Incommensurability and Analysis Situs

by Jonathan Tennenbaum

Printed in The New Federalist, June-July, 1997.

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Incommensurability and Analysis Situs, Part I

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The issue of analysis situs becomes unavoidable, when we are confronted with a relationship of two or more entities A and B (for example, two historical events or principles of experimental physics), which do not admit of any simple consistency or comparability, i.e., such that the concepts and assumptions, underlying our notion of ``A,'' are formally incompatible with those underlying ``B.'' In the case where the relationship between A and B is undeniably a causally efficient one, we have no rational choice, but to admit the existence of a higher principle of lawful relationship (a ``One'') situated beyond the framework provided by A and B as originally understood ``in and of themselves.''

Exactly the stubborn, ``dumbed down'' refusal to accept the existence of such higher principles of analysis situs, lies at the heart of the chronic mental disease of our age. That includes, not least of all, the Baby Boomer's typical penchant for ``least common denominator'' approaches to so-called ``practical politics.'' Antidotes are urgently required.

An elementary access to this problem, as well as a hint atanalysis situsitself, is provided by the ancient discovery--attributed to the school of Pythagoras--of the relative incommensurability of the diagonal and side of a square. This discovery, a precursor to Nicolaus of Cusa's ``Docta Ignorantia,'' could with good reason be characterized as a fundamental pillar of civilization, which ought to be in the possession of every citizen; indeed, the rudiments thereof could readily be taught to school children. Yet, nowadays there are probably only a handful of people in the whole world, who approach having an adequate understanding of it.

In order to appreciate the Pythagorean discovery, it were better to first elaborate a lower-order hypothesis concerning measurement and proportion, and then see why it is necessary to abandon that hypothesis at a certain well-defined point, in favor of a higher-order conception. The hypothesis in question is connected with the origin of what might be called ``lower arithmetic''--as contrasted to Gauss' ``higher (geometrical) arithmetic''--which however is not to deny the eminent usefulness and even indispensibility of the lower form within a certain, strictly delimited domain. On the other hand, the discoveries of the Pythagorean school put an end to what might otherwise have become a debilitating intoxication with simple, linear arithmetic, one not unsimilar to the present-day obsession with formal algebra and ``information theory.''

Linear measure

Already in ancient times, it became traditional to distinguish between three species (or degrees of extension) of geometry within Euclidean geometry itself: so-called linear, plane, and solid geometry. The phenomenon of ``incommensurability'' bursts most clearly into view, when we attempt to carry over certain notions of measurement and proportion, apparently reasonable and adequate for the comparison of lengths along a line, into the doubly- and triply-extended domains of plane and solid geometry. Actually, the problem is already present in the lower domain; but it takes the transition to the higher domains to ``smoke it out'' and render it fully intelligible.

The commonplace notion of measurement and proportion, is based on the hypothesis that there exists some basic element or ``unit,'' common to the entities compared, out of which each of the entities can be derived by some formally describable procedure. In the linear domain of Euclidean geometry--which, incidentally, presupposes the hypothesis, that length is independent of position--this approach to measurement unfolds on the basis of three principles:

First, given two line segments, we preliminarily examine their relations of position, i.e., whether they are disjoint, overlap, or one is contained in the other. Secondly, we superimpose them, by means of so-called ``rigid motion'' (again, an hypothesis!), to ascertain their relation in terms of ``equal length,'' ``shorter,'' or ``longer.'' And thirdly, we extend or multiply a given line segment, by adjoining to it reproductions of itself, i.e., segments of equal length.

By combining these principles, we arrive at such propositions as ``segment B is equal in length to (or shorter or longer than) two times segment A,'' or such more complicated cases as ``three times segment B is equal to (or shorter or longer than) five times segment A,'' and so forth. [Figure 1.] In the case, where a segment B is determined to be equivalent (in length) to a multiple of segment A, it became customary to say, that ``A exactly divides (or measures) B,'' and to express the relationship by supplying the exact number of times that A must be replicated, in order to fill out a length equivalent to B. Where such a simple relationship does not obtain between A and B, it would be natural to direct our efforts toward finding a smaller segment C, which would exactly divide A and exactly divide B at same time (commensurability!). In case we succeed, the ratio of the corresponding multiples of C, required to produce the lengths of A and B respectively, would seem to perfectly express the relationship between A and B in terms of length. So, the proposition ``A is three-fifths of B'' or ``A is to B as three is to five'' would express the case, where we had determined, that A = 3C and B = 5C for some common ``unit'' C. [Figure 2.]

