A NON-ARISTOTELIAN VIEW OF QUANTUM THEORY

Hilgartner & Associates

Joseph DiRienzi

College of Notre Dame of Maryland

ABSTRACT

In this paper the authors explore assumptions used in the formulation of quantum theory, and show how to re-frame quantum theory so as to make it easier to understand, to visualize and to explain.

Although highly successful by both theoretical and experimental criteria, quantum theory has troubled its exponents for some seventy years. We review the character of these difficulties; we trace them to the traditional assumptions encoded in the grammar common to Western Indo-European (WIE) languages, formalized as well as discursive (such as set theory and English); and we re-formulate central constructs of quantum theory, e.g. the Principle of Complementarity, onto a frame of reference based on non-traditional premises. The paper includes a set theory proof which discloses at least one non-standard axiom of quantum theory which its originators demonstrably included, but appeared not to recognize or notice.

Finally, we discuss how to go about putting our proposals to experimental test.

I. INTRODUCTION

In this paper the authors explore some of the problems people report that they experience with quantum theory. We approach these problems from a frame of reference based on the **non-aristotelian premises** proposed by Alfred Korzybski (1879-1950). From this alternative standpoint, we suggest ways to make it possible to understand, to visualize and to explain quantum theory – at the price of re-framing it so as to treat physics in general and quantum theory in particular as "something which humans DO."

Although highly successful by both theoretical and experimental criteria, quantum theory (as currently practiced) has troubled its exponents for almost seventy years. Classical scientists sought to build a complete and consistent model of Physical Reality; some quantum theorists regard that goal as unattainable. Everything about classical physics seems intuitively "right" (familiar, what we've grown used to); quantum theory seems consistently counter-intuitive. Classical physics delivers familiar-seeming deterministic predictions; in the quantum domain, these end up disconfirmed. Quantum theoretic models deliver predictions, in the quantum domain, which survive testing; but these have a probabilistic structure which frustrates and baffles those who aspire to "know" the physical universe, in the traditional sense of this term. Evidently, the observer participates in the process of observing; but no one "understands" just how this participating takes place. In short, physicists agree that quantum theory WORKS - but cannot say HOW and WHY it works.

The authors established an initial connection between quantum theory and our non-standard frame of reference by seeing a match between the pattern of the quantum theory construct of **complementary coordinates** and that of a particular kind of ** ** (a "reversible" **Gestalt,** discussed below).

Other workers may have made similar analogies years or decades ago. Perhaps, due to the apparent "mushiness" of Gestalt terminology, most found such comparisons lacking in rigor and consequently not useful. But the framework surrounding our insight provides unprecedented rigor in this domain. As a first step in a logical analysis of our non-aristotelian theory of human behavior, we had used a set theory notation to state the non-aristotelian premises. We had expressed the overall setting of transacting by means of a Cartesian product space, O ´
E (**organism cross environment**). Then in framing the Postulate of Self-Reflexiveness, we had shown showed that any abstraction, any map, hasshows an internal structure which includes (at least) two components; and we framed these also as a Cartesian product space, Sf ´
Ot (**self cross other**). The presence of these latter two components within any abstraction suggests the notions of **figure** and (back)**ground**, as spelled out in the construct of *Gestalt*. In further developing our set theory calculus of human behavior, we had shown showed that our revised version of the construct of *Gestalt* both implies and assumes our chosen premises. The more recent insight to the effect that complementary coordinates behave like a "reversible" Gestalt, then, makes at least one construct from quantum theory look like it too implies-and-assumes the non-aristotelian premises. That in turn suggests that other key features of quantum theory (perhaps the entire theory) similarly imply-and-assume these non-standard premises.

In the present paper, we examine that supposition in depth.

THE COURSE OF THE ARGUMENT

A. We begin by describing the aspects of the alternative frame of reference which seem most relevant to the present problem.

B. Then we review certain key findings from quantum theory, and begin inter-connecting thesem with our alternative viewpoint.

C. We express our key supposition as two logical hypotheses (ms p. 6), and put them to test by doing a set theory proof (ms pp. 33-46).

D. In the final sections of the paper, we discuss some of the implications of these matters, and consider how we might put our proposals to experimental test.

