The Cosmos and the Physical World
7.1.1.
Every unity, or instance of order, by the very fact of not being the whole universe, divides all ultimate entities into two classes: those it includes and those it excludes. It, as it were, draws a line round its constituents. This is what is meant by saying that order is self-selective. By conferring a certain collective nature only upon those it includes, it separates, or discriminates, or selects them from the rest of the primordial matrix. This matrix, as we saw in the last chapter, is a universal process, composed solely of qualification sequences whose numbers double with each instant generation. Providing a rational account of the universe, therefore, is essentially a matter of discovering, describing and empirically identifying naturally unified sets of qualification sequences within the totality. How, then, are we to set about this task? The ideal method would be to proceed conceptually instant by instant from the Absolute, drawing attention to, and treating of the systems just as they arise. But this turns out to be impracticable, basically because of the sheer numbers involved. After a mere hundred instants - less (as we shall see) than a million million millionth of a second - there exist more than a million million million million million sequences. And this barrier is compounded by two further factors. One is that some of the most important manifestations of order tend to be unobvious. This is chiefly because the constancy grounding the order of such a system, though a collective property to which all component sequences contribute, is a different type of order from that manifested by the sequences themselves. The other is that, because we are human beings, and, as such, ourselves particular modes of synthesis of sequence patterns somewhere deep within the universe, we cannot actually become such primordial non-human systems - least of all whilst engaging in the peculiarly human activity of reflecting upon our inevitably distorted conceptions of them. All this, then, effectively rules out any attempt to discover the genesis and history of natural systems by attempting conceptually to reproduce the universal process as it evolves.

7.1.2.
What method, then, are we to fall back on? Clearly, we must turn to our empirical knowledge for assistance. On the one hand we have our ultimate conceptual schema of qualification sequences - which, of course, tells us that the world as experienced must consist of various associations of these; on the other, we have this experienced world. Our aim must be, then, by constantly commuting between our general ontological schema and our empirical knowledge, to synthesise them into a rationally coherent whole (6.5.2.). But, because our sequences, considered on their own, soon lead into impenetrable complexity, and because our empirical knowledge is encompassed by ignorance and contaminated with error, a method based on the illumination of each by the other must inevitably be a somewhat tentative process. Alternatives present themselves at many junctures, but the scale of this work does not admit of our explicitly weighing the pros and cons of all the prima facie possibilities. In every such case, we have made some kind of prior attempt of this nature, and selected whatever most recommends itself, on the grounds of reason and experience, as true. This method, it will be appreciated, almost wholly confines us to that sub-set of universal sequences which composes 'our' universe - henceforward to be called the Cosmos. But there are two reasons why such a limitation is no very serious defect: firstly, because the Cosmos almost wholly circumscribes our interest; and secondly, because there are reasons (which we shall touch on from time to time) to believe that the Cosmos is by far the most experientially complex part of the universe.

7.1.3.
In this chapter we aim to account for the fundamentals of physics in terms of the elements of the primordial matrix - that is, qualification sequences. Now, although the Cosmos is much greater than that sub-set of sequences within it which we know empirically as the physical world, all its other sequences have ultimately arisen by bifurcation from physical sequences. So that the Cosmos consists of the physical world and all that has derived therefrom. Moreover, we shall find that, as a consequence of the creative nature of physical processes, the ordering selective rules or laws or principles that govern - in effect, determine - the physical world themselves change as a result of these creative developments. Nevertheless, all such changes arise as modifications of a fundamental set of selective rules. These last we shall refer to as the laws of the physical world, and it is with these alone that we deal in this chapter, deferring all consideration of their self-induced modifications till later.

7.1.4.
All qualification sequences bifurcate at every instant, there being always two continuations: an ultimate (X) and a negation (0). All selection, therefore, consists in selecting neither, one, or both of these. We shall assume on empirical grounds, that, once constituted, physical sequences, qua physical sequences, can neither multiply nor be destroyed.¹ This implies, therefore, that at every instant, one X/0 alternative is a physical continuation, and the other, not. And what we are seeking here, are the physical laws that determine this selection. Hence it is the entity to which these laws apply - viz, the physical world, or the totality of physical sequences - which is making the selection on each and all of its member sequences. So that the selection on each sequence is made on the basis of the sequence's belonging to the system, which is itself determined by the selections made on all its member sequences. Hence every member sequence is at once both selector and selected: in a word, each member sequence selectively contributes to the continuation of every member sequence according to the laws of the system. It is this instant by instant conjoint selection that determines the course, and hence the nature, of the physical world. So what are the structural parameters that determine in every case which of the two alternatives is selected? Earlier (6.5.5-6.), we suggested that the key structural parameter is period. We now enlarge on this to assert that the period (or frequency) of X/0 alternation of any sequence is being jointly determined by the periods of all. The physical world is therefore a unified system of periods (or frequencies). What we have to determine, therefore, is the nature of the underlying constancies that, preserved under all these changes of period, render physical sequences a system.

The Noumenal Electron

7.1.5.
All unities are instances of order, or natural modes of togetherness. The most basic mode of physical order is that the physical world consists of qualification sequences: the noumenal correlate of what phenomenal physics falsely conceives as ultimate particles. These sequences all march perfectly in step - a necessary consequence, of course, of their all deriving from a single simple Absolute. More precisely, the mth instant is represented on all the 2^n-1 sequences existing by the nth instant, with m taking all integral values from 1 to n. Now, corresponding to the dual or alternating nature of sequences in our conception, phenomenal physics conceives a fundamental duality in the composition of matter: that of positive and negative electric charge. Here, then, is an obvious tie-up between an empirical phenomenon and an ontological noumenon. We shall therefore equate these, and since - owing to the perfect symmetry of the situation - there is no means of knowing which member of each pair corresponds, we equate, purely for verbal convenience, ultimates with positive charge, and negations with negative. Now, whether a sequence, consisting as it does of a succession of consecutive X/0 alternations, is positively or negatively charged, must depend only on a preponderance of ultimates over negations, or vice versa. As it has so far proved impossible to disintegrate an electron, we shall assume that the electron is an ultimate 'particle'. So that a qualification sequence with a preponderance of Ultimates is a positron, and one with a preponderance of Negations, a negatron - with "electron" reserved for the generic term.

7.1.6.
But physics tells us that charge remains at least broadly constant. This leads us to assume that no physical sequence ever changes from a positron to a negatron or vice versa. However, since we are claiming that it is coordinated change of period that provides the structural basis of the physical world, we must clearly look elswhere than constant period to account for charge constancy. We postulate rather that the ontological reality which does account for it is the ratio of X/0 preponderance: that this is ideally, a fixed - inevitably, in reality, a slightly varying - ratio of duration of consecutive ultimates to consecutive negations. We shall take this preponderance ratio to be the simplest possible, namely 2:1 (or, of course, 1:2). It is not merely considerations of simplicity which have influenced our choice. There are others, which will, I hope, appear as we proceed. But a very basic one we may state at once. The complexity and variety of the Cosmos argue for the most intimately constructive interplay of unity and diversity. And, as I hope the reader will come to agree, it is this ratio which seems best fitted to provide it. Of course, since all possible sequences exist, so must every possible cosmos, with one or more grounded on each value of the preponderance ratio. All these exist; the question is, Which of these values is ours? And the only way this question can ever be answered - with a greater likelihood of truth the greater our ontological knowledge - is the present one: of correlating certain deducible structural consequences of the ratio chosen, with our empirical knowledge.