The paradox of "Euclid's algorithm"

HOW, a practically-minded person would probably ask, might we discover a suitable common divisor C for any given segments A and B? It were natural to first try the shorter of the two lengths, say A, and to seek the largest multiple of A which is not larger than B. If that multiple happens to exactly equal B, we are finished, and can take C = A. Otherwise, we shall have to deal with the occurrence of a ``remainder'' in the form of a segment R, shorter than A, by which the indicated multiple of A falls short of B's length. One possible reaction to this would be, to divide A in half, and then if necessary once again in half, and so on, in the hope that one of the resulting series of sub-segments might be found to exactly divide B. Those skillful in these matters will see, however, why such an approach must often lead to a dead end--as for example when the lengths of A and B happen to stand in the ratio 3 to 5, in which case successive halving of A or B could never produce a common divisor. [Figure 3.]

A much more successful approach, which (at this stage of the problem) represents a ``least action'' solution, became known in later times as ``Euclid's algorithm'': In case the shorter segment, A, does not divide B exactly, we take as next ``candidate'' the remainder R itself. If R divides A exactly, then R is evidently a common divisor of both A and B. Otherwise, take the remainder of A upon division by R--call it R'--as the next ``candidate.'' Again, if R' exactly divides R, then (by working the series of steps backwards) R' will also divide A and B. If not, we carry the process another step further, producing a new, even smaller remainder R'', and so forth. This approach has the great advantage that, assuming a common divisor of A and B actually exists, we shall certainly find one. In such a case, in fact, as the reader can confirm by direct experiments, the indicated process leads with rather extraordinary rapidity, to the greatest common divisor of the segments A and B. [Figure 4.]

The discussion so far, however, leaves us with a rather considerable paradox. For the case, that there exists a segment dividing A and B exactly, the indicated approach to measurement and proportion, provides us with an efficient means to find the largest such common divisor, as well as to derive an exact characterization of the relationship of A to B in terms of a ratio of whole numbers. At the same time, however, some of us might have caught a glimpse of a potential ``disaster'' looming on the horizon: What if the ``Euclid algorithm,'' sketched above, fails to come to an end? It were at least conceivable, that for some pairs A, B, the successive remainders R, R', R''..., while rapidly becoming smaller and smaller, might each differ sensibly from zero.

Within the limits of the ideas we have developed up to this point, we find the means neither to rule out such a ``disaster'' (``bad infinity''), nor to devise a unique experiment which might demonstrate the failure of ``Euclid's algorithm,'' while at the same time providing a superior approach.

Evidently, it were folly to search for an answer within the ``virtual reality'' of linear Euclidean geometry per se. We need a flanking maneuver, to catapult the whole matter into a higher domain.

[To be continued.]

Incommensurability and Analysis Situs Pedagogical Discussion
Part II: Experimental demonstration of incommensurability

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Part I of this paradox appeared in our last issue, dated June 9, 1997.

Moving from singly-extended, linear geometry, to doubly-extended (plane) geometry, provides us with a relatively unique experiment for the solution of the paradox presented above.

Synthetic plane geometry excels over singly-extended linear geometry in virtue of the principle of angular extension (rotation), as embodied by the generation of the circle and its lawful divisions. Among the latter, the square (via the array of its four vertices) is most simply constructed, after the straight line itself, by twice folding or reflecting the circle onto itself.

Having constructed a square by these or related means, designate its corners (running around counterclockwise) P, Q, R, and S. (Figure 1) Our experiment consists in ``unfolding'' the relationship between the two characteristic lengths associated with the square: side PQ and diagonal PR. These two shall play the role of the segments ``A'' and ``B'' in our previous discussion. (Note: the following constructions are much easier to actually carry out, than to describe in words. The reader should actually cut out a square and do the indicated constructions.)

For our purposes it is convenient to focus, not on the whole square, but on the right triangle PQR obtained by cutting the square in half along the diagonal PR. (Figure 2) Note, that the sides PQ and QR have equal length (PQR is a so-called isoceles right triangle); furthermore, the angle at Q is a right angle and the angles at P and R are each half a right angle.

