Can There Be Any Linearity At All?

by Phil Rubinstein

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.

Predictions Are Always Wrong

by Phil Rubinstein

Of late in dealing with the outlook of the population, we often have to face the impact of linearity most directly with respect to the sense of time. This occurs in the form of “can you predict…?”, “can you tell us when…?”, or “your prediction was wrong, it didn’t happen”, etc. All of this reflects a view of space-time that is one of a linear extension, with space as a filled-up box and in effect no concept of time, since time can exist only as change, action, becoming. It is precisely this linearity that simplifies language to dumbness, reduces music to noise and makes all science and geometry of the post-Kepler period incomprehensible.

It is no accident that one can find a nearly completely modern expression of this in Aristotle’s “On Interpretation” — he says first in section III “… verbs by themselves, then, are, nouns and they stand for or signify something…. [T]hey indicate nothing themselves but imply a copulation or synthesis, which we can hardly conceive of apart from the things thus combined.” And then, “we call proposition those only that have truth or falsity in them.” Were this only the ancient outlook of a discredited Aristotle no problem would ensue, but in fact this is the root of the thoroughly modren outlook of Russell, Frege, Carnap, etc. In fact, On Interpretation could be a handbook for information theory. While Aristotle like his modern followers recognized that the thoroughly deterministic outlook that follows from this contradicts the actual choices made by human beings, his resolution is to introduce mere contingency, a kind of randomness, which is allowed to the empty future.

The reality is best grasped by taking an approach rooted in physical economic planning. Begin with a moment in history defined by a resource level determined by an existing science and technology. A horizon can be hypothesized at which the social cost of resources usable at that level of technology would lead to a critical degeneration or inability to maintain capital or labor. That crisis defines the necessary present deployment of advanced technologies to create new scientific breakthroughs. This, however, requires greater density of use of resources, labor and soon, thus, the horizon is changed. Take the example of fossil fuel, nuclear fission, then nuclear fusion. Our present resources may be either stretched to extend the horizon, but that merely worsens the crisis. If we choose to accelerate the use of fission energy, the demand on existing resources USES UP those resources more rapidly. If we plan to achieve fusion, the rate of usage increases.

Thus, the future is changed for present action at each step. The problem then becomes to determine the actual activity required in the present. As this occurs, the relationship between now and the future is constantly altering: that also alters all other activity, allocation of resources, labor, and so on. In this way, the present is itself an incommensurable. It is a perfect example of non-constantly changing action. It is this subjectivity that lies at the root of understanding physical space-time as something both of constantly changed activity of a multiply connected type in the sense of Leibniz.

From this standpoint, one can see that not only is the future causing the present, but that implicit in any hypothesis of this type is an inversion that is assymetric. As the forecast is made it immediately brings us to a new concept of the path of action itself. The relationship of past, present, and future is altered.

This also has implications for language, such as the fundamental role of the subjunctive, and in physics, such as non relativistic relativity and non-statistical quantum theory. That could be raised for future discussion, but at least never let us be caught in Aristotelean conceptions of the future.