Anatomy of Information by Shirleen Stibbe

ANATOMY OF INFORMATION PHILIP A STIBBE

An inquiry into the nature of information

ANATOMY OF INFORMATION

An Inquiry

Philip A Stibbe

  Perpetual Spring

First published 2011 in the UK by Perpetual Spring 126 Rosebery Road, Epsom Surrey KT18 6AA

ISBN 978-0-9533188-0-3

 

This Inquiry could not have been formulated without the unrelenting scepticism and support of my wife Shirleen, to whom it is dedicated.

Contents Section

Page

Terms Of Reference

Exclusively Nothing . . . Change Of State . . . Exclusively Something . . Exclusively Something Or Nothing . Inclusively Something Or Nothing . Something And Nothing . . Division . . . . Inequality Of Proportion . . Equality Of Proportion . . The Limit Of Proportional Inequality

. . . . . . . . . .

2 2 3 3 4 5 7 9 10 15

. . . . . . . . . . . . . .

19 19 21 22 23 25 26 36 41 43 45 47 49 53

. . . . .

56 59 63 66 69

1. The Definition Of Terms 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10

2. The Definition Of Information 2.1 2.2 2.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.5.5

Introduction . . . Private Information . . Mutual Information . . Accidental Information . Published Information . Common Information . Natural Data . . . Natural Information . . Incidental Information . Natural Division . . The Order Of Random Plurality The Order Of One . . The Order Of Two . . The Scalar Of Two . .

3. The Magnitude Of Scale 3.1 3.2 3.3 3.3.1 3.3.2

Asymmetric Division . . . The Limit Of Asymmetric Division . The Scales Of Binary 10 . . The Real Limit Of Symmetrical Division The Real Limit Of Asymmetric Division

4. (overleaf) i

Contents (continued)

Section

Page

4. The Invention Of Zero 4.1

The Supervention Of 0

. . .

79 83 86

. . .

89 91 94

5. The Objects Of Reality 5.1 5.2 5.3

Artificial Information . The Supervention Of Binary 10 The Supervention Of Binary 102

6. The Subjects Of Memory 6.1 6.2 6.3

The Principle Of Apodictic Certainty . The Scalar Of The Macro Sphere Of Binary 10 The Scalar Of Macro Scale Of Binary 2 .

7. The Objects Of Mathematics 7.1 7.1.1 7.1.2

The Question . Positively . Negative .

. . .

97 97 99

Appended Table Of Rules And Lemmas Glossary

Terms Of Reference In his lecture to the Prussian Academy of Science ('Geometry and Experience', January 27, 1921) Albert Einstein includes the precise formulation of a question linking two, otherwise independent, propositions: 'How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?' That mathematics is admirably adapted to the objects of reality, is a matter of human experience: the proposition evidences itself, within the limits of experimental error, upon almost every occasion of individual experimentation. That mathematics is a product of human thought independent of experience, however, is a proposition the evidence for which is of a purely logical order: that it cannot be contradicted mathematically, without begging the seemingly-unanswerable question of what, then, informs the contradiction? In the case of any other subject, inquiry begins with unanswered questions and ends with informed answers; but when information itself is the subject of inquiry, the linguistic formulation of a question necessarily presupposes some, at the very least, of the answers intended to be discovered by investigation. At the very outset we are confronted with a dilemma, which can be resolved only if we presuppose that some part of any question we may formulate is disclosed without question, which is to say unconditionally. In other words, when examining the corpse of information previously shared, we may disclose one horn of our dilemma (an unconditional answer), and a lemma may inform a question, but we may invent neither answers nor questions. The unusual rules of evidence arising from such a presupposition are essentially simple. Firstly, we may make no further presuppositions in the course of our inquiry. Secondly, we may proceed only from answers to questions, and proceed in such a way that any question we may formulate constitutes a singular condition which is both self-evident in a prior answer, and inescapably necessary to account for the cognition of that answer. And finally, when formulating lemmas, we may adopt as unconditional only those answers which describe every possible application of the singular condition they inform. The medium of our inquiry being that of print, we will present any answers admissible within our rules of engagement in the form of printed figures, and subsequently express as lemmas those answers which, by exhausting every other possibility, prove the singular condition they inform. Questions, then, if questions there be, we will print in italics as Rules. So that we may direct the attention of the reader, we will annotate the items of each such category, Figures, Rules and Lemmas, numerically and in the sequence in which they emerge, and we will note as we proceed any special conditions presented by the medium of their expression. As our lemmas are to be formulated verbally we will define, for each annotated lemma and with direct reference to a priorly annotated figure, the terms in which it is expressed. Inevitably, terms defined so closely will assume a highly specialized usage: within the context of our inquiry therefore, a term used in a lemma is intended to take its meaning from that sentence in which it is priorly represented in bold type. If we are able to apply our rules of engagement, then we may hope to reconstruct a living model of the subject of our inquiry, comprising lemmas informed explicitly by figures. And if so, and at the end of our reconstruction a singular and unconditional answer can be expressed by a single lemma, then the rule by which it is disclosed must, of necessity, express that unique normative condition which defines the relationship between that which informs human experience unconditionally, and that purely logical order of mathematics so admirably adapted to describing it.

The Definition Of Terms

1. The Definition Of Terms 1.1 Exclusively Nothing Faithful to our terms of reference, we begin by presupposing nothing, and notate (i.e. symbolize) our presupposition as the content of an empty box, an answer annotated Figure 1, to a question which cannot be framed linguistically without contradiction: what is nothing?

Figure 1

We note that although, in Figure 1, we can see the borders of the box, they are there only to enable us to distinguish the content of the box from the rest of the page. Is that which we have presupposed therefore, Figure 1 without notation? For example:

Nothing (i.e. the absence of information) answers of itself nothing but that, in accordance with our rules of evidence, it precedes the notation (the borders of the box) by means of which it is cognited (i.e. made present) within the borders of the box. 1.2 Change Of State In the meaning of state as an unchanging (i.e. static) condition, the content of the box of Figure 1 represents that which is unchanging prior to (i.e. before) our notation. The singular condition inescapably necessary to represent nothing therefore (i.e. prior to our notation), is the cognition of nothing inside the box post to our notation (i.e. then). Our notation posits (i.e. positions) a singular, static condition. Using if to formulate a singular, conditional preposition posited by our notation in Figure 1, we may (i.e. it is within our rules of evidence so to do) formulate a singular rule: Rule 1: If nothing then nothing ? The adverb 'then' of Rule 1 describes the continuity of the absence of information within the box, conditional upon its continuation therefore, without the box. The absence of information therefore, being contradicted by its continuation, we may stipulate (i.e. admit to our rules of evidence) the exclusive (i.e. only possible) alternative of the absence of information: Lemma 1: There is change of state. We note that within the medium of our inquiry, a singular continuous state can be represented by notation (e.g. Figure 1), but cannot be presented (i.e. made present) by notation. 2

The Definition Of Terms

1.3 Exclusively Something The only possible change from the continuation of nothing in the box of Figure 1, is to there being something in the box. Bearing in mind that the borders of the proceeding Figure are there only to distinguish the content of the box from the rest of the page, upon the occasion (i.e. cause) of a change of state (Lemma 1), we can notate something, i.e:

Figure 2

As expressed by Rule 1, the subject of the conditional preposition may be contradicted by its own continuation then. But the contradictory of a singular continuous state (e.g. Figure 2) cannot be posited prior to the notation of a priorly continuous state (e.g. Figure 1). Without engaging in presupposition therefore, we can offer the reader no analogy, simile or metaphor to account for the contradiction of nothing: our notation may posit, but we may not invent, that which it notates. We must invite the reader therefore, if able so to do, to occasion the contradiction of a priorly-continuous state, in order for the notation of Figures 1 and 2 to exemplify (i.e. represent) that which is presented by the reader. As Rule 1 is informed by change of state (Lemma 1), the condition of Rule 1 may describe the exclusion of something by nothing, or the exclusion of nothing by something, but whichsoever it may be, Rule 1 describes the continuity then of its conditional informant therefore, but not the contradictory of its informant. We note that change of state can be exemplified by notation, but cannot be presented by notation. 1.4 Exclusively Something Or Nothing Albeit change of state can be presented only by the reader, we may observe experimentally (i.e. by mental experience) that by reading Figure 1 and Figure 2 alternately, a change of state is exemplified by translation: literally by moving from 'something or nothing', to the exclusive alternative by which it is contradicted. Since their contradiction can be exemplified by neither state singly (Rule 1), we must, of necessity, suppose there to be an additional agency of information: that we, the readers, present the translation by means of which our notation may exemplify a change of state. The occasion of that translation being, however, the exclusive property of the reader, we will (i.e. choose so to do) use a symbol to notate the self-evident, but otherwise ineluctable, principle of mutual exclusion, and so exemplify in a single Figure, that change of state conditional upon translation, i.e:

Figure 3

In Figure 3 we have configured (i.e. set each against the other) Figures 1 and 2, with a symbol to represent the condition of their exclusive alternation upon the occasion of translation. Of the two continuous states configured in Figure 3 therefore, one else the other (i.e. not both and not neither) may be presented by the reader. Figure 3, however, enables us to represent that exclusive continuity, of one or the other, of two mutually exclusive states which may be determined (i.e. ended) by the reader. 3

The Definition Of Terms

Whichsoever of the mutually exclusive alternatives of Figure 3 may be presented by the reader, can be determined only by the reader. We can, however, characterize the effect (Rule 1) of its determination, as the disclosure of a singular continuous state which presents the exclusive origin of its own continuation, each continuation distinguishable from the next only by its prior exclusion. Continuity becomes the singular, conditional property of that state determining (ending) a translation occasioned by the reader. We will characterize this effect as memory, and admit memory therefore, to be the cause of the continuity of one of two mutually exclusive states, in accordance with: Rule 2: If exclusively nothing then nothing, else exclusively something ? We define memory, then, as the ability to determine a priorly continuous state. With the benefit of memory, when (i.e. upon the occasion of ) there is change of state, then the contradictory of a priorly continuous state precedes its own continuation, according to: Lemma 2: When exclusively something or nothing, then its continuation is the effect of memory. In recognition of the occasion (i.e. translation), of that change of state conditional upon memory, we will define the singular effect of memory as the attribute of the direction of translation, and observe that memory confers upon one, of either the absence of something or the absence of nothing, the property of continuous translation in its own serial (i.e. alternately informed ) direction. 1.5 Inclusively Something Or Nothing We have attributed the continuous serial translation of a singular state, to the memory of a singular direction of translation in which it was priorly informed, which singular property of memory we will define as the translator. The translator, then, is the informant of the direction of the continuous serial translation of that singular state first determined by the reader, and informed therefore in the first place by its contradiction. Which, of two mutually exclusive states determined by the reader, is the informant of the translator, can be determined only by the reader. However, by configuring Figures 1 and 2 symbolically, we have introduced in Figure 3 the notation of two mutually exclusive determinants, of what we will characterize as lateral translation. There being in Figure 3 only one serial alternative to any direction of lateral translation, each of two possible directions of lateral translation can be described prior to the determination of their informants (Rule 1), as 'toward the absence of something' or 'toward the absence of nothing', which exclusive alternatives we will characterize arbitrarily and randomly, as toward-something (i.e. toward the absence of nothing) or toward-nothing (i.e. toward the absence of something). Bearing in mind that the outer borders in the ensuing configuration of Figures 1 and 2 (annotated 4) are there only to distinguish their content from the rest of the page, we can notate that continuous state determined by lateral translation in the priorly-informed direction, as one of either:

Figure 4 4

The Definition Of Terms

In either of the mutually exclusive alternatives of Figure 4, every possible translation but one in the informed direction, is determined by the priorly continuous state: the singular exception being that of continuous translation in the informed direction. That exclusive state priorly determining translation (e.g. Figure 3), therefore, is self-evidently included (e.g. Figure 4) by its own serial contradictory, in every possible direction of translation except that informed by our notation in the first place. And if so, then by alternately presenting two mutually exclusive states, the reader may inform a static attribute of both, that is first posited by either Figure 1 or Figure 2, in accordance with: Rule 3: If something or nothing exclusively then something or nothing inclusively ? Upon the first occasion of the determination of a singular continuous state represented by our notation, the property conferred by memory upon the contradictory of the priorly continuous state, is that of continuous translation in the serial contradictory of the direction then informing the translator. We will characterize as position therefore, that static attribute acquired by memory from the determination of translation in every possible direction informed by the translator. Self-evidently, the represented position of either something or nothing exclusively (e.g. Figure 4), is continuously included, in every possible direction of translation, by its exclusive alternative, according to: Lemma 3: When two mutually exclusive states alternate, a singular position is determined in the serial direction of the continuous state. We will characterize that state (e.g. Figure 4) determining translation in every possible direction but one therefore, as the inclusive state, and the position of the exclusive alternative of the inclusive state, as the contrary of the inclusive state. 1.6 Something And Nothing It follows necessarily from Lemma 3, that the contrary of the inclusive state (i.e. the position of something or nothing exclusively), is the contradictory of the priorly continuous state. There being change of state (Lemma 1) therefore, the effect of memory (Lemma 2) discloses (Lemma 3) one of two (and only two) possible alternatives: (a) The contradictory of something is the contrary of nothing? (b) The contradictory of nothing is the contrary of something? In both case (a) and case (b), and therefore in either case (a) or inclusively case (b): Rule 4: If something or nothing inclusively then something and nothing exclusively ? The singular direction informed by the translator, then, depends upon which of two mutually exclusive states, by constituting the origin of its own continuity in the first place, is supplying what we will characterize an isologue (i.e. that which is indistinguishable in all respects but one) of the position in which its contradictory is determined. In the case of alternating translation, the reader may observe experimentally (e.g. Figure 3) that the direction of translation in which a presently continuous state (whichsoever it may be) is informed, is the serial contradictory of the direction in which it is determined. Depending upon which of the mutually exclusive positions exemplified in Figure 3 is first determined 5

The Definition Of Terms

by lateral translation therefore, the direction informed by the translator is that conferred by cognition upon the contradictory of the presently continuous state, configuratively one of either:

Figure 5

And if so observed in Figure 3, then we may formulate: Lemma 4: When two mutually exclusive states alternate, a singular direction is informed toward the position of the continuous state. In either one of the mutually exclusive alternatives of Figure 5, the notation of that priorly continuous state (something or nothing exclusively) informing the translator in the first place, and the notation of its exclusive contradictory, are so configured that lateral translation (either toward-something or toward-nothing) is presented horizontally. There is, however, no rule governing what we will characterize the orientation of the position determined by lateral translation in the informed direction: Rule 4 is preceded only by the singular condition evident in Figure 4, that every possible translation but one in the direction informed by the translator, is priorly determined. We could as well present Figure 5 (with that one and only self-evident exception (Figure 4) in which the alternatives are not mutually exclusive), in any orientation we care to choose, e.g:

Figure 6

There being two (and only two) mutually exclusive states, however, and there being one (and only one) informed direction of translation in which a singular continuous state can be determined, in order to represent the serial contradictory of the isologue of a singular and presently continuous state, we are obliged to do so horizontally, e.g:

Figure 7

We will, therefore, use the term horizontal to qualify that singular configuration of our notation of two mutually exclusive states, in which the determination of lateral translation (e.g. Figure 7) coincides (i.e. happens both where and when) with the determination of the isologue of the presently continuous state. Howsoever the isologue of a priorly notated state may be orientated therefore, the direction informed by the translator is that in which laterally-horizontal translation is first determined, in accordance with: Lemma 5: When two mutually exclusive states alternate, a singular, laterallyhorizontal translation is determined in the serial direction of the continuous state. But there is a question begged between Lemma 4 and Lemma 5: where is Rule 5? 6

The Definition Of Terms

1.7 Division The problem illustrated by Figures 6 and 7, is that Figure 5 (but not Figure 3) violates the rules of engagement with which we began our definitions of terms. The unconditional alternative of a singular rule expressed by a lemma, must apply immutably to every possible instance (i.e. singular occasion) of the rule it informs: the lemma proves the rule, the rule does not prove the lemma. And in Figure 5, horizontal translation exemplifies only one of many possible lateral translations. If we use arrows to symbolize some of the possible lateral translations allowed by Lemma 4, it is patent that many such directions are possible, e.g:

Figure 8

And if, experimentally, we read any horizontal configuration of Figures 1 and 2 alternately, we can see for ourselves that upon the instance of a change of state, the position informed by our notation alternates with that of its contradictory. The concern of the conspicuously missing Rule 5 is this: prior to the instance of a change of state, lateral translation is continuous in every possible direction informing the translator, e.g. one of either:

Figure 9

And upon the instance of a change of state, lateral translation is determined in every possible direction informed by the translator. Prior to horizontal translation, the only possible singular direction which can be informed by the translator, is the serial contradictory of every possible lateral translation, e.g:

Figure 10

Alternating translation between the horizontally orientated, mutually exclusive alternatives configured in Figure 5, therefore, infallibly answers the question of when a singular lateral translation is determined in the informed direction: it is determined upon the instance of a change of state conditional upon laterally-horizontal translation. But alternating translation fails to disclose where the isologue of the presently continuous state is determined, because in all our figures hitherto, the presently continuous state is then determinable in one (and only one) direction of lateral translation, instead of all but one. If we are to disclose Rule 5, we must so modify our figures that the position presented by the reader, is determinable in every possible direction of translation except that which is informed by our notation. To do so, we require the means to configure in the first place informed by our notation (e.g. Figure 10), that position informed in the direction 7

The Definition Of Terms

denotated (i.e. identified without notation) by the translator. But a singular continuous state and its singular isologue being mutually exclusive, their superimposition, like the principle of mutual exclusion itself, can be represented only symbolically. A symbol is intended to generalize (i.e. represent singularly) every possible isologue of a singular condition for which it is chosen to stand: for example, the graduation marks on a humble school ruler are contrived empirically (i.e. by direct measurement) to symbolize a sub-set of equal divisions of distance. And in order to standardize the measurement of distance, a singular analogue (i.e. exemplar) of distance is chosen, from which the ruler is (ultimately) derived empirically. A ruler, then, represents a standard analogue, the graduated divisions of which may be used to measure any isologue of distance except one: in any direct comparison of a ruler with the standard from which it is derived, it is the standard analogue which measures the ruler, and not vice versa. In Figure 1 we have used a box to notate a singular state which, prior to its notation, is continuous in every possible direction of translation. If therefore, we were to use a symbol to notate the content of the box and not the box, then that symbol would represent the continuity within and without the box, of that singular state notated by the box. With the sole condition that a symbol chosen to notate the content of the box is not the box, then by using that symbol we can posit a singular continuous state without the box, i.e. generalize the position of a singular and priorly continuous state. It is of no immediate significance what symbol we choose, but by published convention we use the symbol 1 to represent the first of anything. We will use the symbol 1, then, to generalize the continuation of any singular state the isologue of which is first determined (Lemma 5) by means of laterally-horizontal translation. Provided that is neither 1 nor the box, it is of no immediate significance what symbol we choose to notate every possible isologue of that primary (i.e. first posited) state, but by published convention we use the symbol 0 for the contradictory of 1. Reconfiguring Figure 5 symbolically, the serial contradictory of every possible direction of translation except the laterally-horizontal, becomes explicit in whichever one of two orders we choose to represent the primary state and its mutually exclusive alternative, e.g: 10 / 01 Figure 11

We note that when we read Figure 11 as representative of two mutually exclusive states, the continuation of the isologue of the primary state represented by 0, is possible in every direction of lateral translation except that in which it is determined by the primary state. We will characterize that singular exception, as translation toward–1 (i.e. toward the primary state). Using round brackets to distinguish the attribute of position from that of direction (e.g. Figure 4), we can describe the only possible lateral direction informed upon the first occasion of determining an isologue of a primary state, as one of either: 1 (0) / (0) 1 Figure 12

As informed by the alternation of two mutually exclusive states, the question of where translation toward–1 becomes the property of the primary state does not arise: it proceeds unconditionally when lateral translation is determined by a singular change of state (Lemma 1). Using superscription to distinguish the isologue of a primary state from the position in which it is first determined, then regardless of which of  or  is the singular property of the translator, we can describe the order of the first determination of the position of an isologue of a primary state informed by lateral translation, i.e: 8

The Definition Of Terms

(1 – 0)0–1 – (0 – 1)1–0 Figure 13

Regardless of which of the continuation, of the absence of something or the absence of nothing (e.g. Figure 1 or exclusively Figure 2), is denotated primary, where two mutually exclusive states alternate, the laterally-horizontal translation occasioning their alternation is divided by the position of the (then) continuous state. But because the said question of where lateral translation is determined does not arise, neither does the rule informed by Lemma 5. The question, informed by Lemma 5 therefore, is not where, but how the translator acquires, in the first place, a singular direction of lateral translation which, upon every possible occasion of its determination, coincides with its own serial contradictory. Other than position, then, what singular attribute of the contrary of the inclusive state is disclosed by Figure 13 but not by Figure 4? 1.8 Inequality Of Proportion In order to configure 'something and nothing' (e.g. Figure 4), as opposed to (i.e. comparable with) the configuration of 'inclusively something or nothing' (e.g. Figure 3), we had first to distinguish from the rest of the page, both of two possible inclusive states (i.e. something and nothing), by supplying the outer borders of Figure 4. By contrast, the notation of Figure 13 makes explicit the condition we were obliged to invite the reader to present: that prior to the first instance of our notation (the box), a singular state is continuous in every possible direction of translation (i.e. unlimited), and post to the first instance of our notation, translation is unlimited in every possible direction except that informed by the alternation of two mutually exclusive states in a singular position. Bearing in mind that the outer borders of the ensuing configuration of Figures 1 and 2 (annotated 14) are there only to exemplify every possible direction of translation post to our notation of a singular and priorly continuous state, it is self-evident, prior to the determination of the priorly continuous state, that the first instance of our notation is lesser than the position of the continuous state it informs, e.g. one of either:

Figure 14

Figure 14 configures two possible isologues of the primary state, either one of which is described by Figure 13 in the order of its determination in a singular position. According to which of the mutually exclusive alternatives of Figure 14 is denotated 1, then upon the first occasion of a serial change of state, the alternation of two mutually exclusive states is generalized symbolically in the order of the cognition of the inequality of proportion, of the content of the box, and the position of its determinant, i.e. one of either: (0 – 1)1–0 / (1 – 0)0–1 Figure 15

If, therefore, we use the symbol < to notate a singular extant (i.e. self-evident) inequality of proportion, cognited prior to its determination by a subject memory (e.g. 9

The Definition Of Terms

Figure 14), and so distinguish inequality of proportion from the direction of the translation by means of which it is determined (see Figure 12), then the order of cognition of a singular, extant inequality of proportion can be generalized in the order of the determination of one of two serially-alternating determinants of a singular direction of translation, i.e: (1 < 0)1–0 – (1 < 0)0–1 Figure 16

We will, therefore, use the symbol – to notate that laterally-horizontal translation, the direction of which is the singular property of memory we have defined as the translator, and the determination of which translation coincides unconditionally with the cognition of a singular, extant inequality of proportion in accordance with: Rule 5: If 1 < 0 then 1 is toward–1 ? In plain words, Rule 5 asks: if the first notation of a singular continuous state is lesser than the position it informs, then the translation by means of which it is determined, is informed in the direction in which the said condition is recognized? Informed by Lemma 5 therefore, Rule 5 discloses how the translator acquires the singular direction informing the position of the primary state when two mutually exclusive and priorly continuous states alternate (Lemma 4). But to inform the condition of Rule 5, the extant inequality of proportion informing the translator is (by definition) determined, leaving unanswered the question of where the translator is informed in the first place. Other than position, then, what singular attribute of two mutually exclusive continuous states is informed by Lemma 5 and not informed by Lemma 4? 1.9 Equality Of Proportion The singular condition of Rule 5 is that the inequality of proportion, of the determinant and the divisor of the primary state, be self-evident prior to the determination of the position it informs, which condition proceeds necessarily from the medium of our inquiry: because, regardless of the shape, size or form of the box we may deploy in the first place to posit the continuity of a singular state, prior to the determination of that state our notation is, necessarily, cognitively lesser than the prior continuity it informs. Provided only that it is not with the symbol 1, we may revisit Figure 1 with any notation we care to choose, and it necessarily satisfies the condition of Rule 5, e.g:

Figure 17

And the singular state posited by our notation (e.g. Figure 17) being continuous in every possible direction of translation but one, when we use 1 to generalize continuity within and without the box, then the effect of Rule 5 is likewise self-evident: the priorly continuous state is tautologically (i.e. defined and redefined) 'toward–1'. But what we cannot do within our rules of evidence, is presuppose the condition of Rule 5, that the contradictory, of the priorly continuous state exemplified without the box, is greater than 1; and if we use 1 to notate both the content of the box and the box, then we cannot generalize the continuous state therefore To account for Rule 5 in both the order of cognition of an inequality of proportion and 10

The Definition Of Terms

within our rules of engagement, therefore, we will use the phrase positive notation to distinguish that which posits a (i.e. any singular) continuous state, whatsoever that positive notation may be and regardless of whether it posits the absence of something or the absence of nothing, from any symbol we agree to use to denotate the prior continuity of the (we note the definite article) posited state. In the order of cognition of a singular continuous state, translation per se (i.e. informed by the spoken word when extant) is indeterminable (i.e. continuous in every possible direction) unless or until informed by positive notation. And post to the positive notation of a singular continuous state, translation is indeterminable in every possible direction except that conferred by memory (Lemma 2) upon the continuity within the box, of the priorly continuous state: which singular condition, of our positive notation being cognitively lesser than the continuous state it notates, we will refer to as that of being isolated in every possible direction of translation but one, i.e:

Figure 18

Figure 17 exemplifies an instance (we note the indefinite article) of a cognited inequality of proportion, informed by one of many (i.e. indefinitely more than one) possible instances of positive notation. We note furthermore, that the isolated informant of a cognited inequality (e.g. Figure 17) may be increased or decreased, by means of translation away from or toward our positive notation. We will, therefore, characterize any singular static instance of a cognited inequality, an instant (i.e. extant instance) thereof. If the first cognited instant of an inequality of proportion, is not determined by a change of state, then translation per se is unlimited in every possible direction but one: the singular exception being A posteriori (i.e. for every possible position of the primary state post to the translation by which it is determined), that singular, serially informed alternative of every possible direction (e.g. Figure 18), in which translation is unlimited. We have defined memory as the ability to determine a priorly continuous state. There being one (and only one) alternative of the priorly continuous state exemplified by Figure 17, and a singular continuous state being unlimited in every possible direction of translation prior to its positive notation, it is of no immediate consequence what positive notation we use to notate the determinant of every possible direction of translation but one, in order to distinguish it from the rest of the page, e.g:

Figure 19

Upon the occasion of the determination of the first cognited instant of an indeterminable inequality of proportion, it is self-evident that the determinant of the priorly continuous state (e.g. Figure 19) is greater than the isolated divisor (e.g. Figure 17) of the priorly continuous state. We will, therefore, characterize the alternation of two possible informants (e.g. Figures 18 or 19) of a singular direction of translation, a pair of instants, and as a pair cannot be singular, as a unary pair of instants, the singularity of which is 11