To compare A (= PQ) with B (= PR), fold the triangle in such a way, that PQ is folded exactly onto (part of) the line PR. Since PQ is shorter than PR, the point Q will not fold to R, but will fold to a point T, located between P and R. (Figure 3) By the construction, PQ and PT are equal in length. Next, note that the axis of folding, which divides the angle at P in half, intersects the side QR at some point V, between Q and R. Observe, that the indicated operation of folding brings the segment QV exactly onto the segment TV.

Observe also, that through the indicated folding of the triangle, the triangular region PVT is exactly ``covered'' by the region PVQ, while the smaller triangle portion VTR is left ``uncovered,'' as a kind of higher-order ``remainder.''

Focus on the significance of that smaller triangle. Note, that in virtue of the construction itself, VTR has the same angles and shape (i.e., is similar to) the original triangle PQR.

Euclid's Algorithm Again

Comparing the original triangle to the smaller ``remainder'' triangle VTR, we can easily see that the former's sides are derived from the latter's by relationships very similar to, though slightly different from, the steps of the so-called Euclid algorithm! (See Part I, in our issue dated June 2, 1997.)

First, in fact, the side RT results from subtracting the segment PT, equal in length to the original triangle's side PQ, from the original triangle's hypotenuse PR. Second, the hypotenuse VR of the small triangle derives from the side QR of the original triangle, by subtracting the segment QV, while the latter (in virtue of the folding operation and the similarity of triangles) is in turn equal to TV, which again is equal to RT. In summary: if the side and hypotenuse of the original triangle are A and B, respectively, then the corresponding values for the smaller triangle will be A' = B - A and B' = A - A'. (Figure 4)

Lurking Paradox

The reader might already notice an extraordinary paradox lurking behind these relationships: Were A and B to have a common divisor C, then that same C would--in virtue of the just-mentioned relationships--also have to divide A' and B'. What is paradoxical about that? Well, the smaller triangle is similar to the larger one, so we could carry out the same construction upon it, as we did to derive it from the original triangle. The result would be a third, much smaller triangle of the same proportions, whose leg and hypotenuse, A'' and B'', would thereby also have to be divisible by the same unit C. And yet, continuing the process, we would rapidly arrive at a triangle whose dimensions would be smaller than C itself!

We are thus faced with the inescapable conclusion, that A and B cannot have a common divisor in the sense of linear Euclidean geometry. The relationship between A and B cannot be expressed as a simple ratio of whole numbers. As Kepler puts it in his ``World Harmony,'' the ratio of A to B is Unaussprechbar--it cannot be ``spoken''; by which Kepler means, it is not communicable in the literal, linear domain. But Kepler emphasizes at the same time, that it is knowable (wissbar), and is precisely communicable by other means.

Evidently, the cognition of such linearly incommensurable relationships, requires that we abandon the notion, that simple linear magnitudes (so-called scalar magnitudes) are ontologically primary. Our experiment demonstrates, that such magnitudes as the ratio of the diagonal to the side of a square (commonly referred to algebraically as the square root of two) are not really linear magnitudes at all, but are ``multiply extended,'' geometrical magnitudes. They call for a different kind of mathematics. What we lay out on the textbook ``number line'' are only shadows of the real process, occuring in a ``curved'' universe. This coheres, of course, with Johannes Kepler's reading of the significance of Golden Mean-centered spherical harmonics in the ordering of the solar system, and in microphysics as well.

Analysis Situs Relationship

The relevant relationship for analysis situs, in the preceeding discussion, is not between the diagonal and side of a square; but rather that between the hypotheses underlying the linear domain, sketched in Part I of our discussion, and the superior standpoint implied in Part II.

A final note: Observe the rotation and change of scale of the smaller triangle relative to the larger. Our experimental transformation of the larger triangle into the smaller, similar triangle, as an inherent feature of the relationship of A to B, already points in the direction of Gauss' complex domain, and the preliminary conclusion, that the complex numbers are ontologically primary--more real--than the so-called ``real numbers.''

(Anticipating what might be developed in other locations: The transformation constructed above, belongs to the so-called ``modular group'' of complex transformations, which are key to Gauss' theory of elliptic functions, quadratic forms, and related topics. Gauss, in effect, reworks the central motifs of Greek geometry, from the higher standpoint of the complex domain.)