II. A NON-IDENTITY-BASED FRAME OF REFERENCE

A. OUR APPROACH

Korzybski suggests that we reject the logical construct of **identity **as the foundation of systematized human knowledge. [1, 2, 3]

In 1965, a research group came together to bring full mathematical rigor to bear on some ongoing efforts to explore this "outrageous" suggestion. As our work proceeded, we eventually framed the issue in more general terms: We recognized that there exist at least two ways to handle the paired (polar, mutually-defining) terms *identity* and *non-identity* at the level of our most basic presuppositions. (a) One can "like" *identity* as a foundation, and "dislike" *non-identity* (the stance of our linguistic forbears – along with most of today's native speakers/writers of **Western Indo-European** (WIE) **languages**, and the majority of the current exponents of WIE logic, mathematics, and science, etc.); or else (b) one can "like" *non-identity* and "dislike" *identity* - an unprecedented stance.

From the beginning of our studies, we had already adopted Korzybski's coherent system or world-view, which centers on a study of how languaging and other forms of symbolizing affect humans and how humans affect their languaging, their symbolizing. In his investigations, Korzybski had reached two end-points: (a) He had brought out into view, and stated in English, the most-fundamental (we might even say **metaphysical**) premises on which his system rests, including three **undefined terms** (**structure**, **order** and **relation**, which in English text we treat as verb-forms rather than noun-forms: *to structure*, *to order*, *to relation*); and three **non-aristotelian** **postulates**, known as **Non-identity**, **Non-allness**, and **Self-reflexiveness**.^{ } (See Note 7 for statements of these in English and in set theory.)

The work already underway in 1965 amounted to a theory of human behavior, based on the non-aristotelian premises and stated in ordinary scientific English. Weproposed set out to do a logical analysis of this doctrine, using a Bourbaki algebraic set theory notation. As the first step in developing a notational calculus of human behavior, we succeeded in explicitly stating the non-aristotelian premises in that notation.[4] In so doing, we provide what amounts to a relative proof: that (by the standards of today's set theory) the non-aristotelian premises of Korzybski qualify as free of self-contradiction or other logical error, if the mathematical theory of sets itself qualifies as free of self-contradiction or other logical error.

When a human adopts novel premises, and so develops hypotheses from an unusual standpoint, s/he faces two likely outcomes: The results may, more or less promptly, disconfirm the hypotheses, and so show these premises as of doubtful value; or, if the preliminary testing does not cast doubt on these unusual approaches, further exploration may show them as opening new possibilities.

So far, our "outrageous" approach has turned out outrageously fruitful.

(a) __Human psycho-social sciences__. The theoretical system framed in our set theory calculus already constitutes a revised and revisionist theory of human behavior, which asks, and answers, this question: "What does a human organism DO that keeps her/him living - more or less intact, and more or less growing - from one moment to the next for a whole lifetime?" In answer, we hold that to say that an organism lives means that s/he generates some kind of survival-oriented **maps** or representations of that territory composed of "what goes on in and around this organism." [5, 6] S/He lives in a condition of fundamental uncertainty, for (in accord with our chosen premises) these maps remain intrinsically inaccurate, incomplete and self-referential. Nevertheless, s/he then guides her/his "doings" or "choosings" by these maps, in the process putting them to test. The **outcome** of this testing may prove more or less **favorable**, from the point of view of the organism. S/He can then judge the maps (hypotheses, assumptions, guesses, etc.) s/he started with, in terms of this outcome. S/He can throw out or otherwise discard those guesses which appear not to have worked (as judged by the outcome), while those which appear to have led to more favorable outcomes s/he can store in such a way that they come up as **expectations** in subsequent relevant encounters. Operating in this manner leads to the most effective survival-behavior yet described: Make one's guesses, test them by acting on them, discard the disconfirmed, and live with the consequences.