7.1.7.
So that: every cosmic sequence is such that whatever its period of alternation, T instants, at any time, the ratio of the dominant primitive (ultimate or negation) to the subordinate in any one cycle is always, for a period of that duration, the closest possible to 2:1. It will, of course, be exactly 2:1 only for periods that are multiples of 3. It is easily calculated from this that the number of consecutive dominants (ultimates for positrons and negations for negatrons) will be either 2n-1, 2n, or 2n+1, and the number of consecutive subordinates, n. We can therefore generalise the form of all X/0 cycles on physical sequences - writing ultimates first - as {2n(±1),n} for positrons and {n,2n(±1)} for negatrons, with the duration of all periods as 3n(±1) instants, where n, the period root, can take all integral values. For example, {35,17} denotes a positron of period 52 instants, and {50,99} a negatron of period 149 instants. To make all this perfectly clear, I append a table for values of n ≤ 6.

PERIOD [3n(±1)]            SUBORDINATE [n]            DOMINANT [2n(±1)]

Force as Selective Effect
7.1.8.
We stated above (7.1.4.) that the physical selection of each sequence is effected by all the physical sequences conjointly. But physics tells us that every particle in the physical world is, at every instant, exerting a force upon every other. This argues that the noumenal correlate of what we conceive phenomenally as force is selective influence. What's more, as many physicists (Maxwell among them) have contended, the cardinal formula of physics must be that for the force between two particles. In which case, a physical world consisting of n sequences can be regarded as consisting of n(n-1)/2 interselecting pairs. And this suggests that most, if not all, of the basic physical concepts can be defined in terms of the intrinsic nature of a sequence and the reciprocal selective influence between one sequence and another. Now, the type of order that individual qualification sequences preeminently embody is what we have termed periodic (6.5.5-6.). And we are contending that what the intersequential influences (forces) are really selecting on any member sequence is its period: 3n(±1). So that the order manifested by the physical world is one of coordination of periods of sequences, half of which are positrons of period {2n(±1),n} and half, negatrons of period {n, 2n(±1)}.

7.1.9.
Interselective influences (forces), we contend, are acting between simples - that is, at every instant. Now, orthodoxy currently sees forces as consisting of three basic kinds: electromagnetic, gravitational, and nuclear. But we are claiming that, at bottom, there is only one kind of force: the coulombic, repulsive between two Ultimates or two Negations, and attractive between an Ultimate and a Negation.² It is now generally accepted that magnetic force is simply a modification of coulombic due to motion of the electrified particles. That gravity is some ten million, million, million, million, million, million times weaker than the coulombic force strongly suggests that the general arrangement of positrons and negatrons very slightly favours attraction over repulsion. This is not too strained an inference in view of the imbalances of motion and position displayed by positive and negative particles in neutral matter. Also to be taken into account are the facts (a) that attraction increases attractive force, whereas repulsion decreases repulsive, and (b) that attractive forces slightly outweigh repulsive: thus, if the physical world consists of n positrons and n negatrons, there will be 2n(2n-1)/2 = 2n²-n interacting pairs. Of these, n² will be attractions, and n²-n, repulsions. As for nuclear forces, the fact that our spatially elementary entities are qualification sequences rather than the intrinsically unchanging particles of mechanistic physics means that we have three extra sets of variables to help us account coherently for phenomena at the nuclear level. I refer to period, to slightly differing charge, and to that displacement in time we call phase. Thus, as is easily seen, and as we shall be showing later, like sequences will attract and unlike sequences repel over a whole period, in a significant minority of phases: a structural ingredient of nuclei, atoms, and molecules whose organisational importance it would be hard to exaggerate. Likewise we see no reason for postulating such mysterious attributes as 'colour', 'charm', 'strangeness', 'truth', 'beauty', etc., confident that a theory in which matter is correctly conceived will be able to explain all physical force in terms of electrical attractions and repulsions.

7.1.10.
Furthermore, we assert that all so-called electromagnetic radiation - from cosmic rays to long radio waves - is nothing but coulombic force. Here, it is worth drawing attention to the self-evident fact that any radiation signal must exert a force on its receiver, since its reception takes the form of changes in the internal motions of the receiver. The whole spectrum of frequencies (equally, of course, periods) arises from the periodic structures of the sources, not from some quasi-material stuff being "propagated through space". What this amounts to is that in this theory all physical interactions between ‘particles’ are coulombic forces ‘travelling’ at the speed of light.

Extrinsic and Intrinsic Change

7.1.11.

Phenomenal physics sees the physical world as consisting essentially of particles in motion; so that the laws of physics are basically laws of motion. Every particle continues in a state of rest or of uniform rectilinear motion - its velocity remains constant - unless influenced by other particles. Such influences are the sole causes of change of motion recognised by physical orthodoxy, which calls them forces, and the changes of motion they produce, accelerations. Force is defined metrically as the product of mass and acceleration (that is, measure of force = mass x measure of acceleration) - mass being the measure of the amount of matter in a body - equivalent, for Newton (who coined the term), to the number of ultimate particles composing the body. This last translates noumenally into the number of sequences which the body contains, so that for any one sequence, force ∝ acceleration. Now, acceleration is defined as rate of change of velocity. But, since our physical conception denies the existence of any nonsensical quasi-substantial spatial medium separating the sequences, what meaning can it give to 'velocity' (rate of change of distance), or, indeed of distance itself? We have stressed in earlier chapters that orthodox physical theory precisely inverts the true order of ontological priority: which is not that spatial change gives rise to intrinsic change, but that intrinsic change gives rise to spatial. Hence, to set the world the right way up we have somehow to derive all such spatial concepts as distance, velocity, acceleration and force from the intrinsic nature of our sequences. We may put this another way. On any one sequence, the selective influence of other sequences will manifest in two distinct kinds of change, which we will call intrinsic and extrinsic. We are claiming that the intrinsic changes consist of changes of period (or, of course, frequency) - increases or decreases as the case may be. These do not exist for mechanistic orthodoxy. The extrinsic changes are those that orthodoxy would see as changes of location and velocity: that is, changes in spatial relationships with the other physical 'particles'. In other words changes in the forces exerted on the sequence by the other physical sequences. But, of course, this second set exists only as changes in that complex pattern of selective influences determining the contents of the first set. And the pivotal structural feature of this whole conception is that intrinsic and extrinsic influences are simply and precisely coordinated.

