We shall not here discuss the inexactitude which lurks in the concept of simultaneity of two events at approximately the same place, which can only be removed by an abstraction.
Albert Einstein
A new distinction between intensive magnitude and extensive magnitude is proposed, in which the two concepts are equally original and equally accessible to our noetic faculty. An attempt is made to instantiate these concepts in the space-time of our experience with a unitary view of space-time and matter. The new concept of intensive magnitude is explicated in the context of special relativity, using devices drawn from chaos theory. The mind/matter dualism is reinterpreted in terms of the marriage of intensification and extensification.
The purpose of this paper is to present - and entertain - a new understanding of intensive magnitude. Intensive magnitude is a scholastic innovation derived from the characteristic adduced by Aristotle to distinguish qualitative beings (qualia) from quantitative beings (quanta): namely, that qualitative beings are capable of possessing a certain quality to a greater or lesser degree, while the greater and the lesser have to do with magnitude. A body can be more or less red while remaining red; it is just more intensely red. However, a person is either six feet tall or he isn't. In modern times, the concept of intensive magnitude has almost entirely lost its autonomy and authenticity. Exact scientific thought persists in defining it against the horizon of extensive magnitude. It has done this for so long that the scientific concept may have finally lost touch with the phenomena we would naturally think of as "intense" or "intensive."
There is something almost pernicious about this tendency. Just consider one of the most recent instances: Grassmann, one of the founders of the vector calculus, carefully distinguishes intensive magnitude from extensive magnitude, assigning the former to ordinary algebra(!) and the latter to the combinatorial art he is about to introduce. This combinatorial art of extension develops into the vector calculus, which gets generalized into the tensor calculus. Whereupon the distinction between tensors and tensor-densities is invoked - totally within the tensor calculus itself - to give physical significance to the distinction between intensive magnitudes and extensive magnitudes.
For the most part, philosophy has abetted this tendency by effectively ruling out the possibility of such a thing as intensive magnitude. What is knowable about nature must either be an extended magnitude itself or allow of being modeled with extended magnitudes. What cannot be so symbolized belongs to the irrational, material, given element of experience that does not allow of being grasped noetically, or in a mathematically exact way, and thus does not really deserve to be called a magnitude at all.
Fortunately, some awareness of intensive magnitude slumbers on in our common consciousness and our common sense. Over the centuries, a concern for this realm of experience has surfaced at odd times and always left its mark. When we hesitate over the words 'further' and 'farther,' or the phrases 'to a large degree' and 'to a large extent,' we are sensing a distinction between intensive and extensive magnitude. When we are struck with a certain inscrutable subtlety in the curious phrase 'remission of sin,' we are catching an echo of the scholastic debate about whether grace could be more or less, remission being the opposite of intension. If we ever struggle to understand the special kind of indeterminacy belonging to the letter-signs employed in algebra, it is because mathematicians in the 17th century never fully succeeded in assimilating Viéte's algebra of intensive magnitudes to Descartes' algebra of extended magnitudes. And many more traces of intensive magnitude could be adduced in our language and common experience.
It is my hope that the new conceptualization of intensive magnitude presented in this paper, though highly abstract, will accord with this largely subliminal awareness. What is more, I believe that it will give an exact statement to what the New Age has been reaching for with its vague notion of "holistic," while it is the lack of such a concept that has been at the root of the dissatisfaction that it feels for modem science. I also contend that an authentic conceptualization of intensive magnitude - something that it may never have had in the modern world - may go a long way toward overcoming some of the impasses in modem thought: the unsatisfactory dualism between matter and space-time, and between matter and mind. And curiously enough, it would seem to show the ultimate emptiness of all things. Finally, if I were to align myself in the Western philosophical tradition, I would be tempted to be very bold and say that Being is intensification.
It is not the purpose of this paper to build a theory around this new concept, or even to lay the foundation for one, but merely to entertain it. After presenting the pure mathematical concept of an intensified space in general, we will try to instantiate it in the space and time of our experience to come up with a unitary view of space-time and matter. We will then reconsider Riemann's argument that the metrical properties of space cannot be fully determined a priori and must consequently be due to the presence of matter, trying to show that the problem of the unification of the gravitation field and the electromagnetic field is the same as the problem of a necessary relation between intension and extension. We will next take the occasion to re-root the concept of intensive magnitude in the philosophical tradition by returning to Kant's analysis of the anticipations of perception. At this point, we will deepen our understanding of intensification by setting up a model with special relativity, using devices drawn from modern chaos theory. The question of temporal intensification will then be broached, and astrology suggested as a means of studying it. All along we will reach until our reaching begins to exceed our grasp, which is perhaps the true way of finding the horizon of a concept.