Circular Action And the Fallacy of ``Linearity in the Small''--Part I

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' (prime sign) and '' (double prime sign) used many times in article
In some of his letters concerning the ``Characteristica Universalis,'' Leibniz notably refers to the virtues of rational methods of entrepreneurial bookkeeping and budget-allocation, as such were originally introduced (according to some credible accounts) by Leonardo da Vinci's collaborator Luca Pacioli. Leibniz remarks, that rational deliberation and discourse should emulate Pacioli's example, in the sense that everything essential to the judgment of any given matter must be accounted for in an ordered fashion, and no steps left out of the argument.

Now, some readers might jump to the conclusion, that Leibniz was advocating some sort of formal, deductive logic. But, stop to consider the following. In any situation, whether in science or war-fighting, the most important aspects that should occupy our attention are the things we don't know, as well as things we do. It would be folly, in attempting to account for any situation, to include only those aspects (so-called ``facts'') of which we have positive knowledge, leaving no room for the singular areas of potential discovery (or surprise) which are the locus of efficient action (change). Those singular areas, on the other hand, are by no means formless or indeterminate. More than 500 years ago, Nicolaus of Cusa gave a most powerful demonstration, after Plato, of how it is possible to know a great deal about what we don't know.

Omission from Reality

Thus, if we omit what Nicolaus of Cusa identified, from our ``accounting'' of reality, then we are falsifying the books. Which is exactly what the Club of Rome did in its ``systems analysis'' model of the world economy, in which scientific and technological progress were brazenly left out. This is the error of those who define ``reality'' solely in terms of their present system of hypotheses, leaving no mental room for the efficient reality of higher hypothesis, which generates an increasing density of singularities in every interval and, in a sense, embodies future discoveries within the present. That act of omission of higher hypothesis, is the plunge downward toward the infinite banality of ``linearity in the small,'' and fascist economics.

To cast some light upon this topic, and upon the fallacy of ``linearity in the small,'' I propose to carry the last two parts' (see New Federalist issues dated June 9 and June 16, 1997) discussion of ``incommensurability and analysis situs,'' a step further. Fresh from the Pythagorean discovery of the relative incommensurability of the diagonal and side of a square, let us turn our attention now to the relationship between the circumference and diameter of a circle. We shall find, that the tactic of folding, which served us so well in the previous case, leads to a rather spectacular failure in the present one. By reflecting upon the deeper (axiomatic) reasons for that failure, we are led to a completely new set of physical ideas, which go far beyond the bounds of Euclidean geometry.

Self-Reflexive Relationship

Recall, that our experimental demonstration of the incommensurability of the side and diagonal of a square, was by no means simply a negative result. The transformation of the larger triangle into the smaller similar triangle as a ``remainder,'' in our construction, seems to provide an exact characterization of the relationship in question, as a self-reflexive relationship of a rather simple type. In a sense, we measured the incommensurability.

Attempting to apply that tactic now to the relationship of the diameter to the circumference of a circle, we might proceed as follows. (Here the same remark as before, is again obligatory: Readers must jump in and work through the constructions themselves.)

First, observe that the diameter of the circle is obtained by folding the circle against itself. Looking at only one of the half-circles defined by that folding, we have a special case of what is sometimes termed a ``lune''--i.e., the figure constituted by any chord of a circle, together with the portion of the circumference enclosed between the endpoints of that chord. For convenience of discussion, I will use the expression ``arc PQ'' (or any other two letters) to designate the circular arc between the endpoints of any given chord of a circle. If we designate the endpoints of the diameter by A and B, we have the lune constituted by the diameter AB and by the circular arc (upper half-circle) arc AB. (Figure 1.)

Next, fold the circle once again upon itself. The result is a second diameter, perpendicular to AB, which intersects AB at the circle's midpoint P. The same second diameter also bisects the circular arc from A to B at a point we shall designate by B', and which at the same time is one of the endpoints of that second diameter. (Figure 1.)

The figure consisting of the two segments AP and PB', together with the circular arc AB', we might perhaps regard as an analogue to the right isoceles triangle in our earlier discussion of the Pythagorean discovery. But now the fun begins.