Distance as Retarded Interaction
7.1.12.
Although physics sees all particles as exerting force on all others at all times, the magnitude or strength of this influence is conceived as varying enormously: more precisely, as the inverse square of the distance(r) apart. That is, force ∝ 1/r². The fact that selective influence (force) varies inversely as the square of the distance, means that all distances are on an equal footing, since the number of possible locations at any distance varies directly as the square of the distance (surface area of a sphere = 4πr²). Now, the distance (r) of B from A is found to be unvaryingly related to the time (t) it takes a 'light' signal to pass from A to B. That is, r = ct, where c is the speed of light (or, of course, electromagnetic radiation of any frequency) equal to 2·998 x 10⁸ m.s^-1; t = t₂-t₁, the time elapsing between emission of the signal (cause) at instant t₁ and its reception (effect) at instant t₂; and r is the distance between the emitter at time t₁ and the receiver at time t₂. So that force ∝ 1/t^2. But, as Burniston Brown has pointed out:
"The only reason why the velocity of light is so important is because it is not the velocity of anything"
(Retarded Action-at-a-Distance, Cortney Publications 1982, p.30). At bottom, it is simply a universal constant for interconverting units of distance and duration. We, in fact, are contending, that what we know as distance is ultimately duration - the time elapsing between a qualified ultimate or negation on one sequence and its selective effect upon another ultimate or negation on some other. Since the smallest possible value of r occurs when t₂-t₁ is 1 instant, the ultimate unit of distance (1 point [or hodon]), must be identical with the ultimate unit of time (1 instant [or chronon]), and hence, the speed with which all forces are transmitted, one point per instant (1 pt.inst^-1). It may be worth pointing out that colloquial speech frequently equates distance with duration, as, for instance, when we say, "It's not far - only an hour by car", or, "She lives just three minutes down the road from me". But the most significant instance of this equation is that astronomical unit of distance, the light-year. Now, phenomenal physics has independent sets of units for distance and time, the only measurement relating them fundamentally being the speed of light; so that if we label our ultimate unit of distance ρ metres, and our ultimate unit of time τ seconds, ρ/τ= c = 2·998 x 10⁸ m.s^-1.

7.1.13.
I have stated, in effect, that distance is the time lapse between cause and effect; the cause being a simple on sequence A and the effect being another, later, simple on sequence B. Also, that the causal relationship between these two simples is essentially one of selector and selected. This calls for clarification. What, as we asserted earlier, is really being selected is the period of a sequence, and this period, as we shall see shortly, is always inversely proportional to the sequence’s absolute speed. So that the selective effect of one sequence on another is its changing the speed (and, hence, the period) of the other – in other words accelerating it either positively or negatively. Hence, the new period is the joint effect of the individual instant accelerations induced by simple on simple during the preceding period. Therefore, what we are calling the effect simple is not itself the effect, but rather the simple upon which the causal simple effects its contribution to the overall effect, which occurs at the end of the effect sequence’s period. We could equally refer to our causal simple as the source, and our effect simple as the receiver, of a force.

Ultimate Units
7.1.14.
In view of the fact that we are claiming our physical sequences to be electrons, and that the smallest distance any two sequences can be apart³, which is also the smallest distance that any sequence can move, is 1 point (ρ metres), it is only reasonable to take this smallest distance as a distance number associated with the spatially elementary electron, namely, either its ‘radius’ or its ‘diameter’. I use quotes here, because, of course, the real electron - namely, the qualification sequence - is not extended: the notion that an ultimate particle could be extended arising from that radically false set of physical conceptions I have dubbed the Fatal Trap. To enquire as to the size of a sequence makes no more sense than to ask what colour it is. I choose the classical ‘radius’(r_e) of the electron (2·818 x 10^-15 m.), in preference to the classical ‘diameter’, as giving the magnitude of r, our fundamental unit of distance, for the following reason. There exists strong evidence that r_e = e²/m_ec² where e is the electron charge, and m_e the electron mass, both of which values have been calculated from experimental data independently of r_e, and there is no point in introducing a ‘2’ into this physically fundamental formula. This gives a value for τ (= ρ/c), of 2·818x10^-15/2·998x10⁸ = 9·400 x 10^-24 s. Hence, 1 second = 1/9·400 x 10^-24 = 1·064 x 10²³ instants, and 1metre = 1/2·818 x10^-15 = 3·549 x 10¹⁴ points.

Velocity x Period = Constant
7.1.15.
Now, forces between particles, as well as varying with distance apart of the particles, also change this distance apart. That is, irrespective of any intrinsic changes they produce in the sequences, forces also produce extrinsic changes - in the form of changes in the magnitude and direction of immediately succeeding forces. And, as I say, the pivotal structural feature upon which this whole theory turns, is that a precise correspondence exists between intrinsic change and extrinsic change. Obviously, if we are conceptually to turn the physical world the right way up by explaining spatial (extrinsic) change in terms of intrinsic change, such a correspondence must obtain. We are saying that the phenomenal concept of force is, in noumenal reality, selective effect. And force is regarded as the cause of acceleration. But acceleration - the extrinsic effect of interselective influence (force) - is change of velocity. Hence, the basic state of external affairs which this force changes is constant velocity. But we are also claiming that the intrinsic effect of the influence is change of period; so that the basic state of internal affairs which this influence (force) changes is constant period. We are suggesting that since what is being intrinsically selected is a period, all extrinsic effect takes place only at the termination of a period. So that if at the end of every period this motion is always of the same magnitude, it follows that the longer the period the slower the motion. In short, velocity(v) ∝ 1/period(T). Hence, v = k/T, where k is some constant. k (=vT) obviously has the dimensions L¹T⁰, and so is a distance. Therefore, if sequences are not to jump points, k must equal ρ, and we have v = ρ/T = cτ/T with all sequences moving at all times at a speed of 1 point per period. Since T = Nτ, where N is the number of instants in a period, it follows that v = ρ/T = cτ/Nτ, or v = c/N. Thus, in our previous examples (7.1.7.), the {35,17} positron has an absolute velocity of c/52, and the {50,99} negatron, an absolute velocity of c/149. Figures 7.1 and 7.2 should help to make this clear.

Two final points: (i)Since the minimum period a sequence can have is two instants (see table, 7.1.7.), it follows that the maximum possible absolute velocity of a sequence is c/2, and therefore that the maximum possible relative velocity between sequences is c. (ii) Since frequency(ν) = 1/period(T), v =ρν, where ν is in Hertz.

Inertia
7.1.16.
Since the effect of the selective influences (forces) from other sequences upon any sequence is to change its period, and hence also its velocity, both of these would remain unchanged were no such influences acting. Another way of expressing this is to say that a sequence's own selective effect upon its continuation is to maintain its present period and velocity. Such autoselection is what phenomenal physics knows as inertia.