The Pure Mathematical Concept of Intensive Magnitude - Intensive magnitudes - or rather, intensified spaces in which intensive magnitudes can be distinguished - may be constructed in either of two ways: either 1) by performing a simple operation on a given, multiply extended, continuous manifold, or 2) at the same time as the multiply extended continuous manifold is itself being constructed, by modifying in a very simple way the well-known procedure for creating a new space as a "product" of two others.
The first way of intensifying a continuous manifold is to run through the given extended manifold with a special act of synthesis that continues each element of the manifold by some other element adjacent to it, or by itself. That is, it is a way of mapping each element onto one of its neighbors and keeping track of all the paths generated in this way.
Imagine a line divided into segments in an arbitrary manner. Ordinary continuity, where the parts are merely butted up against one another, is a special case that results when each segment of the line is continued by itself, mapped onto itself. More generally, however, each segment will be continued by one of the segments adjacent to it. Segments continued in this way will overlap, somewhat in the manner of the strands of a rope, intensifying the one-dimensional space. We assume that it is also possible for two different segments to be continued by the same segment, making them "hypercontinuous" at that part.
The reciprocal operation takes contiguous parts and separates them from one another, making them subcontinuous, or takes the first-order hypercontinuous line and reduces it to ordinary continuity.
This first mapping may be followed by a second which perhaps iterates the same map, perhaps continues the segments yet further according to a new map, perhaps even reverses the first map and returns each segment to its origin. And so forth.
The same line of reasoning can be elaborated to explain the intensification of multiply extended continuous manifolds.
We can accomplish the same thing in the course of constructing the multiply extended continuous manifold itself. In the usual way of constructing a triply extended manifold from two doubly extended manifolds, for example, every point on one of the doubly extended manifolds is conceived to pass over into one of the points of the other manifold in a fully determinate manner. That means that to every point on the one manifold there corresponds a definite point on the other, and vice versa, and also that the lines joining the points of each manifold fill the intervening space without gaps and without crossing one another. Every point on the one manifold is thus a proper function of a point on the other.
We can construct multiply intensified continua at the same time, merely by relaxing this determinacy condition to allow improper or multivalued mapping functions. For now, we will restrict this mapping function to the class of those functions that map a part onto a part of the same scale. Then, for example, two points may map onto the same point (intensification) or the same point may map onto two different points (remission), making the above lines cross or leave gaps. Continua resulting from proper mapping functions may then be understood as continua with minimal or perhaps zero intensification.
If we wanted to, we could probably turn the improper functions into proper ones by means of Riemann surfaces, but this would be to treat the intensified spaces as extensive magnitudes, wherein the parts remain forever exterior to one another, however they are stretched or involuted. Such a procedure would defeat the whole spirit of our enterprise.
In both our constructions, intensification is a kind of coalescence, remission a principle of making discrete, or discretion. These notions seem to be intuitively consistent with the notion of increasing or decreasing intensive magnitude.
In an elastic extended manifold, intensification would result in overlaps or "seams," remission in "tears." Either would necessarily cause a buckling or change of curvature of the manifold. The curvature of a manifold is related to its metric. This suggests that in manifolds that have some principle of cohesion like elasticity, there may be a necessary and intelligible connection between intensification and the metric of extensification.
The notion of coalescence and its opposite, discretion, suggest a "filling" or "emptying" of the space in question - not with something extraneous to the manifold, however, but merely with "more" of the manifold itself. This in turn calls to mind the concept of fractal dimension from fractal geometry, inasmuch as a self-similar jagged line may be thought to partially fill a surface, resulting in a fractal dimension between one and two. It is true that the present concept of fractal dimension has been elaborated in the context of figures and shape (that is, extensive thinking), but there seems to be no reason why it cannot be disengaged from that context and employed in the intensive realm.
For example, we could take one of the Koch constructions and collapse its various segments onto the baseline, causing them to overlap, and then view them as coalesced.