Fold the arc AB' in toward the interior of the circle, creating, as axis of the fold, the segment AB'. Look at the configuration formed between the triangle APB' and the lune consisting of segment AB' and arc AB'. (Figure 2.) The triangle APB' is of a type we have met before--an isoceles right triangle. The relationship of AP to AB' is that of the diagonal to side of a square. Note, that AP is one-half of the original diameter AB. To the extent our previous discussion of the Pythagorean discovery could be regarded as satisfactory, we could say that we ``know'' the incommensurable relationship of AB to AB'. But what about the relationship of segment AB' to arc AB'?

Lunes Not Commensurable

In a sense, the lune formed by AB' and arc AB' is the ``remainder'' which is left when the triangle APB' is removed from the curvilinear figure AP, PB', arc AB'. Now, consider the transformation from the lune AB, arc AB, and the lune AB', arc AB'. Consider the relationship of that transformation, to the transformation we developed in our earlier reconstruction of the Pythagorean discovery. A rather crucial difference comes to light: in our present case, the smaller, ``remainder'' lune is not similar to the original one! While the circular portion arc AB' is half of arc AB, the segment AB' is longer than half of AB, and in fact forms an incommensurable relationship to the same.

To get a clearer insight into what is happening here, carry the construction a step further. Fold the circle a third time onto itself (i.e., fold into a half, a fourth, and now an eighth) to create a diameter which divides arc AB' in half, at a point we shall designate B''. The same diameter bisects the segment AB' at a point P'. (Figure 3.) Now fold arc AB'' toward the interior of the circle, creating as axis the segment AB''. Now, examine the right triangle AP'B'', and the ``remainder'' when that triangle is removed from the figure formed by the segments AP', P'B'' together with arc AB''. That ``remainder'' is the lune consisting of segment AB'' and arc AB''. (Figure 4.)

Examining the circumstances of this second transformation, note that the triangles APB' and AP'B'', while lawfully related, are not similar. Nor, of course, is the lune AB'', arc AB'' similar to either the lune AB', arc AB' or the original lune AB, arc AB. The reader might take a look at Leonardo da Vinci's explorations of this sort of problem, in an elaborate series of drawings.

A Bad Infinity

Those zealous and skillful in this sort of geometry, will find ways to characterize the relationship between AB' and AB'', which are incommensurable, just as AB and AB' were incommensurable, but with a somewhat different relationship. They may suspect, perhaps not without a twinge of horror, that as we continue the series AB, AB', AB'', AB''', the ``degree of incommensurability'' between the original diameter and the ``Nth'' segment in the series, keeps building up!

It is clear, that the segments AB', AB'' are nothing but sides of a square, octagon, 16-gon, 32-gon, etc., inscribed in the circle. What we are doing could be seen, in one respect, as carrying out Archimedes' ``exhaustion principle,'' trying to approximate the circle's area and circumference by polygons of exponentially increasing number of sides. However, it seems fair to say, that our tactic for overcoming the ``bad infinity''--a tactic which in a sense succeeded for the case of the diagonal and side of the square--has ended in a spectacular failure. We don't get ``closure,'' but instead a bewildering array of increasingly complex, incommensurable relationships.

What is the source of the problem? Could it be, for example, that the action of ``folding'' fails to capture the essence of the circle, or what is behind the circle? What have we left out?

P.S. To get a sensuous notion of some of the physical ramifications of the problem discussed here, it is necessary to abandon the armchair. I recommend, as a bare starter, the following ``field'' experiment. While extremely simple, it should provide a first insight into some of issues which Gauss dealt with in his approach to geodesy and measurement in general.

Use a wire, or other means suitably devised, to draw a small arc (say, about 20 cm long) of a circle of radius 10 meters or more. Examine the arc so drawn. If done with precision, the difference of the arc from a straight line-segment is practically imperceptible. How do we KNOW that a discrepancy exists at all, and how might its magnitude be characterized and estimated?

Circular Action and the Fallacy of `Linearity in the Small'--Part II

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Part I of this paradox appeared in our last issue, dated June 30, 1997.

Very often, the greatest obstacle to progress in a given domain, is the tendency to linger within the axiomatics of a failed approach. To get to the heart of the paradoxes presented last week, let us attempt a fresh look at the original problem. The following considerations are ``childishly simple,'' but are no less profound in their implications.

