ones?
What's a 'curved step'?
In normal measurement, a rigid straight rod is used. Bending, or 'curving', that rod will return a 'false measure'. Then too, how much should it be bent? Each unique curve would require a unique bend. 'Walking' around a circle using a compass (or divider) simulates actually walking around a circle, and it does require semi-circular movements (curved steps?) to manipulate the compass: but, the measure it returns is equivalent to a rigid rod measurement. Also, walking around the same circle using smaller and smaller steps, OR, walking around circles of ever increasing size using the same size step, would return a measure of pi equivalent to the one obtained by increasing the number of sides of polygons: i.e., both methods would return an equivalent 'limit'. Walking is simpler: No need for knowledge of polygons. (Good exercise too.)
When some of the earliest contributions to human knowledge are presented, there is a widespread tendency to minimize what the 'ancients' have given us in favor of more modern, post-modern, futuristic, or ultimate theories. How could they be so naive to propose that pi = 3? Yet, I have proposed a rationale for such a result in contemporary terms as well as in terms accessable to them and that might even suggest some new theories.
There is really no reason why pi, or any other 'constant', cannot be viewed in different ways, in different frames of reference, and for different purposes. There is a relationship of the circumference of a circle (or semi-circle) to its radius: that relationship is called pi. The ideal absolute value of pi is a constant. The practical relative value is variable. Both are measures of the same relationship. To avoid confusion, the variable value can be referred to as pii to distinguish it from pi.
The 'constant' relationship is the one that is generally accepted as the 'correct' one. The question may then arise: Why complicate things with a variable value when there is a satisfactory functional constant that has been used for centuries? What value or use could it have? Those questions can only be answered if some possibilities are explored.
There is another property of circles that is not considered very frequently: i.e., its curvature. Given any two circles that are not equal, they have a different curvature. To state that in another way: Every unique circle has a unique relationship (curvature) that is defined by the size of the circle in relation to the size of its radius. Very similar to the relationship I have defined as pii: there would have to be a one to one correspondence of any measure of curvature to the measure of pii as I have defined it.