The Definition Of Terms

attributable to their mutual exclusivity upon the occasion of their alternation in a singular position. We note, however, that the denotation (i.e. 'e.g. Figure 17') of an informant of a singular direction of translation, begs the question of the order of cognition of its informant. When an instant of an indeterminable inequality of proportion is informed in the order of its determination, then which, of two priorly continuous states (e.g. Figure 17 or Figure 19) is the informant of the singular position in which they alternate? In accordance with Rule 3, when two mutually exclusive states alternate, a singular position is determined in the serial direction of the continuous state. Without denotating which, of two priorly continuous states (e.g. Figures 17 and 19) is the precedent of the singular position in which its contradictory is informed, the only possible instant informing the inequality of proportion, of the divisor and the determinant of an indeterminable inequality of proportion, is exemplified by the contradictory, of that continuous state priorly exemplified without the content of the box, i.e. one of either:

Figure 20

We note that, as it is not the ability of memory to determine both a singular continuous state and its exclusive alternative, the determination of either one of the mutually exclusive alternatives of Figure 20, will inform the other. We will, therefore, use the adjective 'secondary' to distinguish the cognition of the inequality of proportion of the divisor and the determinant of a singular and priorly continuous state, from that of the divisor and the informant of a singular and priorly continuous state. Figure 20 configures a unary pair of mutually exclusive instants of the secondary cognition of an extant inequality of proportion. And there being at (i.e. prior to the determination of ) either one of a unary pair of mutually exclusive secondary instants (e.g. Figure 20), no discernible limit of the inequality of proportion informed by the singular position of their isolated divisors, no difference of proportion can be informed by their alternation: they are A posteriori of perfect equality of proportion. If, then, we use the symbol = to notate equality, and regardless of which is the informant and which the determinant of the first instant of its cognition, by randomly (i.e. regardless of order) notating symbolically one of a unary pair of mutually exclusive and priorly continuous states (e.g. Figure 17 or Figure 19), the position of the divisor of the notated state can be generalized in the order of its determination, i.e: (1 < 0)1â&#x20AC;&#x201C;0 = (0 < 1)1â&#x20AC;&#x201C;1 Figure 21

The determination of the divisor of one, of a unary pair of mutually exclusive secondary instants (e.g. Figure 20), will inform the position in which an unlimited lateral translation, occasioned by the reader therefore, was determined in the first place, e.g. one of either:

Figure 22 12

The Definition Of Terms

In accordance with rule 4, when two mutually exclusive states alternate, a singular direction is informed toward the position of the continuous state. Without denotating which, of two priorly continuous states (e.g. Figure 22), is the informant of the translator, then upon the occasion of the serial alternation of two priorly continuous states, the position of the determinant of laterally-horizontal translation, is exemplified by the contradictory (Cf Figures 20 and 22) of the singular divisor of two priorly continuous states, e.g. one of either:

Figure 23

At the first possible secondary instant of the cognition of a singular, extant and indeterminable inequality of proportion (e.g. Figure 22), then, the direction of a priorly unlimited lateral translation, is informed by the singular divisor of two priorly continuous states. We will, therefore, adopt the convention of identifying unilateral (i.e. any singular lateral) translation with the direction in which it is unlimited prior to its determination, in the manner of identifying a wind with the direction from which it blows. Using the symbol t to notate translation per se, and without denotating which of two priorly continuous states is the informant of the translator, a (tautologically) primary unilateral translation, can be described in the order of the determination of one of a unary pair of mutually exclusive binary (i.e. doubled) instants of perfect equality of proportion (e.g. Figure 20), i.e. as one or inclusively the other of: t0 = (0  1)1–0 / (1  0)0–1 Figure 24

The determination of a primary unilateral translation (e.g. Figure 22) therefore, coincides unconditionally with the logical division of an unlimited translation, into two parts of an indeterminable proportional equality, and by logical we intend (and only intend) in a singular direction, i.e: (t0 – t)1–0 = (t0 – t)0–1 Figure 25

Prior to the primary (i.e. first possible) instant of the cognition of an indeterminable inequality of proportion, unilateral translation is unlimited in the direction informed by the determinant of a primary unilateral translation. But at the primary instant (e.g. Figure 22), unilateral translation in the informed direction is limited (e.g. Figure 4) by the contrary of the inclusive state (Rule 4). And having priorly (and randomly) notated symbolically the first possible secondary instant of a singular and priorly continuous state (see Figure 21), the informant of the translator can be generalized in the order of the determination in a singular position (e.g. Figure 23), of the serial divisor of one of a unary pair of binary instants of a singular and priorly continuous state, i.e: (1 < 0)1–1 = (0 < 1)1–1 Figure 26

At the first possible secondary instant of a singular and priorly continuous state (e.g. Figure 17 or Figure 19), every possible unilateral translation in the direction informed 13

The Definition Of Terms

by the translator, is unconditionally lesser than that primary unilateral translation the determination of which informed the translator in the first place, i.e: (0  1)1–0 > (1  0)1–1 Figure 27

Figure 27 therefore, describes the order of cognition of the inequality, of a primary unilateral translation the direction of which is informed at the (tautologically) primary instant of a primary state, and a singular serial translation in the informed direction, determined unilaterally at the first possible secondary instant of the primary state, i.e: (t0 – t)1–1 = (t0 – t)1–0 – (t0 – t)1–1 Figure 28

We have defined the primary state as the continuation of a singular state the position of which is informed by that isologue first determined by means of laterallyhorizontal translation (e.g. Figure 22). Regardless of which of two mutually exclusive and priorly continuous states, therefore, we may generalize symbolically as 1, having done so, and there being no discernible limit of the primary state (e.g. Figure 18), the position of the primary state is exemplified A posteriori by the perfect equality of proportion, of every possible singular unilateral translation in the informed direction, and its own serial contradictory determined at the first possible secondary instant of the primary state, i.e: (0  1)1–1 = (1  0)1–1 Figure 29

We define a dimension as that property of an analogue which affords empirical measurement in a singular direction, of its every possible isologue. A singular serial dimension, of the singular position, of a singular and priorly continuous state determined in the singular direction we have notated symbolically, is measured by the translator at the first possible secondary instant of that singular state the prior continuity of which we first denotate 1, i.e: (10)0–1 > ((0 – 1)0 / (1 – 0)0 ) Figure 30

Figure 30 describes, within the rules of engagement of our inquiry, both the order of cognition of a singular, extant and indeterminable inequality of proportion, and the proportional division by the translator of one of a unary pair of mutually exclusive binary instants of perfect equality of proportion. The symbol / we have adopted to represent mutual exclusion therefore, exemplifies the division by the primary state, of a singular and continuous translation, into two parts of perfect equality. We note that the proportional equality, of a unary pair of mutually exclusive divisors of a singular binary instant (e.g. Figure 23), and therefore the singular condition of Rule 5, could not be defined without the generalization, by means of what we will characterize our secondary notation 10, of the primary state and its exclusive alternative, begging the question therefore of which, of two priorly continuous states, is the informant of the translator. The answer to that question, however, being by definition the singular property of memory, by virtue of our random, symbolic notation of a unary pair of secondary instants of perfect equality of proportion, we can formulate the conjunction of an exclusive alternative to Rule 5, with our first formulation of Lemma 5: 14

The Definition Of Terms

Lemma 5: and where two mutually exclusive states alternate, a singular, seriallyhorizontal translation is determined in the lateral direction of the continuous state. The measure of its every possible isologue being the property of the translator, and there being no determinable inequality of proportion between two secondary instants of the primary state, the translation towardâ&#x20AC;&#x201C;1 informed by memory in the first place can be attributed only to a prior and innate limit of the memory of the reader. 1.10 The Limit Of Proportional Inequality In order to disclose the singular condition of Rule 5, we have placed no preconditions upon the nature of an isolated state but that it is lesser than the continuity it notates, and no preconditions upon the priorly continuous state but that of being finite (i.e. determinable by a subject memory). Informed by our secondary notation, however, the condition of Rule 5 (where and when 1 is lesser than 0), would seem, contrary to our rules of evidence, to presuppose the empirical condition, such that the prior continuity of a primary state denotated 1 is measurably equal to its determinant prior to its determination. But let us suppose instead, on the evidence of a unary pair of mutually exclusive instants of the secondary cognition of an extant inequality of proportion (e.g. Figure 23), that translation per se is an unconditional property of both an isolated state and the priorly continuous state it notates. By way of example, compare Figure 23 with:

Figure 31

In both Figures 23 and 31 we have configured the positive notation (the outer borders), of one of every possible informant of the order of the inequality of proportion, of a translation the direction of which is informed alternately, and a translation the direction of which is determined alternately (see Figure 28). By means of unilateral, horizontal translation from one to the other of two mutually exclusive binary instants of perfect equality of proportion, we can exemplify, in both Figures 23 and 31, that inclusive alternation described by Rule 4, of 'something and nothing'. And by means of unilateral translation logically toward or away from our positive notation of any singular binary instant (e.g. Figure 23 or Figure 31), we can exemplify the inequality of proportion, of any two binary instants informed by a singular divisor of a singular continuous state. We will, therefore, use the phrase uniform change to describe that property of a singular divisor of any two binary instants of a singular continuous state which, upon the occasion of logical translation, changes continuously in every possible direction of lateral translation. We note that at any binary instant but one informed by logical translation away from Figure 31, the divisor of the continuous state, and therefore that singular dimension informing the translator, decreases (i.e. lessens), but the inequality of proportion, presently informed by our positive notation of one, of every possible outer border of the continuous state, is constant (i.e. unchanging). The singular exception is informed where the ability of a subject memory to determine uniform change, is exceeded, e.g. one of either:

The Definition Of Terms

Figure 32

At which singular secondary instant, there being no discernible outer border, the inequality of proportion informed by the isolated state is indeterminable. We will, therefore, characterize that singular secondary instant at which uniform change is last informed by translation logically away from our positive notation, the inclusive limit of the memory of the translator. And we note that at any secondary instant but one informed by translation logically toward Figure 31, the divisor of the priorly continuous state, and therefore that singular dimension informing the translator, increases (i.e. greatens), but the inequality of proportion, informed by our positive notation of the outer borders of the priorly continuous state, is constant. The singular exception is informed when the ability of a subject memory to inform uniform change is determined: upon the occasion of which, the determinant of the last informed inequality of proportion is, self-evidently, not lesser than the inclusive limit of the memory of the translator, e.g. one of either:

Figure 33

We will, therefore, characterize the determinant of that singular secondary instant at which uniform change is last informed by unilateral translation toward our positive notation, the exclusive limit of the memory of the translator. Figures 32 and 33 configure the determinant of the greatest possible inequality of proportion informed by a subject memory (the inclusive limit), and the determinant of the smallest possible inequality of proportion informed by a subject memory (the exclusive limit), as unary pair of mutually exclusive and perfectly equal divisors of a singular binary instant, of the secondary cognition of a singular, extant and indeterminable inequality of proportion. The determination of one of the divisors of a singular binary instant, will inform the order of cognition of a singular, secondary instant of a singular and priorly continuous state, and disclose the order therefore, of the informant of the translator. If, however, we denotate symbolically one of the divisors exemplified in both Figures 32 and 33 (i.e. both the box and the content of the box), then we cannot denotate the priorly continuous state. We will, therefore, posit the inclusive limit of a subject memory where and when the smallest possible inequality of proportion determinable by a subject memory, is informed by the uniform change of the translator at the instant of its determination, i.e. in the exclusive limit of a subject memory, e.g. one of either:

Figure 34

Figure 34, then, exemplifies one of a unary pair of mutually exclusive secondary instants of a singular and priorly continuous state, which, upon the occasion of its denotation, 16

The Definition Of Terms

will inform the position in which its exclusive contradictory is determined, and therefore the order of the informant of the translator. We note, however, that upon the occasion of the denotation of either one of the mutually exclusive alternatives notated positively by Figure 34, the inequality of proportion informed by the (then) singular isolated state is indeterminable. We will, therefore, posit the exclusive limit of a subject memory where and when the greatest possible inequality of proportion determinable by a subject memory, is informed by the uniform change of the translator at the instant of its determination, ie. in the inclusive limit of a subject memory, e.g. one of either:

Figure 35

Self-evidently, there can be no logical limit of the proportional equality of a unary pair of divisors of a singular binary instant of a singular, extant and indeterminable inequality of proportion. We have, however, already posited a finite limit of inequality, as the greatest possible translation informed by the uniform change of the divisor of a singular binary instant prior to the instant of its determination. We will, therefore, use the term moment to characterize that which is determined between any two instants of a singular and priorly continuous state. If we attribute the greatest possible moment of the uniform change of the divisor of a singular binary instant (e.g. Figure 35), to the combined (i.e. added) uniform change of the inclusive and the exclusive limit of a subject memory, then that which is determined between two secondary instants of a singular and priorly continuous state, is the moment of the greatest possible unilateral translation (t0) determinable by a subject memory. In which case, the greatest possible moment of a singular, unlimited logical translation occasioned and determined unilaterally by a subject memory, is exemplified by the instantaneous (i.e. between two secondary instants of a singular and priorly continuous state) determination of the divisor of a singular binary instant, e.g. one of either:

Figure 36

In order, therefore, to distinguish between the inclusive and the exclusive limit of a subject memory, we will characterize the divisor of any singular binary instant determined by the unilateral horizontal translation of a subject memory, the vertex of a subject memory (e.g. Figure 33, not Figure 36). The vertex of a subject memory, then, is the conditional informant of the greatest possible constant inequality of proportion determinable by a subject memory, and the determinant of the contradictory therefore, of the exclusive limit of a subject memory. We have used the symbol 0 to generalize every possible isologue of the primary state at the instant of its determination, of which generalization, our positive notation of the vertex of a subject memory exemplifies merely one of every possible inclusive limit of memory per se. Regardless of the shape, form or nature of the exclusive limit of a subject memory, the informant of the translator is the greatest possible moment of the uniform change of the vertex of a subject memory, generalized A posteriori as:

The Definition Of Terms

(0  1)0 = (1  0)0 Figure 37

We note, however, that if we attribute the finite limit of inequality exclusively to the unilateral translation of a subject memory (i.e. the isolated state is static), the inequality of proportion of the vertex and the exclusive limit of a subject memory is indeterminable. And if we attribute the finite limit of inequality exclusively to the moment of the uniform change of the vertex (i.e. a subject memory is static) then, self-evidently (e.g. Figure 31) and by definition, the inequality of proportion of any two secondary instants of a singular and priorly continuous state is indeterminable. But if we attribute the finite limit of inequality to the combined uniform translation T of a subject memory and a singular isolated state which is not an isologue of the vertex of a subject memory, then the moment of the combined uniform change, of the divisor of a singular binary instant and the limit of a subject memory by which it is determined, is exemplified A posteriori by the instantaneous alternation of a unary pair of mutually exclusive and perfectly equal instants (see Figure 30), of a singular, extant and indeterminable inequality of proportion, e.g:

Figure 38

And if so, then upon the occasion of the instantaneous determination in a singular position, of either one of a unary pair of mutually exclusive instants of perfect equality of proportion (e.g. Figure 38), and regardless of the order of its prior determination, the vertex of a subject memory is A Priori (i.e. for every possible instant at the moment of its determination) perfectly equal (see Figure 28) to its instantaneous determinant, i.e: (10 – 1)0 = (10 – 1)1 Figure 39

And although we cannot, without presupposition, attribute the direction of a primary translation unilaterally, the conjoined alternatives of Lemma 5 constitute a proposition (i.e. a question to which there two and only two possible answers): Rule 5: If 1 < 0 then 1 is toward–1 else 0 is toward–1 ? In plain words, Rule 5 asks: when two mutually exclusive alternatives are disclosed by a singular condition, then if the condition is not informed, the exclusive alternative is unconditional? And if so, and if the contrary of a positively notated state is not greater than its determinant, then upon the occasion of its determination, a singular moment of translation is informed in the serial direction of the continuous state, according to: Lemma 6: Between any two instants there is one and only one moment of translation.

The Definition Of Information

2. The Definition Of Information 2.1 Introduction It is the burden, of both our emerging model and the science of empirical measurement, that an analogue and that which it measures in a singular direction of translation are, within the limits of experimental error, perfectly equal. Self-evidently and by definition, however, an instant of a singular and priorly continuous state is (the tautology is deliberate) static, and no two determinable instants of a singular and priorly continuous state can be of perfect equality of proportion, unless both are divided by a singular, static, isolated state which is not an isologue of the limit it determines, one such example of which we have notated positively as:

Figure 40

We have, therefore, to distinguish two kinds of informants contributing to our inquiry: that which is static A posteriori, and that which is static A Priori. As the reader's is the only possible occasion of a translation by means of which change of state can be informed by our notation, we will qualify (i.e. distinguish verbally) with the adjective 'published', that which is, therefore, static A Priori. That which is published, then, includes our positive notation (e.g. Figure 40), our secondary notation (i.e. 10), and the written symbols and language by means of which positive notation and secondary notation is annotated and denotated by the publisher, as exemplified by our inquiry in its entirety. As the publisher of our inquiry can know neither when nor where a reader other than the publisher may translate that which is published herein, we will confine our use of the phrase a subject memory to refer exclusively to the memory of the publisher, and with that qualification, note that where and when Figure 40 exemplifies a singular instant of an extant and indeterminable inequality of proportion, it is informed in the first place by the determination of a primary unilateral translation occasioned by the subject memory. And the direction of which translation being, both self-evidently and by definition, the singular and exclusive property of the subject memory, we will qualify as 'private' that which is static A posteriori. 2.2 Private Information According to Lemma 4, and therefore prior to the recognition of that inequality of proportion informing the singular condition of Rule 5, the direction, of a primary unilateral translation determined at Figure 40, is informed by denotating the position of our positive notation (i.e. the box and not the content of the box), e.g:

Figure 41

Upon the occasion of informing the position of a divisor of a singular and priorly continuous state, the subject memory can predict (i.e. is informed A posteriori), with what we will define as apodictic certainty (i.e. without presupposition), that the informant of direction (the content of the box and not the box) and the position it informs, are of perfect equality of proportion. 19

The Definition Of Information

We have defined the vertex of a subject memory as the conditional informant of the greatest possible inequality determinable by a subject memory. The determinant of the finite limit of inequality being the exclusive limit of the subject memory, we will, randomly and without secondary notation, qualify the greater, whichsoever it may be, of any two mutually exclusive states informing a singular instant (e.g. Figure 40), the 'primary' state. Our positive notation being static A Priori, that which is normative (i.e. providing an instance) of the primary state is defined by that singular moment of a primary unilateral translation of the subject memory, the determination of which informs one of the exclusive alternatives of Rule 5, with the apodictic certainty, that the static divisor of the exclusive limit of a subject memory and the divisor of the primary state, are of perfect equality of proportion, i.e: t0 = (t – t0)0–1 / (t – t0)1–0 Figure 42

At any primary instant of the primary state, the subject memory can predict, with apodictic certainty, that upon the occasion of determining such an instant, the perfect equality of proportion of the vertex and the primary state will be informed A posteriori, by the determination of one of two mutually exclusive divisors (e.g. Figure 33) of a singular binary instant, e.g. (and bearing in mind our caveat on outer borders) one of either:

Figure 43

We have attributed (Lemma 6) the finite limit of inequality to the combined uniform translation T of a subject memory and a singular isolated state which is not an isologue of the vertex of a subject memory. Upon the occasion of informing the equality of proportion of the vertex and the primary state, a condition normative of the translator is informed A posteriori, in the order of the determination of one of unary pair of mutually exclusive and perfectly equal divisors (e.g. Figure 43), of a singular divisor, of a singular binary instant (cf Figure 33), as: T = (t0 – T )1–0 Figure 44

We note that the proportional equality occasioning the condition normative of the translator (e.g. Figure 40) coincides with the determination of the (otherwise) indeterminable equality of proportion of the vertex and the exclusive limit of a subject memory. At any primary instant of the primary state, a singular condition normative of the translator is, presently and privately, informed with apodictic certainty by the prior determination of one of two mutually exclusive informants of a finite proportional equality, in accordance with: Rule 6: If 0 is T toward–1 then 0 = 1 ? In plain words, Rule 6 asks: if the determinant of the contradictory a primary state is equal to the primary state, then upon the occasion of its determination, a singular serially-horizontal translation is determined A Priori? 20

The Definition Of Information

2.3 Mutual Information In accordance with rule 6, whichsoever of a unary pair of determinants of singular binary instant may be the vertex of a subject memory, at a primary instant of the primary state, a condition normative of the translator is informed by the determination of the inequality of proportion of the vertex and the inclusive limit of a subject memory. Rule 6, therefore, posits a question informed by our secondary notation: if the equality of proportion of the vertex and the exclusive limit coincides with the primary instant of the primary state, then at what prior instant (i.e. when ad infinitum) could the proportional equality normative of the translator be informed in the first place? We note, however, that at any primary instant informed by our positive notation, the stasis (i.e. the condition of being static) of our positive notation, is informed without the determination of the primary state. Our positive notation being static A Priori therefore, we can exemplify every possible primary unilateral translation of a subject memory, the determination of one of which coincides with the stasis of the subject memory, e.g:

Figure 45

Our positive notation and the subject memory being static prior to the primary instant of the primary state (e.g. Figure 45), the only possible inequality of proportion informed without the determination of the primary state, is that of a primary unilateral translation, the determination of the moment of which is normative of the primary state, and (Lemma 5) a uniform unilateral translation of the divisor of the primary state, determined at the primary instant of the primary state. At which instant, the equality of proportion of the vertex and the exclusive limit of the subject memory, being indeterminable within the finite limit of inequality, a singular instant normative of the translator is informed with apodictic certainty in the order of the smallest possible inequality (see Figure 39) determined therefore by the exclusive limit of the subject memory, i.e: (t – t0 )0–1 > (t0 – T )0–1 Figure 46

An instant normative of the translator coinciding with an instant normative of the vertex of a subject memory, we will define as a datum (i.e. a unit of information) the smallest possible inequality normative therefore, of both the vertex of a subject memory and the translator. A datum, then, coincides with the instant of the recognition (i.e. secondary cognition) of a divisor of the primary state, upon the occasion of informing that position of the divisor in which it is static A Priori: the singularity of which position is priorly informed (e.g. Figure 45) for every possible position of the primary state, prior to the instant of its determination. At any primary instant of the primary state, a singular datum is informed by determining the position of one of a unary pair of divisors (e.g. Figure 43), of a singular divisor, of a singular binary instant. At which primary instant, the informant of that singular position being our positive notation, private information is defined by: that datum the present and private determination of which is normative of the vertex of the subject memory, according to: Lemma 7: Between any two mutually exclusive instants of a finite proportional equality, there is one and only one moment of translation. 21

The Definition Of Information

The order of the determination of a primary unilateral translation therefore, is defined by informing the position of the informant, of every possible position of a randomly-denotated primary state, i.e: (t0 – T )1–0 = (t0 – T )0–1 Figure 47

If, then, there is an occasion other than that of the subject memory, of a unilateral translation determined by our positive notation, the subject memory can predict (Rule 6), with apodictic certainty, that regardless of where and when that translation may be determined, the smallest possible inequality determined by the reader, will inform that singular dimension of the primary state we have defined as the translator. And if so, then any single putative reader may predict with mutual (i.e. shared by two memories) certainty, that where and when a primary instant of the primary state is informed by the subject memory, the direction of the translator will be toward the primary state. 2.4 Accidental Information According to Lemma 7, a singular moment of translation normative of the vertex of a subject memory is determined instantaneously, upon the occasion of informing one of two mutually exclusive instants of a determinable proportional equality. There is a question begged therefore, by our secondary notation: if an instant normative of the vertex coincides with a primary instant of the primary state, then at what prior instant (when ad infinitum) could an inequality of proportion be informed by the vertex? We have stipulated that the order of the alternation of a unary pair of mutually exclusive continuous states, cannot be described without generalizing both a primary state and its exclusive alternative: their alternation is exemplified prior to our symbolic representation of the order of their alternation. We note, however, that the singularity of a datum is attributable to the unconditional equality of proportion, of a divisor of the primary state and the position of the divisor of the primary state, prior to the instant of its determination. Whatsoever inequality of proportion may be informed at a primary instant of the primary state, is limited by the prior accidence (i.e. superimposition) of a unary pair of data (e.g. Figure 45) of perfect equality of proportion. If, in order to account for the order of mutual exclusion exemplified by translating our secondary notation (see Figure 47), we postulate (i.e. admit as necessary but not selfevident) that a datum normative of private information is informed by a singular divisor of the primary state prior to the instant of its determination, then the determination of the moment of a unilateral translation normative of the primary state, and the determination of the moment of a uniform unilateral translation normative of the vertex (e.g. Figure 45), coincide. In which case, if we particularize (i.e. reduce to a point) every possible primary unilateral translation presently and privately determined by a putative reader (e.g. Figure 45), we can generalize the translation per se of the divisor and the position of the divisor (the box and the content of the box), in the order of the recognition of the singular informant an accidental pair of data, at the primary instant of the primary state, i.e: t = (t0 – T )0–1 – (t0 – T )1–1 Figure 48 22

Accidental Information

Figure 48 describes a singular, uniform, unilateral translation determined instantaneously upon the occasion of the accidence of a unary pair of data of perfect equality of proportion, at a primary instant of the primary state. As it is self-evident that we cannot inform the stasis of our positive notation prior to determining a unilateral translation, we will so stipulate, that when an instant normative of private information is informed by translation, then it is informed by a singular divisor of the primary state prior to the determination of the primary state. In order to distinguish a datum normative of private information, from a datum mutually normative of the direction of translation, we will qualify the former as a primary datum. The occasion of private information being the determination of one of an accidental pair of primary data, the reader, whatsoever or whomsoever the reader may be, can predict, with apodictic certainty, that that which is determined at a primary instant of the primary state is the perfect equality of proportion (see Figure 47) of a unary pair of primary data normative of the condition of the translator (i.e. Rule 6). The equality of proportion of a unary pair of primary data normative of the condition of the translator, however, being determined prior to the primary instant of the primary state, we will define as natural intelligence: that which presently and privately determines the equality of two mutually exclusive divisors of a singular, extant and indeterminable inequality of proportion. The order of cognition, of the smallest possible inequality determined in the exclusive limit of a subject memory, being the contrary of the cognition an instant normative of the vertex (see Figure 48), the subject memory can predict that regardless of where and when, upon the occasion of informing the position of the vertex and the position of the divisor of the primary state, that which is presently and privately definitive of private information, is mutually definitive of the condition normative of the translator, in accordance with: Rule 7: If 0 = 1 then 1 / 1 is towardâ&#x20AC;&#x201C;1 ? In plain words, Rule 7 asks: if the divisor of a primary state is equal to the determinant of the divisor of the primary state, then upon the occasion of their alternation, a singular serially-horizontal translation is informed A Priori? And if so, then mutual information is defined A Priori by: that proportional equality the present and private determination of which is normative of the translator. 2.4.1 Published Information We have stipulated that one, of an accidental pair of primary data of perfect equality of proportion, upon the occasion of its determination without the determination of the denotated primary state, mutually informs a condition normative of the translator. But in order to exemplify the rule informed by Lemma 7, we have particularized every possible unilateral translation normative of the primary state, prior to informing an instant normative of the vertex of the subject memory. There is a question begged therefore by mutual information: how can the conditional equality described by Rule 7, of the primary state and the vertex, be informed by the determination of the equality of an accidental pair of primary data, the equality of which is normative of the vertex in the first place? We note, however, that while (i.e. unless or until) the primary state can be denotated as the greater of two unequal states, the particularization of a primary unilateral translation (e.g. Figure 45) identifies (i.e. makes identical in all respects but one) the 23

Accidental Information

divisor and the position it informs, prior to informing the position of the primary state, e.g:

Figure 49

And if the primary state can be denotated verbally prior to its determination, then so too can the position informed by the divisor of the primary state. We will, therefore, use a noun, 'dragon', to identify the divisor of the primary state exemplified by Figure 40, with the position it informs; and so denotate with the published word, the position of the primary state and the position (the box and not the content of the box) of the divisor of the primary state, e.g:

Figure 50

We note that upon the first occasion of identifying the position of a static divisor of the primary state, that which informs the position of the divisor and the primary state, informs the noun. In which case, and there having been no prior common (i.e. shared by more than two memories) identification of the divisor of the primary state, a singular instant normative of the noun coincides with a primary instant normative of the primary state. We invite any putative reader, therefore, to determine that position of the divisor normative of the noun 'dragon' without determining the primary state, by rotating our positive notation of the divisor (Figure 40) until informed by the position of the divisor exemplified in the following Figure (annotated 51):

Figure 51

As read by the subject memory (and so determining the need for the qualification 'putative'), Figure 51 exemplifies a position of the divisor which is not normative of a noun. At which instant it is evident to the reader (i.e. self-evident to the publisher), that the inequality of proportion, presently and privately informed by the position of the divisor of the primary state, is necessarily perfectly equal to the inequality of proportion priorly informed at the primary instant normative of the noun 'dragon': which observation may be verified (i.e. recognited with apodictic certainty) by continuing the logical rotation of our positive notation (e.g. Figure 51), without determining the primary state, until the position of the divisor of the primary state is informed by the recognition of a singular secondary instant definitive of the (we note the definite article) dragon therefore. And if so verified by the reader, then an instant definitive of the dragon, coinciding with an instant mutually normative of the vertex therefore, Figures 50 and 51 describe the positions of a unary pair of mutually exclusive and perfectly equal secondary instants of a singular, extant and indeterminable inequality of proportion, for every possible limit of the rotation of the primary state and the singular divisor of the primary state. There being, however, no possible limit of the rotation of the primary state but that privately informed by the recognition of its divisor, and there being, at any such instant, no 24

Accidental Information

discernible difference between the rotation of the primary state and the rotation of the singular divisor of the primary state, we will use the phrase rotational stasis to qualify that static condition, of the position of the primary state and the position of the divisor of the primary state, presently and privately informed at every possible instant of the recognition of any singular position informed by the divisor of the primary state. At the first moment of rotation determined by the reader at Figure 51, and there having been no prior mutual identification of the position informed at that instant by the divisor of the primary state, that which is presently and privately determined (by natural intelligence therefore), is the instantaneous accidence of a unary pair of mutually exclusive instants of perfect equality of proportion. If, then, we use the noun 'unicorn' to identify the position informed by the divisor of the primary state at the secondary instant exemplified by Figure 51, the reader may observe, by continuing the logical rotation of the divisor of the primary state, a unary pair of mutually exclusive primary data of perfect equality of proportion, the accidence of which is determined instantaneously at a moment of rotational stasis informed by the recognition of any singular secondary instant definitive of a noun. And if so, then the moment of rotational stasis definitive of a noun, coinciding logically with an instant normative of the noun, the order of its recognition coincides A Priori with the order of its cognition, i.e: (10)0–1 < (10)1–1 Figure 52

Accidental information then, is defined by: that which is presently and privately informed by the mutual identification of natural intelligence. The effect, however, of particularizing every possible unilateral translation normative of the primary state, is that we cannot predict (Rule 1), without presupposition and so with apodicitic certainty, that upon the occasion of informing a primary datum, the direction informed by the translator coincides with that direction of the combined uniform translation by means of which it is informed in the first place: and unlimited logical translation, if not toward the finite limit of inequality, is indeterminable, i.e: Rule 8: If 1 / 1 is toward–1 then t is toward–1 ? In plain words, Rule 8 asks: if the determinant of the prior continuity of an undetermined primary state is equal to a singular divisor of the primary state, then the moment of translation between them is indeterminable? 2.4.2 Common Information We have attributed the singularity of a primary datum mutually normative of a noun, to the determination of one of a unary pair of mutually exclusive primary data of perfect equality of proportion, in the order of the recognition of its exclusive alternative. The reader therefore may observe (e.g. Figures 50 and 51) that, at an instant normative of a noun, that which informs the noun depends upon the orientation of the divisor of the primary state, at a moment of rotational stasis informed presently, privately and exclusively by the reader. The conditional phrase of Rule 8 therefore, describes a question begged by the published word: if private information is defined by determining one of an accidental pair of mutually exclusive primary data at the primary instant of the primary state, then at what prior instant (when ad infinitum) could the orientation of an exclusive alternative of an 25

Accidental Information

extant datum, be mutually informed in the first place? For example, of what (if anything) is the following primary instant (annotated Figure 53) mutually normative but of the primary state?

Figure 53

In order to disclose the answer to the question, posited in Figure 53 by the primary state therefore, we invite the reader to determine the smallest possible unilateral translation of the divisor of the primary state exemplified by Figure 53, which would be required in order for the equality, of the divisor and the singular position it informs, to be measured by any means other than by the divisor per se. For example, as informed subjectively (i.e. presently and privately by this reader), the specified translation can be represented as one of either:

Figure 54

Following our convention of notating translation with the direction in which it is unlimited, we note that the specified translation t1 (e.g. Figure 54) may be determined at either one (but not both) of two points of transition (i.e. logical change) between the primary state and its divisor. There being two possible directions of the translation t1, we will notate as P that singular point of transition at which t1 is, presently and privately, determined by the reader, and its exclusive alternative as P'. The direction of the translation t1 determined by the reader at P is, of necessity and by definition, private information. Our positive notation being static A Priori, however, we will use the phrase straight line to denotate the smallest possible unilateral translation determinable between any two extant points of transition. The straight line between any two isologues of a singular position informed by our positive notation (e.g. Figure 54) exemplifies a singular unilateral translation, privately-determined in one of two possible directions of unilinear (i.e. logical and in a straight line) translation, i.e. either one of: P'  P = P  P' Figure 55

Regardless, then, of which of two possible points of transition may be notated P by the reader, the greatest possible finite unilinear velocity of the unilateral translation of a singular divisor of the primary state, can be represented (e.g. Figure 54) as either one of two accidental straight lines determined between two extant points of transition, i.e: P  P'  P Figure 56

And while it is self-evident that neither of the directions represented in Figure 56 is that of the finite unilinear velocity of the unilateral translation t1 exemplified in Figure 54, we can represent every possible direction of t1, by drawing an imaginable (i.e. conceivable but not extant) sphere centered at P', i.e. one of either: 26

Accidental Information

Figure 57

And if the reader can imagine such a sphere (Figure 57), then the reader may, presently and privately, verify (i.e. prove experimentally) that upon every possible occasion of determining t1 at P, the direction of t1 is informed at P', i.e: V1 = t1 â&#x20AC;&#x201C; P' Figure 58

And if so, then there being one (and only one) extant divisor of the primary state, the determination at P of the unilinear velocity V1 (Figure 58) exemplifies the transposition (i.e. exchange by translation), of a position informed A Priori by the divisor of the primary state, and the position of the divisor of the primary state informed A posteriori by memory, in one of two (and only two) possible directions of translation, i.e. one of either:

Figure 59

Having informed A posteriori the rotational stasis of the divisor of the primary state and memory at Figure 53, then upon the occasion of informing t1 unilaterally (e.g. Figure 54), a singular point of transposition P'' (e.g. Figure 59), of the divisor of the primary state and the position it informs, is determined in a straight line between two centres of the rotational stasis of our positive notation, i.e: P' ď&#x201A;Ž P'' = P'' ď&#x201A;Ź P Figure 60

The unilateral translation t1 being informed without subjective rotation, Figure 60 generalizes a unilinear difference, between the subjective centre of the rotational stasis of our positive notation, and the point of transition P determined, but not informed, at P. And although the direction of t1 is not represented in Figure 60, we can represent the said unilinear difference by drawing the great circle of an imaginable sphere centered (e.g. Figure 59) at P', i.e. either one of:

Figure 61

There having been no prior mutual identification of the position informed by the divisor of the primary state (e.g. Figure 53), Figure 61 generalizes the greatest possible 27

Accidental Information

unilinear difference determinable at the point of transposition P'' of two perfectly equal positions informed by unilateral translation without subjective rotation, as the radius r of an imaginable sphere of diameter d (see Figure 57), i.e: V1 â&#x20AC;&#x201C; d = r Figure 62

Having, then, determined the unilinear velocity V1 at P, the smallest unilinear translation necessary to measure the perfect equality of an accidental pair of positions informed A posteriori without subjective rotation, can be represented as one of either:

Figure 63

We have stipulated that no two determinable instants of a singular and priorly continuous state can be of perfect equality of proportion, unless both are divided by a singular, static, isolated state which is not an isologue of the limit it determines. We note, however, that the point of transposition P'' (e.g. Figure 63) and the subjective centre of the rotational stasis of the singular divisor of the primary state, define the greatest possible straight line determinable between the point of transition P' and the said centre. The reader, then, having randomly chosen one of two possible points P of transition at which to determine the unilinear velocity V1 (i.e. Figure 54), may inform that unilinear difference (see Figure 62) between two positions determined by their accidence, by rotating the singular divisor of the primary state (e.g. Figure 63, whichsoever it may be), about the subjective centre of the rotational stasis of memory and the primary state, until the accidence of P' and P in straight line (see Figures 60, 61) is determined by the recognition of a primary datum mutually normative of a noun, i.e: r = t1 â&#x20AC;&#x201C; d Figure 64

At which instant (e.g. Figure 63) the mutually-informed centre of the rotational stasis of our positive notation and memory, and the point of transposition P'' of two perfectly equal positions, are unconditionally accidental, i.e:

Figure 65

We invite every possible reader therefore, to inform the greatest possible unilateral translation mutually determinable instantaneously in a straight line, by rotating a primary datum mutually normative of a noun in Figure 65, about our mutually-informed centre of the rotational stasis of memory and the primary state, and logically in either one of two possible directions of rotation, until a position priorly and privately informed without 28

Accidental Information

rotation (e.g. Figure 53), is informed by the accidence in straight line, of our common point of rotation, and the point of transposition P'' of a binary pair of mutually exclusive positions of perfect equality of proportion, e.g. one of either:

Figure 66

At which instant (Figure 66) the unilinear difference (see Figure 64) between our common point of rotation and the common point of transposition P'', being equal to the unilinear difference between the said centre and the point of transition P, the greatest possible finite unilateral translation determined instantaneously by the reader, is represented by a straight line drawn on the surface of an imaginable sphere of radius r, e.g. one of either:

Figure 67

In order, therefore, to distinguish instantaneous unilateral translation in a straight line (e.g. Figure 67) from unilinear velocity (e.g. Figure 57), we will define as distance: that finite logical translation in a straight line necessary to determine the extant equality of two positions of the primary state - so that we may mutually-inform the difference between the subjective limit of instantaneous unilateral translation in a straight line, and unilinear translation per se. The greatest possible distance determined instantaneously between the points of transition P' and P (e.g. Figure 67), and regardless of the direction of rotation in which the accidental equality of a unary pair of mutually exclusive primary data may be informed, is tautologically equal to the semi-circular arc c of the great circle of an imaginable sphere of radius r. And our common point of rotation being static A priori, then the smallest possible unilinear difference between two positions of perfect equality, is unconditionally equal to the radius r of a virtual (i.e. of imaginable dimensions) sphere, determined instantaneously between our common point of rotation and the common point P'' of their transposition. And we note that while the invention of a virtual sphere breaches the rules of evidence with which we began our inquiry, the unconditional inequality of two finite translations does not. If, then, we use the colon to represent that inequality, the subjective limit of instantaneous unilateral translation in a straight line, is described by the ratio (per se) commonly symbolized as , ie: c:r =  Figure 68

In order to distinguish the finite limit (Figure 68) of instantaneous unilateral translation in a straight line, from the conventional definition of  and the empirical accuracy with which  may be measured when expressed as a number, we will define the former as the speed S of the instantaneous unilinear transposition, of two positions of an extant equality of proportion, as informed by a singular divisor of the primary state when determined in either one of two possible directions of unilateral translation (e.g. Figure 54) 29

Accidental Information

without subjective rotation, i.e: S = t1 : r Figure 69

We have, however, defined unilinear distance as that logical translation in a straight line necessary to determine the perfect equality of two positions; and the perfect equality of two positions informed by unilateral translation, is determined instantaneously (e.g. Figure 63) at the subjective centre of the rotational stasis of memory and our positive notation. The finite unilinear difference, between the subjective centre of the rotational stasis of memory and our positive notation and either one the points of transition P or P', being equal (see Figure 64) to the unilinear difference between our common point of rotation and the common point of transposition P'', then upon every possible occasion of informing the static accidence, of two positions of a finite equality informed by a singular divisor of the primary state, the smallest possible distance normative of the unilinear velocity V1, is tautologically equal to the combination of a distance dx equal to the diameter of an imaginable sphere of radius r, and the unilinear difference dy equal to r, determined in a straight line between the point of transition P or P', and the subjective centre of the rotational stasis of our positive notation, e.g. one of either:

Figure 70

We will, therefore, deploy the symbol + commonly used to represent addition (i.e. combination), to notate the subjective centre of the rotational stasis of memory and our positive notation, and so generalize the sum (i.e. combination) of two distances commonly normative of the translator prior to the mutual identification of natural intelligence, informed by the instantaneous accidence of the stasis A Priori of a singular the divisor of the primary state, and a perfectly equal position informed A posteriori by memory, i.e: V1 = dx + dy Figure 71

Regardless of the shape, form or nature of the exclusive limit of a subject memory, at any extant instant privately recognized, by virtue of prior experience, without subjective rotation, the distance determined (but not informed) at the subjective centre of the rotational stasis of memory and the primary state (e.g. Figure 70), defines the instantaneous unilinear velocity of a singular divisor, of two perfectly equal positions of a singular primary state, ie: V0 = dx â&#x20AC;&#x201C; dy Figure 72

And if so (Figure 72), then the reader may verify experimentally that there is one (and only one) moment of the rotation of our positive notation and memory, at which the combination, of the unilinear velocity V1 and the instantaneous unilinear velocity V0, can be determined in a straight line, e.g: 30

Accidental Information

Figure 73

And if so, then prior to our mutual identification of natural intelligence, and at the first possible instant privately-informed by the recognition of that primary datum (e.g. Figure 53) the position of which is informed A posteriori, the distance determined (e.g. Figure 57) between a binary pair of perfectly equal divisors, of a singular divisor (the primary state) of a singular binary instant, is unconditionally equal to the distance informed by the sum of: V0 + V1 = dx Figure 74

We note, however, that when determined at the subjective centre of the rotational stasis of our positive notation without subjective rotation (e.g. Figure 73), V1 and V0 are logically accidental, ie: V0 â&#x20AC;&#x201C; V1 = dx Figure 75

Upon every possible occasion of recognizing a primary datum the position of which is priorly informed without subjective rotation, the greatest possible combined uniform translation of memory and our positive notation, is unconditionally equal to the distance determined at the subjective centre of the instantaneous logical rotation of a singular divisor of the primary state, i.e: T = V0 + V1 / V0 â&#x20AC;&#x201C; V1 Figure 76

Which translation (Figure 76), being equal to the distance informed (e.g. Figure 67) by the instantaneous transposition of two positions of an extant equality of proportion, a datum (we note the singular) privately normative of both the translator and the vertex of memory, is predicated (i.e. informed with apodictic certainty) by a singular dimension of a primary datum informed A posteriori without subjective rotation, e.g. one of either:

Figure 77

There being no possible occasion of the instantaneous logical rotation of our positive notation but that of a subject memory, in order to distinguish the instantaneous logical rotation of our positive notation, from the moment of the rotational stasis of memory and our positive notation, we will define the former as logical spin. Prior to the first possible occasion of the mutual identification of natural intelligence, the logical spin of our positive notation about a common point of rotation is determined by the recognition of the instantaneous accidence, of two positions of an extant equality priorly informed at the common point of their transposition. 31

Accidental Information

While the moment of the rotational stasis of memory and our positive notation therefore, cannot be privately-informed prior to the determination of the primary state, nor mutually-informed by determining the primary state, we can generalize the distance between two perfectly equal positions of memory and the primary state at the common point of their transposition (e.g. Figure 70), as the sum of two perfectly equal unilinear differences (see Figures 71, 72, 73), i.e: dx = (dz – dx) + (dz – dx ) Figure 78

And while the direction of translation, in which the speed S of the instantaneous unilinear transposition of two equal positions is determined by their accidence, is unconditionally private information, by virtue of the conventions we have adopted (e.g. Figures 11, 12, 13) we can generalize both the distance (Figure 78) and direction of translation per se at the moment of their instantaneous accidence, i.e. where I represents the instant informing two equal positions of a common divisor of the primary state, the distance informing the translator may be described by the generalized sum† of two perfectly equal unilinear differences, determined instantaneously toward–1, i.e: d1 =

 (10  1)

I 0

Figure 79

Our positive notation being static A Priori, and the divisor of the primary state (e.g. Figure 53) being self-evidently not an isologue of the limit of the speed S (see Figure 69), the publisher can predict with apodictic certainty that, upon every possible occasion of the determination without subjective rotation, of the equality of two equal positions, a distance unconditionally and self-evidently greater than d1 (Figure 79) will commonly-inform the orientation of a singular divisor of the primary state, e.g:

Figure 80

The reader, then, may verify that upon the occasion of informing the greatest possible distance between two points transition defined by the static accidence of a binary pair of positions (e.g. Figure 54), the logical spin of the translator is privately-informed by a single dimension equal to d1, of a common divisor of the primary state, which dimension therefore, we will characterize the vertex of a common divisor of the primary state, according to: Lemma 8: When V1 is toward–1 then d1 is toward–1. In plain words, Lemma 8 states: the singular moment of translation between any two instants of a finite equality, is continuously equal to the unilinear velocity determined in the limit of their instantaneous accidence. †

Stibbe, Shirleen (1989) 'Exponential Sums with Generalized Divisor Functions' - PH.D. Thesis (Royal Holloway and Bedford New College, University of London). 32

Accidental Information

2.4.3 Natural Data According to Lemma 8, in the limit of the instantaneous accidence of two instants of a determinable equality, a singular (Lemma 7) and continuous moment m of translation towardâ&#x20AC;&#x201C;1 is equal to the continuity in a straight line, of the combined unilinear velocity V1 privately-determined randomly by the reader at one of two possible points of transition P, and the unilinear velocity V0 determined at the subjective centre of the logical spin of the translator, i.e: m = V0 + V1 Figure 81

There is a question begged therefore by private information: if, prior to our mutual identification of natural intelligence, the informant of the translator is a continuous difference determined instantaneously between two points of transition, then at what prior instant (i.e. when ad infinitum) could that direction of V1 we have defined as the singular property of memory, be privately-informed in the first place? We have, however, already stipulated that the stasis A Priori of our positive notation cannot be informed A posteriori without the occasion of unilateral translation. The determinant of a primary unilateral translation t0 (see Figure 24) occasioned by the reader being a singular common divisor of the primary state, then upon each and every occasion of informing the stasis of our positive notation, the greatest possible distance determined at the speed S, of the instantaneous unilinear transposition of two positions of a finite equality, can be represented as the radius equal to d1 (see Figure 79) of the great circle of a virtual sphere, e.g. one of either:

Figure 82

The determination of a unilateral translation t0 (e.g. Figure 82) coinciding then, with the subjective centre of the rotational stasis of memory and our positive notation, the smallest possible distance determined by the instantaneous unilinear transposition of two positions of the primary state, is unconditionally greater than d1, e.g:

Figure 83

We will, therefore, define the smallest possible laterally-horizontal distance (see Figures 11, 12) determined instantaneously at the common point of transposition (e.g. Figure 83) of two positions of a finite equality, as the horizon of memory. We invite the reader, then, to inform the smallest possible difference, between the subjective centre of the logical spin of the translator (e.g. Figure 83), and the transposition of two positions of the primary state, by rotating the vertex of memory, logically about our common point of transposition, in the only possible direction in which a laterally-horizontal distance d0 greater than d1 can be mutually-informed without the subjective rotation of the vertex of a 33

Accidental Information

common the divisor of the primary state, i.e:

Figure 84

And although there are two possible directions (e.g. Figure 84) in which the logical spin of the translator may be informed, the reader may verify that, upon the occasion of informing a primary datum mutually normative of a noun, the rotation of the vertex of the common divisor is momentarily opposed to the logical rotation of the vertex of memory? And if so, then we will characterize the direction of that momentary opposition as counterlogical, i.e: S â&#x20AC;&#x201C; (S â&#x20AC;&#x201C; 1) = 1 â&#x20AC;&#x201C; 0 Figure 85

The representation therefore, of a singular instant at which laterally-horizontal translation is privately informed without subjective rotation, requires the counterlogical rotation of the common divisor of the primary state exemplified by Figure 84, until the combined speed (Figure 85) of two perfectly equal and opposite unilinear translations, is determined at the subjective centre of the logical spin of the translator, e.g:

Figure 86

While the moment of the logical spin of the translator cannot be privately-informed therefore, the publisher can predict, with apodictic certainty, that upon every possible occasion of the determination of a primary translation t0, a distance equal to or greater than d0 will be commonly-informed in the limit of the speed S of two perfectly equal and opposite unilateral translations, e.g. one of either:

Figure 87

And if so, then the reader may observe A posteriori (e.g. Figure 87), by determining the greatest possible singular unilateral translation between two points of transition, the counterlogical rotation of our mutual horizon, informed instantaneously by a unary pair of primary data of perfect equality. While the moment of the counterlogical spin of our mutual horizon cannot be privately-informed therefore, the distance determined in the limit of the finite accidence of two mutually exclusive instants is, self-evidently and necessarily, equal to or greater than the distance d0 (e.g. Figure 86) priorly informed instantaneously in the limit of the unilinear transposition of two positions of perfect equality. Which difference of proportion, being informed at the singular centre of the logical spin of the translator and the counterlogical spin of our mutual horizon, is, by virtue of our secondary notation, commonly represented symbolically by the generalized sum of: 34

Accidental Information

d0 

 (10)

I 0

Figure 88

Prior to (e.g. Figure 17) our mutual identification of natural intelligence then, and there being no other occasion for the subjective rotation of the vertex of memory, the subjective accidence of our mutual horizon and the vertex of a common divisor of the primary state, privately-informs two unconditionally unequal dimensions d1 and d0 of a singular divisor of the primary state, e.g:

Figure 89

Which dimensions (e.g. Figure 89) therefore, commonly-inform the orientation of a unary pair of primary data, at the first possible instant of the mutual identification of natural intelligence. We have defined as natural intelligence, that which presently and privately determines the proportional equality of two mutually exclusive divisors of a singular, extant and indeterminable inequality of proportion. We will, therefore, define our mutual horizon and the vertex of a singular common divisor of the primary state, the rotational stasis of which commonly determines the orientation of one of a unary pair of primary data, an accidental pair of natural data. The first possible instant mutually normative of a noun, then, is commonlyinformed without subjective rotation, by an accidental pair of natural data. We note therefore, that a primary datum commonly normative of a noun, having been be mutuallyidentified once (i.e. on one and only one prior occasion), is thereafter (i.e. post to) the occasion (cause) of the accidence of a pair of natural data. In order to distinguish a noun mutually-identified with a primary datum the orientation of which is informed by an accidental pair of natural data (e.g. Figure 50), from a noun mutually identified with a primary datum the orientation of which is informed by the subjective rotation, of a accidental pair of natural data priorly occasioned by a primary datum normative of a noun (e.g. Figure 51), we will qualify the former as naturallyinformed. A naturally-informed noun, then, is commonly identified at a singular instant, at which the perfect equality of two positions of the primary state informed by unilateral translation, is determined by an accidental pair (e.g. Figure 89) of unconditionally unequal dimensions, of a singular position of the primary state, i.e: 10 – 11 = 0 – 1 Figure 90

At any instant of the mutual identification of natural intelligence with a naturallyinformed noun therefore, that which is normative of common information is the orientation of a singular primary datum, informed by an accidental pair of natural data without subjective rotation, in accordance with: Rule 9: If 1 – 1 = 0 then 10 – 1 = 1 ? In plain words, Rule 9 asks: if the informant of a primary state is equal to the determinant of its prior continuity, then the distance and direction in which the primary state is informed is equal A posteriori to the distance and direction in which it is determined? 35

Accidental Information

There is a question begged therefore by common information: if, at every possible instant mutually normative of a naturally-informed noun, the perfect equality of two positions of a singular primary state (Rule 9) is determined by a singular position of the primary state, then how can the equality predicating the formulation of our secondary notation (i.e. Figures 79, 88, 90) be informed in the first place? 2.4.4 Natural Information According to our emerging model of information, equality per se is privatelyinformed in the first place, by the contradiction of two mutually exclusive divisors of a singular binary instant. We have, however, stipulated that an instant privately normative of the primary state is informed prior to the determination of that state. Having defined memory as the ability to determine a priorly continuous state, we invite the reader to inform that equality of proportion predicating the formulation of our secondary notation, by determining the primary state exemplified by Figure 53, e.g. (and bearing in mind our caveat on outer borders):

Figure 91

Figure 91 is intended to exemplify the superimposition of that singular and continuous state presented privately, as for Figure 2, by the reader, upon a position of the primary state informed (e.g. Figure 53) by a singular common divisor at a singular primary instant normative of the primary state. On this occasion, however, there being A posteriori no discernible difference, between the determinant of the primary state and the informant of the primary instant thereof, the isologue, of the singular position in which two mutually exclusive states may alternate, is provided A Priori by the primary state. Which position was priorly informed, by the determination of the contradictory of the primary state in the order of cognition described by Figure 79. But on this occasion (Figure 91), the position of the primary state has been informed by the instantaneous determination of one of two perfectly equal positions of the primary state, i.e. by natural intelligence. Upon the occasion of the determination of that position, therefore, the inclusive state presented by the exercise of memory, is necessarily greater than the position it determines. The contrary of 0, being lesser than the contradictory of 1 priorly determined at a singular primary instant normative of the primary state, then at the moment of its determination, the direction informed by the translator (Rule 5) is toward–1. But in accordance with Rule 8, the continuous moment of unilinear translation toward–1 is indeterminable. At any such instant we may inform two equal and opposite translations toward–1 and so predict the equality of 1 and 0, or we may inform the equality of 1 and 0 and so predict two equal and opposite translations toward–1. The question begged by common information is how can we inform both? The answer is to be found in the artifice (i.e. drawing) of Figure 53, by virtue of which the accidence of a pair of natural data informed at a primary instant of the primary state, is determined at a singular, common point of rotation on our mutual horizon (e.g. Figure 91). Having identified the translations t0 and t1 with the directions in which they are unlimited, we will extend that convention to the contrary (Rule 4) of 0, and so define a 36

Accidental Information

plane as: that which is presently and privately informed by the determination of an accidental pair of natural data and a singular common point of rotation. Upon the occasion of informing, by determining the primary state, the inequality of two planes of translation, that which is privately-informed is the plane in which the translation t1 was priorly limited in either one of two possible directions of unilateral translation (e.g. Figure 54) by the logical spin of the translator, i.e. the plane of t1 in which the priorly continuous unilinear velocity V1 was informed, randomly, by the reader: m1 = t1 + t1 Figure 92

The velocity V1 normative of the translator being unconditionally private information, in order to mutually-inform the equality of 1 and 0 we must, of necessity, ask the reader to inform Rule 2 in the absence of a common divisor of the primary state, the conditions of which are difficult to configure. But there being, prior to the determination of the plane of t1, no possible informant of a change of the position of memory, we will represent the effect of memory in the absence of a common divisor of the primary state, with the counterlogical spin of our mutual horizon, about the point of origin of the primary state informed subjectively without a change position, and prior to informing the instantaneous transposition of two equal positions of the primary state, e.g:

Figure 93

In accordance with our convention, Figure 93 exemplifies the accidence of the contraries of the planes of t1 and t0, in the order of cognition represented by the formulation of our secondary notation in Figure 90. The reader, then, may verify that in the continued absence of a common divisor of the primary state, and unilateral translation in the plane of t0 being unlimited in every possible direction of translation, the planes of t1 and t0 limited by the counterlogical spin of our mutual horizon, are perfectly equal, e.g:

Figure 94

While the equality of the planes of t1 and t0 (Figure 94) cannot be determined, and every possible direction of translation being tautologically towardâ&#x20AC;&#x201C;1, unilateral translation towardâ&#x20AC;&#x201C;1 is continuous in two unconditionally incidental (i.e. accidental and coincident) planes. Prior to informing the plane of t1, and without the subsequent determination of the primary state, the incidence of unilateral translation and the moment of continuous unilinear translation, is unlimited unless or until determined by an occasion of the stasis of a subject memory, e.g: 37

Accidental Information

Figure 95

While Figure 94 exemplifies every possible point at which, prior to informing the translator, unilateral translation in two incidental planes may be determined, Figure 95 exemplifies the point, at which the determination of one of two incidental planes, informs a singular change (Lemma 1) of the direction of continuous translation per se, from that of unilaterally toward–1, to that of the particular moment (Figure 95) of continuous unilinear translation toward–1. At any such point, informed therefore (see Figure 21) in the contrary of the order of cognition represented by the formulation of our secondary notation in Figure 90, the perfect equality of an incidental pair of planes of translation is extant (Figure 95) A Priori, i.e: 11 – 1 = 10 Figure 96

If, then, Figure 95 exemplifies the subjective centre of the moment of the rotational stasis of memory and the primary state, and there being no occasion to inform the logical spin of the translator, we can represent the counterlogical spin of our mutual horizon in the plane of t0, with any circle greater than a point we care to draw, e.g:

Figure 97

The limit therefore, of the greatest possible moment of continuous unilinear translation in the plane of t1, is defined tautologically by the ratio of the circumference C and the radius r0 of the great circle, of virtual sphere determined at the particular moment (e.g. Figure 95) of continuous unilinear translation toward–1, i.e: m1 = C : r0 Figure 98

Prior to informing the unilinear velocity V1, and so prior to the first possible instant normative of translation per se, the greatest possible finite unilinear translation in the plane of t0 can be represented on our mutual horizon at any point (e.g. Figure 95) at which a singular change of the direction toward–1 informs the subjective centre of the rotational stasis of the primary state, e.g:

Figure 99 38

Incidental Information

At which point (e.g. Figure 99) therefore, the greatest possible finite straight line in any single plane of unilinear translation is tautologically equal to the diameter d0 of the great circle of virtual sphere of radius r0, e.g. one of either:

Figure 100

The greatest possible finite unilinear translation toward–1 particularized at the subjective centre of the rotational stasis of memory and the primary state, is equal to d0? And if so, then the limit of the combined uniform translation T of a subject memory and the primary state, is equal to the sum of two unconditionally equal, opposite and incidental unilinear translations determined in the plane of t1, e.g:

Figure 101

And while we cannot, prior to determining the primary state and in the absence of a common divisor of the primary state, inform the plane of t1, by virtue of our secondary notation, the generalized sum of two perfectly equal unilinear differences determined toward–1 (e.g. Figure 101), is commonly informed (see Figure 79) by the incidence of a binary pair of positions of perfect equality, i.e: binary 10 = r0 + r0 Figure 102

In the limit (see Figure 98) of the particular moment of continuous unilinear translation toward–1, the distance determined (but not informed) by unilinear translation in two planes, is exemplified by the sum, of the combined uniform translation of memory and the primary state, and a singular primary translation t0, e.g. one of either:

Figure 103

And while we cannot inform the particular moment of continuous unilinear translation toward–1, the greatest possible incidental unilinear translation determinable in any two planes, is described tautologically by the proportional difference (cf inequality of 39

Incidental Information

proportion) of two unconditionally unequal distances, i.e: t0 : T = 1 : binary 10 Figure 104

When, therefore, a primary unilateral translation is determined on our mutual horizon at the common point of transposition of two positions of perfect equality (e.g. Figure 70), then in the order represented by the formulation of our secondary notation in Figure 90, a singular change of direction at that point, defines the moment of the first possible cognition of a translation per se, e.g. one of either:

Figure 105

At which common point of transposition (Figure 105), and prior to the first possible instant normative of private information, translation per se is unconditionally incidental in two planes of unilinear translation, ie: t = t0 Figure 106

The incidence of t and t0 (Figure 106) being determined by translation, the first possible translation normative of private information is necessarily that of a singular divisor of the primary state; and in accordance with Rule 1, the sum of two distances commonly normative of the translator (see Figure 71), cannot be informed without a change of the position of memory. We note, however, that in the case of the particular moment m1 of continuous unilinear translation in the plane of t1 (see Figure 98), the stasis A Priori of our positive notation is informed A posteriori (e.g. Figure 97), by the determination of a primary unilateral translation we have attributed to the reader, at a common point of rotation represented artificially by our positive notation, i.e: m0 â&#x20AC;&#x201C; m1 = m1 Figure 107

Which particular case (Figure 107) coinciding with the determination of a primary unilateral translation t0 (e.g. Figure 95), and as we can represent the counterlogical spin of our mutual horizon with any circle greater than a point we care to draw (e.g. Figure 105), then whensoever determined at the common point of transposition of two perfectly equal positions, the combined uniform translation T of memory and the primary state, is tautologically equal to the sum of two perfectly equal and opposite translations, in two unconditionally incidental planes of unilinear translation, i.e: m0 = m1 + m1 Figure 108

That which predicates common information therefore (Rule 9), is the stasis A posteriori of a complementary (i.e. adding to a singular) pair of incidental planes of 40

Incidental Information

unilateral translation, limited by the counterlogical spin of our mutual horizon about a singular, common point of rotation. At any such point informed in the order of the cognition of translation per se (e.g. Figures 93, 105, 97, 95), the greatest possible finite unilinear translation in the plane of t0, is described tautologically as the radius r0 of the great circle of a virtual sphere of diameter d0, i.e: r0 = d0 – r0 Figure 109

And although the counterlogical spin of our mutual horizon in a single plane, can be represented artificially in only one of two incidental planes of unilinear translation, our positive notation (e.g. Figure 97) is nonetheless extant. We will, therefore, define as incidental information that which is informed by any singular change of the direction of continuous unilinear translation toward–1. The distance equal to d0 (see Figure 88) between an incidental pair of perfectly equal positions of the primary state therefore, is described by the sum of the unilinear differences of two incidental translations in two planes of unilinear translation (see Figure 103) when determined by the particularization of continuous unilinear translation in the plane of t1, according to: Lemma 9: The binary complement of 1 is 0 – 1. We cannot express Lemma 9 with our secondary notation, because that which it informs (i.e. the condition of 0 = 1 – 1 predicted by Rule 9) is, by definition, not extant at any instant normative of that priorly continuous state we have notated 1. Nor can Lemma 9 be expressed without verbal qualification, because without the adjective binary the distinction, between an accidental pair of unilinear differences normative of the translator (see Figure 79), and the particular sum of two perfectly equal unilinear differences determined in any single plane (see Figure 102), is lost. The equality predicating the formulation of our secondary notation, however, being that of an incidental pair of perfectly equal positions of the primary state, that which is predicated by the determination of incidental information is, by definition, natural information, e.g:

Figure 110

2.5 Incidental Information Figure 110 is intended to illustrate that the moment m0 of incidental unilinear translation in two planes, is a special condition presented by the medium of our inquiry, and not, therefore, natural information. There is a question begged therefore, by the order of our secondary notation: how can that equality predicted by Rule 9 be commonly informed by our positive notation, prior to informing the equality of two non-incidental positions of the primary state? We may, however, mutually observe (e.g. Figure 110) that a primary datum naturally-normative of a noun, is privately-informed prior to the instant at which it is 41

Incidental Information

mutually identified with a noun: the quality of plurality (per se) extant in the instantaneous accidence of a unary pair of mutually exclusive primary data, is presented naturally prior to the determination of the primary state, by the primary state and its isolated divisor. Which divisor being singular at every possible primary instant of the primary state, we will nominate (i.e. name) the first datum, of any unary pair of mutually exclusive primary data extant in Figure 110, that may be privately determined by the reader at the moment m0 of their instantaneous accidence, 'one'. Upon the occasion of determining one of a unary pair of mutually exclusive primary data exemplified in Figure 110, the plurality of 'two' is informed by the mutual identification of natural intelligence with a singular noun. In order, then, to distinguish the singularity of a noun (e.g. 'two') from the plurality (we note the definite article) it nominates, we will qualify such nouns with the adjective simple (i.e: not composite in respect of parts). And the reader may verify that the extant plurality of mutually exclusive states commonly-informing Lemma 3, may be mutually identified with the simple noun 'two', prior to the determination of the primary state. While the exclusive limit of a subject memory cannot, without the determination of the primary state, be informed privately, we note that when we represent the great circle of a virtual sphere artificially (e.g. Figure 97), the plane of t0 is not merely represented by our positive notation, but is presented by the medium of our inquiry. Within which special conditions, and when the moment m0 of incidental unilinear translation in two planes is determined in the plane of t1 (see Figure 107), the dimensions of any two planes of unilinear translation limited by the counterlogical spin of our mutual horizon (e.g. Figure 103), are described A posteriori by our secondary notation in the finite ratio, of: binary 11 : binary 10 Figure 111

Our mutual horizon being one of every possible pair of accidental natural data, and having attributed the limit of the smallest possible finite inequality of proportion (cf proportional difference) to the private and exclusive limit of the memory of the reader (see Figures 32 and 33), we will define the plane of our positive notation and the exclusive limit of memory, when artificially informed at the moment m1 of continuous unilinear translation towardâ&#x20AC;&#x201C;1 (i.e. Figure 95), an incidental pair of virtual (i.e. effective by virtue of natural information) data. There is a question begged therefore by the medium of our inquiry: how can the plurality (see Figure 111) normative of the order of two planes of unilinear translation, be commonly-informed prior to the mutual identification of a primary datum naturally normative of a singular noun? To which the simple answer is that it cannot. We note, however, that within the special conditions of the medium of our inquiry, translation per se is the private property of the reader, and upon any occasion of the stasis of a subject memory that is greater than a point (e.g. Figure 110), an accidental pair of natural data is predicated by the determination of an incidental pair of virtual data, e.g:

Figure 112 42

Incidental Information

2.5.1 Natural Division The question begged by the order of our secondary notation, of how that equality generalized by the linguistic qualification (binary) of our secondary notation may be commonly informed, is answered by comparing Figures 110 and 112: our positive notation being lesser than the plane in which it is presented, and the plane in which it is presented being a property of the medium of our inquiry, the proportional difference of two unconditionally unequal distances (see Figure 104) is, by virtue of the medium of our inquiry, commonly presented (e.g. Figure 112) in the indeterminable ratio of: m0 + m1 : m1 Figure 113

Whence (Figure 113) the question begged by the medium of our inquiry: the order of the plurality of an incidental pair of virtual data, cannot be informed prior to our mutual identification of the plurality of two mutually exclusive primary data, in the order of cognition (see Figure 90) of a unary pair of primary data naturally-normative of a simple noun. We have, however, defined the vertex of a singular common divisor of the primary state as the lesser, of an accidental pair of natural data predicated by the determination of incidental information. At every possible instant normative of private information, the vertex of a common divisor of the primary state is the scalar (i.e. informant) of the inequality of two dimensions, of a subjectively static plane of translation (e.g. Figure 112) presented by our positive notation, e.g:

Figure 114

By virtue of the prior and mutual identification of a primary datum naturallynormative of the noun 'dragon' therefore, the reader may observe in Figure 114, by rotating the vertex of the common divisor of the primary state, experimentally in either one of two possible directions of logical rotation, that there is one (and only one) direction of logical rotation in which a singular instant naturally-normative of a noun (see Figure 88) can be informed: the singularity of which instant is extant in the perfect equality, of a unary pair of mutually exclusive primary data unified (i.e. made singular) instantaneously, by their combined rotation about a point commonly-informed in the greater plane (see Figure 100) of a virtual horizon, the orientation of which horizon is priorly-informed by the borders of a page of our inquiry, e.g:

Figure 115

And if so, then by informing the common point of rotation of a unary pair of mutually exclusive primary data, the reader may recognize a primary datum normative of 43

Incidental Information

the simple noun 'two', but not normative of any other noun: which non-verbal recognition, of an instant privately-normative of two therefore, we will characterize as the specialization of a singular primary datum. At any instant of the specialization of a primary datum (e.g. Figure 53), and so at the first possible instant normative of private information, the quality of plurality is extant in the subjective moment of the rotational stasis of the primary state and a singular divisor of the primary state? And if so, then regardless of whether the primary unilateral translation informing the quality of plurality is the property of memory, or of the divisor of the primary state, or of both, a division of the primary state is informed by the incidence, in the plane of our positive notation, of a pair of asymmetric (i.e. recognizably different) planes of translation per se. Upon the occasion, and whatsoever that occasion may be, of a primary instant naturally-normative of a specie (i.e. that which is specialized) of our positive notation, the quality of plurality informed by the instantaneous accidence of a unary pair of mutually exclusive primary data, presents a complementary pair of asymmetric planes of unilinear translation. We invite the reader, therefore, to inform the scalar of the unconditional inequality of an accidental pair of natural data exemplified in Figure 53, by rotating the common divisor of an incidental pair of virtual data, logically and in the direction privatelyinformed by the translator, about the common point of rotation of a unified pair of mutually exclusive primary data, until the subjective rotational stasis of the primary state and the said divisor, is informed by the recognition, of a singular primary datum specialized privately, prior to the mutual identification of natural intelligence, e.g. one of either:

Figure 116

There being no possible natural determinant of the logical spin of the translator (e.g. Figure 34), the reader may observe experimentally (e.g. Figure 114) that the combined rotation of a unified pair of primary data, is determined instantaneously (e.g. Figure 115), by the counterlogical rotation of a virtual horizon in the presented plane of our positive notation. The presented plane being lesser than the exclusive limit of memory, the greatest possible finite unilinear distance, limited therefore by an incidental pair of virtual data, is represented at the common point of rotation of a unified pair of primary data, by the diameter d0 lesser than binary 11 (see Figure 88), of a circle determined (but not informed) in the presented plane of our positive notation, e.g:

Figure 117

As we have defined that which is informed by the determination of incidental information as natural information, Figure 117 represents the natural division of the plane of our positive notation, by one of two recognizably-different, but perfectly equal, complementary pairs (we note the plural) of asymmetric planes of unilinear translation. 44

Incidental Information

The medium of our inquiry, therefore, exemplifies two asymmetric models of translation per se in two unconditionally unequal dimensions of the single plane of our positive notation, each of which models is scaled by a virtual datum and a natural datum, e.g:

Figure 118

Our caveat on outer borders is that they are there only to distinguish the content of the box from the rest of the page: the limits of the virtual data represented in Figure 118 are those of the single plane in which they are drawn, and necessarily, therefore, lesser than the exclusive limit of memory. But at every possible instant artificially-normative of common information, the division of two planes of unilinear translation, by a laterallyhorizontal translation occasioned unilaterally by the reader, is normative of 'more' than two planes of translation per se? And if so, then regardless of what the finite unilinear dimensions (i.e. magnitude) of the exclusive limit of a subject memory may be, the plurality of planes of unilinear translation determined at the subjective centre of the rotational stasis of memory and the primary state (Lemma 9), can be commonly denotated linguistically and without our secondary notation, by using the non-exclusive contrary 'fewer', of the self-evident, comparative noun 'more' (cf 'greater'), in accordance with: Rule 10: If two, then one fewer than two ? In plain words, Rule 10 asks: if any one extant instant naturally-normative of a simple noun determines two static and incidental planes of unilinear translation, then one plane fewer than two is exemplified by incidental unilinear translation in the plane of our mutual horizon? The subjectively-static accidence of a pair of natural data informed alternately in Figure 118 exemplifies the asymmetry of two incidental pairs of virtual data. There being, however, no discernible difference between the counterlogical spin of our virtual horizon and the indeterminable moment of the rotational stasis of the memory of the reader and the primary state, then we can mutually define as abstract information, that which is naturally informed by the instantaneous determination of one of two asymmetric and incidental pairs of virtual data. A singular question therefore, informed by the medium of our inquiry, is this: of what singularity is the noun 'one' normative, at an instant commonly-normative of the plurality of two, but not normative of the plurality of more than two? 2.5.2 The Order of Random Plurality To the question (Rule 10) informed by the linguistic inversion of a comparative noun, our models of natural information have supplied two (but only two) answers: a) the extant singularity of a primary datum specialized at the subjective centre of the rotational stasis of the primary state, by determining the combined logical rotation of an accidental pair of natural data, and 45

Incidental Information

b) the extant singularity of a primary datum specialized at the subjective centre of the rotational stasis of the primary state, in the limit of the instantaneous transposition of two positions of a subject memory. But the order of the determination of an incidental pair of perfectly equal positions of the primary state described linguistically by Lemma 9, is predicated by neither accidental information nor incidental information: we have already stipulated that no two determinable instants of a singular and priorly continuous state can be of perfect equality of proportion, unless both are divided by a singular, static isolated state which is not an isologue of the limit it determines. By virtue of the medium of our inquiry, however, a plurality of more than two unequal instants can be represented randomly in the plane of our virtual horizon, e.g:

Figure 119

And by virtue of the prior and mutual identification of a singular primary instant naturally-normative of a noun, we can ask the reader to inform two instants exemplified in Figure 119, by means of unilateral translation between a non-incidental pair of primary data mutually normative of the dragon (we note the definite article), and so isolate one instant fewer than two. But in order to inform a pair of non-incidental instants normative of the dragon, we have first to inform a singular instant normative of the dragon. The reader may verify that whichsoever instant may first be normative of the dragon in Figure 119, and regardless of the direction of unilateral translation, what we cannot inform is one instant fewer than two prior to informing a non-incidental pair, in what we will, therefore, define as the cycle (i.e. rotational incidence) of one fewer than two instants naturally normative of a noun. In accordance with Rule 10, whensoever two instants mutually definitive of a specie of our positive notation are informed discretely (i.e. discontinuously) in the cycle exemplified by Figure 119, one instant fewer than two is extant, but not determinable. The linguistic transformation of a verb to a noun being informed per se, however, we may invite the reader to inform one instant fewer than two, by means of any two logical unilateral translations (we note the generalized plural of a singular noun). Upon the occasion of which, the reader may observe the isolation of that instant first-normative of a discrete pair of instants. Which first-normative instant, having been priorly determined by the reader, we may mutually predict (but not verify) that one more than two incidental pairs of instants normative of the dragon (e.g. Figure 119) will be determined in the random order of any two logical unilateral translations presently and privately informed in the random cycle of one fewer than two instants. And if so, then by common and written agreement (i.e. common rote), the plurality of one more than two pairs of non-incidental instants determinable in the random cycle of one fewer two, may be nominated 'three'. In order to distinguish the random order of any two logical unilateral translations, from the plurality of two mutually-informed naturally by a discrete pair of a single specie of our positive notation, we will define the former as a simple quantity. A simple quantity, then, is predicated by the discrete plurality, of finite pairs of a single specie of our positive notation, when determined in the random cycle of one fewer than two instants naturally normative of a noun. 46

Incidental Information

And while two (and only two) incidental pairs of the specie 'dragon' can be determined in the random cycle of one fewer than two (e.g. Figure 119), post to the invention of our common rote, the simple quantity of three discrete pairs is informed in the logical order prescribed thereby verbatim, i.e. "one pair, two pairs, three pairs" (the quotation marks are deliberate). In the logical order of a priorly cited (i.e. written) common rote therefore, the simple quantity of one dragon more than two dragons (we note the generalized plural of a singular noun) informed discontinuously in the random order of two, is extant in Figure 119, i.e: "one dragon, two dragons, three dragons". Which simple quantity of "one" more than two, being indeterminable prior to the invention of our common rote, we will, with the singular exception of the simple noun 'two', qualify as abstract any noun or noun phrase with which the generalized singularity of a plural noun (e.g. the specie 'dragon' of more than two 'dragons') is qualified linguistically, in accordance with: Rule 11: If one fewer than two then one more than two ? In plain words, Rule 11 asks: the plurality of one more than two is informed unconditionally? In order to distinguish an abstract noun qualifying the generalized singularity of a plural noun, from an abstract noun qualifying the generalised plural of a simple noun, we will define the latter as a number (e.g. 'three' dragons). And there being no shortage of simple nouns with which to cite by common rote, the plurality of one more than "three" numbers recited verbally, and of one more than "four" numbers, and of one more than "five" numbers, the need to distinguish a simple number of numbers, from the abstract qualification of the generalized plural of a singular noun, can be deferred linguistically ad infinitum. We are required, however, by the rules of engagement with which we began our inquiry, to account for how a simple number of one more than two can be informed without abstract qualification. 2.5.3 The Order Of One According to our emerging models of information, our common rote prescribes verbatim the order (i.e. "one, two, three") in which the simple quantity of three dragons may be commonly-informed in the cycle of one fewer than two, by nominating any 'one' dragon isolated by any two logical, unilateral translations. Which logical order, being published information and extant in the order in which numbers are written, is mutually extant in the verbal order of their recitation. On the authority of our common rote therefore, we may represent three random, unequal instants of a primary datum not mutually normative of a noun, in a single position, e.g:

Figure 120

Without the benefit of any noun to be qualified in the order of "one, two, three", 47

Incidental Information

however, the denotation of 'three in Figure 120', instead of being mutually normative of a simple quantity, would inform 'three' as a complex abstract noun, definitive thereafter of nothing but the continuity of Figure 120. Similarly, without the benefit of a noun with which to qualify a pair, we cannot, with apodictic certainty, predict that the determination of two incidental pairs by one discrete pair fewer than two, as mutually-informed unilaterally in Figure 119, is mutually extant in Figure 120. But we may invite the reader to identify in Figure 120, by means of unilateral translation in any direction, a discrete pair naturally normative of a single specie of our positive notation. And as we cannot inform one pair fewer than two prior to informing two pairs (we note the abstract noun phrase), we further invite the reader to count (i.e. nominate in the order of our common rote), by means of logical unilateral translation, every possible discrete pair exemplified in Figure 120, as specialized presently and privately by the reader. Self-evidently, there can be no logical limit of the numbers countable cyclically in Figure 120. We note, however, that as we cannot inform a discrete pair of a specie prior to informing the specie, the logical direction of a continuously countable cycle, is informed upon the count of 'one'. Having attributed the noun 'translation' to the linguistic transformation of verb mutually-informed per se, we may ask the reader to count the discrete unilateral translations required to verify, with apodictic certainty, which pair in Figure 120 first informed both a specie and the logical direction of the cycle. And if the reader is able so to do, then on the authority of our common rote we may commonly, with apodicitic certainty, nominate as 'four' the fewest possible discrete unilateral translations normative of one pair fewer than two, in the cycle consequential upon (i.e. occasioned by) the random choice of any one, of many (i.e. more than four) possible directions of translation in which to inform a specie. But what we cannot do, privately or otherwise, is determine more than two incidental pairs in the said cycle, and nowhere in Figure 120 is the a simple quantity of more than four, extant. If, however, the abstract plurality of discrete unilateral translations counted cyclically in Figure 120, could be counted ad infinitum in simple numbers, the simple quantity of discrete unilateral translations would coincide with the simple quantity of numbers cited in the logical order of rote: the logical order prescribed by rote, would coincide with that logical direction of the cycle chosen randomly by the reader. The linguistic transformation of an abstract noun to an adjective being mutuallyinformed per se, we may further invite the reader to inform the logical order of rote, by tautologically enumerating (i.e. qualifying verbally) the ordinance (i.e. position), of discrete unilateral translations counted cyclically in Figure 120 (e.g. 'first', 'second', 'third', 'fourth', 'fifth' et cetera). And if the reader is able so to do, we may commonly verify (but not predict), that the first possible occasion of informing one pair fewer than two in any consequentially logical cycle of a privately-informed specie, coincides unconditionally with the fourth of four logical unilateral translations. While we cannot, therefore, predict that Figure 120 will be normative of a simple quantity of three, as the first and the fourth of four discrete unilateral translations are unconditionally incidental, we may, with the certainty of linguistic tautology, verify that Figure 120 is normative of the simple quantity of 'one' fewer than four non-incidental unilateral translations. 'One' fewer than four, commonly identifies the ordinance (third) of the sum of three discrete, non-incidental unilateral translations priorly counted logically. In which case, we may invite the reader to inform counterlogically, the ordinance (second) of two fewer than four; and so the ordinance (first) of that discrete unilateral translation nominated 'one' in the first place. Upon the occasion, then, of informing the ordinance of the first of four unilateral translations in the consequential cycle of three non-incidental unilateral translations, the 48

Incidental Information

ordinance of three incidental unilateral translations is extant? And if so, the direction of that incidental pair of unilateral translations informing the ordinance of the first of four, being unconditionally opposed to that unilateral translation nominated 'one', we may commonly verify (but not predict) the ordinance of two equal and opposite unilateral translations, priorly determined in Figure 120 in the consequentially-logical order of one, by the instantaneous transposition of two asymmetric virtual data. Post to the citation of our common rote, and with the benefit of the singular abstract noun 'one', we (commonly) may count the sum of privately-informed, common points of the instantaneous transposition (e.g. Figure 120) of asymmetric virtual data, in the order of any one non-incidental unilateral translation informing the consequentially-logical cycle (e.g. Figure 120) of three. There being no discernible limit of the primary state, and no discernible difference between the primary state and the exclusive limit of a subject memory, we may represent symbolically, three asymmetric virtual data, in the presented plane of our virtual horizon, i.e: 111 Figure 121

The question therefore, of how a number can be informed without abstract qualification, can be deferred ad infinitum, by counting the simple quantity q of complementary planes of unilateral translation normative discontinuously of the primary state (e.g. Figure 121), and extant in any consequential cycle of three common points, of the instantaneous transposition, of two perfectly equal positions of the primary state, when nominated in the counterlogical order of their ordinance, i.e: q =

 (10)

I 1

I 1

Figure 122

2.5.4 The Order Of Two The question deferred by nomination expresses the dilemma with which we commenced our inquiry: we cannot, linguistically and without contradiction, inform the scalar of 'one' prior to informing that which is scaled by the scalar of 'one'. We note, however, that a cycle of three common points of the instantaneous transposition of two perfectly equal positions of the primary state, can be exemplified by publishing three asymmetric primary data (we note the generalized plural of a singular noun), normative of neither a singular noun nor of a single specie of our positive notation, in a singular position, e.g:

Figure 123

The primary data of Figure 123 being self-evidently distinctive (i.e. unlike), we invite the reader to verify that in Figure 123, and in any cycle of three non-incidental discrete unilateral translations (e.g. Figures 119, 120, 121, 123), the determinant of any 49

Incidental Information

fourth of four discrete unilateral translations, coincides with the determinant of that first unilateral translation predicating the logical direction of the cycle. And if the reader is able so to do, then we may represent the smallest possible translation predicating the logical direction of the cycle, as any one of three possible unilinear translations, e.g:

Figure 124

Our positive notation being static A Priori, 'one' of the straight lines in Figure 124 exemplifies the instantaneous unilateral translation of the vertex of the reader, determined in one of two (and only two) possible directions of unilinear translation. On the authority of our common rote therefore, we may verify (but not predict), that regardless of which of many possible directions may be chosen randomly upon the count of 'one', in every possible logical cycle of three simple (i.e. neither incidental nor accidental) translations, the fourth of four is preceded by two changes of that chosen direction, the first of which changes is predicated by the determination of two simple translations, e.g:

Figure 125

If, therefore, we were to generalize the number two symbolically, then by superscribing the determinant of any two simple translations determined in the random order of one (e.g. Figure 125), we could generalize the position of any one of three instants at which the first change, of a privately-chosen direction of unilateral translation, is predicated in the consequential cycle of three simple translations? And if so, then providing that the symbol we choose to denominate (i.e. un-name) the number two is simple (individual) and, as the accidence of 1 and 0 generalizes every possible position of a subject memory, is neither 1 nor 0, it is of no immediate significance what symbol we invent to represent our primary (i.e. first) number, but by common convention we use the written Arabic 2. The linguistic transformation of a noun to a verb being informed per se, we invite the reader to number the determinant of the second of any two simple translations exemplified by Figure 123, with our primary number. Which one of three possible instants is numbered 2 by the reader, is unconditionally private information. We note, however, that by numbering the ordinance (second) of any two simple translations (e.g. Figure 125) counted in the random order of one, the second of two simple translations predicates a singular change of the direction of unilateral translation. The direction of the fourth of four discrete unilateral translations (e.g. Figure 124), coinciding unconditionally in a single plane with the direction of the first of three simple translations, the reader may verify that two and only two changes of direction can be informed by translation in any single plane of the consequentially-logical cycle of three simple translations. 50

Incidental Information

Every possible change of direction in a single plane, being informed by two simple translations, and our primary number 2 being normative of both a singular position of the vertex of memory and the quantity (see Figure 122) of two simple translations, we may, in accordance with our convention, superscribe that complementary pair of positions of the primary state, the instantaneous transposition of which was priorly determined upon the count of 'one', in a priorly uninformed position, i.e: (10)2–2 = (10)0 Figure 126

The informant of each change of direction being the determinant of two simple translations, counting changes of direction, informs simple translations in two's (we note the generalized plural of a simple number). The reader therefore may observe in Figure 123 that the position, of the determinant of every second of two simple logical translations, changes progressively (i.e. in the direction of the consequential cycle). And if so, then if we number every possible second-of-two positions in the consequentially-logical cycle of the ordinance of 2, the simple quantity of many determinable directions of unilinear translation in the plane of our virtual horizon, can be represented symbolically as an extant quantitative sum, i.e: q = 2+2+2 Figure 127

And by reciting in the verbal order prescribed by common rote, the simple quantity of numbers, represented (Figure 127) in either one of two possible logical directions of the ordinance and quantity of two numbers, we may commonly quantify (cf qualify) the abstract noun 'many', linguistically, i.e: 'there are "six" determinable directions of unilinear translation informed by our positive notation in Figure 123'. But by numbering every possible second-of-two positions in the consequentially logical cycle of three simple translations, the priorly progressive cycle of the ordinance of 2, is then extant in every possible direction of the random order of 'one' simple translation. As two, and only two changes of an informed direction of translation can be determined in the consequentially random cycle of 2, however, we may commonly verify that the informant of every two changes of direction is unconditionally (Lemma 9) the determinant of the 'third of three' non-incidental unilateral translations. Wheresoever, therefore, n nominates the number of non-incidental unilateral translations counted in any logical cycle of three simple translations, n tautologically nominates the third of three equal divisions (see Figure 126) of the primary state, i.e: n = 2+2+2 / 2 Figure 128

Upon the occasion of the fourth of four simple translations nominated in any cycle of the random ordinance of 2 therefore, a singular change of the logical direction of translation is determined (but not informed) instantaneously, in the priorly-informed order of one change of direction fewer than two, defined by the generalized sum of: 1

 (2  1)

= (2)1

I 0

Figure 129

Any reader, then, may verify presently and privately that prior to informing the 51

Incidental Information

second of three primary data mutually normative of a noun (e.g. Figure 119), or the second of three pairs naturally normative of a single specie of our positive notation (e.g. Figures 120, 121, 127), or the second of three positions informed asymmetrically by generalized primary data (e.g. Figures 123, 126), the continuity of unilinear translation in a single plane is, without the determination of the primary state, indeterminable, i.e. as expressed numerically: 2–1 =1 Figure 130

But, if we use the symbol  to represent every possible point exemplified in Figure 125, at which the continuity of unilinear translation in a single plane is informed discontinuously by unilateral translation in the plane of our virtual horizon, then the fourth of four discrete unilateral translations, coinciding unconditionally with that of the first change of direction, is defined by the quantity of changes of direction, counted in pairs (we note the generalized plural of an abstract noun), i.e: 2+2 =

 2  (2  1)

I 0

Figure 131

We therefore invite the reader to verify that, there being no discernible change of the singular position of the primary state, in every possible cycle of three generalized primary instants, the random ordinance of two changes, of the direction of unilateral translation in the plane of our virtual horizon, is the consequential scalar of the second of any two virtual data (e.g. Figure 118) nominated alternately in the logical order cited by common rote, i.e: 1

 n  (n  1)

= (n)2

I 0

Figure 132

But, in Figure 132 the linguistic contradiction, for the resolution of which we invented our secondary notation in the first place, is merely restated by nomination: how can more than one second-of-two positions generalized quantitatively by our primary number, be described as singular without inventing a complex abstract noun with which to denotate the multiple (i.e. plural ) of a simple number? We note, however, that by virtue of the numerical expression of the contradiction (Figure 132), a combination of numbers and language affords the means to resolve the contradiction, by commonly identifying the random ordinance of 2 in the logical order of our common rote, i.e: where A is an imaginable 'area' (Figure 132) and x the number of n simple translations nominated cyclically in the order of our common rote, the plane of our virtual horizon is defined tautologically, in accordance with: Rule 12: A =

 x  ( x  1)

I 0

But we have stipulated that the random order of 2 cannot be commonly-informed prior to the determination of a fourth discrete unilateral translation in the consequentiallylogical cycle of three simple translations. There is a question posited therefore by our common rote: how can the complementary sum of one and two be commonly informed by positive notation prior to informing the complementary sum of two and two? 52

Incidental Information

2.5.5 The Scalar Of Two The difficulty presented by combining the flexibility of language with the logical order of rote is this: while the nominated ordinate qualifying the generalized plural of a singular noun can be quantified by reciting numbers in the verbal order of a common rote, the invention of a singular noun, in the case of rule 12 'area', to denotate the multiple of a simple quantity (see Figure 132), merely redefines the primary state without distinguishing a divisor which is not an isologue the limit it determines. As exemplified by Rule 12, the numerical definition of 'an area' describes both a box and the content of the box without distinction, and prior to the invention of numbers, no area can be defined without either extant denotation (e.g. pointing at it) or by phrasing a mutually-informed possessive noun (e.g. "our home field", "your home farm", 'a page of our inquiry'). Our rules of evidence, therefore, require us to distinguish numbers normative of simple quantities by virtue of tautology, from extant pluralities naturally normative of our common rote. In order to draw that distinction, we will qualify as complex (i.e. of more than one part) any nominated plurality x qualifying an abstract noun, which noun affords no verbal distinction between the ordinance the number 2 and the quantity of 2 numbers recited by rote. And in accordance with that stricture, we will qualify as real (i.e. naturally informed) any number predicated by an extant representation of the quantitative difference y between two complex numbers when recounted (i.e. counted again) in the order opposite to that of their citation by common rote, in the consequentially-counterlogical (see Figure 122) order of one, i.e: xâ&#x20AC;&#x201C;n = y Figure 133

We have defined a dimension as that property of an analogue which affords empirical measurement in a singular direction, of its every possible isologue. Every possible first and fourth of four simple translations, privately nominated by rote in any logical cycle of three, being subjectively incidental, their perfect equality of proportion measures one dimension of a single plane of incidental unilateral translation? We note that the equality of two incidental unilateral translations is a private property of the memory of the reader, commonly-informed by the ordinance of four simple translations counted in the logical order of rote. The common scalar of one dimension therefore, is any fourth simple translation in the cycle of three, measuring the first of four. Having privately-informed the incidence of a first and a fourth simple translation counted in the random order of one, then by recounting non-incidental unilateral translations in the counterlogical order of one, we may commonly quantify two dimensions more than one, priorly and privately determined by two changes of the logical direction of unilateral translation. But without informing the logical incidence of more than one incidental pair of unilateral translations we cannot, without presupposition, inform the equality definitive of any more than one dimension. The reader may observe experimentally, however, that when counted logically in the random order of 2, there is one and only one informant of every two changes of the direction of unilateral translation. We may, with apodictic certainty, represent two non-incidental dimensions of the primary state (see Figure 121), informed logically in the random order of two changes of the direction of unilateral translation, in the quantitative proportion of: 2:2+2 = 1:2 Figure 134 53

Incidental Information

The reader may also observe experimentally, the subjective constancy (i.e. the quality of remaining unchanged), of that direction of unilinear translation informed discontinuously (see Figure 131), in the cycle of three simple translations. There being no discernible difference between the subjective limit of our virtual horizon and that singular position of a subject memory at which two discrete unilinear translations in a single direction are determined, Figure 134 represents the division, of one of two accidental and symmetrical (i.e. indistinguishable) planes of unilateral translation, predicated by denominating the scalar quantity of two unconditionally equal unilinear translations, determined by the singular informant, of every two changes of the direction of unilateral translation, when counted in the consequentially-logical order of 1, i.e: 1 = 2×1/2 Figure 135

The nth of any n more than three simple translations counted in any logical cycle of three therefore (see Figure 132), describes the scalar of the asymmetric division by the plane of our virtual horizon (e.g. Figure 118), of a singular unilinear translation between two perfectly equal positions, predicated quantitatively in the indeterminable proportion of one fewer than two unilinear dimensions of a single plane, i.e: x : x×x = 1:x Figure 136

In order, then, to distinguish the symmetrical division (see Figure 135) of one, of a complementary pair of asymmetric planes of unilateral translation (e.g. Figure 118) predicated in the random order of 2 (see Figure 134), from the uniform asymmetric division of its exclusive alternative (Figure 136), we will use the symbol ÷ to represent the latter, and so qualify as fractional (i.e. informed by division) any real number z denotating an extant, quantitative difference, of simple translations counted between any two numbers x more than y, and x more than 1 priorly nominated, in the logical cycle of three generalized unilinear translations in a singular and serially-informed direction, i.e: z = 1 / (x ÷ x2 ) – 1 / (y ÷ y2) Figure 137

Accordingly then (Figure 137), the reader may verify privately (e.g. Figures 124, 122, 121, 120, 119), that prior to the authorship of our common rote, any two changes of that logical direction of unilateral translation priorly informed randomly in any cycle of three generalized primary data, predicate the scalar r0 (e.g. Figure 100) of the speed S of the instantaneous unilinear transposition of a binary pair of virtual data (see Figure 118) of perfect equality, i.e: binary 10 – 1 = 2 – 1 Figure 138

The ordinance and scalar quantity of two symbolized by our primary number being informed naturally therefore, we will qualify as rational (i.e. informed by ratio) the scalar of the real number 2, when determined instantaneously by the second of every two changes of direction counted in the random order of one, in the consequentially-logical direction informed by the denomination 1, of the first of three simple translations, in every possible cycle of three generalized primary data, i.e:

Incidental Information

binary 10 : 1 = 2 : 1 Figure 139

But in any cited common rote, a simple quantity of complex numbers extant in the logical order of their ordinance, can be counted. And any complex number can be nominated, by recounting in the logical order of rote, the extant quantity of numbers from any priorly nominated number redefined tautologically as the 'first one', i.e: q:1/q = q:1 Figure 140

And while the invention of symbols to denominate complex numbers may breach our rules of evidence, the ratios of the simple quantities of which they are normative (Figure 140) do not, i.e: 1/1÷n = n:1 Figure 141

The reader may verify with apodictic certainty, that the quantitative difference (see Figure 133) of any two complex numbers cited linguistically (written) by any common rote, can be numbered simply by counting extant translations determined instantaneously between them, in the presented plane of a virtual horizon, in either one of two logical directions, i.e: z = n–1 Figure 142

Consequentially, therefore, any real number of instants nominating the order of binary 10 (see Figure 122) is, tautologically, the scalar (see Figure 102) of our indeterminable virtual horizon, i.e: z

d0 = binary  (10) z z 1

Figure 143

And the scalar (Figure 143) of 2 radii of a virtual sphere, is the scalar of the counterlogical spin of our mutual horizon in (but not limited by) the presented plane of our virtual horizon, determined at the particular moment (see Figure 98) of subjectively continuous unilinear translation (Lemma 2) in a singular serial direction, i.e: C = m1  r0 Figure 144

But there remains the question informed numerically by Rule 10: how can the difference of two complementary sums, be informed positively without a complex abstract noun (e.g. 'area', 'moment') to denotate the complementary sum of one greater than (cf fewer than) the complementary sum of two, because (i.e. consequential upon our prior definitions): 2×1/2 = z×1÷z Figure 145

The Magnitude Of Scale

3 The Magnitude Of Scale 3.1 Asymmetric Division In Figure 118 we have configured two asymmetric models of translation per se which, post to the invention of a common rote, are normative ad infinitum of those scalar quantities nominated verbally by numbers when qualifying the generalized plural of the noun 'translation'. Each model singly represents two incidental and perfectly equal positions of the primary state, the instantaneous transposition of which is informed by determining the logical spin of an accidental pair of natural data in one of two possible directions of unilinear translation. Counting each such instantaneous transposition therefore, tautologically quantifies the scalar of its own, asymmetric limit, normative of an exponentially (i.e. uniformly) increasing difference (Rule 12) between the scalar of two equal positions, and that which is scaled by two equal positions. The difficulty presented alternately by Rules 10 and 11 is that we cannot, without inventing a complex abstract noun (e.g. Rule 12), distinguish the quantitative ratio defining the speed S of the instantaneous unilinear transposition of a symmetrical pair of positions (Rule 10), from the quantitative ratio defining the natural division of the moment of the indeterminable, counterlogical spin of a virtual horizon (Rule 11). We note, however, that drawing in a single plane enables the representation of three straight lines exemplifying, in the absence of any primary data, the smallest possible finite translations determined instantaneously, in that cycle of three generalized primary data priorly exemplified in Figure 123, i.e:

Figure 146

There being no accidental pair of natural data in Figure 146, the reader may observe that we can, if we will, mutually-identify an artificial datum (we note the singular) mutually normative of the primary state, with a complex abstract noun. Prior to doing so, however, the reader may also observe, by translating the vertex of memory randomly in any one of six possible directions informed by the stasis A Priori of our positive notation, the constancy of that position of memory (whichsoever it may be) informed by every 2 changes of the first-informed direction of translation, in a consequentially-logical cycle of the moment of the subjective rotational stasis of memory and the primary state? And if so, then at any one subjective centre of the rotational stasis of memory and the counterlogical spin of our virtual horizon, informed by unilateral translation in the plane of our drawing (i.e. Figure 146), the direction of unilinear translation informed discontinuously toward–1, is then continuously toward–1. The direction, chosen randomly by the reader therefore, of the first of any four discrete unilinear translations counted in Figure 146, being unconditionally toward–1, then upon the count of four, the incidence of two unilinear translations logically toward–1, and the subjective continuity (Lemma 2) of unilinear translation toward–1, commonly informs the continuity of logical unilinear translation in one dimension of the plane of our drawing. While we cannot determine the moment of the counterlogical spin of our virtual horizon therefore, we can commonly verify that in any extant logical cycle of three generalized unilinear translations (cf Figures 146, 123), the continuity of unilinear translation in one dimension of the primary state, is informed serially in the order of two discontinuous unilinear translations toward–1. In the absence of any information but that of an artificial datum (i.e. Figure 146) 56

The Magnitude Of Scale

and that supplied by common rote, we may count discrete unilinear translations commonly normative of the verbal order of rote, exemplified in the consequentially-logical cycle of two discontinuous unilinear translations towardâ&#x20AC;&#x201C;1, and so commonly verify the incidence of the fifth of five and the second of two discrete unilinear translations in the plane of our drawing, and of the sixth of six and the third of three, and of the seventh of seven and the fourth of four, normative ad infinitum of continuous unilinear translation in one fewer than two (Rule 10) serially-discontinuous dimensions of a single plane? And if so, then the plane of our virtual horizon being indeterminable in Figure 146, and the primary state being the exemplar but not the limit of singularity, we will define the virtual limit of continuous unilinear translation in any single plane, as the complement (i.e. that which informs the singularity) of two serially-discontinuous planes of incidental unilinear translation, i.e: 1 = (2 Ă&#x2014; 2) / 2 + 2 Figure 147

Having priorly invented our secondary notation to denominate the self-evident, but otherwise ineluctable, quantity and position of two serially-discontinuous and equal (accidental) planes of unilinear translation privately-informed A posteriori (Lemma 3), counting four discrete unilinear translations in any consequentially-logical cycle of three, nominates the complement of two accidental planes of incidental unilinear translation, on what we will define as the exponential scale of binary 10, i.e: binary (10)n = (2)n Figure 148

Consequentially, in either one singly, of our two asymmetric models (Figure 118) of translation per se, the symmetrical division (see Figure 147) of the virtual limit of continuous unilinear translation in the plane of our positive notation, is doubled (Figure 148), by each nomination n recited in the logical order of rote. Without the instantaneous alternation therefore, of the ratio and the proportion of an accidental pair of natural data, the real number three can be naturally-informed neither rationally nor fractionally. We note, however, that the virtual limit of an artificial datum (i.e. Figure 146) being the exclusive limit of a subject memory, but there being no discernible difference between the exclusive limit of a subject memory and the primary state, the symbolic representation (Figure 147) of the complement of two serially-discontinuous planes of incidental unilinear translation, unifies the ordinance of the first and fourth, of any four unilinear translations nominated in a single plane. In the consequentially-logical order of 1 dimension of the plane of our positive notation (i.e. Figure 146), we can count the extant plurality of one more than two unconditionally unequal dimensions, informed discontinuously in the cycle of three subjective centres of the rotational stasis of memory and the primary state, exemplified logically (cf Figure 122) in the verbal order of our common rote; and so privately-inform the real number (see Figure 133) of discrete unilinear translations required to determine the consequentially-counterlogical spin (cf Figure 77) of our positive notation. With the sole provision therefore, that in breach of our rules of evidence the reader is primed (i.e. priorly informed), with a simple symbol priorly adopted by common convention to denominate the quantity and ordinance of the third of three numbers cited by rote (i.e. the ordinal number 3), then the symmetrical division of the virtual limit (see Figure 147) of continuous unilinear translation in any one of two serially discontinuous planes, is defined by the stasis A Priori of our positive notation in the simple quantitative ratio of: 57

The Magnitude Of Scale

3–2:2–1 = 1:1 Figure 149

The only extant equality of proportion determinable in Figure 146, however, being that informed by the unification of discrete pairs of incidental translations, then if, in the absence of an accidental pair of natural data and without breaching our rules of evidence, we are to denominate the real number three, then we have first to denominate that unilinear translation which coincides with the second of two counted in the random order of one, i.e: 5–2:2–1= 3:1 Figure 150

In which case (Figure 150), the real number 3 is defined fractionally on the scale (see Figure 136) of thirds (we note the qualification of the quantified ordinance of three simple numbers), by the discrete proportional difference of five thirds and two thirds, of the complement of 2 (see Figure 132) accidental planes of incidental unilinear translation, i.e: 5×1÷3 – 2×1÷3 = 3×1÷3 Figure 151

Figure 151 describes the real number 3 as the discrete proportional difference of two complex numbers, defined numerically (but not informed) in the quantitative ratios of 5 : 1 thirds and 2 : 1 thirds of the virtual limit of continuous unilateral translation in a single continuous plane. Which continuous plane being indeterminable ad infinitum, in order to inform the real number 5 we have first to denominate the seventh of seven unilinear translations enumerated in the (then) consequentially-logical order (see Figure 145) of the quantity of 2 accidental positions of the primary state, i.e: 7–2:2–1 = 5:1 Figure 152

In which case (Figure 152), the real number 5 is defined fractionally on the scale of fifths, by the discrete proportional difference of seven fifths and two fifths of the complement of 2 accidental planes of incidental unilinear translation, i.e: 7×1÷5 – 2×1÷5 = 5×1÷5 Figure 153

And in order to inform the real number 7 on the scale of sevenths, we have first to denominate the ninth of nine unilinear translations. There being no logical limit of our ability to invent simple symbols with which to quantify the xth. of x unilinear translations enumerated in the logical order of the instantaneous transposition of 2 accidental positions of the primary state, there can be no shortage of real numbers subsequently informed by the extant quantitative difference of two fractional numbers, i.e: q × (1 ÷ (q – 2)) – 2 × (1 ÷ (q – 2 )) = 1 Figure 154

By the simple experiment of allowing q in Figure 154 to represent any rational number but 2, the reader may verify that the nomination x, of the second of any pair of incidental unilinear translations counted in the presented plane of Figure 146, 58

The Magnitude Of Scale

tautologically defines the alternation, of the proportional scalar of two fractions, of the virtual limit of continuous unilateral translation in one fewer than two (Rule 10) accidental and symmetrical planes, and the rational scalar of two accidental planes of incidental unilinear translation, i.e: ((x)2 – x / x ) : 1 = x – 1 : 1 Figure 155

And as the proportional scalar of the symmetrical division (Figure 155) of the virtual limit of continuous unilateral translation in any single plane (see Figure 154), is doubled (see Figure 148) in the quantitative ratio of each denominated number quantifying the ordinance of our common rote, the rational scalar (x – 1), of the complement of two accidental planes of incidental unilinear translation informed discontinuously toward–1, is defined tautologically on the scale of our primary number, i.e: q × (1 ÷ (q – 2)) – 1 = 2 × (1 ÷ (q – 2 )) Figure 156

The reader may verify (Figure 156) that when counted verbally in any consequentially-logical cycle of three (e.g. Figure 146), the extant plurality of discrete translations (per se) normative of the number n qualifying the generalized plural of the noun 'translation', is tautologically normative of the simple quantity q symbolized conventionally by the denominator of any n more than 2. And if so, then the verbal count, of any four generalized unilinear translations exemplified in the cycle of three by a single artificial datum (i.e. Figure 146), being normative of the rational scalar (see Figure 155) of the complement of two accidental planes of incidental unilinear translation, tautologically informs a common model of a singular and exponentially-increasing divisor of the said complement, normative of real numbers on the exponential scale (see Figure 148) of binary 10, i.e: binary (10)n – 1 =

 (10)

z 0

Figure 157

There is a question therefore, informed by the presentation of an artificial datum (i.e. Figure 146) in a single plane, of how the exponential divisor of the complement of two accidental planes of continuous unilinear translation, can be nominated in the order of 2 (see Figure 132) by a real number prior to the denomination of the complement of two serially-accidental positions of the primary state represented (Figure 157) by our secondary notation? 3.2 The Limit Of Asymmetric Division The difficulty confronting our rules of evidence is that, by virtue of drawing an artificial datum in a single plane, a unary pair of planes of translation normative of the complement of two, can be described numerically on the exponential scale of binary 10 with the unconditional certainty of tautology. But because no two dimensions of the common divisor of the primary state represented in Figure 146 can be equal, and while the counterlogical spin of our virtual horizon cannot be determined, then except by the mutual identification of natural intelligence with a complex abstract noun (e.g. 'triangle'), the 59

The Magnitude Of Scale

singularity of the plane presented by the medium of our inquiry is not extant in Figure 146. And when formulating lemmas we may adopt as unconditional only those answers which describe every possible application of the singular condition they inform. We note, however, that by virtue of the very cause of our dilemma, we can represent those subjective centres of the rotational stasis of memory and the primary state exemplified discontinuously in Figure 146, in a singular and continuous position of a single and continuous plane, i.e:

Figure 158

With the benefit of neither a number nor a noun (e.g. 'points') with which to qualify the extant plurality (Figure 158) of a specie of our positive notation, the only singularity exemplified therein is that of the primary state: consequentially, therefore, every possible direction of translation exemplified by Figure 158 is towardâ&#x20AC;&#x201C;1. While we cannot determine the counterlogical spin of our virtual horizon, the continuity of both the position and the plane exemplified in Figure 158, can be informed by neither the incidence nor the accidence of any two planes of translation. Prior to the invention of language, there being no means of informing the subjective equality of an incidental pair of discrete translations, whatsoever equality of proportion may be privately-informed by Figure 158, can be neither mutually- nor commonly-informed. We have, however, stipulated that in order to inform the stasis of our positive notation, we have first to determine a primary unilateral translation t0 in the plane of t1. The limit of unilinear translation in the plane of t1 being the counterlogical spin of our mutual horizon (see Figure 98), then while the counterlogical spin of our virtual horizon cannot be determined, the moment of the rotational stasis of memory and our virtual horizon is, necessarily, either equal to or greater than the counterlogical spin of our mutual horizon. The plane of t0 being that presented by the medium of our inquiry, then post to the invention of both language and drawing in a single plane, we may ask the reader to determine the counterlogical spin of our virtual horizon, by informing the rotational stasis of the plane exemplified in Figure 158. Upon the occasion of which (the reader may verify by every possible means) there is one and only one position of memory at which that rotational stasis can be informed, i.e:

Figure 159 60

The Magnitude Of Scale

And if so, Figure 159 exemplifies the counterlogical spin of our mutual horizon in the plane of our virtual horizon: in which single plane therefore, the greatest possible logical translation between any two centres of the counterlogical spin of our mutual horizon, may be represented in either one of two possible directions of rotation, by any one fewer than two radii of an extant circle drawn in that plane, e.g:

Figure 160

While we cannot determine the counterlogical spin of our mutual horizon, therefore, Figure 160 presents a finite distance informable by unilinear translation toward or away from the centre of the counterlogical spin of our mutual horizon. We have stipulated that there is only one position of memory at which the stasis and the singularity of the plane of translation exemplified by Figure 158 can be informed. But because our extant presentation (i.e. Figure 160) of the distance r (cf Figure 67) informs the greatest possible unilinear translation in a single plane toward or away from the centre of the counterlogical spin of our mutual horizon, we can draw two centres of the counterlogical spin of our mutual horizon, at which the singularity of the said plane may be informed, i.e. either one of:

Figure 161

An incidental pair of perfectly equal virtual data, being mutually exclusive, cannot be drawn in a single plane. In Figure 161 however, we can distinguish the finite distance r determined logically at the speed of the instantaneous unilinear transposition of two perfectly equal positions of memory (see Figure 102), from the conventional informants of the ratio ď °, i.e: rď&#x201A;´S = c Figure 162

There being two possible directions toward the centre of the counterlogical spin of our mutual horizon informed (see Figure 161) by a non-incidental pair of virtual data, we cannot mutually-inform the unilinear velocity (cf Figure 58) of their transposition. But nor could we draw such a pair prior to mutually identifying a privately-informed position of memory (see Figure 159) at which a single plane of unilinear translation is exemplified by 61

The Magnitude Of Scale

more than two possible radii. Having defined abstract information as that which is naturally informed by the instantaneous determination of one of two incidental pairs of virtual data, and the distance r informed towardâ&#x20AC;&#x201C;1 being equal to the distance r determined towardâ&#x20AC;&#x201C;1 (see Figure 161), the sum of the finite distances determined in a single plane, in order to inform two perfectly equal positions of memory, may be represented by any diameter d of the extant circle (i.e. Figure 160) predicating the counterlogical spin of our mutual horizon. The publisher, therefore, may represent a single exemplar of the said diameter, i.e:

Figure 163

There being no discernible difference between the stasis A Priori of our positive notation and moment of the rotational stasis of memory and the primary state, then by determining the counterlogical spin of our virtual horizon, in either one of two possible directions of the logical rotation of the exclusive limit of memory, the reader may recognize in Figure 163 a singular divisor of the primary state which is an isologue of the limit by which it is determined. And if so, then in the special case of Figure 163, we can draw a model of two serially-continuous dimensions, of that single plane of continuous unilinear translation priorly exemplified discontinuously by Figure 146, and limited by the counterlogical spin of our mutual horizon, i.e:

Figure 164

Post to the invention of both our common rote and our symbolic denominators of simple numbers therefore, we can commonly prove categorically (i.e. without conditional clauses) in accordance with Pythagoras's theorem, that in the special case of Figure 164, the symmetrical division of a common divisor of the singular divisor of the primary state, is informed by the simple addition (cf Figure 132) of two asymmetric planes of unilinear translation, the logical continuity of which planes is informed by the moment of the rotational stasis of our positive notation and the counterlogical spin of our mutual horizon, i.e: A / 2 = ((r)2 + (r)2 ) ď&#x201A;¸ 2 Figure 165

The Magnitude Of Scale

We note that the categorical proof of the perfect equality expressed by Figure 165 cannot be delivered without the use of complex abstract nouns. The extant area, however, of the common divisor of the singular divisor of primary state (see Figure 164), being selfevidently greater than either one of its asymmetric divisors, is normative of the simple difference of two extant areas, i.e: A â&#x20AC;&#x201C; (r)2 = (r)2 Figure 166

Consequentially, therefore, the symmetrical division of the singular divisor of the primary state (see Figure 164), is informed in the constant proportion of: (r)2 : A = r : d Figure 167

The singular divisor of the primary state (see Figure 164) being the only isologue of the limit by which it is determined, then by informing the distance r with every possible real number z, the numerate (i.e. primed with both our common rote and our symbolic denominators of simple numbers) reader can verify categorically, ad infinitum and without a complex abstract noun (cf Figure 131), that the asymmetric division of the primary state informed by the alternation an accidental pair of natural data (see Figure 118), is limited towardâ&#x20AC;&#x201C;1 by every possible radius of a singular virtual datum, in the constant ratio (cf Figure 145) of the moment of the rotational stasis of memory and the primary state, to that of the counterlogical spin of our mutual horizon, informed quantitatively in the order of our primary number, i.e: 1 / (z)2 : 1 / (z + z)2 = (2)2 : 1 Figure 168

An incidental pair of perfectly equal virtual data can be neither drawn in a single plane, nor informed prior to their determination. Furthermore, we have stipulated (Rule 1) that, with the exception of the conventional adoption of an extant empirical datum definitive tautologically of a 'unit' of 1 named dimension, a singular datum (cf a unary pair of primary data) can neither inform nor be informed: stasis and translation are mutually exclusive. And if the ordinance of our primary number cannot be informed prior to the denomination of the simple number two, then Pythagoras's theorem begs the question of which, of simple numbers or complex numbers, defines the other? 3.3 The Scales Of Binary 10 To the discovery of drawing in a single plane we can attribute, in the order of their subsequent invention, written language, the symbolic denomination of simple numbers, plane geometry, complex numbers, and the mathematical proof of the special case of the right-angled triangle normative of Pythagoras' theorem. But the verbal order of our common rote being extant in its recitation, post to their invention the distinction, between the names of simple quantities and their symbolic denominators, is lost, together with the order of their invention: we can nominate symbolic denominators verbally in the logical order of rote. And because we can nominate the denominators of simple numbers, we can prove categorically that the symmetrical division of the divisor, of the common divisor, of the singular divisor of the primary state exemplified in Figure 164, is informed by the simple 63

The Magnitude Of Scale

addition of two extant asymmetric areas, i.e: 1 / 2 = (x + x)  x  2 + (x + x)  x  2 Figure 169

But where the symmetry of the division expressed by Figure 165 is predicated by the counterlogical spin of our mutual horizon, our numerical generalizations of distance inform a virtual symmetry: Figure 169 predicts a perfect equality which cannot be predicated by a static, asymmetric and non-incidental pair of areas, i.e: 1 / (A – (x)2) = 1 / (x)2 Figure 170

By informing the distance r (see Figure 164) with every possible simple number (i.e. Figure 169), the numerate reader may verify the constancy of that static area A of the plane of our virtual horizon, informed in the quantitative proportion of the counterlogical spin of our mutual horizon and the moment of the rotational stasis of memory and the primary state, i.e: (x + x)  x  2 : A = 1 : (2)2 Figure 171

Figure 171, therefore, predicts the symmetrical division of the common divisor, of the singular divisor, of the primary state presented in Figure 164, the virtual symmetry of which division we can represent by drawing a virtual circle limited toward–1 by a square (Figure 170) of side x, i.e:

Figure 172

And we can prove categorically that: A = 2  (x)2 Figure 173

While we can determine neither the moment of the rotational stasis of memory nor the counterlogical spin of our mutual horizon therefore, and because we can prove the limit (Figure 173) of the complement two asymmetric divisors of a singular virtual datum (see Figure 172), we can likewise prove that the ratio (cf Figure 167), of the single moment M (Lemma 7) of the rotational stasis of the plane of our virtual horizon, to the counterlogical spin of our mutual horizon (see Figure 162), is expressed quantitatively as: M : 2  (x  S) = 2 : 1 Figure 174

The Magnitude Of Scale

Which ratio (Figure 174) being that of the diameter to the radius of any circle we may care to draw, we can prove categorically that the limit of the symmetrical division of a singular virtual datum is defined absolutely (i.e. independently of numbers) in the simple ratio of: Cd : cr = d:r Figure 175

Presented with the categorical proof (see Figure 173), of the absolute limit of a single plane of unilinear translation, the numerate reader can return to Figure 146 and verify, with absolute certainty, that prior to the denomination of simple numbers, the complement of two accidental planes of unilinear translation informed discontinuously in the consequentially logical cycle of three generalized unilinear translations, is described unconditionally as: binary 10 – r = 1 Figure 176

By means of the simple transposition of a distance and a direction, we may transpose the complement d of two distances, and the binary complement (Rule 9) of 1, i.e: binary 10 = r + r Figure 177

Consequentially therefore, the exponentially increasing scale of the real complement of two planes of incidental unilinear translation, is predicated proportionally, in the verbal order of nominating the exponential scale of binary 10 (see Figure 148), i.e: (d)n = 1 /

 (10)

0 n

n 1

Figure 178

With the proviso therefore, that we do not distinguish simple numbers from generalized divisor functions (Figure 178), the discrete increase predicted by Rule 12, of that static plane of our virtual horizon mutually-informed by Figure 159, is predicated on the exponential scale of binary 10 by the symmetrical division, of both the circumference C of the circle of radius r and the complement of two squares of side r, by the singular diameter d0 (see Figure 88) of the counterlogical spin of our mutual horizon, i.e:

Figure 179

The divisor of the complement of two generalized squares being the divisor of the complement of two hemispheres (Figure 179), the difference between the binary complement of two asymmetric planes, and the real complement of two planes of 65

The Magnitude Of Scale

incidental unilinear translation, is lost, i.e: 2    r  2  (r)2 = S Figure 180

To the numerate reader therefore, Pythagoras affords a categorical proof of the continuation of a virtual plane, increasing exponentially in proportion to the scale of binary 10, and predicated (Rule 12) by denominating the order of 2 virtual planes of unilinear translation in the verbal order prescribed by common rote. There being no logical limit of our ability to count, there can be no logical limit of our ability to inform the discretely-increasing scale (see Figure 178) of the real complement of two incidental planes of unilinear translation. We have, however, already defined the finite limit of an accidental pair of perfectly equal divisors of a singular binary datum (see Figure 39) as the exclusive limit of the memory of the reader. We may invite the reader therefore, to determine the finite limit of the accidence of a binary pair of virtual data, by means of unilateral translation toward the centre of the singular virtual datum exemplified by Figure 179. And if the reader is able so to do, then with the proviso that we maintain the distinction between our secondary notation and real numbers, the limit of the exponential increase predicted indeterminably (Rule 12) by common rote on the exponential scale of binary 10, is continuously equal to the square of the diameter d0 (see Figure 177) of the circle of radius r0, according to: Lemma 12: binary (10)2 =

10  (10  1)

I 0

Which square (Lemma 12), being neither presented nor represented in Figure 179, we will define as the virtual limit (see Figure 118) of the moment of the rotational stasis of our mutual horizon and the plane of our virtual horizon, i.e: binary (10)2 = d02 Figure 181

Which virtual limit (Figure 181), being continuous regardless of the position of the reader, we have represented without brackets. How, then, other than as the product of human thought independent of experience, could such a square be informed? 3.3.1 The Real Limit Of Symmetrical Division It will not have escaped the reader's attention that by formulating Lemma 12, we have violated the rules of engagement with which we began our inquiry. But the violation, rather than owing to a lack of rigour or vigilance on our behalf, is borrowed by the condition of numeracy: we are conditioned to identify simple quantities normative of simple numbers, with the conventionally-adopted symbolic denominators of simple numbers. We can enumerate in the verbal order of rote, those symbolic denominators nominated verbally by rote. Having thereby lost the distinction, between the ordinance of a simple number and the simple quantity of numbers counted logically by rote, the numerate reader may verify categorically (Lemma 12) that by equating our secondary notation 10, with any simple quantity identified with the symbolic denominator x of a simple number n, we inform the 66

The Magnitude Of Scale

sum of two virtual squares (see Figure 181), quantitatively, on what we will define as the n-ary scale of binary 10, i.e: d02 =

 n  (n  1)

I 0

Figure 182

And having simplified (i.e. informed quantitatively) the sum of two virtual squares (Figure 182), then for every simple number n of more than 2, the rational scalar of two accidental planes of incidental unilinear translation (see Figure 155) is predicated by the simplified ratio of two unconditionally unequal virtual squares, i.e: x  (x – 1) : (x  1) = ( x – 1) : 1 Figure 183

Conditionally, therefore, every denominator x of a simple number n of more than 2 (Figure 183), commonly predicates a complex number z, which we inform on the n-ary scale of binary 10, in the simplified ratio of: (x – 1) : 1 = z : 1 Figure 184

With the ability to inform the difference of two simple quantities without translation (Figure 184) therefore, we can verify numerically, that any complex rational number z commonly informed on the n-ary scale of binary 10 (cf Figure 145), generalizes the division of the real complement (see Figure 177) of binary 10, i.e: d0 = z  1  z Figure 185

If, therefore, by mutual agreement, we choose to equate our secondary notation with any simple quantity of more than two, identified without translation by the symbolic denominator x of a simple number n, then we mutually nominate the scalar quantity of the complement (see Figure 176) of binary 10, i.e: binary 10 – z = 1 Figure 186

And consequentially, a virtual scalar 10 of the plane of our virtual horizon, is commonly identified with that simple quantity priorly identified without translation, by the symbolic denominator x of n, i.e: 10 = z + 1 Figure 187

There can be no quantitative difference, therefore, between the virtual scalar (Figure 187) of a virtual plane, and that simple number n priorly denominated x, i.e: 10 – x = x – n Figure 188

But the numerate reader is conditioned to identify the absence of a quantitative difference, with the symbol 0. In which case (Figure 188) the numerical difference 67

The Magnitude Of Scale

expressed by two nominators, of the denominator x of the simple quantity of n numbers counted verbally in the order of rote, is equal to 0, i.e: 10 – n = 0 Figure 189

If, therefore, by common convention, we identify our secondary notation with any simple quantity of more than two, then we commonly inform 0 as the numerical difference of two identical virtual planes, one of which is informed proportionally on the tautologically-enumerated scale (Figure 187) of n nths of 10, counted in the uniformlyincreasing order of 1 nth of 10, i.e: 1  10 = 1  n Figure 190

The exclusive alternative of which plane (Figure 190) we have defined numerically (see Figure 189). But consequentially, we have also defined a generalized divisor 1 nth, of the virtual scalar of the plane of our virtual horizon (see Figure 187), in the simplified ratio of: z + 1 : 1 = 10 : 1 Figure 191

And having informed 0 quantitatively (see Figure 189), then by borrowing the simplified sum of any two unconditionally unequal virtual squares defined in the logical order of rote on the n-ary scale of binary 10 (see Figure 178), the quantitative difference of two squares is defined rationally in the ordinance of: z  (10)0–(n+1) = (10)0–n – (10)0–(n+1) Figure 192

Which difference of two squares (Figure 192), when nominated in the logical order of a common rote, decreases ad infinitum in the constant ratio of: (10)0–(n+1)  z + 1 = (10)0–(n+1) : 1 Figure 193

But the quantity z (see Figure 192), of the multiple divisions (Figure 193) of the virtual scalar of the plane of our virtual horizon, being informed without translation by the symbolic denominator of a simple number, then where z is a rational number (see Figure 184) commonly normative of 1 nth fewer than 10 nths (see Figure 190), the n-ary scale of binary 10 is normative of a virtual plane increasing exponentially (cf Figure 190), as the simplified sum of the borrowed squares of 10, counted logically on the proportional scale of: (10)z = 1  1  (10)0–(n+1) Figure 194

Having proved (Figure 174) the ratio, of the counterlogical spin of our mutual horizon to the counterlogical spin of the plane of our virtual horizon at the moment M of their rotational stasis, then by means of counting the real number (Figure 194) of the ordinants of 10, in the opposite of the logical order rote, we count the uniformlygeneralized squares dividing the virtual scalar binary 10 of the plane of our virtual horizon, in the inverse (i.e. opposite order) of the scale of their proportional increase? 68

The Magnitude Of Scale

And if so, then we commonly inform the finite limit of the borrowed squares of 10 (Figure 194), predicated numerically (see Figure 183) by the difference 0 of two perfectly equal squares, on any commonly-adopted scale of n = 10, in the ordinance of 2, i.e: n-ary 102 = 100 Figure 195

We note, however, that every possible simplified sum, of the borrowed squares of 10 counted logically on the n-ary scale of binary 10 (see Figure 193), is necessarily lesser than 0. Logically, then: binary 102 > 0 > n-ary 102 Figure 196

There being no possible limit of the exponentially-decreasing ratio, of any two borrowed squares of 10 counted logically on the n-ary scale of binary 10 (see Figure 193), in order to distinguish the virtual limit (Lemma 12) of the counterlogical spin of our mutual horizon, as determined (but not informed) by unilateral translation toward the subjective centre thereof, from the finite limit of symmetrical division (see Figure 195), as informed without translation by mathematical induction (i.e. inverse logic), and the latter being informed categorically by real numbers, we will qualify the latter as the real limit of symmetrical division. But in order to comply with our rules of evidence, we have first to account for how simplified ratios conditionally predicating real numbers (see Figure 184), can be informed prior to informing the proportional difference (see Figure 190) normative of 1 nth of 10 nths. 3.3.2 The Real Limit Of Asymmetric Division The question of how the exponentially-increasing scale of binary 10 (see Figure 148) may be informed proportionally in the logical order of rote, is a question we have already answered: having (in the first place) denominated the primary state as 1, and its contradictory as 0, then by transposing a distance and a direction (see Figure 176), we inform binary 10 as the sum of two radii, of a circle delimiting (i.e. marking) the counterlogical spin of our mutual horizon in the plane of our drawing (see Figure 159). Prior to the mathematical inversion of 1 and 0, binary 10 describes the sum of two radii of an extant circle, as the diagonal (see Figure 179) of a virtual square, which square doubles exponentially (see Figure 148), in proportion to that discrete increase of binary 10 informed numerically by nominating the real complement of two radii, in the logical order of a common rote. And by counting the squares of binary 10 in the inverse of the verbal order of rote, reader may verify that except by unilateral translation, the virtual limit (Lemma 12) of the symmetrical division of the plane of our virtual horizon, cannot be determined within the limits of 1 and 0. The conditional identification of 0 as the numerical difference of two equal quantities therefore, transforms the question asked by Rule 9 into an axiom (i.e. a proposition affording no verifiable alternative) i.e: 'Rule 9: the distance informed by a binary pair of symmetrical data is 0.' Axiomatically, the symbolic representation (see Figure 121) of three asymmetric virtual data exemplified discontinuously in the plane of our virtual horizon (e.g. Figure 120), constitutes of itself three, symmetrical virtual data presented in the said plane, the 69

The Magnitude Of Scale

counting of which informs the unary scale (n = 1) of binary 10 independently (i.e. regardless) of direction, i.e: 111 = one two three / three two one Figure 197

The ordinance of the simple number two being informed by every possible verbal order of our common rote, then Rule 11 is informed by Figure 121? And if so, then the question posited by common rote, of how the complementary sum of one and two can be commonly informed by positive notation prior to informing the complementary sum of two and two, is answered: by representing (e.g. Figure 121) subjectively-equal positions of the primary state on the unary scale of binary 10, and not the singular and greater contrary of the primary state. Figure 121, however, and therefore Rule 11, affords no distinction between one more than two, and many more than two unilinear translations, the counting of which in the consequentially-logical cycle of three (see Figure 123) may be privately normative of the logical order of rote. But the limit of our ability to present positive notation within the limits of 1 and 0 being the private property of the reader, we may represent the smallest possible quantity of symmetrical virtual data normative of many more than two, i.e: x = 11111 Figure 198

Figure 198 therefore, exemplifies an extant analogue of a finite translation in the plane of our virtual horizon, which translation increases uniformly as the sum of n – 1 generalized unilinear translations, informed tautologically in either of two directions, by counting virtual data verbally in the logical order of a common rote, i.e: x2 = n – 1  1 Figure 199

There being no logical limit of our ability to invent simple numbers, there can be no logical limit of the analogue (e.g. Figure 198) of a unilinear translation informable by common rote on the unary scale of binary 10. And by means of recounting counterlogically in the opposite of the order prescribed by rote (i.e. count-down), the reader may verify that the quantitative difference of 1 (cf Figure 188), between any two virtual squares (Figure 199) so nominated, defines the difference of 0 and 1 as the square of the constant unilinear difference determined between them, i.e: 0 = 1–1–1–1–1–1 = 0 Figure 200

Self-evidently, we cannot represent the binary complement (–1) of 1, without first representing 1. If, however, as in Figure 123, we were to represent the unary scale of binary 10 with self-evidently distinctive generalized data priorly normative, by virtue of conditioning, of a logical order, then we could inform the finite unilinear difference between x2 and 0 informed by mathematical induction (Figure 200), by counting virtual data in a singular direction of unilinear translation, e.g: 70

The Magnitude Of Scale

A+B+C+D+E+F Figure 201

Axiomatically, nominating the unary scale of binary 10 in the order of A = 0 (Figure 201), informs a translation in plane of our virtual horizon, that increases uniformly as the sum of n generalized unilinear translations (cf Figure 199), in singular logical order prescribed alphabetically, i.e: n2 = x2 Figure 202

If we are pre-conditioned linguistically, therefore, to identify simple quantities with the symbolic denominators of simple numbers cited by common rote, then every denomination by the numerate reader of the squares of n (e.g. Figure 201) nominated in the inverted order of the unary scale of binary 10 (see Figure 182), informs a discrete distance, as the sum of the sides the squares determined within the limits of 0 and n, i.e: binary 10 = x Figure 203

And if so, then the real complement of two radii (see Figure 177) descriptive of the diagonal of a virtual square lesser than binary 102, increases on the unary scale of binary 10, independently of direction, in the finite ratio (see Figure 111) of: d0 + r0 : d0 Figure 204

Axiomatically, excepting only by equating our secondary notation with a simple quantity of numbers, the mathematical inversion of 1 and 0 cannot be limited by 0. But equating our secondary notation with a simple quantity of more than two numbers, informs the n-ary scale of binary 10 in the order of 2 (see Figure 195), and not the unary scale of binary 10: and by informing the unary scale of binary 10, we inform the square of the binary complement (â&#x20AC;&#x201C;1) of 1 in the order of 0, and not the square of binary 10. Consequentially, the scalar of the translation on which we commonly inform the generalized representation of an extant plurality of virtual data (e.g. Figure 198), is unconditionally lesser than the scalar of binary 10. In order to distinguish the exponential scale of binary 10 (i.e. 10 = 2) on which unilateral translation (see Figure 120) privately informs the serial continuity of three virtual data, from the unary scale of binary 10 (see Figures 121, 201) on which the distance of unilinear translation in the plane of our virtual horizon is informed by mathematical induction, we will qualify the former as the macro (i.e. greater) scale of binary 2. The finite ratio (Figure 204) normative of the macro scale of binary 2, which may be determined, but not informed, by unilateral translation toward or away from the subjective centre of the counterlogical spin of our mutual horizon in the plane of our virtual horizon (see Figure 159), cannot be commonly-informed except by superscribing the order of 2 (see Figure 148), and so cannot be informed prior to the discovery of drawing in a single plane. We note, however (see Figure 69), that the macro scale of binary 2 is that on which we mutually identify nouns, and therefore language, verbally. As language may be mutually informed verbally prior to the invention of writing, the simple quantity, of one or inclusively many more than two numbers, but not the symbolic denominators thereof, may 71

The Magnitude Of Scale

be commonly informed linguistically prior to the discovery of drawing, as the tautological sum "two" per se of "one" and "one", plus the complement of any nominated quantity of two or more simple numbers, according to: Lemma 11: The binary complement of 0 is 1. Excepting only by virtue of pre-conditioned numeracy therefore, that axiom of mathematics predicated by the difference 0 of two equal numbers cannot, without breaching the rules of evidence of our inquiry, inform a real number. Had we not, of necessity, stipulated that an instant normative of private information is informed prior to the determination of the primary state, we might invite the reader to verify the prediction of Rule 9, and so inform the virtual limit of binary 10, and the finite limit of the macro scale of binary 2 therefore, by the exercise of memory. But having so stipulated, we are required by our rules of evidence to account for how the mathematical inversion of 1 and 0 can be informed logically without informing the n-ary scale of binary 10. Which question we have already answered (Lemma 11): it cannot. In which case, we are obliged to ask the question, what part of mathematics is, after all, a product of human thought independent of experience?

The Invention Of Zero

4 The Invention Of Zero The first rule of engagement with which we commenced our inquiry is that we may invent neither answers nor questions, and accordingly, we have justified the invention of numbers by identifying the tautological verbal correspondence, of those quantities normative of numbers, and those quantities of numbers of which numbers are consequentially normative. But in doing so, we have disclosed a proposition which, it would seem, we must either accept or deny without evidence: while the transposition of a distance and a direction can be informed by mathematical induction, an extant translation informed naturally by determining one of a unary pair of specialized virtual data, is indeterminable. But suppose that, having drawn the smallest possible finite translations (i.e. Figure 146) determinable instantaneously, in that cycle of the particularization of translation per se (i.e. Figure 158) at one of three subjective centres priorly exemplified in Figure 123, we were to notate the random order of two, using simple symbols normative of a logical order informed literally (i.e. by writing) prior to the invention of numbers, e.g: A B C Figure 205

By virtue of the disposition (i.e. conditional arrangement) of the superscription of our positive notation (Figure 205), and the reader's being (tautologically) literate, instead of inviting the reader to inform a consequentially-logical cycle of translation in Figure 205 privately and randomly (i.e. heuristically), we may direct the reader to inform translation per se in alphabetical order, with the apodictic certainty that doing so will inform a logical (singular) direction of cyclical translation. Hypothetically prior to the invention of numbers, but post to the invention of written language, by disposing the superscription of our positive notation the verbal order of our alphabet will commonly-inform one of two (and only two) logical directions of cyclical translation, priorly chosen randomly by the publisher (i.e. Figure 205); the exclusive alternative of which direction, is: B A C Figure 206

Our positive notation being static A Priori, and there being in Figures 205 or 206 no extant point of the particularization of translation, then howsoever we may dispose our superscription, by translating as directed by the publisher, we may commonly identify the discrete division of translation per se on the unary scale of binary 10, when represented in the plane of our virtual horizon as: t = A B C A Figure 207

And as each superscription of our positive notation identifies a change of direction, we may commonly inform the sum of the discrete divisions of translation per se on the unary scale of binary 10, when represented as: 73

The Invention Of Zero

t = (A

B) + (B

C) + (C

Figure 208

Self-evidently, there can be no limit of the logical cycle of translation informed by the verbal order of our alphabet. We can, however, direct the reader to inform our positive notation (e.g. Figures 205 or 206 inclusively) in the opposite of the verbal order of our alphabet, and if able so to do, inform the sum of: t = (C

B) + (B

A) + (A

Figure 209

And self-evidently, there can be no limit of the logical cycle of translation informed in the opposite of the verbal order of our alphabet. There being no determinable difference between the sums (Figures 208, 209) of the discrete divisions of t informed on the unary scale of binary 10, we may equate them in the order of their identification, i.e: (A

B) + (B

C) + (C

A) = (C

B) + (B

A) + (A

Figure 210

Howsoever we may dispose our superscription therefore, and whichsoever of the discrete divisions of t may be identified literally thereby, by the process of elimination, the continuity of the plane of our virtual horizon is informed by the equal and opposite translations identified as: C A = A C Figure 211

In which case (Figure 211), we may equate the sums of the remaining divisions of t in the order of their identification, as: (A

B) + (B

C) = (C

B) + (B

Figure 212

Prior to the invention of a common rote normative of more than three numbers, then while the plurality normative of "three" discrete divisions of t may be informed heuristically (see Figure 121) on the unary scale of binary 10, the distance between a unary pair of perfectly equal virtual data remains indeterminable, i.e: (A

B) = (B

Figure 213

Unless, that is, we can commonly identify the distance translated cyclically in any two opposite directions of rotation. If, then, we were to invent a word with which to nominate the contradictory of 1 verbally, we could express the proposition of Rule 9 both literally and mathematically, i.e: (A

B) + (B

A) = 0

Figure 214

The evidence, that such an invention would breach our rules of evidence, is extant (see Figures 205, 206), but the invention of "zero" makes the difference of more than two equal and opposite translations axiomatic, because it nominates every possible position of 74

The Invention Of Zero

the memory of the reader informed by the stasis A Priori of our positive notation, as the sum (Figure 214) of the discrete divisions of t (see Figure 210) informed on the unary scale of binary 10 (see Figure 211): the discrete distance translated cyclically in the plane of our virtual horizon is axiomatically equal to the discrete distance determined cyclically in the plane of our virtual horizon, i.e: (A

C) + (C

A) = 0

Figure 215

By virtue of the invention of zero, the distance determined cyclically in the plane of our virtual horizon, is continuously equal to zero, i.e: A

A / B

B / C

C = 0

Figure 216

But we have stipulated that the incidence, of our positive notation and the position of the memory of the reader informed by the stasis A Priori of our positive notation (e.g. Figure 205), cannot be informed prior to the determination of a primary unilateral translation t0 initiated by the reader. In order to inform the cycle directed by the publisher, therefore, the reader has first to determine the incidence two unconditionally unequal translations in the plane of our virtual horizon. The invention of zero, however, enables the literal description of the mathematical sum, of two unequal moments of translation (see Figure 108) determined in the plane of the combined uniform translation T of memory and the primary state, prior to informing the first possible cycle of translation directed by the publisher, i.e: m0 + m1 = m1 Figure 217