Rather than fixate on the special case of the relationship between the diameter and circumference of a circle, I propose to examine, more broadly, the relationship of any circular arc, to any straight line segment. Consider the proposition, that no circular arc, no matter how small, could ever coincide with a straight line segment. By reflecting on the evidence for such a proposition, we might gain some new insight into the inner nature of the ``creature,'' whose existence is suggested by our difficulties in reconciling the diameter with the circumference of a circle.

Geometry in the Small

To this purpose, construct a circle with center P and radius r, and imagine an ``extremely small'' circular arc with endpoints A and B. How small? Consider, for example, the tiny arc obtained by successively folding the circle upon itself (successive halving) 100, or even 1000 times! (Figure 1.) Reflecting on the nearly unimaginable smallness of the angle and arc length involved, the question should pose itself: Do the constructions of Euclidean geometry remain valid and applicable at such extraordinarily small length- (or angle-) scales? At what point do entirely different physical principles confront us, when we pursue the ordering of our Universe (the ``Cosmos'') down toward the ``infinitely small?''

Without attempting to address that issue directly at this point, let us first assume, that the Euclidean constructions preserve at least a certain degree of relative adequacy for the length-scale we are dealing with. In that case, we can easily evoke the necessary existence of a tiny discrepancy between the circular arc AB and the line segment AB, as follows.

By an additional act of folding, generate a diameter which cuts the circular arc in half, while at the same time halving the line segment AB, at a point we shall call C. (Figure 2.) Note, that triangles PAC and PBC are both right triangles; in fact, they are superimposed under the indicated act of folding. Assuming the constructions of Euclidean geometry are applicable at this scale, the sides of these triangles, or rather the squares on those sides, are related by Pythagoras' famous theorem: The square on the hypotenuse PA is equal to the sum of the squares on the sides AC and PC. Note, that PA has a length equal to the radius of the circle, while PC must necessarily be smaller by some small, but distinct ``quantum.''

Close, But Not Quite

Why? The length AC, which is half of AB, while extremely small, is still distinctly greater than zero. Hence, the square on AC also has a non-vanishing magnitude, and since the square on PA is the sum of that tiny square on AC and the square on PC, we must acknowledge that the square on PA is slightly larger in area that the square on PC. The inescapable conclusion is, that PC is shorter than PA--by an exceedingly small, but nevertheless distinct and implicitly calculable quantum. Thus C cannot lie on the circle's circumference, but rather is slightly separated from it on the inside of the circle.

Since C lies on the line segment AB, that separation at the same time represents a distinct ``gap'' between the straight-line segment AB and the circular arc, even when the ``gap'' is hardly perceptible to sense perception. Evidently, the existence of that ``gap'' is a persistent, irreducible feature of the relationship between circle and straight line.

But, what should we say if the assumptions of Euclidean geometry were demonstrated to break down at the given, relatively microscopic scale? The existence of diffraction and refraction of light, for example, might be taken as strong evidence to the effect, that such a breakdown cannot be avoided. In that case, the very existence of such a singularity (the breakdown) were sufficient to establish the non-linear character of the circular arc!

On several accounts, however, the discussion so far hardly suffices to dispel a certain uneasiness. Indeed, we have rather increased it.

Limits of Euclidean Geometry

Does not the character of our attempts to characterize the relationship of circle to straight line, in a sense, display the conceptual limits of Euclidean geometry itself? In other words, although the circle and straight line are acknowledged as forms in Euclidean geometry, their existence and their relationship cannot be accounted for within Euclidean geometry, except in a negative way.

The mere determination of discrepancies or gaps, even at a potentially ``everywhere dense'' array of locations, does not define the relationship positively. No array of singularities in and of themselves, no matter how densely we try to ``pack'' them, could ever ``add up'' to the process which is generating them. From that standpoint, the proposition, that ``the circle is a polygon with infinitely many sides'' might well be suspected of being nothing but a sophistical trick, a brazen attempt to evade the issue posed by Parmenides' paradox. Evidently, in order to account for what lies behind the circle, we have to go outside the domain of Euclidean geometry--not to ``non-Euclidean geometry'' in the usual mathematicians' sense, but to something very different.