Acceleration Quantified
7.1.17.
We have seen above that force produces acceleration, and that this acceleration is inversely proportional to the square of the distance (r) between cause and effect; that is, a ∝ 1/r² or a=k/r². Can we determine the value of the constant, k? Since k = ar², it must have the dimensions L¹T^-2 x L² = L³T^-2. But our fundamental 'length' is ρ, and our fundamental time τ, so that we can put k equal to Kρ³/τ², where K is some constant, giving a = Kρ³/τ²r² m.s^-2. If we assume that the effect simple is given an acceleration of 1 pt.inst^-2 by a causal simple 1 point distant, then, since 1 point =ρ m. and 1 pt.inst^-2 = c/τ m.s^-2 , at a distance of 1 point: c/τ = Kρ³/τ²ρ² or ρ/τ² = Kρ/τ², whence K=1. Hence, on this obvious assumption, a=ρ ³/τ²r² = c²ρ/r² m.s ^-2. Notice also, that if r metres = n points (that is, r = nρ), a = c²ρ/n²ρ² = c²/n²ρ m.s^-2. But 1m.s^-2 = τ/c pt.inst^-2. Therefore, a =τ c²/cn²ρ = cτ/n²ρ = 1/n² pt.inst^-2. That is, at n points distant from the effect sequence, a causal simple imparts to one simple of the effect sequence an increase or decrease of velocity of 1/n² pt.inst^-1. Since every sequence remains stationary over a period, it follows that the increase or decrease of the sequence's velocity after the lapse of a whole period will be the vector sum of these instant changes in velocity, each of which will itself, of course, be a resultant of the changes in velocity imparted to the sequence at each instant by a member simple of all the sequences of the physical world. In formal terms, after one period, a = Σ_N[Σ_M-1(1/n²)] pt.inst^-1pd^-1, where n = distance in points, M = number of qualification sequences in the physical world, N = number of instants in the one period, and Σdenotes vector summation. Note that sequence velocity = c/N m.s^-1 = 1pt.pd^-1.

Systemic properties of distance qua distance
7.2.1.
We have introduced the noumenal conception of distance as a temporal relation obtaining between two physical sequences: the time lapsing between an instant selector on one sequence, and the instant at which, on the other, it exerts its selective effect (7.1.13.). Now, although this defines the essence of distance, the basic attribute of spatial relationships, at least two considerations tell us that a great deal more than this must be comprehended in these. Firstly, what is it which determines the particular value of the time lapse (t₂-t₁)? Certainly, nothing in the two sequences alone. So far as these in isolation are concerned, t₂-t₁ could be any value. Their distance apart depends on previous motions. All sequences move at the rate of one point per period; and since the selective effect of one sequence on another is to change its sequential period, it follows that such effects must change the rate of change of distance between sequences, and hence the magnitude of subsequent effects, inversely proportional as these are to the square of the sequences’ distance apart, So, clearly, the present distance apart of any two sequences must be a consequence of a succession of previous changes, each involving all physical sequences, going right back to their origins as physical sequences - when, obviously, additional kinds of selective parameter must be implicated. Secondly, every time a footballer kicks the ball, this straightforward act changes not only the distance relations between the kicker and the ball, but between the ball and every other body in the physical world⁴, quite irrespective of any forces they may be exerting on it. What is it, then, that so coordinates these distances that, in effect, change of one means change of all? What is it which exercises these collective constraints, and so welds these myriad time lapses into a single system? It is to these questions respecting systemic constraint that we first turn – deferring the closely connected subject of the origin of physical sequences till later.

7.2.2.
All qualification sequences exist, but the only factor which determines whether they exist as physical sequences is whether or not they conform to the selective parameters which effectively determine the nature of the physical world. At any instant on a physical sequence one of its two continuations does so conform, and one does not. Now, according to the present theory, one general attribute of the physical world is that every physical sequence contributes, at every instant, to the selection of the physical continuation of every other (7.1.4.). Further, that this selective share of any sequence in determining the particular continuation of any other varies from instant to instant, being proportional to 1/t², where t (= t₂–t₁) is the time lapse between the causal or selective instant (t₁) on the one sequence, and the effect or selected instant (t₂) on the other - always bearing in mind that, as explained earlier (7.1.13.), what we are here and henceforward calling the effect simple is not itself the effect, but rather the sole simple of the effect sequence through which the t₁ simple contributes to this effect. We call r the distance between the two instants, where r points = t instants, or, in S.I. units, r metres = c x t seconds, where c (= 2·998 x 10⁸ms^-1) is a universal constant. And since all sequences move within the system at a rate of one point per period, changing a sequence’s period must change its rate of motion. So that the selective effect of one physical sequence on another is necessarily dual: intrinsic (change of period), and extrinsic (change of rate of change of distance): this latter, of course, changing the magnitude of the immediately subsequent selective effect of the one on the other. Now, what primarily makes the physical world a system, is that this increase or decrease of distance of the effect sequence from the causal (and, of course, from what we have said, every sequence exists in the role of both cause and effect to every other at all times) does not merely affect the distance between the sequence pair directly involved, but changes the distance of the effect sequence from almost every other physical sequence. So that, in effect, change of distance with respect to one, entails change of distance with respect to all. This implies that all distances are in some way mutually entailed: that each is part of the one system, so that, in reality, it is always this one system that is involved in every distance change. Our immediate object, then, is to determine the general nature of this system.

7.2.3.
Distance is essentially time lapse between causal instant on sequence A and effect instant on sequence B. And, as we have noted, it is obvious that the particular value of the time lapse in no way depends upon the intrinsic natures of these instants. And one can extend this non-dependence to the individual sequences: any two sequences, so far as they alone are concerned, could be at any distance from one another. Cause/effect time lapse – that is, distance - arises from the attribute of every physical sequence, as such, to participate in selecting the X/0 continuation of every other physical sequence, qua physical sequence. Hence, without this mutually selective attribute of physical sequences, distance as such would not exist. However, the fact that every physical sequence is implicated in the particular succession of periods – and hence distance changes - of each, does not, in itself, imply that all these distance changes are systematically coordinated: that distance changes effected by one alter the distance relations (and hence distance changes) of all the others in some comprehensively ordered fashion. Whence, then, does the systematic ordering of distances between physical sequences arise? It arises from the nature of distance as such, which nature, as we have seen, is essentially a duration, or time lapse, between an instant event on one physical sequence and a causally connected instant event on another. Although such event pairs give rise to distance per se, the systemic attributes of distance owe nothing to the particular natures of such events, and may hence be studied in abstraction from them.

7.2.4.
Distance between physical sequences is no other than time lapse between selective cause and selected effect. So when we assert that the systemic properties of distance arise from distance as such, clearly time lapse is centrally implicated. And since time lapse is a general property of the simples composing a single qualification sequence, we would expect to find these systemic properties, or rational entailments, to be already in evidence here; but obviously in a far simpler form than in cause/effect time lapses involving vast numbers of physical sequences. In every qualification sequence, every qualified Ultimate or Nullity has a position in time relative to every other simple: as t instants prior or subsequent to this other. Since distance apart is, at bottom, time apart, our system of distances is essentially no more than the intersequential elaboration of this system of temporal positions manifested in the single sequence. And just as in a single sequence, every simple, irrespective of its particular intrinsic nature, has a temporal position within the system of instants; so here, at any instant, every sequence has a distance position relative to every other within the whole system of cause/effect time lapses. We call such systemic distance positions, locations. And just as, for every single sequence, any particular temporal relation is the same for all sequences, irrespective of their particular natures, so here, these locations remain constant, irrespective of the particular sequences which happen to be occupying them at any instant. For the duration of a period each sequence remains at the same location and then moves to one at one point distant. Distance (in points) between location A and location B will then mean the time lapse (in instants) between a causal instant at location A and its effect instant at location B.