This amounts to saying that, according to our theoretical system, human organisms appear CAPABLE of functioning by the logic of science. In the face of the intrinsic uncertainty of map-making, we achieve the feat of continuing to survive by relying on the self-correcting structure of human experiencing.[7, 8]

But of course, any human or any group of humans can at any moment subvert this self-correcting.[9] To manage that, one need only say (or more likely, non-verbally take the attitude, without saying it aloud): "I don't make any guesses – I see only what's REALLY THERE." Thereafter, if and when the outcome of one's endeavors turns out unfavorable, one will not judge one's guesses as disconfirmed, nor throw any out. Instead, one will defend them against all comers, thereby arranging to guide one's subsequent "doings" or "choosings" by already--discovered error. But (on one level or another) one already knows that, in survival terms, this will prove less than effective. To an onlooker, the ensuing **self-defending** behavior appears rigid, or fixated, or neurotic, or psychotic, or in general warranting application of her/his favorite diagnostic terms.

(b) __Biology__. Our theoretical system also delivers a revisionist theory of biology[10] (which includes (i) a formal, notational, definition for the term **living**, (ii) a novel accounting for the origin of living organisms[11], (iii) a revised, notationally-based accounting for biological evolution, and (iv) an explicit theory of how the human species gains its living in the biosphere).

(c) __Physics__. Although not primarily a physical theory, it also suggests revisions in modern physics.[12, 13]

(d) __Foundations of logic and mathematics__. When in 1971 we turned to the topic of logical foundations, however, we encountered a major obstacle: a new kind of self-contradiction. [14] It arises between (i) the presuppositions underlying the "content" of our notational theory (specifically, views developed through the rejection of identity in any guise or form, explicit or tacit, as expressed in the Postulates of Non-identity, Non-Allness and Self-reflexiveness) and (ii) the presuppositions which underlie the notation itself (specifically, the mathematical theory of sets of the WIE tradition, which among its premises includes the modern **Logical Axiom of Identity**).

Nothing we tried held out any prospect of avoiding or otherwise getting around that self-contradiction. Eventually, we concluded that no evasion could succeed or suffice. We set out to abandon the mathematical theory of sets, and all other WIE languages, discursive as well as notational, and to generate a language which starts "from the very beginning" - whatever that may mean, in this context - from the non-aristotelian premises of Korzybski, in which to continue our investigating. Within a few months of adopting this unsettling resolve, we uncovered a fundamental relationship between grammar and assumptions, and disclosed what, from our frame of reference, amounts to an untenable assumption encoded in the grammar of WIE languages such as English or the mathematical theory of sets.[15] We summarize these findings in the next section.

These unexpected findings opened the way for us to derive a grammar from Korzybski's premises[16, 17] - so far as we know, the first DERIVED grammar. After doing that, we generated a "Let's Keep Track of What We Say" notation on that derived grammar; translated much of the set theory work into the alternative notation; and in 1974-5, took stock again. Since then, we have not remained idle.

B. QUANTUM THEORY FROM AN ALTERED FRAME OF REFERENCE

In the present paper, we propose to utilize this altered frame of reference, and to address issues raised by quantum theory from a standpoint based on non-identity. We expect this to make clear HOW and WHY quantum theory works, as a theory in physics.

We find two main strands in quantum theory - one conservative and traditional, and the other innovative. We suggest that the languages - mathematical as well as discursive - used by the quantum physicists encode the traditional strand, and interfere with their innovative theorizing. These theorists - like the rest of the exponents of WIE scientific method or scientific thought - have built up their various quantum theories in part by unawarely generalizing on the linguistic structure of the WIE languages. In so doing, they grant a **privileged position** to the grammar common to these languages, and to the presuppositions encoded in that grammar.[18] But in order to branch off from Newtonian physics and frame a quantum theory at all, these workers also deviate from and reject some part of the traditional presuppositions encoded in the WIE grammar, and thereby develop the innovative strand. (They do this in particular by relaxing certain of their classical expectations, e.g. letting go of the surmise that certain physical constructs, such as position and momentum, occupy the role of *attributes* of "things" – physical bodies – by re-assigning these constructs to the role of *abstractions*, generated when a human observer employs an operator which extracts ‘information’.)

The resulting unrecognized contradiction between the traditional and the innovative aspects of quantum theory has prevents prevented the practitioners from "understanding" what their experiments show, and their theories say.

HYPOTHESES:

We offer the hypothesis (A) that there exists an innovative strand within quantum theory which presupposes and implies a tacit version of the Postulate of Non-identity.

We offer the further hypothesis (B) that modifying quantum theory so that the entire edifice explicitly rests on the Postulate of Non-identity, etc., will lead to advantages in the theoretical, experimental and explanatory aspects of the theory.