Anticipating that event, let us return once more to our ``super-small'' circular arc, and look at the matter from another flank. What is the change, when we go along a circular arc from point A to point B? Recall Eratosthenes' method to estimate the circumference of the Earth. That method was based on observation of a change of angle of sighting, when we observe the Sun from two different points on the Earth's surface. In our present case, suppose, for example, that a very distant star happens to be located at a certain time ``directly overhead'' at A (i.e., along the continuation of the ray from P to A). That same star would appear slightly off the zenith (overhead direction) as seen from B at the same moment. (Figure 3.) Off by how much? As Eratosthenes noted, the angular displacement from the zenith would be equal to the angle which PA makes with PB at the center of the circle (or the Earth).

Now consider an arbitrary observation point C on the arc AB. Changing the position of C, we see that the star's displacement from the zenith increases at a constant rate as we move from A to B along the circular arc. (Figure 3.) Does this not suggest a completely different approach to the comparison of a circular arc to straight line, than we have taken up to now?

At any point C on the circular arc, construct the perpendicular to the radial line PC, otherwise known as the ``tangent.'' Compare the direction of the tangent at A with that of the tangent at B. Evidently, the angular change in direction is again equal to the angle formed by PA and PB at the center of the circle. However small that latter angle might be, as long as it is has a distinct non-zero magnitude, the same angle will be re-emerge as a change in direction of the tangent at A as compared with the tangent at B (for example, as determined by sightings along the tangents onto the celestial sphere). (Figure 4.) Note, that for the case of a straight-line segment, as opposed to a circular arc, the ``horizon'' or direction of motion does not change.

Aha! Are we not close to a much more direct, more fundamental characterization of the discrepancy between the circular arc and any line segment?

"Rate of Change"

Consider, the implications of the idea of a variable displacement along the circular arc. Insofar as the tangent represents a direction of motion, or alternatively a ``horizon'' for the point C, the tangent at C changes its direction at a constant rate as C moves along the circular arc from A to B. Might we not, as a preliminary hypothesis, take that rate of change as the measure of the relationship between the circular arc and any given straight line? And might we not take the notion of ``constant rate of change'' as an appropriate basis for redefining the existence of the circle itself, and even the entire domain of geometry?

Indeed: The notion of ``rate of change'' has no existence within Euclidean geometry! Introducing that notion ``from outside,'' means a fundamental, axiomatic revolution in mathematics. Note, that the act of ``redefinition'' of geometry in the indicated way--which, of course, remains to be richly explored--has no assignable ``length'' or other scalar magnitude. We are back to analysis situs.

Consult Nicolaus of Cusa's ``Docta Ignorantia,'' particularly section 13 of the first book, where Nicolaus has the beautiful figure of a manifold of circles of varying curvature. (Figure 5.) The notion of an ``interval'' between differing rates of change (curvature), has opened up a new pathway toward an intelligible representation of the relationship between the circumference and diameter of a circle.


by Phil Rubinstein

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It is often the case that mathematicians, scientists, and their followers are able to see anomalies, paradoxes, and singularities, but maintain appearances by limiting such incongruities to the moment or the instant or position of their occurrence, only to return immediately to whatever predisposition existed in their prior beliefs, mathematics, assumptions. It is precisely this error that allows linearization in the small, in the typical case through reducing said singularities to an infinite series. In fact, in even the simplest cases, as we shall see, the singularity, anomaly or paradox requires every term in the pre-existing system to change, never to return to its prior form.

There is nothing complex or difficult in this. Let us take the simplest example. Construct or imagine a circle with the two simple folds we have used before. Now, construct the diameter and its perpendicular bisector giving us four quadrants. Now, take the upper left hand quadrant and connect the two perpendicular radii by a chord at their endpoint. If we consider the radius of the circle to be 1, we have a simple unit isosceles right triangle. Thus, from previous demonstrations, the chord connecting the two legs of the right triangle is the incommensurable square root of 2. Now, rotate the chord or hypotenuse until it lies flat on the diameter, or, alternatively, fold the circle to the same effect. The anomaly here is quite simple. Not only is the ratio of the chord to the diameter of the circle incommensurable, but the question arises: where does the end point of the chord touch the diameter? How do we identify it? From the standpoint of integral numbers and their ratios, this position cannot be located, neither can it be named within that system. This, despite the fact that if we take all the ratios of whole numbers between any two whole numbers, or ratio of whole numbers, we have a continuity, that is, between any two, there are an infinite number more. What, then, is the location? Is there a hole there or break? While this has often been the description, this is clearly no hole! By the simplest of constructions, we have the location, exactly. Our chord does not "fall through," its end does not "fall into a hole"!

