The Trisection of Angles

Appendix 2a - Measurement

The fundamental problem is "measurement". Except for geometry, exact measurement has been "proven to be impossible" - implicitly - by the introduction of the concept of "incommensurable" lengths. Some lengths can be defined exactly only in terms of "transcendental" numbers, which themselves can not be defined exactly in terms of non-transcendental numbers. The measures represented by transcendental numbers can only be approximated by other numbers to any degree of accuracy required. Calculus "hedges" the problem with a concept of "limit" arrived at by successive approximations from two directions using "infinitesimals" that are "un-measurable".

The fundamental problem of measurement is that it is a comparative process that requires a reference. Measurement can be characterized merely as 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 show that they are exactly equal; when they are unequal it becomes necessary to express the inequality in terms of some reference. After all, what does unequal mean? Is it bigger, longer, fatter, . . . ?

Simply stating the primary standard or reference of inequality may be sufficient to satisfy the intuitive sense of inequality. If "a" is larger than "b", any fool could see it; just put them together. If they are not equal, then they are unequal. The problem of measurement comes about when the question: "How much?", is asked. To give an exact answer to that question, a third element is required that specifies an exact relationship by which both can be measured in the same way.

What appears to be the simplest "measure" is not as simple as it appears to be unless it is interpreted by our intuitive sense in the simplest intuitive way. Take "length" as an example. The simplest intuitive notion of length is a straight line, i.e., the "shortest distance between two points", yet that too depends on "how you look at it". Even if that could be accepted as a "definition", it still involves a combination of other intuitive senses, i.e., equality, distance, time, motion, et cetera. In addition, the "circularity" implied by the term shortest cannot be entirely removed by using other terms because "circularity" seems to be a way of combining intuitive notions by "rolling them around in our mind".

In order to communicate, some "common ground" must be established by assuming that such "problems" are resolved, or are moot, or irrelevant. In this case it is assumed that a straight line is "given". Is it possible to measure its length exactly?

Notice that posing such a question assumes the establishment of "common agreement, not only of the terms used, but also of the concepts that they represent, on some level of understanding, including the "intuitive" one. In fact, it is impossible to exclude the intuitive understanding of the concepts and their relationships. Without them, there would be no "meaning". Thus, intuitive notions, especially "gut feelings", should not be ignored, especially when "something seems wrong". Is it lack of understanding of the terms or concepts or the possible relations between them? Or is it disagreement? Or both?

So that there will be as little misunderstanding about my position on exact measurement as possible, I will make the following assertions:

1. Given any line, it has an exact measure.
2. Given any line, it can be measured exactly in terms of the exact measure of some unit of measure.
3. Given any two lines, they can be measured exactly in terms of the same exact measure applied to each in exactly the same way.

That's what exact measurement means to me. Though stated only in terms of linear measure, those assertions are equally applicable to any measured quantity. They represent my intuitive understanding of the terms, the concepts they represent, and their relationships. To deny any of these propositions is to deny that exact measurement is possible.

It should be noticed that I did not "hedge" or qualify any of the terms with a "theoretical" - "practical" distinction. That distinction is not relevant to this discussion. This discussion applies equally well to both "types" of measure. It is not my intention to "taint" the concept of "theoretical exactness" by saying that, for the present purpose, it is the same as "practical exactness", or any such thing. If anything, I would say that the "practical" should not be "sneered at" and underestimated. In any case, the first and second statements adequately account for any qualifications that can be ascribed to the exact measure or unit of measure.

I know that these propositions are not generally accepted. In fact, there is a mathematical "proof of impossibility" with regard to the third proposition, and a large body of mathematical knowledge has been built on the resulting "incommensurables".

Before tackling the theoretical fallacy, I would like to point out that I have no difficulty or disagreement with their analysis or conclusions, IF I INTERPRET THEM AS PRACTICAL LIMITATIONS OF EXACT MEASUREMENT. Mathematicians, however, claim that they have proven the "theoretical impossibility" of finding, for example, a common measure between the side and diagonal of a square. That's in direct disagreement with my third proposition. The disagreement is serious, or not, depending on how it's viewed.

See for an example Curious Construction - 1

If it is impossible to find a common measure, what does measurement mean? If it is theoretically impossible, what does exact measurement mean?

If a common measure cannot be found, then measurement is impossible! Exact measurement is either possible, or impossible; it cannot be both. Perhaps in the way of mysticism the contemplation of impossible possibilities could represent a mystical paradox whose contemplation could yield a deeper understanding in the form of a deeper paradox. However, in a rational context, things are either possible or impossible, but not both.

Suppose I propose a small change in the mathematicians' conclusion with regard to the side and diagonal of a square by suggesting that what has been proven is that the common measure between them cannot be specified in the present number system as an integer. Why claim more than that?

The inability to specify a given length or its unit of measure exactly in a given system in a particular way, does not necessarily imply that it cannot be specified exactly in that same system in a different way, or that it cannot be specified exactly in a different system. An inability to specify something certainly does not prove that it does not exist. After all, what was it that could not be specified?