To test hypothesis (A), we shall explore the relations between quantum theory and Non-identity. In so doing, we not only shall use standard, identity-based WIE languages, such as English and the mathematical theory of sets, but also shall use a contrasting viewpoint based on the non-aristotelian premises, and which requires that we use **non-identity reasoning.** Briefly stated, non-identity reasoning argues from *finding "things" not-identical with* (or "*different from"*) *one another*, and also *not-identical with themselves*.

(Here we will present a thumbnail sketch of our non-standard viewpoint; but to develop it rigorously, explicitly, in detail, lies outside the scope of the present paper.)

C. ASSUMPTIONS, GRAMMAR AND LANGUAGE

We shall discuss quantum theory in terms of **assumptions**.

Our assumptions concerning the term *assumptions* (a **multiordinal** inquiry – see [19]) include the following:

a) We do not restrict this the term *assumption* to the domain of explicitly stated verbal propositions. We include structures which remain non-verbal and perhaps non-verbalizable. For example, many people do know and can state in words the visual "rule of inference" known as **overlay**, which expresses the generalization that, within someone's visual field, when the outline of one object partially covers up the outline of another object, the covering object stands closer to that person than does the covered one.[20] But nobody needs to state this generalization in words in order correctly to use this rule to estimate relative distances. Visually speaking, the cup overlays the saucer, and most sighted people can pick up the one they reach for, even if they cannot SAY how they "know" how to do that. Their mainly unstated, visually-based assumptions about relative distances survive practical test.

b) We also discuss **tacit** assumptions - presuppositions that, we say, shape someone's behavior (including her/his theorizing) but which at a given date during her/his own era seem "obviously true", trivial, or otherwise do not arise as topics of discussion. At a later date, however, operating in terms of our own explicit and tacit assumptions, we may find that we cannot account for the relations between that person's behavior (including theorizing) and our own unless, by inference, we attribute these presuppositions to that person. Neither Newton nor Galileo, for example, discussed the velocity of light as a theoretical topic. But when Einstein, from his relativistic viewpoint, examined Newtonian physics, he found that he could interconvert between the Galileo transformations (which Newton had appropriated) and the relativistic transformations of Lorentz and Einstein by replacing the term for the known, finite velocity of light with a term with an "infinite" value. This mathematical relationship leads to the logical inference that Newton tacitly and unawarely assumed that light has an "infinite velocity" (or otherwise stated, that its velocity has no upper bound).

Our scrutiny of quantum theory starts with an analysis of some hidden assumptions incorporated into the languages used to develop it.[21, 22, 23] In particular, we shall use non-identity reasoning to disclose some of the presuppositions encoded in an identity-based WIE grammar.

Any language has to have some kind of grammatical structure, and that structure incorporates tacit assumptions (or "implicit and unstated agreement(s)") which, in effect, direct its speakers/writers - e.g. us - to form a particular class of Gestalten. Specifically, they direct us to attend to certain aspects of what goes on in and around us (the figure of the Gestalt) and to ignore others (the (back)ground of the Gestalt). They also direct us to segment along linguistic lines that which we do attend to, what we do notice (in brief, to segment the figure of that our Gestalt). We who use the language

cut nature up, organize it into concepts, and ascribe significances as we do, largely because we are parties to an agreement to organize it in this way - an agreement that holds throughout our speech community. The agreement is, of course, an implicit and unstated one, BUT ITS TERMS ARE ABSOLUTELY OBLIGATORY; we cannot talk at all, except by subscribing to this way of organizing and classifying data decreed by the grammar.[24]

With reference to visual processing, for example, by the time we "SEE" something, "what we see" already bears the imprint of these "unstated agreement(s)" that Whorf speaks of (or what others call "our linguistic habits").

To get a sense of the particular assumptions built into the grammar common to the WIE discursive and mathematical languages (such as English or set theory), look first at vocabulary. In a big dictionary, for example, entries labeled as** nouns** and **verbs** vastly outnumber the aggregate total of the entries for the other **parts of speech**. The vocabularies of our mathematical languages, the authors argue, may consist ONLY of cognates of nouns and verbs (with no other parts of speech): for example, **quantities** or **things** (noun-cognates, e.g. *3* or *x* ), and **operations** or **relations** (verb-cognates, e.g. *equals* or *not-* ).