Now, we find the typical effort is to say, yes, there is a strangeness here, but we can make it as small as we like. By constructing a series of approximations, we get a series of ratios that get closer and closer. Fine, one might say, but still, what is the description or number by which we designate the location? Well, comes the answer, the infinite series description can be substituted for the place or number, and everything in this description is itself a number, or ratio of numbers. Thus, we have reduced the problem in fact and located the continuum on our diameter. One may reflect that, as simplified as this is, it is essentially the point made by Cauchy, etc., although in a different context.

In the calculus of Leibniz, the differential or limit exists as the area of change which determines the path of physical action. Cauchy reduces that physical reality to a mere calculation, by substituting an infinite approximation, or series for the limit, or area of change. What is lost is simply that reality which determines the physical action, and thus the ability to generate the idea of lawful change as a matter of physics.

But, does the anomaly go away? Clearly, it does not. To identify the actual position, which exists by construction, with a series that is infinite, endless and made up of precisely components proven NOT to be at that position, does not solve the anomaly. The position exists, is different, and remains singular.

In fact, much more follows. Label the left end of the diameter A and the location where the chord and diameter meet B. We will label the intersection of the diameters O. We can now ask what happens if we move back along the two lines, the chord and diameter. Let us say we move from B towards O, the center of the circle. Since the end point B of the chord is incommensurable with the diameter, if one subtracts any rational distance towards O, the position reached is still not commensurable, and this is so for ANY rational distance from B all the way to A. So, every position so attained is likewise incommensurable, as many as there are rational numbers. If I attempt to subtract an incommensurable amount (e.g., by constructing an hypotenuse and folding it), one has not solved the problem but merely used a position unlocatable by integral numbers or ratios of them. In fact, we now have a new infinity of these unlocatable positions back on the diameter.

This process can be looked at in the following manner. Is the position at the end of the chord greater than, less than, or equal to a given position back on the diameter? If we take also any position obtained by subtraction as above, do we attain a position greater than, less than, or equal to a rational number on the diameter? In fact, it is impossible to express the answer to these questions! One may attempt to say that an infinite series is as close to, but always less than, some arbitrary distance, but unless one knows beforehand the position, one can never know whether we have passed the position, or are not there yet. The concept of predecessors or successors or equivalence is inoperable, inclusive of whole number cases.

Since this occurs as has been shown, everywhere on the two lines, the only solution is to change the conception of number, measure, or position for every position on the diameter and chord. To simply add "irrationals" will not do, since this will leave us with inconsistency everywhere: in effect, a line made up of locations that cannot be compared.

The problem expands to a critical point with the addition of the relation of the diameter to the circumference. We must change the concept of number for every position. In this case, integers, rationals become a case of a changed number concept or metric. Properly understood, rather than attempting to linearize the discontinuity, we should say every position on the line has "curvature." This becomes more transparent if we think of Cusa's infinite circle as in fact the ontological reality of the so-called straight line. Only such a "straight" line could contain the positions cited above, could be everywhere curved, and yet a line.

How did this occur? An anomaly was shown to exist. To incorporate that anomaly's existence requires a full shift in hypothesis. More especially, any linear construction is not an actual hypothesis, since it is unbounded and open ended, its extension is always arbitrary. To exist, an hypothesis requires, conceptually, "curvature," that is, change which identifies its non-arbitrary character. That is its hypothesis. That is, what exists in the anomaly in the small is a reflection of its characteristic actions, its hypothesis. There are no holes, no arbitrary leaps. Now, of course, this leaves open the question -- what other changes, hypotheses may be reflected requiring further hypothesis. It is no mystery that any line, or segment of a line existing in a universe of such action will manifest those actions down to its smallest parts, and do so for each such action.

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The preceding article is a rough version of the article that appeared in The American Almanac. It is made available here with the permission of The New Federalist Newspaper. Any use of, or quotations from, this article must attribute them to The New Federalist, and The American Almanac.

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