In order to distinguish the zero sum (i.e. Figure 215) of equal and opposite translations informed cyclically in the plane of our virtual horizon, from the zero difference of unequal incidental translations determined cyclically in the plane of our virtual horizon (i.e. Figure 216), we will, for future reference, subscribe the publisher's notation of the cycle (i.e. Figures 205 or 206 inclusively) with the letter u for unilateral translation, and (to avoid the ambiguity of the lower case l), v for unilinear translation, e.g: Au / Av = 0 Figure 218

While the distinction between simple and incidental unilinear translation cannot be informed in the plane of our virtual horizon (Figure 218), nor represented on the exponential scale of binary 10 except by the invention of more than three numbers (see Figure 174), according to Lemma 7, at each and every change of direction superscribed by the publisher in Figures 205 or 206, the uniform translation T of memory and the primary state, is equal to the zero sum of two moments of translation we have subscribed with our secondary notation, i.e: m0 â&#x20AC;&#x201C; m1 = m0 + m1 Figure 219

Prior to informing a logical direction of cyclical translation in Figure 205 or Figure 206, the zero sum (Figure 219) of unilinear translation in the plane of our virtual horizon 75

The Invention Of Zero

(see Figure 159) informs one (i.e. Figure 205 or 206) of a unary pair of virtual data of perfect equality of proportion. Being commonly literate and numerate, then it should be possible for we the readers of our inquiry, to discover how literacy and numeracy become superimposed, and that part of mathematics therefore, that is not a product of human thought dependent upon experience? 4.1 The Supervention Of 0 We have endeavoured, unsuccessfully hitherto, to apply our rules of engagement rigorously. We have disclosed how a language may be mutually informed, and how, by virtue of a common language, simple numbers may be mutually informed: and how, therefore, the logical order of a common rote may be informed heuristically in the random order two, by a complex abstract noun qualifying the plurality of one-or-many more than two instants mutually normative of a noun (Rule 11). But it is not at Rule 11 that literacy and numeracy overlap. The noun 'two' is excused the qualification 'abstract' because the plurality of two is unconditionally extant at every possible instant normative of the primary state, which state we have denotated symbolically as 1. The symbolic denominator 2 being informed by both the plurality and the ordinance of two numbers cited by rote, noun and symbol denotate an extant quantity interchangeably. Consequentially, whatsoever quantity n may be represented numerically, it is contradicted by the quantity of 1 more than n, informed on the macro scale of binary 2 by the constancy of the direction and singularity of the primary state. According to our heuristic model of information therefore, mathematics is a product of the contradiction informed by human experience, of the instantaneous transposition of a distance and a direction, and the informant of the contradiction is the memory of the reader. In accordance with Rule 9, the distance and direction in which the primary state is informed, being equal A posteriori to the distance and direction in which it is determined, by reading our secondary notation toward–1 on the unary scale of binary 10, the trinary (n = 3) scale of binary 10 is informed as the sum of the squares of the binary complement of 1, i.e: binary 11 = 3  (–1)2 Figure 220

And accordingly, by reading our notation (Figure 220) of the trinary scale of binary 10, the finite limit of the diameter of the counterlogical spin of our mutual horizon (e.g. Figure 159), is informed in the plane of our virtual horizon on the quarternary (n = 4) scale of binary 10, i.e: trinary 11 = (1 + 3)  (–1)2 Figure 221

And so on, indefinitely, until the invention of zero: because post to the invention of zero, at any subjective centre of the rotational stasis of memory and the primary state informed by our positive notation (e.g. Figure 95), the zero sum of equal and opposite unilateral translations in the plane of our virtual horizon, coincides with the sum of two equal and opposite moments of translation (see Figure 219), i.e: m0 – m1 – m0 + m1 = 0 Figure 222

If, therefore, we notate Figure 159 alphabetically, then upon the occasion of 76

The Invention Of Zero

informing the stasis A Priori of our positive notation, zero (Figure 222) commonly defines one of three possible pairs of particular points (see Figure 161) at which the absence of translation per se (i.e. t = zero) is informed, e.g:

A C B

Figure 223

In the absence, then, of an accidental pair of natural data to inform the translator, at any particular point at which the zero sum of translation in the plane our virtual horizon is informed by our positive notation, translation in any direction requires a change of the subjective centre of the counterlogical spin of our mutual horizon (e.g. Figure 223). As, however, the stasis A Priori of our positive notation cannot be informed prior to the determination of a primary translation t0 initiated by the reader, in order to inform the logical cycle A, B, C, the reader has first to determine t0 at one of three possible particular points notated by the publisher. Informing both the stasis of our positive notation and a particular point notated A, requires the instantaneous transposition of a pair of particular points (e.g. Figure 223) normative of "zero"? And if so, then whichsoever particular point is first informed by the reader, in order to inform the direction of the cycle, the reader has first to inform the disposition of the publisher's notation, by unilateral translation within the finite limit (see Figure 221) of binary 10, in one of two possible directions. We will, therefore, generalize as X any particular point first informed by the reader in Figure 223, and Y as the second, in order to demonstrate (i.e. evidence linguistically) that which is exemplified in Figures 205 and 206: the distance X – Y of incidental unilinear translation required to inform the disposition of the publisher's notation, cannot be informed logically as zero, prior to informing the cycle Y – Y. The consequence of which deduction (i.e. subtraction) is exemplified logically in either Figure 205 or Figure 206: the reader can count (see Figure 122) the smallest possible number of discrete, logical unilinear translations, required to inform the determination of translation (i.e. Y – Y = zero), in the plane of our mutual horizon. Having, defined the speed S of the instantaneous unilinear transposition of any two positions of an extant equality of proportion (see Figure 69), a singular and subjectively continuous moment of translation toward–1, informed discontinuously at any one of three particular points (e.g. Figure 223), is transformed by zero into a virtual distance, determined in the plane of the logical cycle Au, Bu, Cu, between the subjective centre of the counterlogical spin of our mutual horizon, and each in turn, of the particular points superscribed by the publisher: which virtual distance can be represented positively as the radius r0 of any two circles (see Figure 223) we may care to draw, e.g:

The Invention Of Zero

Figure 224

The virtual distance between any two static artificial data informed heuristically in Figure 205 or Figure 206, being continuously equal to one fewer than two radii of a circle of diameter d0 (e.g. Figure 223), the distance of unilinear translation determined in a single plane, and limited by zero upon the count of four, is described mathematically on the quarternary scale of binary 10 (cf Figure 104), as: X – Y = 0 – (1 + 3)  r0 Figure 225

From the invention of zero, we can commonly infer (i.e. determine categorically) by the logical addition of the radii (Figure 225) of the great circle of a virtual sphere of diameter d0, the quantitative equality of: quarternary 11 – 1 = binary 11 Figure 226

And the greatest possible finite inequality (see Figures 204, 111) of two unconditionally unequal dimensions of the primary state, is informed quantitatively on the unary scale of binary 10, as: binary 11 : binary 10 = 3 : 2 Figure 227

Quarternary 11, however, cannot be informed prior to equating our secondary notation with a simple quantity of one-or-many more than four numbers cited by common rote. Literacy and numeracy overlap at zero, post to which invention there can be no Lemma 10 informing Rule 10, according to: Lemma Zero: When 11 – 1 = 10 then 10 = 2. In plain words, Lemma Zero states: two equal planes of logical unilinear translation can be neither determined nor informed. The invention of zero, then, supervenes (i.e. displaces) the answer to that singular question priorly informed by the medium of our inquiry: the finite difference between any two incidental planes of unilinear translation being 'zero', defines the singularity of both. We note that while the verbal evolution of a language and the invention of an alphabet predicating the written forms thereof, fall prior to (and so beyond the scope of) our inquiry, provided only that an alphabet constitutes one-or-many more than two unique denominators of the medium (e.g. sight, sound, smell, taste, touch, or other unknown to the publisher) by which it is shared, the order prescribed by the citation of an alphabet, is extant in its recitation. The invention of the zero sum of more than two equal and opposite unilinear translations in a single plane, then, being that part of mathematics that is, after all, the only product of human thought independent of experience, we are obliged by our rules of engagement to ask, is mathematics admirably adapted to the objects of reality? 78

The Objects Of Reality

5 The Objects Of Reality At this critical juncture of our inquiry, it is perhaps expedient to review the nature of the objects of reality in our emergent model of information. We have defined an object of reality as any singular instant that may be naturally normative of a noun. We have stipulated no preconditions upon the origins of such instants. Accordingly, with one exception, objects of reality cannot be normative of abstract nouns: the exception being, if coined within a language, those numbers commonly informed heuristically, by an extant plurality of one-or-many more than two instants naturally normative of a singular noun. Although the invention of the complex abstract noun 'zero' is self-evidently in breach of our rules of evidence, the distinction, between the macro scale of binary 2 on which we mutually inform simple quantities, and the unary scale of binary 10 on which we delimit simple numbers in the order of 1, could not otherwise be commonly inferred. In order to discover if mathematics is admirably adapted to the objects of reality, we must persist with that mathematical model of real numbers, as naturally-informed by artificial data through the medium of our inquiry. 5.1 Artificial Information In the course of our inquiry hitherto, we have mutually identified that special condition of the medium in which it is presented, as the ability it affords the publisher to represent discontinuous objects of reality, in an order in which they are logically continuous. In Figure 121, for example, we have generalized the subjective continuity of virtual data, in the order in which they are informed heuristically: and Figure 121 constituting one-or-many more than two objects of reality, Rule 11 is informed naturally by Lemma 11 prior to the first possible occasion of informing the zero sum of many translations in the plane of their determination. With the proviso therefore, that we do not equate our secondary notation with a simple quantity of fewer than four numbers, the scalar difference of the unary and the quarternary scales of binary 10, can be exemplified in either one of two possible directions of logical translation, as increasing unilinearly, e.g: binary 111 = 1 + 2 + 4 / 4 + 2 + 1 Figure 228

Which uni-dimensional scalar of magnitude (Figure 228), being both selfevidently and necessarily lesser than macro scale of binary 2, we will qualify as the micro scale of binary 10. There being no logical limit of simple symbols we may draw to denominate simple numbers, then with the additional proviso that we do not redefine 0 as 'zero', the uniformly-increasing fractional difference of the unary and n-ary scales of binary 10, can be exemplified indefinitely on the micro scale of binary 10, e.g: binary 11111 = 1 + 2 + 4 + 8 + G / G + 8 + 4 + 2 + 1 Figure 229

The unary scale of binary 10 (Figure 229), therefore, exemplifies an extant analogue of a unilinear translation in the plane of our virtual horizon, which can be described mathematically as a distance (see Figure 203) which increases uniformly within the limits of 1 and 0 as n â&#x20AC;&#x201C; 1, where n nominates a simple quantity of artificial objects (1's) 79

The Objects Of Reality

counted verbally. The micro scale of binary 10, therefore, can be exemplified by the symmetrical division of any real analogue of unilinear translation (e.g. abacus, adding machine, computer) which can be described mathematically (Rule 12) as increasing exponentially in the random order of 2, e.g. in Figure 229: binary 11111 = (G –1)2 / (G – 1) Figure 230

With the said provisos, therefore, as the distance (n – 1) exemplified on the unary scale of binary 10 (Figure 230), increases logically in the order of rote, the binary complement of 0 increases uniformly as the square of the binary complement of 1, and the micro scale of binary 10 increases exponentially in the order of 2, as the sum of the squares of the binary complement of 1. We note that the micro scale of binary 10 is predicated rationally (cf Figure 178), without the expression of multiples of simple numbers, as the exponential sum (see Figure 157) of the divisors of binary (10)n – 1. If, therefore, we generalize the plurality q represented by any simple number counted on the micro scale of binary 10, the complement of binary 10 is described discretely, as: binary (10)n = q2 Figure 231

Every possible complement of binary 10 then, being equal to a numerical square (Figure 231), the binary complement of 1 can be represented as the diameter d0 (see Figure 204) of one, of any two circles of radius r0 we may care to draw, e.g:

Figure 232

As exemplified by any extant analogue of the micro scale of binary 10 therefore, the finite limit of binary (10)n can be represented within the limits of experimental error, as the sum of two squares of side r0, i.e:

Figure 233

From which finite limit (Figure 233) we may deduce the velocity of unilinear translation, informed naturally on the micro scale of binary 10 by the determination of either one, of a unary pair of perfectly equal virtual data, i.e: q2 / 2 = r02 Figure 234

And if so (Figure 234), the finite ratio, of the moment of the rotational stasis of the 80

The Objects Of Reality

incidence, of the plane of our virtual horizon and the counterlogical spin of our mutual horizon (see Figure 174), can be commonly-informed mathematically as: M : 2  r0 2  S = 2 : 1 Figure 235

And accordingly (Figure 235), the position of memory being static A Priori, the moment of the logical spin of the plane of our virtual horizon can be commonly informed mathematically as: M / 2 = r0 2  S Figure 236

That which is described mathematically on the micro scale of binary 10 (Figure 236), then, can be exemplified within the finite limit of binary (10)n, by an extant analogue drawn in the plane of our virtual horizon, i.e:

Figure 237

Figure 237 is an artificial representation, drawn faithfully on the model of Figure 233, of the finite limit of the logical spin of our virtual horizon, as described mathematically by Figure 236. And we can demonstrate categorically, that the virtual limit binary 102 of binary 10 (Lemma 12), is equal to sum of two real squares drawn to scale in the plane of our virtual horizon, i.e:

Figure 238

In order to distinguish real objects (e.g. Figure 238) naturally-informed by drawing, from that which is informed mathematically, we will qualify the former as artificial information. The virtual limit binary 102 of binary 10, as informed artificially by Figure 238 on the micro scale of binary 10, can be commonly described as the sum of two lesser squares drawn in the single plane presented by the medium of our inquiry, i.e: binary (10)n = r02 + r02 Figure 239

We note the difference between the virtual limit binary 102 informed (Lemma 12) by mathematical induction on the macro scale of binary 2, and the limit (Figure 239) informed on the micro scale of binary 10 by simple addition: on which latter scale, the diameter of the counterlogical spin of our mutual horizon (cf Figure 163) can be represented artificially, as:

The Objects Of Reality

Figure 240

The singular moment of translation between any two instants of a finite equality, being continuously equal to the unilinear velocity determined in the limit of their instantaneous accidence (Lemma 8), then upon the occasion of informing of any two of three hemispheres exemplified in Figure 240, we naturally inform the artificial divisor of two hemispheres, as the semi-circular arc of a finite sphere, divided symmetrically by the plane of our mutual horizon, i.e:

Figure 241

Subjectively, therefore, Figure 241 exemplifies the logical continuation in the plane of t, of the diameter of a finite sphere, artificially informing the translator at a single point at which translation per se (see Figure 10), in every possible finite plane, is particularized, i.e:

Figure 242

Figure 242 exemplifies an indeterminable isologue of the finite limit (see Figure 237) of binary 102, informed naturally by the determination of either one (but not both), of a unary pair of hemispheres of a virtual sphere, the subjective diameter of which sphere is conditional upon the exponential increase or decrease (see Figure 239) of the binary complement of 1 the order of 2, i.e: binary 10 = r0 + r0 Figure 243

And if so, then Figure 242 exemplifies an artificial isologue of a lesser isologue (Figure 237) informed rationally towardâ&#x20AC;&#x201C;1, of the subjective moment of the rotational stasis of memory and our positive notation, i.e:

Figure 244

The Objects Of Reality

There being no logical limit of the micro scale of binary 10, then the diameter of a virtual sphere (see Figure 243), informed artificially on the micro scale of binary 10, changes discretely ad infinitum as the diagonal, of an abstract square increasing in the order of 2 (see Figure 118): which sphere, therefore, we will qualify as the macro sphere of binary 10. And we note that while there can be no logical limit of the micro scale of binary 10, the limit of any extant analogue of unilinear translation that can be described mathematically as increasing exponentially in the random order of 2 (see Figure 234), is conditional upon its artificial representation: any such artificial memory is, tautologically, bi-dimensional (i.e. of two dimensions). With the provisos therefore, that we neither redefine 0 nor identify our secondary notation with any simple quantity, we can formulate that which informs Rule 10, in accordance with: Lemma 10: binary (10)n / 2 = r02. In plain words Lemma 10 states: the finite limit of every possible sphere is subjectively informed in the plane of its own great circle. 5.2 The Supervention Of Binary 10 The question asked by Rule 10 is of the continuity of a simple plurality of planes of unilinear translation, evidenced by any extant instant of a single object of reality informed by translation. The self-evident answer (Lemma 10) to that question defines a numerical relationship, between an artificial memory informed on the micro scale of binary 10, and the macro sphere of binary 10 informed by human experience: provided, that is, human experience includes that of both written language and pre-conditioned numeracy. The reader may deduce from Figure 239 the scalar difference, of an artificial memory informed on the unary scale of binary 10, and that finite limit, of the great circle of the macro sphere of binary 10 informed by human experience: where n – 1 is a distance informed by counting virtual data represented by objects of reality on the unary scale of binary 10, the micro scale of binary 10 increases rationally as the sum of the exponential divisors (see Figure 178) of 1, e.g: 11111 =

1 (10)

0 n

n 0

Figure 245

And there being, on the micro scale of binary 10, no objects of reality but those ratios of simple quantities (Figure 245) expressed numerically in a logical order of 2, the diameter, of any great circle of the macro sphere of binary 10, as limited by any extant artificial memory that can be described mathematically as increasing in the random order of 2, being tautologically equal (Lemma 10) to the side of the square of n, can be exemplified by a Figure scaled by any circle we may care to draw within the limits of our virtual horizon, e.g:

Figure 246

The Objects Of Reality

And there being no logical limit of the micro scale of binary 10, the finite limit of the unary scale of binary 10 can be described mathematically (see Figure 245) as the sum of the exponential divisors of an abstract square of side (n –1) i.e: (n –1)2 =

1 (10)

0 n

n 0

Figure 247

The micro scale of binary 10, therefore, affords an indefinitely flexible model of the bi-dimensional limit of the macro sphere of binary 10: post to the extant representation of which model (e.g. Figure 246), the divisors (n – 1)2, as described rationally on the micro scale of binary 10 (Figure 247), can be drawn to scale, i.e:

Figure 248

Within the limits of experimental error, therefore, Figure 246 represents the continuity of the macro sphere of binary 10, limited by the plane of our virtual horizon when n is 2, i.e:

Figure 249

And if so, then a real object informed artificially, and subjectively normative of the macro sphere of binary 10 (Figure 249), can be represented within the limit of an abstract square (Lemma 12) by any single circle greater than a point, that we may care to draw within the finite limit, of an artificial memory represented by the plane of our virtual horizon. But having drawn any such circle, our ability to represent the exponential divisors of binary (10)2 as real objects (e.g. Figure 248), is limited by our ability to discern the difference between a circle and point: within which limits, the greater the circle we draw, the more the exponential divisors of binary (10)2 we can present. The mathematical description of the micro scale of binary 10 (see Figure 247), however, is of a finite memory which can be exemplified artificially only by the unary scale of binary 10, to which there is no logical limit: Figure 247 describes the bidimensional limit binary (10)2 of an abstract memory, increasing uniformly as the sum of its exponential divisors, in the ratio of the diameter and the radius of a circle. But when an object of reality subjectively-normative of the macro sphere of binary 10 is presented in the plane of our virtual horizon (e.g. Figure 249), the micro scale of binary 10 (see Figure 247) describes the divisors of an abstract memory decreasing exponentially, in the constant ratio of the abstract squares of the diameter and the radius of circle, i.e: (n)2 : (n / 2)2 = 4 : 1 Figure 250

The Objects Of Reality

In the absence of every possible object of reality, that which is described mathematically on the micro scale of binary 10, is the indeterminable constancy of the ratio, of the moment of the logical spin of the plane of our virtual horizon, to the counterlogical spin of our mutual horizon. The informant of which ratio being that singular, natural datum we have defined as our mutual horizon, then by means of the rotation, in either one of two possible logical directions, of that positive notation of the virtual limit of the counterlogical spin of our mutual horizon exemplified in Figure 179, we can mutually inform a singular moment of rotational stasis at which the plane of our mutual horizon coincides with the plane of our virtual horizon, i.e:

Figure 251

In Figure 251 we may mutually observe the bi-dimensional limit of the macro sphere of binary 10 (e.g. Figure 248), supervened by a circle predicated on the micro scale of binary 10 when n = 2, by the moment of the rotational stasis of the incidental planes of our mutual and virtual horizons. And if so, then we can demonstrate categorically that the limit of our virtual horizon is constantly equal to the ratio of a semi-circular arc c (Figure 251) of the great circle of the macro sphere of binary 10, to the diagonal of the square of binary 10, i.e: M : S = binary (10)2 : binary (10)2 / 2 Figure 252

The reader may recognize in Figure 252 a mathematical description, of that contrary of the inclusive state we formulated linguistically (Rule 4) as 'something and nothing exclusively', as the logical continuation of an abstract bi-dimensional plane binary (10)2, the side of which (and not the diagonal) is limited by the ratio of the circumference and the diameter of a virtual circle, at the moment of the incidence of two planes of unilateral translation, i.e: M = S+S Figure 253

The only possible evidence of which limit (Figure 253) is its supervention (e.g. Figure 251), by the drawing of a circle subjectively normative of the macro sphere of binary 10. With the provisos attending the formulation of Lemma 10, therefore, we can infer by simple addition, that the limit of the macro sphere of binary 10 is equal to the ratio of the circumference and the diameter of its own great circle, i.e: M = C:d Figure 254

The limit of the macro sphere of binary 10 (Figure 254) being informed artificially by a real circle (see Figure 251), then we can deduce that at the moment M of unilinear 85

The Objects Of Reality

translation between any two perfectly equal planes, the exclusive contrary of the macro sphere of binary 10, is proportionally equal (see Figure 174) to the difference between a real circle normative of a virtual sphere, and a greater circle described mathematically (Figure 254) at that singular moment M of unilinear translation commonly represented as ď °, i.e: M â&#x20AC;&#x201C; M/2 = M/2 Figure 255

And if so, then we can re-formulate Rule 4 mathematically, as: Rule 4: If M / 2 then M ? In plain words, Rule 4 asks: the diameter of every possible sphere is equal to twice the distance determinable at the speed of instantaneous unilinear translation? 5.3 The Supervention Of Binary (10)2 As expressed by the re-formulation of Rule 4 (above), it may appear that we can demonstrate, categorically, that the exclusive contrary of something and nothing, is an extant sphere of finite but indeterminable proportions. We note, however, that the relationship expressed by Rule 4, of mathematics and objects of reality, and every possible categorical demonstration thereof, either informs or is informed by the unary scale of binary 10: that which is described by Rule 4 is the subjective continuity of a real plane presented by the medium of our inquiry, in which a circle is the artificial object, and not the subject, of the noun 'memory', and the continuity of the moment of translation is the effect (Lemma 2), and not the cause, of the subject memory. But if the zero difference of two equal and subjectively static planes of unilinear translation is informed at the moment M / 2 of the rotational stasis of our mutual and virtual horizons (e.g. as in Figure 223), then for every possible real circle that may be normative of the macro sphere of binary 10, a virtual circle is determined at the moment M of the instantaneous unilinear transposition, of one of two incidental pairs of virtual data. The diameter of which lesser circle, defines a virtual distance (see Figure 224) determined between any two, perfectly equal virtual data, i.e:

Figure 256

The diameter of which lesser circle (Figure 256) is equal to the side of a square lesser than binary 102, i.e:

Figure 257

The Objects Of Reality

The side of which lesser square (Figure 257), being equal to the radius of the macro sphere of binary 10 (see Figure 256), the lesser square is mathematically equal to: r02 = binary (10)2 / 4 Figure 258

According to Lemma Zero, therefore, at the moment of informing the zero sum of any more than two equal and opposite unilinear translations in a single plane (e.g. Figure 257), the speed of unilinear translation in that plane (see Figure 138) is numerically equal, to the square of the radius of the macro sphere of binary 10, i.e: r0 2 = 1 Figure 259

And thereafter, and in the absence of any real objects normative thereof, discrete instances of the uniform exponential increase of the great circle of the macro sphere of binary 10, are described (see Figure 247) in the logical order of our common rote, in the ratio (see Figure 250) of the abstract squares of the diameter and the radius of circle, i.e: C/2 = Ď&#x20AC;Ă&#x2014;n Figure 260

We note that Figure 260 describes a relationship of mathematics to an abstract memory, and to no object of reality but the plane presented by the medium of our inquiry, which proposition the reader may verify by supervening the macro sphere of binary 10 with two mutually exclusive divisors of a singular secondary instant, informed on the macro scale of binary 2, by the unilateral determination of the primary state. Provided that we do not redefine the binary complement of 0 as 'zero', we can formulate in plain words that relationship, of the unary and micro scales of binary 10 normative A posteriori of Lemma 10, as: Rule 0: No more than two equal and opposite unilinear translations can be determined in a straight line ? Rule 0 asks a question, disclosed by any extant analogue of the unary scale of binary 10 informed by human experience: how, other than by the invention of the complex abstract noun zero, can the subjects of an abstract memory be related mathematically to objects of reality?