7.2.5.
It cannot be too strongly stressed that this location system is purely abstract. Distance as such cannot exist without the physical sequences; it is a property arising, as we have seen, from certain relations – pertaining to mutual selection from the totality - between these sequences. But as its further, systemic, properties do not depend on any further properties of the sequences, they may, for purposes of understanding, as I have said, be considered in abstraction from the particular sequences in which they collectively inhere. In reality, this distance system exists only implicitly in the coordinated, mutually constraining, changes of distance between the physical sequences. We are merely rendering explicit the constancies underlying this system of constraints; abstracting them, for purposes of understanding, from the substantial realities in which they collectively inhere. Since every physical sequence is at all times located somewhere within this one system of constraints, we may regard belonging to this distance system as an extrinsic general attribute of every physical sequence by virtue of its ongoing selector/selected relationship with every other. We now proceed, by means of the abstract concept of location, to derive the nature of this system of time lapses between instant selective event on one sequence and instant selected event on some other, without any reference to the particular events concerned.

The Line

7.2.6.
Consider, then, a causal event at instant t₁ of sequence A and an ‘effect’ event at instant t₂ of sequence B, and let the two locations be r points distant, where r is numerically equal to t₂-t₁. And, so far as purely abstract considerations are concerned, r can be as large as we please. Now consider a third location, n points distant from A, and r-n points distant from B. n can take all integral values from 1 to r-1, with r-n taking the corresponding set of values r-1 to 1. We call the series of locations constituting the locus of n, a straight line (or sometimes just a line); whose terminations, r instants apart, are the spatial locations A and B. Each location defining this line is at a fixed distance from every other. Since r can take any value, and n any value < r, it follows, by necessary implication, that a straight line has many general properties relating to distance and change of distance. All these are expressions of its systemic nature: of the unity – born, like all unities, of repetition - informing its diversity.

7.2.7.
One of these properties is that, in relation to any arbitrarily selected location (O) on a line AB r points long, there are always two locations p points from it (p = 1,2,3 ….) one (x₁) closer than O to A (further than O from B), and one (x₂) closer than O to B (further than O from A). So that there are two sets of p points, the x₁ set and the x₂ set. We say that these two corresponding sets are in opposite directions from the O location: one set in the A direction (towards A and away from B) and one set in the B direction (towards B and away from A). We can, if we wish, call one direction (the A direction, say) positive and one direction (the B) negative. Then every point on the line has a unique directed distance ±x from the O location, and, as a necessary implication, a unique directed distance from every other. As one would expect, since the location O is arbitrary. This appears an analytical truism only because the straight line is formed on the instant by instant additive structure of the qualification sequence, from which, however indirectly, the arithmetical mind has abstracted the whole number series. The straight line is an ordered system of points because its structure necessarily implies the existence of such unchanging distance relationships under all changes of spatial location. In short, the constant distances between all locations necessarily implies that the changes in the distances of a sequence from all locations consequent upon its movement from one location to another, will, through the maintenance of some general relational constancy, be completely interdependent. All such constraints are necessary implications of the nature of the line as defined above (7.2.6).

The Grid

7.2.8.
So far, then, in this consideration of the systematic nature of Cosmic distance, we have confined ourselves to the straight line. And though the Cosmos is not a straight line its distance relations are nonetheless ultimately grounded on it - as plainly evidenced by the linear nature of 'light'. Consider again the arbitrary point O on our line, which we will now call the line OX, where X is some other point on this line. One can think of a second line OY, intersecting the first line in the common location O, such that there is perfect reciprocity both between the two lines, and between both pairs (+ and -) of directions. We say that two lines in this relation of perfect reciprocity about their point of intersection are orthogonal. But the point O is arbitrary, just as which of the two lines is OX (the initial line), and which OY, is arbitrary. So that every point on both lines could equally be our point of intersection. In this way an indefinitely large grid of lines is generated, defining a plane. Every grid location (that is, intersection) has an identical set of distance relations. O X and OY, then, could be any pair of orthogonal lines of this grid. Of the others, half are orthogonal to OX, and half are orthogonal to OY. The former set have no common location with OY, or with one another; and the latter, no common location with OX or with one another. Grid lines possessing no common location are said to be parallel. Hence, a grid possesses four fundamental directions: the positive and negative x directions, applying to OX and all lines parallel to it, and the positive and negative y directions similarly relating to OY. Movement in either of the OX directions (± x) entails no movement in either of the Y; and reciprocally for the OY directions (± y). Every location on the plane is so many points distant, either in the positive or negative direction, from both OY and OX It is therefore uniquely specified for any given O by this number pair, which we write as (x,y). There are certain matters arising from the distance between grid intersections which we shall defer until we have dealt with grids of grids, which we call lattices.

The Lattice
7.2.9.
Because it contradicts no property of the plane, a third line (OZ) exists at the O location orthogonal alike to OX and to OY. This generates an OX-OZ plane and an OY-OZ plane alike identical in all respects to the OX-OY plane, with the three planes existing in a relationship of perfect reciprocity. Since all orthogonal intersections must, in themselves, be identical, it follows that a third orthogonal line arises at every location on all these three planes, thereby creating a three-dimensional lattice of locations. The planes thus created either intersect in a common line or do not intersect at all. There are thus three parallel (non-intersecting) sets of planar grids; an OX-OY set, an OX-OZ set, and an OY-OZ set. And there are six basic directions: positive and negative in the X, Y, and Z directions: ±x, ±y, and ±z. Every lattice point is thus uniquely specified for a given O by a number triplet (x,y,z). Change of location in any one of these three pairs of directions involves no change in the other two.

The Non-Existence of Higher Dimensional Spaces.

7.2.10.
Because it does contradict a property of the plane, no fourth orthogonal line can pass through the O location. Let OW be such a fourth hypothetical line. Then the plane OZ-OW must intersect the plane OX-OY in a common line, POQ say. And, because POQ is in the plane OZ-OW, to which both the lines OX and OY are orthogonal at O, OX and OY must both be orthogonal to POQ. But, by virtue of the fact that POQ is also in the plane OX-OY, there are three lines in this plane, OX, OY, and POQ, mutually orthogonal at the location O. But it is a basic property of the plane that only one coplanar line can be orthogonal to another at any location on that other. Hence there can exist no such hypothetical fourth mutually orthogonal line. This carries the implication that our three-dimensional lattice is the most complex system of spatial locations possible. ‘Space’ is saturated. The postulation of n-dimensional systems, where n can take any positive integral value, is, of course, based on nothing more real or rational than adding the requisite number of terms to the Pythagorean metric for 3-dimensional space. G.F.B. Riemann (1826–1866), who pioneered this line of thinking was careful to call his symbolic creations ‘manifolds’. He saw them for what they truly were: purely mathematical creations. But as mathematics ever more comprehensively hijacked physics – to the latter’s tragic detriment - these manifolds became transformed conceptually into higher dimensional ‘spaces’ possessing at least equal reality with our everyday space of 3-dimensions, and spawning a limitless world of pseudo-profundities for the awestruck contemplation of the simple minded.