We could also reconceptualize the Cantor set within the sphere of intensive thinking. Instead of removing the middle portion of a segment, for example, and continuing to remove the middle portion of the successively smaller segments that remain, until we produce a kind of dust, we could take contiguous segments and remove them to a distance from one another.
This gives us a kind of natural measure of intensification. Accordingly, we provisionally make use of similarity dimension and measure intensification with the expression logN/log(1/r) , where N is the number of parts and r is the ratio of part to whole.
It should be apparent that we do not have to iterate the reconstructions to infinity in order to give meaning to the coalescence or discretion of a space, as we must for their extensive counterparts. However, higher iterations do give a kind of depth to the coalescence and should also be a component in our measure of intensification. Whether the intensification is directly proportional to both the dimension number and the number of iterations making up the depth is a matter that we will consider shortly.
Physical Space and Time of our Experience - What would it mean for the space and time of our experience to be intensified?
We maintain that space is a three-dimensional manifold intensified by energy. The parts of space do not simply endure in time in some pre-given state of contiguity. They too must be continually remapped back onto themselves or each other as continuations. And in order for a region or "extent" of space to have any coherence, they must be remapped together, or at once. This is the function of energy. We cannot take it for granted that ordinary continuity prevails in the spatial continuum any more than we can take it for granted that a Euclidean metric obtains in a certain region of space. We do not even have any a priori reason for thinking that the intensification of space is homogeneous. In fact (as we will argue), tangible matter is itself the result of a special local intensification of space.
We further maintain that the successive spatial mappings that make up the time of our experience form a manifold that is also intensified by energy. In order for a "stretch" of time to have any coherence, the successive remappings must be recursively related by the same function. In order for there to be any duration in the space of our experience, as well as succession, this recursive function must help bring about a repetition of the same spatial intensification state. But we have no reason for assuming a priori that this repetition is direct; it may be the outcome of a multiply periodic repetition cycle, what we call temporal intensification, since it is the superposition of two or more cycles at the same time. Again, this entire cycle may be a stretch of nonhomogeneity in a higher order repetition cycle, in which case we might call it a temporal "body."
Energy is intensification, and it is its nature to intensify a region of space to infinity, to make more and more parts of the space coalesce at finer and finer scales. But other regions of space have their respective principles of intensification too, and the same region may be participating in many different intensification schemes, each wanting to go its own way and tending to curb the intensification scheme the local energy would follow. So there is a self-renewing internal principle of intensification and a counteracting nonlinear restraining principle. Modern nonlinear dynamics has taught us to model such a scenario with a one-dimensional recursive map. Then, various possibilities arise depending on the balance of these principles. Sometimes the local energy will be overwhelmed and the intensification will necessarily assume its lowest possible state; usually it will produce a cyclical succession of intensification states.
But the situation is not quite so simple. The emerging succession of intensification states also constitutes a manifold. Energy works no less to intensify this manifold to infinity as well. To intensify temporal moments is to turn a simple cycle into a multiply periodic one. But just as each region of space may have to participate in, and conform to, several different spatial intensification schemes at different scales, so each emerging intensification state may have to belong to several different schemes of temporal intensification, several different cycles, corresponding to the larger picture that the given region is part of. Thus energy's tendency to multiply the articulations of temporal intensification to infinity is counteracted by the competing claims of other schemes to make them intensify according to some other scheme. In a manner analogous to the above, the result is a certain limited, multiply periodic scheme of successive intensification states.
Thus, spatial intensification and temporal intensification reciprocally influence one another.
We might call this cyclical succession of states a clock. A useful technical definition of a clock was given by Weyl: "If an absolutely isolated physical system (i.e., one not subject to external influences) reverts once again to exactly the same state as that in which it was at some earlier instant, then the same succession of states will be repeated in time and the whole series of events will constitute a cycle. In general such a system is called a clock. Each period of the cycle lasts equally long." Weyl adduced this as an objective criterion for equal lengths of time.
It is a truism that without change there would be no time. But what we are looking at in our intensification scenario is the most basic and fundamental energy process there is in a certain region of space. No change takes place between succedent intensification states. There is no time flowing on continuously in which the intensification happens. Instead, it is the recursive repetition itself that generates time. Furthermore, when we have a recursively periodic cycle of intensification states, these must all be thought of as coalesced into a single moment. We must remember that we are not dealing with extensive repetition in which each new stage is exterior and distinct from its predecessor. Instead, we are proposing an intensive repetition in which each "successive" step encloses and carries along the former steps, at least until the first intensification state in the cycle, the one most deeply enclosed, begins to relax, as it must do as long as the temporal intensification component of the local energy is curbed in its attempt to intensify to infinity, which would be tantamount to rendering all the intensification states simultaneous with one another, in which case there would be only an endless moment.
