CAN YOU SOLVE THIS PARADOX?

by Jonathan Tennenbaum

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?

{(To be continued.)}

CAN YOU SOLVE THIS PARADOX?

by Jonathan Tennenbaum

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'<cm

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.