A Plurality of Lattices?

7.2.11.
Consider a hypothetical straight line joining (x₁,y₁,z₁) and (x₂,y₂,z₂), where x₁≠x₂, y₁≠y₂, and z₁≠z₂. Our lattice axes OX, OY, and OZ have no initial orientation in space, a notion which in our system is meaningless since there exists no absolute space for the lines OX, OY, and OZ to be oriented to. All orientation is with respect to the initial line: for the lattice lines, either parallel, or orthogonal to these parallels. But this initial line might have been that containing the points (x₁,y₁,z₁) and (x₂,y₂,z₂), and a lattice developed upon this base. And this, of course, would apply equally to any two points in our original lattice. But, further, it would apply to any two points in any lattice developed from any two points in the original lattice. And equally from any two points in any lattice so developed, and so on indefinitely. Now, it must never be overlooked that these point lattices have, in themselves, no substantial reality. Any such lattice is an abstracted collective property of the totality of physical sequences, having no reality whatsoever apart from these. And only one such lattice exists, this single system of distance relations being abstracted from the reality of qualification sequences. Which of all the infinity of abstract lattices it is, is obviously irrelevant, since each of these, as the original lattice, are identically one. When we come in the next section to cosmic origins, we shall see how this whole system of distance relations, based on orthogonality, arises as an abstracted attribute of the sequences collectively. Here, we need do no more than remind ourselves that distance relations manifest in two different ways. Firstly, through interselective effect: both in time lapse between cause and effect (∝r, where r points = t instants such that in absolute units r/t =1, and in our units, r/t = c, the so-called velocity of light) and in magnitude of effect (∝1/r²). And, second, through the motion of sequences at the rate of 1 point per period of N (N≥2) instants. In the first, nothing substantial passes between the two sequences concerned; the effect instant occurs t instants subsequent to the causal instant, and that is all. And, as for the second, as we hope to make clear later in defining the accommodation to concrete reality of the abstract ideal, the movement of sequences will involve only the locations created by the lattice intersections. So that, on both counts, the purely conceptual creation of such non-orthogonal lines and their point locations, while perfectly legitimate simply as a development of our abstract orthogonal lattice, unlike this lattice, in no way furthers the elucidation of relations between the components of physical reality.

Distance Between Lattice Points
7.2.12.
Hence, although actual distances are systematised within only a single lattice, we are still left with the basic question: What is the distance of any lattice location from any other? This question is best answered by considering the two-dimensional grid, and then extending this answer to the lattice. Referring to Figure 7.3 below: BC (of length a points) and AC (of length b points) are two grid lines intersecting orthogonally at C. It is required to find the distance AB (of length c points). To do this, we draw BD perpendicular to AB, to meet AC produced in D. Using Euclidean geometry, it is a simple matter to prove that triangles ABC, DAC, and DBA are equiangular, and hence have corresponding sides which are proportional. From this it follows that the sides of triangle DAC are b/a x, and those of triangle DBA c/a x the corresponding sides of triangle ABC. Now, BD = BC + CD. Therefore, c/a x c = a + b/a x b, Multiplying each term by a, we have: c x c = a x a + b x b, or c² = a² + b². Whence c = √(a² + b²).

7.2.13.

Hence, the distance between any two grid points (intersections) can always be expressed as the square root of the sum of the squares of two grid line distances. Now, what assumptions are we making here, and can they be justified from our definition of a grid of points? Consider any two gridlines OX and OY intersecting at point O, and of lengths p points and q points respectively. Then, we can think of a third line of length r points joining X and Y. (We shall return in a later section devoted to the accommodation of the ideal in the real to the difficulty posed by the fact that r will not, in general, be an exact number of points.) Now, since all grid intersections are, from the very nature of their origin, interchangeable in all respects, and, moreover that there is complete reciprocity between the two grid directions, it follows that exactly the same XOY situation can be reproduced eight times at any point in the grid. In effect, we have the one triangle XOY which can be oriented eight ways at any point in the grid, with OX and OY as gridlines at right angles to one another. It is identically the one triangle since its internal measurements consisting of the lengths of its sides (p, q, and r points) and the magnitudes of the angles (changes of direction) between them are identically one in all cases; all that is changing are its external relations with other parts of the grid. But, now, let us suppose our unit length is not one point but n points. OX and OY will be p units and q units of length as before, but this time their absolute lengths will be pn points and qn points respectively. There is nothing to make the distance XY anything other than r units of length, or rn points, whatever the value of n. So, once again we have identically the one triangle, so far as the magnitude of its angles and the lengths of its sides (pn, qn, and rn points) are concerned, but this time with n able to take any integral value for any particular triangle. Finally, if our initial grid line had been XY, our lines OX and OY would no longer be grid lines, but their orthogonality would still hold good. They would then correspond to the orthogonal lines AD and AB in Figure 7.3.

7.2.14.

Hence, in our (x,y) notation, the distance r_p between any two points of intersection in an orthogonal planar grid is given by r_p = √[(x₂-x₁)² + (y₂-y₁)²]. The reader will, of course, know this as the Pythagorean relationship, and it can easily be extended to our lattice in the form of r = √[r_p² + (z₂ - z₁)²], so that r = √[(x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²] gives us the distance between any two points (x₁,y₁,z₁) and (x₂,y₂,z₂) in our abstract lattice of locations, where every location is such a point.

7.2.15.

There are certain matters arising from this distance which it would be as well to clear up here. The first relates to a hypothetical straight line joining (x₁,y₁) and (x₂,y₂). If there is such a line of locations at point intervals, it will, indefinitely extended in both directions, intersect every grid line non-orthogonally. And since the two grid locations are arbitrary, a host of such locations and lines would exist. A further set of lines and locations would then arise from distance relations between these additional locations, and so on indefinitely. Now, it must not be overlooked that this whole system of distance relations is abstracted from the reality of qualification sequences. And distance arises in two ways from the sequences. First, through interselective effect: both in time lapse between cause and effect (∝r), and in magnitude of effect (∝1/r²). And, second, through the motion of sequences at the rate of 1 point per period. In the first, nothing travels point by point between sequence and sequence; the effect instant occurs r instants after the causal instant, and that is all. So that we can correctly state that sequences are always at some location or other, but not that individual locations exist in their own right; they are, as we explained earlier (p.10) merely a convenient way of drawing attention to the constancies underlying the Cosmic sequence's collective distance relations. And, as for the second, as we shall make clear in the two following sections, the movement of sequences will involve only the locations created by the grid of orthogonal lines. So that, on either count, the conceptual creation of such non-orthogonal lines and their point locations, while perfectly legitimate simply as a development of our abstract orthogonal grid, unlike this grid, in no way furthers the elucidation of physical reality.