The Subjects Of Memory

6 The Subjects Of Memory It would seem that any possible answer informing Rule 0 must, of necessity, be informed A Priori. The advantage afforded by pre-conditioned numeracy, however, enables the representation of those finite exponential divisors, of an abstract of memory described mathematically within the limits of binary 102, on the scale of any singular, real circle artificially normative of the macro sphere of binary 10. And at every possible instant of a naturally-informed object of reality which is an isologue of neither a circle nor a straight line, a singular artificial memory is informed by a plane (e.g. Figures 50, 51), the dimensions of which are limited by an accidental pair of unconditionally unequal natural data, e.g:

Figure 261

A circle being not an isologue of the dragon exemplified in Figure 261, then by drawing a circle in the plane of the dragon, the diameter of which circle (see Figure 80) exemplifies the combined logical spin of a unified pair of mutually exclusive primary data, we can represent the finite limit, of a virtual sphere informed by their instantaneous transposition, on the macro scale of binary 2 (e.g. Figure 119), i.e:

Figure 262

Which virtual sphere (Figure 262), being self-evidently lesser than the macro sphere of binary 10, we will qualify as the micro sphere of binary 2. But, if we now draw a square in the plane of the dragon, the diagonal of which square is equal to the diameter of the micro sphere of binary 2 (cf Figure 257), then we can exemplify the virtual limit d02 of binary 10 (Lemma 12), scaled by an object of reality greater than a point, that is an isologue of neither a circle nor a square, e.g:

Figure 263

The limit of binary 102 being a virtual sphere, then an artificial, bi-dimensional memory represented on the unary scale of binary 10 (see Figure 220), can be related mathematically to an object of reality (Figure 263) on the trinary scale of binary 10, by an abstract subject of memory (Lemma 11) of constant, but indeterminable proportions, i.e: trinary 11 = binary 102 Ă&#x2014; binary 10 Figure 264

If the reader, therefore, can imagine a cube described by Figure 264, then the finite limit of the macro sphere of binary 10, can be informed mathematically (Figure 264) on the 88

The Subjects Of Memory

macro scale of binary 2, at the first and every possible instant mutually normative of an object of reality. And if so, then if we return to the first figure of our inquiry (Figure 1, Page 2), the reader may recognize an object of reality (a square) in the plane presented by the medium of our inquiry, dividing an abstract subject of memory (a cube) which can be represented fractionally (see Figure 137) on the quarternary scale of binary 10, i.e: trinary 11 / 2 = binary 103 / 2 Figure 265

We note that the mathematical description (Figure 265) of the symmetrical division of an abstract cube, is precise (i.e. not subject to experimental error), and the proportions thereof immutable (i.e. not subject to change). Figure 265 is a mathematical description of one of two halves, of a singular subject of memory (the content of the box of Figure 1), informed naturally on the macro scale of binary 2 by the dimensions of a real square, i.e: binary 103 / 2 = 22 Figure 266

The diagonal of which square (Figure 226) being equal to the diameter of a virtual sphere, the question we are addressing is, how can the difference, between the micro scale of binary 10 informed on the scale of the macro sphere of binary 10 (e.g. Figure 1), and the micro scale of binary 10 informed on the scale of the micro sphere of binary 2 (e.g. Figure 263), be informed? I.e:

Figure 267

6.1 The Principle Of Apodictic Certainty In accordance with our re-formulation of Rule 4, the order of trinary 11 (see Figure 264) normative mathematically of the content of the box of Figure 1, is mutually extant in the plane of our virtual horizon, by virtue of the momentary difference (see Figure 255) of a static position informed by an object of reality, and the perfectly equal (and static) position of the subject memory it informs. We have attributed the order of two halves, of a cube informed rationally on the trinary scale of binary 10, to the scalar quantity of three planes extant in our drawings. And by virtue of our pre-conditioned numeracy, the sum of two halves of singular subject of memory is informed fractionally (see Figure 265) on the quarternary scale of binary 10, on which the dimensions of an object of reality (see Figure 266) are described in the order of 2. In the case of Figure 1, the diagonal of the square of 2 being equal (see Figure 233) to the radius of the macro sphere of binary 10, the continuity of an abstract volume can be described mathematically on the quinnary (i.e. n = 5) scale of binary 10, as the sum of the cubes of the diameter of the micro sphere of binary 2, i.e: binary 103 / 2 = 4 Ă&#x2014; r03 Figure 268

If, in the absence of any information but that of Figure 268, the reader can imagine 89

The Subjects Of Memory

a singular volume (we note the complex abstract noun), then that which is informed mathematically by Figure 268 is precise, and the proportions thereof immutable. And if so, then the finite limit of the macro sphere of binary (10), as informed by Figure 233 within the limits of experimental error, can be informed precisely on the macro scale of binary 2, as: binary 103 = 2 × 4 × r03 Figure 269

At the moment M / 2 of the rotational stasis of our mutual and virtual horizons therefore, the virtual limit binary 102 (Lemma 12) of the counterlogical spin of our mutual horizon, is precisely equal (Figure 269) to the sum of four squares of side r0, e.g. one of either:

Figure 270

The diameter of a virtual sphere, informed artificially by our positive notation (e.g. Figure 249) at the moment M / 2 of the counterlogical spin of our mutual horizon, is mathematically equal to the side of the bi-dimensional limit by which it is determined, i.e: binary 102 = 4 × r02 = 2 × d02 Figure 271

The limit of the singular moment M (Lemma 7) of continuous unilinear translation in a single plane therefore, is described precisely and immutably on the nonary (i.e. n = 9) scale of binary 10, as: binary 102 × 2 = 8 × r02 = d03 Figure 272

Inside (i.e. within) which limit (Figure 272), the n-ary scale of binary 10 affords a precise model (see Figure 260) of the sum of the symmetrical, bi-dimensional divisors (see Figure 118) of a singular abstract subject of memory, informed rationally by real numbers on the macro scale of binary 2, i.e: (n – 1)2 / 2 = binary

 S  (10)

n 0

Figure 273

While the primary state continues outside (i.e. without) the limit d03 (see Figure 272) of the moment M of continuous unilinear translation in a single plane therefore, the micro scale of binary 10 affords an indefinitely flexible model (see Figure 246) of bi-dimensional multiples of binary 103 (see Figure 269), informed by complex numbers on the octernary scale of binary 10, i.e: x × binary 103 = n × (8 × r02 ÷ d03 ) Figure 274

According to Lemma 10 therefore, on the macro scale of binary 2, the radius of the 90

The Subjects Of Memory

macro sphere of binary 10 (see Figure 273) is precisely equal to the side of the mathematical square of 8, as nominated in the verbal order prescribed by common rote, i.e: r02 = 1 × (8 × r02 ÷ d03 ) Figure 275

And accordingly (Figure 275), we can re-formulate Rule 5 mathematically, as: Rule 5: While the primary state continues 1 = r02 ? In plain words Rule 5 asks: the singularity of multiples of complex numbers is the subject of memory? The re-formulation of the proportional inequality (Rule 5), of the micro scale of binary 10 informed inside the limit of the moment of continuous unilinear translation in a single plane, and the micro scale of binary 10 informed outside the said limit, being precise and immutable, formulates the principle of apodictic certainty with which the subjects of memory are related mathematically to artificial objects of reality. 6.2 The Scalar Of The Macro Sphere Of Binary 10 We have defined the conditional substantive 'while' of our re-formulation of Rule 5 (above) as 'unless or until', from which we may infer that a proportional inequality informed numerically, and the principle of apodictic certainty arising therefrom, is subject to the continuity of the memory of the reader; which proposition can be verified with apodictic certainty by the exercise of memory; but the absolute verification of which, requiring as it would the determination of memory, is unavailable. In order to distinguish that certainty subject to the continuity of the memory of the reader, from that certainty predicated by the continuity of a common, written language, we will qualify the latter as mathematical certainty. We have stipulated, however, that the singularity of the primary state is privately informed by natural intelligence prior to the determination of the primary state, at a singular instant normative of both the primary state and the translator (Rule 7). The continuity of the primary state and of its symbolic representative, constitute in isolation the real number of two objects of reality, the subjective singularity of each of which is, in accordance with rule 5, disclosed toward–1 on the octernary scale of binary 10, in the ratio of the binary complement of 0 to the square of the radius of a virtual sphere, e.g: 1

Figure 276

Figure 276 exemplifies the moment of the rotational stasis of our mutual and virtual horizons, informed (see Figure 204) on the macro scale of binary 2 at the moment M / 2 of unilinear translation in a single plane. In order to exemplify the macro sphere of binary 10 on the macro scale of binary 2, therefore, we have only to delimit that plane of our virtual horizon exemplified in Figure 276, inside the virtual limit binary 102 (Lemma 12) of the counterlogical spin of our mutual horizon (see Figure 271), e.g: 91

The Subjects Of Memory

1 Figure 277

As a consequence of which delimitation, and regardless of the inequalities of proportion informed by our positive notation, Figure 277 exemplifies three artificial objects of reality, the subjective singularity of each of which is predicated mathematically inside the limit d03 of the moment M (see Figure 272) of continuous unilinear translation in a single plane. The reader may verify that at the moment of the incidence of our mutual horizon and our delimited virtual horizon (Figure 277), the logical spin of our positive notation (see Figure 236) defines a virtual circle, i.e. one of either:

Figure 278

The diameter of which virtual circle (Figure 278), being equal to the side of the finite limit of the counterlogical spin of our mutual horizon, the said limit can be drawn to scale outside the moment of the logical spin of our positive notation, i.e. one of either:

Figure 279

The delimitation of the plane of our virtual horizon inside binary 102 (e.g. Figure 277), scales the moment of the counterlogical spin of our mutual horizon (Figure 279), in the proportion of the radius and the diameter of a virtual circle, predicated mathematically (see Figure 275) on the nonary scale of binary 10, i.e: 22 : 23 = 4 : 8 Figure 280

The diagonal of the square (Figure 279) being equal to the side of binary 102 (see Figure 271), then by virtue of the extant representation (see Figure 279) of the square 23 predicated mathematically on the nonary scale of binary 10 (Figure 280), the great circle of what we have defined as the macro sphere of binary 10 (see Figure 244), can be drawn on the macro scale of binary 2 outside the square, i.e. one of either:

Figure 281

The Subjects Of Memory

In accordance with Rule 5, the cube 23 predicated on the octernary scale of binary 10 (Figure 281), exemplifies the finite limit of the macro sphere of binary 10, as a singular subject of memory disclosed inside the limit d03 (see Figure 272) of the moment M of continuous unilinear translation in a single plane, i.e: 22 = 23 / 2 Figure 282

Mathematically, therefore, on the macro scale of binary 2, the finite bi-dimensional limit of the macro sphere of binary 10 (Figure 282), is the symmetrical divisor of the unified sum of four cubes (see Figure 268) of the radius r0, of a circle of diameter d0, scaled by any artificial delimiter of the plane of our virtual horizon that is lesser than binary 102, e.g:

Figure 283

The moment of the counterlogical spin of the square 22 is exemplified inside the cube of 2, as two multiples of the greatest possible speed of the momentary unilinear transposition, of any two of three possible hemispheres (e.g. Figure 283), constituted a virtual sphere by the consequential stasis, of the third of three possible hemispheres limited, at the moment M / 2 of unilinear translation in a single plane, by the circumference of a singular circle, i.e. one of either:

Figure 284

It is mathematically certain, therefore, that the greatest possible straight line informed toward–1 by any isolated instance of our positive notation (e.g. Figure 276), is equal to the semi-circular arc c of the great circle of the macro sphere of binary 10, i.e: c / 2 = 22 × S Figure 285

Mathematically, therefore, at the moment M of continuous unilinear translation in a single plane (e.g. Figure 284), the semi-circular arc of the great circle of the macro sphere of binary 10, is precisely (cf Figure 249) equal to: c = 23 × S Figure 286

At the moment of the rotational stasis of the artificially delimited plane of our virtual horizon (e.g. Figure 284) therefore, the counterlogical spin of our mutual horizon (a singular natural datum) is the consequential scalar of two virtual hemispheres, the 93

The Subjects Of Memory

momentary transposition of which, when nominated alternately in the order of 1, defines the macro sphere of binary 10 as a singular subject of memory, unified in the order of 1 and 0, i.e. one of either:

Figure 287

The circumference of the great circle of the macro sphere of binary 10 increases uniformly on the macro scale of binary 2, in proportion (cf Figure 204) to the discrete mathematical multiples of a hemisphere (see Figure 286) informed on the octernary scale of binary 10, inside the virtual limit of the counterlogical spin of our mutual horizon (Figure 287), i.e: C = n × 2 × 23 × S Figure 288

We note, however, that the mathematical model of discrete bi-dimensional divisors of the cube of 2 (see Figure 275) predicating the principle of apodictic certainty (see Figure 282), is necessarily informed inside the finite bi-dimensional limit 22 of the macro sphere of binary 10 (see Figure 287), begging the question of how a singular subject of memory can be informed by complex numbers (Rule 5) without the prior generalization of the cube of d0. 6.3 The Scalar Of The Macro Scale Of Binary 2 In order to comply with the rules of evidence of our inquiry, we have distinguished simple nouns verbally-normative of simple quantities by virtue of tautology, from complex nouns qualifying the plurality of abstract numbers, by requiring real numbers to be predicated by an extant quantitative difference, recounted counterlogically between any two numbers cited by common rote. The consequence of which stipulation is that, when we equate our secondary notation with any simple quantity of numbers, the mathematical inversion of 1 and 0 informs the sum of 10 = n squares of the binary complement of 1, scaled rationally by 10 –1 real numbers within the limits of 0 and 1, i.e: n-ary 10 =

 (10  1)

n 0

Figure 289

We note that n-ary 10 (Figure 289) is informed by the simple addition (cf multiplication) of the increasing quantity, of uniformly-generalized squares that diminish (i.e. decrease uniformly) in the logical order of our common rote, when expressed as the divisors of the binary complement of 0, i.e: 1 / (n)2 = 1 /

 (10  1) n 0

Figure 290 94

The Subjects Of Memory

To the numerate reader therefore, a singular, bi-dimensional divisor of a self-evident but unlimited singularity, increases on the n-ary scale of binary 10, as n multiples of the generalized square of n –1, i.e: n-ary 102 =

10  (10  1)

n 0

Figure 291

The extant representation of which square (Figure 291) on the macro scale of binary 2 (see Figure 281), constitutes the virtual, and unconditionally subjective scalar of the macro sphere of binary 10. In order, then, to exemplify a common scalar the macro scale of binary 2, we have only to delimit the plane of our virtual horizon with a square, at an instant mutually normative of an object of reality that is neither a cube nor a sphere, e.g:

Figure 292

Where the binary complement of 0 nominates the primary state mathematically (see Figure 290), Figure 292 exemplifies three objects of reality, informed rationally (cf Figures 277, 280) in the decreasing order of two asymmetric divisors of the primary state. At which instant, the logical spin of an object of reality defines the semi-circular arc of a circle, in one of three possible planes of unilinear translation, i.e:

Figure 293

The moment of the rotational stasis, of our mutual horizon and the consequentially counterlogical spin of the square (Figure 293) in a singular direction (cf Figure 284), is exemplified outside the square, i.e:

Figure 294

At which moment of rotational stasis (Figure 294), the diagonal of the square, informed in a single plane outside a virtual sphere unified by memory, exemplifies the greatest possible straight line determinable in a single plane of unilinear translation, when scaled by an object of reality that cannot be an isologue of a singular subject unified by memory, i.e:

The Subjects Of Memory

Figure 295

At the moment (M / 2) of the incidence of our mutual and virtual horizons then, the greatest possible distance determinable on the macro scale of binary 2 (Figure 295), is equal to the diameter of a circle defined in a virtual plane informed outside a virtual sphere, i.e:

Figure 296

The square of the diameter of which circle can be drawn (within the limits of experimental error), on the macro scale of binary 2, in the plane of the counterlogical spin of our mutual horizon, i.e:

Figure 297

Which greater square (Figure 297), being equal to the virtual limit of the counterlogical spin of our mutual horizon (Lemma 12), exemplifies binary 102, commonlyinformed by three objects of reality scaled by the dragon. The great circle of what we have defined as the macro sphere of binary 10 (see Figure 244), can be drawn on the macro scale of binary 2 inside binary 102 (cf Figure 281). The lesser square (Figure 297) therefore, exemplifies the square of binary 10, informed rationally on the macro scale of binary 2 by the difference of two squares, i.e:

Figure 298

On which scale (Figure 298) binary 10 is exemplified by the semi-circular arc c of a virtual sphere, the diameter of which sphere is equal to the side of the square of binary 10. Mathematically, therefore, the micro scale of binary 10 (see Figure 247) describes the divisors of the bi-dimensional limit binary 102 of the counterlogical spin of our mutual horizon, decreasing exponentially in the constant ratio (see Figure 250) of the square of the diameter of the macro sphere of binary 10, to the square of binary 10, i.e: 96

The Subjects Of Memory

M / 2 = 2 / c2 Figure 299

When scaled by any singular object of reality that cannot be an isologue of a singular subject of memory, the virtual limit of the counterlogical spin of our mutual horizon (Lemma 12), is described precisely and immutably on the macro scale of binary 2, as: binary 102 = M × c2 Figure 300

Figure 300, then, exemplifies the question, and not the answer, that we have made the subject of our inquiry: what, then, is the scalar of a singular object of reality? 7. The Objects of Mathematics We have attributed the principle of mathematical certainty to the continuation, independently of any single memory, of a common language in which both the order of simple numbers cited by rote, and the logical direction of translation in which they are read, are predicated by the word-order of the language in which they are written. The consequence of which is that, when the tautological quantity of eight simple numbers counted counterlogically, is identified with their symbolic denominators (the pre-condition of numeracy), every possible number but 2 is either a multiple or a fraction of a multiple, of a square of 8 unified by memory, whence that tautological structure normative of the logic of mathematical induction. We have noted that the limit, of any extant analogue of unilinear translation that can be described mathematically as increasing exponentially in the random order of 2, is conditional upon its artificial representation in two dimensions, e.g. by the keyboards of such adding machines as abacuses, pianos and computers, or by the artificial representation of the unary scale of binary 10, in the order of 1 fewer than 2. The axiom contended by our rules of evidence, therefore, is that logical unilinear translation in a straight line, can relate to or be related by real numbers, because on the octernary scale of binary 10, the difference (see Figure 228) between the complement of binary 10 and 1, is 7, i.e: binary 103 – 1 = 111 Figure 301

The condition of numeracy proceeding from the invention of drawing in a single plane, the question informed ultimately by the medium of our inquiry is this: in what fashion, way, or mode, does a mathematical model expressing the relationship of straight lines to the subjects of memory, relate to the objects of reality? 7.1 The Question The question, so conspicuously missing between Lemma 4 and our first formulation of Lemma 5, and which does not arise from the determination of the primary state, is exemplified by Figure 301: in the absence of an object of reality that is not an isologue of the limit by which it is determined, is 1 the binary complement of 0, or is 1 the square of the binary complement (–1) of 1?

The Objects Of Mathematics

According to our model of information, the answer to that question depends upon whether the diameter of the macro sphere of binary 10 is informed proportionally inside the virtual limit binary 102 of the moment of the static incidence of our mutual horizon and the plane of our virtual horizon (e.g. Figure 279), or informed rationally outside the said limit (e.g. Figure 276). Having attributed the radius of the macro sphere of binary 10 to the plane presented by the medium of our inquiry, we may ask the reader to inform the macro sphere of binary 10, proportionally and inside the virtual limit binary 102, by informing that plane as a singular object of reality (a 'page'). The page of our inquiry being neither a circle nor a square, then while it continues as a singular object of reality, the finite bi-dimensional limit of the macro sphere of binary 10 (see Figure 287), is mathematically equal to: M Ă&#x2014; c2 = 22 Figure 302

Upon the occasion of informing the moment M / 2 of the rotational stasis of a virtual horizon informed by an object of reality (the page) and the counterlogical spin of our mutual horizon, the reader may experience the transition of the scale of binary 102, from that of the micro scale of binary 10 predicated by mathematical induction, to that of the macro scale of binary 2 predicated mathematically (Figure 302) by a real number, i.e: M Ă&#x2014; c2 / 2 = 2 Figure 303

In which case (Figure 303), at the instant of informing the page, the greatest possible straight line between the position of the observer (cf 'reader') and the subjective centre of the rotational stasis of memory and the primary state, is equal (see Figure 285) to the radius of the great circle of the macro sphere of binary 10, the diameter of which circle is predicted mathematically as the diagonal of 22. Which circle, as it cannot be a subject of memory prior to an instance of its supervention, constitutes what we will qualify a mathematical object. The evidence of the transition of scale, from and to that informed by our positive notation and that informed by the page on which it is written, is necessarily private information. We may, however, ask the reader if there is any single object of reality greater than a point, other than a page of our inquiry, a cube, or a sphere, informing a virtual horizon? 7.1.1 Positively According to Lemma 10 the observer, in order to identify an object of reality meeting the criteria (i.e. plural conditions) of the question, had first to identify a singular object of reality, by translating the subjective centre of the rotational stasis of memory and the primary state from the page, to a point of transposition inside the macro sphere of binary 10, of a unary pair of mutually exclusive primary data. Prior to identifying the critical object (i.e. that object of reality meeting our multiple criteria), the logical spin of the translator (Lemma 8) is privately-informed by the vertex of a singular common divisor of the primary state. While we cannot inform the moment of the rotational stasis of memory and the primary state therefore, the point of transposition of a unary pair of mutually exclusive primary data coincides with the subjective centre of the rotational stasis of memory and the primary state, inside the limit 98

The Objects Of Mathematics

(see Figure 300) of binary 102. Prior to informing the unconditional inequality of an accidental pair of natural data, and the orientation therefore of an object of reality, and at a singular point of the particularization of translation per se, the radius of a singular object of mathematics we have defined as the micro sphere of binary 2, is (see Figure 81) continuously equal to: m Ă&#x2014; c2 / 2 = 2 Figure 304

According to Lemma 8 therefore, that vertex of an object of reality privatelyinforming the logical spin of the translator, is mathematically equal to the diameter of the micro sphere of binary 2, i.e: d1 = m Ă&#x2014; c2 Figure 305

In accordance with Rule 8 therefore, an indeterminable transition of the translator, from the scale of the macro sphere of binary 10 predicated on the macro scale of binary 2 (see Figure 303), to the scale of the micro sphere of binary 2 predicated on the micro scale of binary 10 (Figure 305), is informed by determining the moment of the combined logical spin of an accidental pair of natural data, at a singular instant normative of the critical object of reality. The evidence, however, that the said instant is not a subject of memory, requires the smallest possible straight line between the observer and the subjective centre of the rotational stasis of memory and the primary state, to be informed by an object of reality. We invite the observer, therefore, to experience the indeterminable transition, by determining the difference, between the critical object and the page of our inquiry. Upon the present and private occasion of which the transition, informed by the determination of two objects of reality, exemplifies the instantaneous inversion of two mathematical objects (cf Figures 303, 304) scaling the virtual limit binary 102 (Lemma 12). The evidence, however, that the inversion of scale is not a subject of memory, requires the reader (cf observer) to inform the inverse of that prior and determinable transition from the macro scale of binary 2 on which our positive notation is informed artificially, to the macro scale of binary 2 on which a page of our inquiry exemplifies a singular object of reality. And upon the occasion of the transition from observer to reader, the instantaneous logical inversion of binary 10 on the octernary scale may be self-evident, i.e: Lemma 0: Memory is the contradictory of the binary complement of 0. In plain words, Lemma 0 states: the greatest possible finite unilinear translation in a single plane is informed by the instantaneous inversion of objects of reality on the octernary scale of binary 10. And if so, then the rest belongs to the anthropocentric history, of the science of empirical measurements which approximate, within the limits of experimental error, those mathematical objects that are so admirably adapted to the virtual subjects of memory. 7.2 Negative See Rule 1. 99

Table of Rules and Lemmas

Rule

If nothing then nothing ?

If exclusively nothing then nothing, else exclusively something ?

If something or nothing exclusively, then something or nothing inclusively ? If something or nothing inclusively then something and nothing exclusively ?

no.

Lemma

There is change of state

When exclusively something or nothing, then its continuation is the effect of memory.

When two mutually exclusive states alternate, a singular position is determined in the serial direction of the continuous state.

When two mutually exclusive states alternate, a singular direction is informed toward the position of the continuous state.

When two mutually exclusive states alternate, a singular laterally-horizontal translation is determined in the serial direction of the continuous state, and a singular serially-horizontal translation is determined in the lateral direction of the continuous state.

Between any two instants there is one and only one moment of translation.

Between any two mutually exclusive instants of a determinable proportional equality, there is one and only one moment of translation.

If 1 < 0 then 1 is toward–1 else 0 is toward–1 ? 10,18

If 0 is T toward–1 then 0=1? If 0 = 1 then 1 / 1 is toward–1 ?

6,15

If 1 / 1 is toward–1 then t is toward–1 ?

When V1 is toward–1 then d1 is toward–1.

If 1 – 1 = 0 then 10 – 1 = 1?

The binary complement of 1 is 0 – 1.

If two then one fewer than two ?

If one fewer than two then one more than two?

A =



x  ( x  1) I ?

binary (10)n / 2 = r02.

The binary complement of 0 is 1. binary (10)2 =

I 0

10  (10  1)

I 0

zero When 11 – 1 = 10 then 10 = 2. No more than two equal and opposite translations can be determined in a straight line ?

Memory is the contradictory of the binary complement of 0.

Glossary

A posteriori

for every possible position of the primary state post to the translation by which it is determined

A Priori

for every possible position of the primary state at the moment of its determination

abstract information

that which is naturally informed by the instantaneous determination of one of two incidental pairs of virtual data

accidence

superimposition

accidental information

that which is presently and privately informed by the mutual identification of natural intelligence

admit

include as evidenced

analogue

exemplar

apodictic certainty

without presupposition

artifice

drawing

artificial information

naturally informed by drawing

asymmetric

recognizably different

prior to the determination of

axiom

a proposition affording no verifiable alternative

because

consequential upon our prior definitions

bi-dimensional

of two dimensions

binary

doubled

categorically

without conditional clauses

cited

written

cognited

made present

cognize

observe

coincides

happens both where and when

common

shared by more than two memories

common rote

common and written agreement

complement

that which informs the singularity of

complementary

adding to a singularity

complex

of more than one part

configured

set against each the other

consequential upon

occasioned by

constancy

the quality of remaining unchanged i

Glossary

constant

unchanging

criteria

plural conditions

critical object

that object of reality meeting our multiple criteria

datum

a unit of information

decreases

lessens

deduction

subtraction

delimiting

marking

demonstrate

evidence linguistically

denominate

un-name

denotated

identified without notation

determinant

that which determines

determine

end

dimension

that property of an analogue which affords empirical measurement in a singular direction, of its every possible isologue

direction

the singular effect of memory

discretely

discontinuously

disposition

conditional arrangement

distinctive

unlike

empirically

by direct measurement

enumerating

qualifying verbally

exclusive

only possible

exemplify

represent

experimentally

by mental experience

extant

self-evident

finite

determinable by a subject memory

fractional

informed by division

generalize

represent singularly

identifies

makes identical in all respects but one

imaginable

conceivable but not extant

immutable

not subject to change

incidental information

that which is informed by any singular change of the direction of continuous unilinear translation towardâ&#x20AC;&#x201C;1 ii

Glossary

increases

greatens

indeterminable

continuous in every possible direction

infer

determine categorically

informant

that which informs

inside

within

instance

singular occasion

instant

instance extant

instantaneous

between two secondary instants of a singular and priorly continuous state

isologue

that which is indistinguishable in all respects but one

literally

by writing

logical

in a singular direction

macro

greater

magnitude

finite unilinear dimensions of

mathematical induction inverse logic may

it is within our rules of evidence so to do

memory

the ability to determine a priorly continuous state

micro

lesser

moment

that which is determined between any two instants of a singular and priorly continuous state

multiple

plural

mutual

shared by two memories

mutual information

that proportional equality the present and private determination of which is normative of the translator

natural information

that which is predicated by the determination of incidental information

natural intelligence

that which presently and privately determines the equality of two mutually exclusive divisors of a singular, extant and indeterminable inequality of proportion

nominate

name

normative

providing an instance of

nothing

the absence of information

numerate (adjective)

primed with both our common rote and our symbolic denominators of simple numbers iii

Glossary

object of reality

any singular instant mutually normative of a noun

occasion

cause

once

on one and only one prior occasion

opposed to

comparable with

outside

without

particularize

reduce to a point

per se

informed by the spoken word when extant

plane

that which is presently and privately informed by the determination of an accidental pair of natural data and a singular common point of rotation

posits

positions

post to

then

postulate

admit as necessary but not self-evident

precise

not subject to experimental error

predicated

informed with apodictic certainty

predict

is informed A posteriori

primary

first posited

primary state

that singular state the isologue of which is first determined by means of laterally-horizontal translation (denotated 1)

primed

priorly informed by common information

prior to

before

private

subjective

private information

that datum the present and private determination of which is normative of the vertex of the subject memory

progressively

in the direction of the consequential cycle

proposition

a question to which there are two and only two possible answers

qualify

distinguish verbally

randomly

regardless of order

rational

informed by ratio

real

naturally informed

recognition

secondary cognition

recounted

counted again iv

Glossary

secondary notation

serial

alternately informed

simplified

informed quantitatively

specie

that which is specialized

state

an unchanging condition

stipulate

admit to our rules of evidence

straight line

the smallest possible unilateral translation determinable between any two extant points of transition

subjectively

presently and privately by the reader

sum

combination

supervenes

displaces

symmetrical

indistinguishable

tautological

defined and redefined

thereafter

post to

therefore

prior to our notation

towardâ&#x20AC;&#x201C;nothing

a direction informed by the absence of something

toward-something

a direction informed by the absence of nothing

transition

logical change

translation

movement

translator

that singular property of memory informing direction

transposition

exchange by translation

unilateral

any singular lateral

unilinear

logical and in a straight line

verified

recognited with apodictic certainty

verify

prove experimentally

vertex

the informant of the greatest possible finite inequality of proportion

virtual

of imaginable dimensions

when

upon the occasion of

while

unless or until