Thus the succession of intensification states does make up a clock, but it is a clock that makes time instead of merely telling it.
It might be objected that at some smaller scale in this region there might be different cyclical processes taking place, shorter ones that could measure the longer ones in question. But we have already made the assumption that the internal energy has intensified to the greatest depth possible. The only way to create a faster clock would be to introduce more energy.
We could have dealt with spatial and temporal intensification together in one argument if we had considered intensification in a four-dimensional space-time continuum, but that might have obscured the different roles of spatial and temporal intensification. There would be no coherence to space at all if there were no time, and there would be no time if no space, because there could be no recursion of intensification states, no succession and duration together, no simultaneity or moment. Thus time is not just another coordinate in a four-dimensional system.
Instead of trying to construct a limited intensification out of a dynamical balance between selfexcitation and damping, as we did above, we might have tried to describe it in terms of a Poincaré recurrence, although it would have to be an elaborated version of one to allow for the improper functions that are used in the mapping. However, the idea behind a Poincaré recurrence seems more in keeping with extensive thinking, remapping all the points in a manifold. The dynamical procedure that we did use also has the advantage that it gives a meaning to the "elasticity" or coherence of the continuum, which will be important when we discuss the influence of intensification on the space-time metric.
To recapitulate: We have tried to point the way to a unitary view of matter and the space-time continuum. In a sense, we have constructed matter out of nothing, as a special intensification of the space-time continuum. This is in keeping with the vision of field theorists like Einstein, whose goal has been to understand matter as a kind of "knottiness" in the space-time continuum. However, their approach seems to be totally dictated by extensive thinking, which makes it hard for them to understand why the "knot" held together in the first place. This difficulty is solved as a matter of course in the approach outlined here.
The a priori Determination of the Metrical Properties of Space - Riemann seems to have been the first to really worry about the metageometrical problem of constructing multiply extended manifolds instead of merely taking geometrical spaces as given. In this way he discovered that the metrical properties of a three-dimensional manifold were not fully determined by general concepts of quantity. From this, he concluded that the metrical properties of the space of our experience would have to be due to the empirical content of that experience, that is, its material content rather than its formal characteristics. So matter would have to provide the binding forces to hold the parts of space together. Different densities of matter locally would result in some parts of space being bound together more "tightly" than others, thus producing different spatial metrics, different curvatures, and thus different geometries.
Brilliant and revolutionary as this argument is, there is one major unexamined assumption behind it, deriving from its overall Kantian orientation. Even though Kant calls quality as well as quantity a mathematical category (as opposed to the dynamical categories relation and modality), Riemann takes it for granted that quantity is the only category that could be an a priori determinative of the geometry of space, the only category that would have any bearing on whether space was Euclidean or not, so that if some geometrical characteristic is not due to the formative influence of quantity, it must be assigned to the empirical realm.
However, if matter/energy is the intensification of space-time in the manner outlined above, so that space can still be understood as an "elastic" continuum, then the intensification of space-time is a seam, which necessarily causes a buckling or curvature of the space, thus affecting its geometrical characteristics. In other words, the intensification of space-time is necessarily connected to its extensification. Thus we can already see, in an intuitive way, a path leading towards the unification of the electromagnetic field (intensification) and the gravitational field (extensification). The question is, to what extent are the intensification properties and the metrical properties of this unified field deducible from purely formal characteristics of their mathematical description, and what is left over to be given in experience?
It seems that we have to deepen Riemann's investigation. We must first see whether the intensification characteristics of our space - the general nature of the mapping - are fully determined by general concepts of quality. This argument is parallel to Riemann's. Then we must see whether the conditions due to quality, taken together with those due to quantity, more fully narrow the possibilities for our three-dimensional space than the conditions due to quantity alone.