We suggest that functionally speaking, we speakers/writers distinguish between these two grammatical categories by tacitly applying one of Aristotle's "Laws of Thought," namely, the "Law of Identity," which says, "What is, is", or "A is A" "C is C". Thus, we regard the terms we label as nouns as "identical with themselves," and the ones we call verbs as "not-identical with themselves." The significance of this way of generating or distinguishing between these two kinds of terms will become more apparent shortly.

These two kinds of terms occupy a key role not only numerically but also grammatically: To form a complete sentence, a speaker or writer in a discursive language does not have to use any of the other parts of speech, but must combine at least one noun or noun-phrase with at least one verb or verb-phrase, e.g.

The cat grins. ;

or, to form a **well-formed formula** in a mathematical notation, must combine at least one quantity with at least one operation, or at least one thing with at least one relation, e.g.

x = 3 .

Not-A .

We suggest further that we speakers/writers of WIE languages usually assign a cosmic significance to these grammatical conventions: That which we notice – as dictated by the assumptions encoded in our grammar – we cut up into two kinds of "pieces." However, we do not notice – or we decline to admit – that we had anything to do with this segmentation. In other words, as native speakers/writers of WIE languages, we assume that, independent of any observer or any observing, "the world" or "reality" – the non-verbal – REALLY DOES consist of two "types", different from and incommensurate with one another: one static-and-unchanging, the other more or less evanescent. (Then we marvel at the "pre-formed harmony" between Language and Reality: How convenient that the vocabulary of our Language consists of two main kinds of terms, with just the right linguistic "attributes" to represent the two non-verbal aspects of such a Reality!) Otherwise stated, we **project** the structure of our grammar onto the Cosmos. And in so doing, we presuppose a fundamental **dualism**, with the Cosmos divided into two parts: an immaterial or mystical side (verb-like, and suggested by terms such as *soul* or *spirit* or *mind*), contrasted against a material or physical or "real" side (noun-like, and suggested by terms such as *body* or *the physical* or *matter*).

In our discursive languages, then, we signify these static-and-unchanging "things" by self-identical nouns or noun-phrases, and the more or less evanescent "relations" by not-self-identical verbs or verb-phrases. In our mathematical languages, we designate the "fixed entities" by our self-identical quantities or things (e.g. *x* or *3* ) and the "non-fixed aspects" by not-self-identical operations or relations (e.g. *equals* or *not-* ). And we expect to find nothing "left over" after doing so. We act as if (mental) Language fitted (physical) Reality flawlessly.

To complete this analysis, let us now bring forth a tenet of non-identity reasoning which holds that **explicitly to distinguish** (or to make a distinction) **between** some A and some B amounts to **explicitly positing their non-identity**. Contrariwise, the notion of **failing to distinguish between** some A and some B amounts to **tacitly positing the identity** of that A with that B . According to this tenet, when a commentator demonstrates that a person, or a frame of reference used by a person, fails to make a particular distinction, then our commentator has provided grounds for inferring that that person, or that frame of reference, relies on what we call** the postulate of tacit identity**. And by the non-aristotelian premises, any such reliance amounts to a **fundamental theoretical error**.[25:243-4]

Consider the matter in this light. The grammar of the WIE languages (e.g. English) requires their speakers/writers explicitly to make a large number of distinctions: (a) between singular and plural instances of the "thing" designated by a count noun, which we indicate by the affix for grammatical number – *three* cow*s *– or between the possible relative quantities of "stuff" designated by a mass noun, which we indicate by using certain adjectives, demonstrative adjectives, etc. – *a little* water; (b) between past, present, future, etc., which we indicate by selecting the tense of a verb ("time", e.g. *he speaks*, *he spoke*,* he will speak*, etc.); (c) between different locations, which we indicate by using certain adverbs, prepositional phrases, etc. ("space", e.g. *here*, *there*, *in the kitchen*, etc.); and so on.