Is that any different from saying that the square root of two is an exact number that cannot be specified exactly by integers, fractions, or decimals? A transcendental number refers to an exact number or length or an exact relation. It cannot be specified exactly. So what? No, that could be serious; let me rephrase that. If it cannot be specified exactly in a particular way or in a particular system, so what?

Viewed in this manner, transcendentals wreak no havoc on my intuitive sensibilities. "Incommensurables" still bother me, but I translate that to unspecifiables and carry along the qualifications that they are so only because they are being considered in a limited frame of reference or from a limited point of view.

I don't think that the intuitive sensibilities of mathematicians are any different from mine. They did not appear happy with many of the results they felt compelled to accept. There appeared to be no other way. Practical considerations alone led them to devise measures that would allow them to proceed, despite the limitations, to all sorts of wonderful discoveries. They were even able to reestablish the intuitive notion of exact measure by use of "infinitesimals" and "limits". To me that is an implicit recognition of an unspecified unit of measure within a limited frame of reference wherein the notion of exact measure is derived from the "limits" within the system.

How does an infinitesimal differ from an unspecified unit of measure? That question is rhetorical. I see no difference. They are merely different ways of representing the same concept in a particular context. I asked that question to introduce another, more interesting one. How does an infinitesimal differ from a point?

I have already briefly discussed the concept of a point in terms of the difficulty in conceptualizing one without also conceptualizing its representation. I also pointed out that the concept of position carries with it some connotation of dimensionality in the relationship of a point to its point of reference to establish its position. Even the most abstract concept of a point carries with it some dimensional connotation that can be reduced no further than the difference between the pure concept and its representation.

Suppose I propose that in a frame of reference in which measurement is being considered, points have dimension. Would that be heresy? Points without dimension could still be allowed to exist in another more abstract frame of reference; they could even be allowed to exist in the more limited frame of reference to emphasize some relationship that is not one of dimensionality. Points could be defined as infinitesimals, or perhaps it would be better to define infinitesimals as points that are exhibiting some of their dimensional characteristics.

Suppose the infinite universe were being considered in terms of time, space, and motion. The question as to whether or not points have dimension may never arise. Then again, someone might ask; "How many angels can dance on the head of a pin?" I could point out that the current discussion is not concerned with the relative size of angels or the space they occupy, that in the present frame of reference they have no dimensionality. As absurd or as serious the question may be taken to be, it can be answered from the same frame of reference in which it arose. I could say: "An infinite number", that is the same number as the number of points that can occupy the same position at the same time. I could also assume that the questioner posed the question because he did not "jump the gap" between his frame of reference and the one under consideration. Exercising patience, I could say, "If you would tell me the size of angel you had in mind ... " I'm sorry; did I say "angels"? I meant angles. No matter, they both take up the same amount of space in the physical universe.

Enough metaphysics and philosophy. I hope that little excursion helped to pass along some concept of what I mean by a "frame of reference" and a "point of view within a given frame of reference". The fact that misunderstandings and disagreements can be a result of views from different frames of reference, or the result of different views within the same frame of reference, is fairly obvious. Less obvious, but equally true, is the fact that there can be an interface between different frames of reference, the fact that the interface can be viewed as a common frame of reference in which common points of view can be established to resolve the differences of the different frames of reference so represented. Even less obvious than that is the fact that these facts can be represented geometrically as an intersection, with the points of intersection representing common points of view from which to view the common elements, each of which can be representing their own frames of reference with other points of view - an iterative process.

What does this have to do with geometry? To be more blunt about it, what does this have to do with that "crazy" triangle, GVL, that makes its appearance in a modified bisection? I honestly don't know. Some strange shift in reference seems to be taking place, and I am doing my best to try to understand it.

It is more difficult to understand and explain the relative measures of three lines and their included angles than it is to understand and explain the relative measures of two lines and their included angle, which in turn must be understood and explained in terms of relative measure. If my third proposition were generally accepted, this discussion would be unnecessary.

Since my third proposition is not generally accepted and since it has been mathematically "proven" to be an impossibility, I must assume that the second, perhaps even the first, proposition is not generally accepted because they are NECESSARILY related.

It is for these reasons that I feel that I MUST begin with the simplest and purest concepts that are relevant to the present discussion. I shall therefore start with a pure line. I ask you to conceive of one. I suggest, if you really try to do so, that your concept of a pure line and mine will be the same. I really mean the same, equal, identical in all essential attributes. I believe that. I also believe that you do not. How could it be possible? There are so many differences in our education, background, and experiences. It cannot be possible. But, I will try yo prove to you that it is. In fact, I have already made the first attempt to do so by asking you to honestly try to conceive of a pure line, thereby proving it to yourself. I appealed only to your intuitive senses, which are essentially the same as mine.

How else could such a proposition be proven or disproved?

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