One feature of our approach deserves to be singled out for special comment. Riemann quite properly thinks of the parts of the manifold as preceding the whole and needing to be bound together, since he is thinking in terms of extension. But in the case of intensive magnitude, the parts of the continuum do not precede the whole in experience. It is true that we have allowed them to do so in our construction of an intensified space, but even there we stipulated that these parts were participants in larger intensification schemes; they were not understood to enclose smaller ones. Intensive magnitude is a truly "holistic" concept.
Replanting Intensive Magnitude in the Philosophical Tradition - The question is whether the above construction of intensive magnitude is more than just an image in terms of extensive magnitudes (that is, merely a model), but instead represents its authentic character ontologically speaking. We can use a straight line to represent not only a length but also a number; if we divide the line into equal parts, it is a multitude of equal parts and looks like a multitude of parts. But if we use a line to stand for one of the general numbers of algebra, and justify this use by saying that the number of parts of the line is indeterminate until a measure is assigned, then the line is a mere symbol of a general number, it represents it by the convention we have introduced but it does not look like a general number. How does the construction of intensive magnitude represent it? Is intensive magnitude as here presented something that can be seen for what it is noetically? That is, can we stand back and look at it in the mind's eye, or can we only receive it as a given, inseparable from perception itself?
The procedure of representing intensive magnitudes with straight lines goes at least back to the 1400s, if not further. These lines could even be mixed with lines representing extensive magnitudes in the same figure. They represented intensive magnitudes no differently than the other lines represented extensive magnitudes. The intensive magnitude was understood to be just as directly intuitable and seen in the mind's eye as the extensive magnitude. And at that time there was still a tacit understanding that intensive magnitudes were just as "real" as extended magnitudes, and totally distinct from them.
But after Descartes identified the real of nature with extension, and others such as Galileo distinguished secondary qualities from primary ones (understanding qualities to be due to the modification of our sense organs - themselves material and extended - and their intensities the degree to which they were modified), intensive magnitudes lost their place. This is when the representation of the late Renaissance began to turn into the modeling of modern times. In so far as the formless element of our experience could be understood, it could only be understood symbolically. As a result, we now understand the intensity of light, for example, as the amplitude or vertical displacement of a certain wave.
Unlike his immediate predecessors, Kant did try to retain a formal synthetic principle of quality, and did make this principle responsible for intensive magnitude. Nevertheless it has seemed to some that it was Kant's own characterization of intensive magnitude that led to its extinction as a directly intuitable mathematical concept not requiring the mediation of extensive magnitude, in that he made it an anticipation of perception and not of intuition, a principle of the immediately given in general rather than of the synthesis of the immediately given. You could not do mathematics with it because you could not get away from the merely empirical, except with a general concept that can not be constructed in manifold ways in the intuition in the same way that a straight line can be drawn in different sizes.
It is not entirely clear that this is Kant's own view. In any case, here we will try to interpret him in the opposite way, primarily because of the remarkable example of Viéte, who sought to create an algebra of intensive magnitude in the late 1500s.
To be brief, Viéte thought that the Greek mathematicians had had an algebra of intensive magnitudes. He called the objects of this algebra "species," which is the Latin term for the subject in which qualities inhere. These species were "functions" of the things being sought in a problem, which were the greater and the lesser. A typical "function" might be the sum of the greater and the lesser; however, Viéte understood this sum to result in the greater and lesser together, coalesced into a unity. (It is pertinent to remark that Viéte seems to have considered his species to be a restoration of Plato's eidetic "numbers," which are constructed by the operation of the One on the Indeterminate Dyad - the greater and the lesser, at the same time.) These were not numbers, but qualitative analogues to numbers, and Viéte constructed a logistic, that is, a calculus of the rules of calculation for these curious objects, which he represented by letters.
If we were to return to a Kantian context, then we might say that Viéte's concept of species is the true schema of quality with respect to time, just as number is the schema of quantity with respect to time. The concept of species is time content or the filling of time.
Much could be said about the effect of these ideas on the deep structure of algebra, but what matters to us here is whether there is a true spatial counterpart for the logistical concept. Is there an object in spatial intuition that properly corresponds to Viéte's species in the same way that length corresponds to number? Presumably, it would have to be something that reflected the property of coalescence.
So the question becomes, can we look at the two lines in an original way and see them coalesce into one instead of remaining an aggregate? This is analogous to seeing a line as a number. According to Kant, the only a priori quality that all magnitudes have is their continuity. Then, can we see the quality of the magnitude in general, namely its continuity, increase or decrease in a direct and nonsymbolic way? Above, we have tried to show how this is possible with the new concepts "hypercontinuity" and "subcontinuity."