7.3.1.

We are now in a position to understand how Cosmic sequences originate and continue their existence as Cosmic sequences. A principal manifestation of distance is as time lapse between instant cause (t₁) on sequence A and its instant effect (t₂) on sequence B. But, as we noted, there is nothing in the two sequences considered in isolation from the rest of the Cosmos, to particularly associate t₁ on A with t₂ on B. This association is a consequence of where sequences A and B are, distance wise, within the whole system of Cosmic sequences - a consequence of their relative spatial locations at the causal instant, and the motion of sequence B subsequent to this. Now, these locations are somewhere on our abstract 3-dimensional lattice, and our immediate concerns are a) to describe exactly what this lattice has been abstracted from, and b) to explain, in general terms, how sequences attain their locations within it.

7.3.2.
We assume that Cosmic sequences, qua Cosmic sequences, do not bifurcate; that is, at every instant, only one of the two continuations of a Cosmic sequence is part of the Cosmos. Hence, there is no growth of the Cosmos by way of branching sequences. We postulate instead that all Cosmic sequences arise, instant by instant, by budding off a single parent sequence. This sequence is part of the Cosmos in that it gives rise to every Cosmic sequence and also defines the zero Cosmic location. But because, as this zero location, it cannot move, it plays no part in causal relations, and therefore keeps the same form eternally. Clearly, it must be perfectly symmetrical in Ultimates and Negations; and in view of its fundamental nature we give it the simplest possible form: X→0→X→0→X→0→, the first X being The Absolute. And because, also, the 2:1 ratio is built fundamentally into Cosmic structure, we believe that all positrons must begin as {2:1} sequences and all negatrons as{1:2}sequences. Further, that, as a consequence, all electrons, negative and positive alternating, leave the parent sequence at a speed of c/3: that is, one point every three instants.

7.3.3.
In this theory, the first Cosmic sequence - an electron - began on the second instant of time (taking The Absolute as the first), completing its first period on the fourth instant of time at a location of one point from the Cosmic zero as defined by the {1:1} sequence. Its second cycle took place at a location of two points from the origin, and was completed on the seventh instant of time. Clearly, during its nth cycle, provided that autoselection (7.1.16.) alone were operating, the sequence would be at a distance of n points from the origin, and would complete this nth cycle at the 3n+1^st instant of time. Notice that n, the number of points from the origin, is also the number of cycles, including the present cycle, subsequent to the sequence's bifurcation from the parent sequence. For purposes of exposition we shall first consider the artificial situation where no intersequential influences are acting; that is, where motion is due solely to autoselection. The real state of affairs will then be given by the instant by instant superimposition of the intersequential influences upon this autosequential mode of distance change.

7.3.4.
In the purely hypothetical absence, then, of intersequential influences, motion due solely to this autosequential mode would take the sequence in a straight line, one location per 3-point period, further from the zero location, or distance origin. In which case, the first sequence would, on the nth instant of time, have completed (n-1)/3 cycles, and therefore be (n-1)/3 points distant from the origin if this number is an integer, or at the immediately higher integral number of points if it is not. So that, in general, we can say that motion arising solely from this autoselective cause results in a Cosmic sequence having completed [n-(t-1)]/3 = (n-t+1)/3 cycles, where n is the present instant and t the instant of the sequence's Cosmic origin. Thus, at the 100th instant of time, the Cosmic sequence (a positron) which began at the 27th instant would have consisted of (100-27+1)/3 = 24 ²/₃ 3-instant cycles and would therefore have been at a location of 25 points in a straight line from the origin.

7.3.5.
This tells us how far, consequent upon this autosequential mode of distance change only, sequences would be from the Cosmic origin, or zero location. But how far will they be from one another? Each sequence at every instant defines a location on one of the three straight lines intersecting orthogonally at the Cosmic origin; and each has moved from the origin to this location in steps of one point every three instants. It can do this in one of six directions, and the only non-arbitrary order for the first six sequences would be +/-, +/-, +/-. Clearly, which direction we call plus and which minus is immaterial: equally, which we label X, which Y, and which Z. Since positrons (p) and negatrons (n) alternate, and the first sequence (beginning on the second instant of time) must be a negatron, we assert on purely nominal grounds that the first six Cosmic sequences originated in the following order: -X(n), +X(p), -Y(n), +Y(p), -Z(n), +Z(p). But 2 collinear directions, 3 mutually orthogonal lines, and 2 kinds of electron are involved, and the number of ways these three parameters can be combined is 2x3x2 = 12. So that, a second group of six, an exact repetition of the first, would arbitrarily associate each of the two linear directions with one of the two types of electron: positive(+) with positron(p) and negative(-) with negatron(n). We therefore reject this in favour of a reversal of direction for the second group, making the first twelve sequences: -X(n), +X(p), -Y(n), +Y(p), -Z(n), +Z(p), +X(n), -X(p), +Y(n), -Y(p), +Z(n), -Z(p). The whole 12-instant cycle then endlessly repeats itself.

7.3.6.
But, of course, in reality, this autosequential mode is only one of the two determinants of distance relations. Concurrent with, and, as it were, superimposed upon it, is the intersequential, so that the motion of the sequences is always a resultant of these two. We have seen that the autosequential mode always produces a distance change (velocity) of 1 point/period (ρ metres/period of Nτ seconds); and that at every instant the intersequential mode imposes upon this a change of velocity of ρ²c/r² m.s^-1. (=1/n² pt. inst^-1, where n points = r metres.). Under the action of these intersequential forces the velocities of the sequences will begin to change both in magnitude and direction.Their periods will change from 3 instants, and they will deviate from motion directly away from the origin. At the completion of each period the effect of these accelerations induced by intersequential influence, is to change both the magnitude and direction of the existing velocity. Hence, just as the autosequential mode of distance change now maintains speeds other than c/3, so it maintains directions other than the original six. To repeat: the autosequential mode maintains the velocity's status quo; the intersequential mode changes it. Inertia plus impressed force, in short.

REAL AND IDEAL: A RATIONAL ACCOMMODATION

7.4.1.
In section 7.2. we dealt with a system of distance constraints operating throughout the physical universe. This requires that, irrespective of the particular forces acting at any time upon the individual qualification sequences of which the physical world is composed, the distance changes between every such sequence and every other resulting from the operation of these intersequential forces are precisely coordinated. In short, change of distance from one as a result of the operation of a force, implies a coordinated change of distance from all as a consequence of all physical sequences belonging to an unchanging system of distance constraints. And we saw that such distance coordination entailed that all sequential motions must be from location to location in an unchanging 3-dimensional lattice of locations. In this lattice, every location defines an intersection of the three mutually orthogonal basic spatial directions (x,y,z,) as defined by the origin of the physical world (7.3.5.). This implies that every lattice location is one point distant from six other locations. And hence - as indeed must be implied by the very nature of distance (7.1.12.) - the smallest distance any sequence can move is one point.. Now, in purely abstract mechanistic conceptions of ‘space’ there is no quantum of distance, nor of direction. In theory, both locations and directions can be infinitesimally close to one another; each is ideally continuous. This, of course, leads to all manner of wholly insoluble problems - the inevitable consequence of naively projecting upon the objective world our subjective conceptions of it. In this ideal abstract conception, ‘space’ has no special directions or indivisible unit distances; nor is there any empirical evidence that it does. So that, our task here is to explain how quanta of distance and direction, as required by our abstract location lattice, are systematically preserved so far as ‘particle’ motions are concerned, in such a way as to present no contradiction with the equivalence of all distances and directions as empirically established.