Therefore, Kant should not be understood to be showing the impossibility of intensive magnitude as a viable mathematical concept. Instead, it is in his work that the concept can be picked up again in an authentic way and replanted in the tradition.
Intensive Magnitude and Relativity - As we have explained above, the rest energy of a body may be understood as spatial intensification. In a preliminary and informal way, we resolved this intensification into two elements: the dimension number and the number of intensification states contained in the moment. In order to begin to see whether this distinction has any physical significance and consistency within the context of present day physics, we will try to interpret the relativistic effects of uniform motion in terms of intensification.
Above, we also made the assumption that it is the nature of energy to intensify its space recursively to infinity if left to its own devices, although this tendency is continually counteracted by the differing degrees of intensification in the neighboring regions of space and the resistance they provide. The result is a limited amount of intensification taking place through a multiply periodic recursion of intensification states. The intensification is thought to generate time. We came to this result by employing a one-dimensional, nonlinear recursive map. We used a one-dimensional map because it seemed that repetition has only a single directed sense. Furthermore, we argued that one complete cycle of the recursion takes place within a single moment, and in fact defines the moment. At this point, it would be helpful to consider the well-known bifurcation properties of such a map.
Consider the logistics map In+1 = vIn(1-In). We know that its bifurcation properties are universal for maps that generate point sets that lie on curves having a quadratic maximum, so that we can use it as a model. Let In be the initial intensification state as measured by the dimension number, and In+1 the subsequent state. (We do not want to use t standing for time as a subscript, because the different values of n correspond to different states within a moment, at least as viewed from a system at rest relative to the moving body.) Let v be the parameter controlling the steepness of the parabolic hump.
From the study of this map it is known that for v less than 1, successive iterations of the map go to zero; greater than 4, it goes to negative infinity. As the parameter increases within this range, the map goes through a variety of behaviors: first, stable points with increasing value until just under 3, whereupon the last such point bifurcates and there are two stable points for a while, which both grow as the parameter increases; then the two become four, and so forth, with successive bifurcations until a value is reached where the number of points has become infinite.
Now, remember that the intensification state and the number of repetitions in the recursion are due to the energy that makes the body in the first place. The energy of the body may be increased kinetically by setting it in motion or directly by absorption. The two specifically relativistic effects in these cases are an increase in momentum in the first case, and an analogous increase in mass or inertia in the second.
We will try to see if these increases are related to a change in intensification reflected in the recursive map above, by assuming that the controllable parameter is equal to the uniform velocity in the first case and the absorbed energy in the second case.
First let v equal the uniform velocity of a body relative to us. Clearly we must understand its motion to be a kind of energy "flux" in which at least some of the parts are being repeated or continued within each moment by different parts than they would be if the body were at rest. When v is less than the minimum value, then successive iterations of In will go to zero. We read this to mean that there will be no relativistic addition to the energy below this value, and no change in the intensification state or recursion scheme.
When v exceeds that minimum value, then there would be an increase in the intensification state that would add to the traditional nonrelativistic value of the kinetic energy. Furthermore, since the intensification is equal to logN/log(1/r), and it does not seem reasonable that the number of parts is increasing by the change in intensification, we can assume that log(1/r) is decreasing, and consequently that the ratio r of part to whole is increasing. This could be interpreted to mean that the overlapping is becoming greater and the overall length of the body is contracting.
When v reaches the next critical value, the intensification state bifurcates and assumes two values, which would indicate an oscillation between two intensification states within the moment itself. Since these two values are still within the same moment to the observer at rest with the moving system, it must appear to us that his time interval has dilated in some way. Meanwhile, the length of the body continues to contract. And as v continues to approach the third critical value, the body contracts even more, although the dilation of the time interval remains the same.
This line of reasoning can be continued until v approaches the speed of light, at which time the ratio of part to whole must approach unity, and consequently, the overall length must shrink to zero, although the number of iterations within the moment must grow to infinity, and thus the moment must appear to us to dilate to infinity.
It would be fairly easy to calculate the rate at which bifurcations would have to occur in order to be in rough accord with special relativity, using the law by which time dilates as a function of v as a consequence of the Lorenz transformation on the one hand, and the fact that the values of v corresponding to a bifurcation have to converge to the speed of light. This ratio could then be used to help discover the recursive rule itself, since the rate of convergence is now known to be universal for large classes of function.
For example, we could quickly rule out the simplest quadratic map, In+l = vIn(1-In), because at speeds less than 1/3.57 of the speed of light there would be no permanent relativistic effect in terms of shortening, and between 1/3.57 and 3/ 3.57 there would be contraction but no dilation. Between 3/3.57 and 3.36/3.57 there would be time dilation as well. These changes do not accord with the manner in which time dilates; namely, by an amount roughly equal to v2/2c2. But more importantly, the convergence rate here is the Feigenbaum constant, which cannot agree with the ratio of the calculated time dilations. And by ruling out this map, we also rule out any map having a quadratic maximum. But there are other convergence constants calculated for maps having other types of maximums.
One strange result of this model is the discontinuous nature of the time dilation. However, we must remember that beyond a certain point the bifurcations start occurring so quickly that the result would be a nearly continuous growth in the time dilation.
The other very strange result is the lack of any relativistic effect beneath a certain velocity, but this could very well be compatible with experience, since the relativistic effect is hardly detectable until the velocity becomes considerable. We might also speculate that the variable luminosity of certain stars may be related to an oscillation between two different intensification states caused by a considerable velocity of recession from us.
Next, let the parameter v be the absorbed energy. Presumably, when the body absorbs more energy of a certain kind from the outside, its intensification of the space increases and the rest mass of the body increases. It seems natural to ask anew Einstein's original question of how the inertia of the body depends on its energy content. Here it is very tempting to consider the upper limit of the parameter to be the amount of energy the body could absorb before its ratio of mass to radius turned it into a "black hole." Since time also stops at this point, we might suppose that the recursion had assumed an infinite number of stages again, as it did for motion at the speed of light.
At the other end of the scale, there would seem to be a least amount of energy that the body could absorb that would actually account for an increase in intensification. It is all too tempting to try to relate this to Planck's quantum of action for that kind of energy.
In between, there are the successive bifurcation points that result in newly constituted moments involving more and more recursively related intensification states coalesced together. It should be apparent that we would try to invoke quantum theory to fix the parameters in this model, but this is much too complex to go into at this point in our speculations.
Einstein's "kinematical" derivation of the Lorentz contraction was a vast improvement over previous dynamical derivations that made ad hoc assumptions about forces, although it did not rule out the possibility of such forces. But even though it was logically impeccable, it left some dissatisfaction as to whether there was a special cause involved.
Furthermore, Einstein's derivation is not totally kinematical. The principle of relativity is kinematical, but the principle of the constancy of the speed of light involves very subtle assumptions. Here we only point out that light is an event in the electromagnetic field, and we have tried to constitute this field from a principle of intensification. Again, it is interesting to note that the intensity of light was the example of intensive magnitude par excellence throughout its history.
Nevertheless, the argument we have just presented does not yet try to find a deeper ground for the facts established by relativity in the concept of intensive magnitude. On the contrary, it attempts to gain a deeper insight into intensification by assuming these facts (and perhaps the facts of quantum theory as well).
However, there were two unexpected consequences of the argument, which, if they could be experimentally confirmed, would point to an insufficiency in the two principles of relativity as they presently stand. These were the inference that there was a minimum velocity at which relativistic effects began, and the inference that time dilation changed discontinuously. Of course, it is also possible that intensification requires a recursive map of more than one dimension to be correctly modeled, that we have incorrectly resolved the total intensification into components, etc. None of these would invalidate our basic approach, however.
We wish to acknowledge in passing that Kant in his youth produced a very curious, speculative argument claiming that there was a minimum and maximum velocity wherein a body could maintain its vis viva. Similarly, there was a minimum and maximum mass between which the vis viva was proportional to the mass.
Intensive Magnitude and the Mind/Matter Dualism - At this point, I would like to reach directly for the horizon - making no pretense of systematic reasoning, but arguing directly by analogy with the physical ideas of intensification and extensification presented so far - and speak about mind.
Just as tangible matter consists of a special local intensification of space resulting in a nonhomogeneity of the spatial continuum, so mind consists of a special local temporal intensification which is tantamount to a nonhomogeneity in the moment, a temporal "body," a self. Extensive spatial singularity results from intensive spatial nonuniformity, existence from introversion. Extensive temporal singularity results from intensive temporal nonuniformity, intentionality from recursion. This self is no more and no less real than material body.