7.4.2
We saw earlier (7.1.17.) that a sequence’s acceleration after one period is, a = Σ_N[Σ_M-1(1/n²)] pt.inst^-1pd^-1, where n = distance in points, M = number of qualification sequences in the physical world, N = number of instants in the one period, and Σ denotes vector summation. And that the sequence’s new velocity is given by adding this vectorially to the existing velocity. Let the components of this acceleration, or change in velocity (Δv) along the three lattice directions (x,y,z) be Δv_x, Δv_y, Δv_z . And let the components of the current velocity on these axes be v'_x, v'_y, and v'_z. Then the component of the new velocity in the x-direction is v_x = v'_x ±Δ v_x, according to whether Δv_x is in the same or the opposite sense to v'_x , and similarly for v_y and v_z. So that the magnitude of this present velocity (v) = √(v_x²+v_y²+v_z²). Let its direction be that which makes acute angles of α, β and γ with the x, y , and z axes respectively; so that cos α = v_x/√(v_x²+v_y²+v_z²). , and similarly for v_y and v_z. Now, the distance moved by the sequence in this new direction and at this new velocity is 1 point. Hence the component ideal distances moved along the x, y, and z axes will be respectively cosα, cosβ, and cosγ: that is, v_x/√(v_x² + v_y² + v_z²), v_y/√(v_x² + v_y² + v_z²), and v_z/√(v_x² + v_y² + v_z²) points. It will be noted, then, that all changes in distance from the end of the preceding period to the end of the current period can be given entirely in terms of distance components – themselves functions of velocity components – along the three spatial axes.

7.4.3.
Consider any sequence at lattice location (x,y,z). This will be its real location, but its ideal location will be x ± Δx, y ± Δy, z ± Δz, from the spatial origin, where Δx, Δy, and Δz are less than one ideal point. These ideal distances then change, at the end of the period, to x ± Δx ± v_x/√(v_x ² + v_y ² + v_z ²), y ± Δy ± v_y/√(v_x ² + v_y ² + v_z ²), z ± Δz ± v_z/√(v_x ² + v_y ² + v_z ²)}. Now, if Δx and v_x/√(v_x ² + v_y ² + v_z ²) are in the same sense, and their sum is greater than or equal to1 point, then the sequence moves by one actual point either in the positive or the negative x direction (away from or towards the spatial origin). If their sum is < 1, or if they are in opposite senses, then the actual location remains the same, although, in the latter case, the ideal location may now be in the opposite direction from it. Similarly for y and z. Now, at each change of period, one of three outcomes occurs for each of the three lattice directions: the real location changes by 0, +1 or -1 points. This provides 3³ = 27 possible outcomes. 1 (= 1x2⁰) of these will be that the sequence remains where it is; 6 (= 3x2¹) will be that the sequence moves by one point along one of the axes; 12 (= 3x2²) will be that the sequence moves by one point simultaneously along two of the spatial axes - that is, effectively along the diagonal of a unit square; 8 (= 1x2³) will be that the sequence moves by one point simultaneously along all three spatial axes - that is, effectively along the diagonal of a unit cube. Its new ideal x location will then be either x + Δ₁x, x - Δ₂x, (x-1) - Δ₃x, or (x+1) + Δ₄x, where the subscripts denote new and different values of Δx. And similarly for y and z.

7.4.4.
But there is another quantisation involved in distance, which we have not so far dealt with: that of distance between locations. In general, this distance, r = √[(x₂ -x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] will not be an exact integer, but will lie somewhere between two integers, r₁ and r₂ (= r₁ + 1). Now, in real, concrete terms distance apart in points is time apart in instants - the time elapsing between a causal instant of sequence A and an effect instant of sequence B. So that in saying that the distance between two locations is √√[(x₂ -x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] points, we are really saying that the time lapse between a causal instant of a sequence at the (x₁,y₁, z₁) location and the effect instant of a sequence at the (x₂, y₂, z₂) location is √√[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²] instants. So that in talking of fractions of a point we are really talking of fractions of an instant. Thus, when we say, for example, that the distance of the (18,-7,2) location from the location arbitrarily taken as (0,0,0) is √377 = 19.416 ...points, we are also, more fundamentally, saying that the corresponding time lapse is 19.416 ... instants: that is, between I9 and 20 instants. In talking thus of fractions of instants, we are, in effect, giving each simple event a duration of 1 instant; in which case the signal arrives at the (18,-7,2) location some time after the beginning of the 19th instant event subsequent to the causal instant event, and prior to the advent of the 20th such event - that is, during the 1 instant duration of this 19th subsequent simple event. Now, of course, in reality, where there is no change there is no lapse of time, and there are no changes within a simple event by definition. In real terms, therefore, what the above implies is that the effect instant in this case is 19 instants subsequent to the causal instant. So that in general, where location B is between r and r+1 ideal points from location A, the effect instant of a sequence located at B will be r instants subsequent to the corresponding causal instant of a sequence located at A. The real distance between sequences at these locations is thus r points, and the force of the causal instant at A on the effect instant at B ∝o1/r².

7.4.5.
With these cosmic rules in operation, despite the fact that, at any location, sequential movement and intersequential influence is organised entirely around the three spatial axes, no directions are specially favoured. Nor does the fact that sequences must always be at a lattice intersection give rise to anything more than minute differences between real and ideal; and, in any case, since every direction has its opposite, and neither one is preferred to the other, any such differences must tend to cancel out.

 

NOTES & REFERENCES

1.The so-called annihilation of sub-atomic particles seems to me most easily explained by the clamping or fusing together of an electron and a positron (see note 3. below), Pair production is their subsequent separation by an external force attracting the one and repelling the other.
2.There appears to be no necessary reason why the correspondence should be this way round, although it is obvious that, were it the other, with like charges attracting, and unlike repelling, the Cosmos would be greatly impoverished structurally - simply because there would then be far less complex mingling of unity and diversity. So that, if there is another Cosmos - as presumably there is - where this obtains, it seems very unlikely that any highly complex localised structures could arise within it.
3. In our noumenal theory, there is no spatial medium, and hence no question of two sequences ‘occupying the same space’. At the same time “occupying the same space” is only mechanese for “possessing the same spatial relationships”. And on our theory there is nothing at all to prevent two sequences, if brought together by sufficiently strong attractive forces, possessing these.
4. Unless he ‘bent it’ into the arc of a circle, in which case it would not change its distance from a body situated at the centre of that circle.

 

 back to home

chapter 6

chapter 8

Theory of the Universe