Being ultimately a temporal intensification of spatial intensification states, mind is localized around its matter although not necessarily exactly coincident with its boundaries. The boundaries of the material body are actually the "intersection" of the spatial coalescence that would be due to the temporal intensification scheme alone and the coalescence that would be due to the spatial intensification scheme alone. Since a change in temporal intensification necessarily results in a change in the spatial intensification, and vice versa, we see that mind and matter interact and are simply different manifestations of the intensification principle of energy.
A nonhomogeneous temporal intensification represents a higher order recursion of spatial intensification states. Recursion is a kind of feedback loop. Here we identify recursion with selfconsciousness. The lowest level recursion defines the moment and thus is not self-consciousness even though there is feedback. This is the kind of consciousness possessed by inanimate matter. Higher level recursions are ways of intensifying the manifold of moments and thus may be understood as ways of structuring the time of an inner awareness. This is the deeper meaning of Kant's synthetic unity of apperception. It is clear that there are many higher degrees of consciousness and self-awareness, depending on the nature of the higher order recursion.
Outside of the nonuniform local intensifications of the temporal moment that we call selfconsciousnesses, there is no thought in the sense of self-awareness, although there is mind in the larger sense, since there is temporal intensification of the single moment, even in the regions of space outside of material body.
Just as the intensification of space-time that we call matter produces a curvature of that space, so the local intensification that produces the self produces a "buckle" in its own "extended" temporal field of a synthetic unity of concepts. This "buckling," and the awareness of self that we have through its means, is a mood, a state of mind. So just as every electromagnetic field is accompanied by a gravitational one, and vice versa, so every thought is accompanied by a mood, and vice versa.
The buckling caused by the local temporal intensification, or consciousness, is communicated to the temporal intensification taking place in the regions surrounding this consciousness, causing a buckling there as well. Thus the surrounding space can bear the impress of a mood and carry it, although the surrounding space cannot have the mood in the same way as a self-consciousness. Reciprocally, this buckling can affect the buckling of other local self-conscious intensifications, causing them to experience the mood in their own way.
Thus, as a large body like the planet Mars intensifies its space in its own characteristic way, it reaches out to touch us with a gravitational field. But as it intensifies its temporal moments in its own characteristic way, thinking its angry thoughts and becoming what it is, it also reaches out to touch us with its temporal field, its mood. And we have no choice but to feel what it feels in our own characteristic way.
In order to study this effect, we have no choice but to become astrologers.
Note: The ideas presented in this paper were sharpened considerably in numerous conversations with John Townley and Ellen Black. At the first Matrix Neo-Astrology Conference, I learned that Michael Erlewine had been pursuing analogous inquiries, though with a more psychological orientation, and contact with him has helped me embed these ideas in a larger context.
References
Einstein, Albert. On the Electrodynamics of Moving Bodies. 1905.
Grassmann, Hermann. Die Ausdehnungslehre von 1844.
Kammerer, Paul. Das Gesetz der Serie (The Law of Seriality). 1919. Although this book is not explicity cited in this paper, some of Kammerer's ideas on persistence causality and the abating of series are embedded deep within it. The author has translated it and is collaborating with John Townley on a semi-popular book on Kammerer's thought.
Kant, Immanuel. Critique of Pure Reason.
Kant, Immanuel. Gedanken von der Wahren Schaetzung der Lebendigen Kraefte. (Thought on the Correct Evaluation of Living Forces). 1747
Maier, Anneliese. Das Problem der Intensiven Groesse in der Scholastik. 1939.
Mandelbrot, Benoit. The Fractal Geometry of Nature. 1983.
Riemann, Bernard. Ueber die Nypothesin, welehe der Geometrie zu Grunde liegen. (On the Hypotheses that Lie at the Basis of Geometry.) 1854.
Viéte, Francois. Introduction to the Analytic Art. 1591. The brief interpretation of this work included in the present paper is being elaborated more fully by the author in a work in progress entitled Restorers of the Lost Art.
Weyl, Hermann. 1918 (first edition). Space-Time-Matter.
This article was first published in the Matrix Journal 1990 Summer/Fall - Volume 1 Issue 1