Nothing in the grammar of the WIE languages, however, REQUIRES any speaker/writer to distinguish between the words s/he generates and that which the words designate (if anything). Let us put this in terms of the* ***map-territory analogy**, which holds that to say an organism **lives **signifies that it generates some kind of *map(s)* of that* territory* composed of "what goes on in and around our organism": Nothing in the grammar of the WIE languages requires language-users to distinguish between *map *and *territory.* We have no specially-named parts of speech, no reserved prepositions, adverbs, etc., no prefix, suffix or infix – no grammatical arrangement at all – which makes the map-territory distinction and which every speaker HAS to use in every sentence. In terms of the principle stated above, that amounts to saying that the grammar of the WIE languages tacitly identifies map with territory. By tacitly regarding the match between our languaging and the territory it refers as "perfect," we WIE speakers/writers display – at levels we rarely even notice, much less question – the assumption that our neuro-linguistic maps provide "perfect knowledge" of the territory. But – at least on the levels we remain aware of – we recognize that no human legitimately has access to "absolute certainty" or "perfect knowledge."^{ }

If even one of our neuro-linguistic maps actually did qualify as identical with the territory, that would mean that it satisfied a completely accurate and exhaustively complete one-to-one relation, so that every point of the map represented one and only one point of the territory, and that no point of the territory went un-represented.^{ } We usually recognize that this relation does not hold, nor would we want it to – we would find such maps uselessly detailed and insufficiently organized and focused. We do not commonly recognize, however, that a map we regard as perfectly identical with the territory in principle has no "room" in it for any kind of representation of the map-maker/observer, and so appears "perfectly objective" – what someone referred to as an "immaculate perception."

In other words, the presuppositions which underlie the spoken and written, discursive and notational, WIE languages – including the current versions of quantum theory – violate both the Postulate of Non-identity and, the Postulate of Self-reflexiveness and so systematically, transparently eliminate the observer from consideration.[26] In so doing, they both allow the possibility of starting from already-discovered error, and disallow taking into account the effects of what one has just done on one's here-now situation and on one's proposed next actions. (See Note 7.)

Granted, the exponents of relativity and quantum theory introduced the construct of "taking the observer into account," and had some initial successes in actually doing so.^{ } But they did so at the cost of representing the observer as an "unmoved mover," who by observing the observed produced alterations in the observed (or at least in the observation) but none in the observer. Both relativity and quantum theory account in quantitative terms for these alterations in the observed (or the observation). But the theories remain inconsistent – they do not show the process of observing as altering the observer, and so in effect they eliminate the observer from consideration in yet another a non-traditional way. Thus what they give with one hand they take away with the other.[27]

1. ROUGH CROSS-CHECK

The authors see no reason for our readers to accept these assertions just on our say-so. To check out the first assertion **logically** and at the level of "language" – the supposition that we WIE speakers/writers establish the distinction between noun and verb by means of the "Law of Identity" - extract a noun-form or noun-cognate from these trial sentences and insert it into the spots within the "Law of Identity" reserved for noun-forms: Say or write

3 = 3 .

or

A cat is a cat.

The resulting locutions, by WIE standards, appear **well-formed** or **grammatical**.

Then (without straining to set up some kind of special context to validate otherwise unacceptable "sentences"), extract a verb-form or verb-cognate from these trial locutions, and insert it into the "Law of Identity" in a similar fashion.

= = =

or

Grins is grins.

The resulting locutions do not, by our standards, appear well-formed or "grammatical". No one, we contend, would so consider such expressions. We speakers/writers of WIE languages do not regard it as proper to place a verb-cognate in the spots in Aristotle's "Law of Identity" reserved for noun-cognates.

In other words, this rough check does not cast doubt on the hypothesis that we distinguish between *nouns* and* verbs* by tacitly applying Aristotle's "Law of Identity."

Our second assertion - the supposition that we speakers/ writers of WIE languages tacitly assign a cosmic significance to these grammatical conventions, or in other words, that we unawarely project the structure of the WIE grammar onto the non-verbal, thereby secretly awarding to that grammar a privileged position - does not deal with the ways terms or symbols within a single frame of reference relate to one another. Instead, this assertion has an **empirical** focus, and we cannot test its validity by subjecting it to logical scrutiny. We must find some other way to check it out. This we propose to do self-reflexively, by examining our own text.

The way we frame our second assertion embeds some suppositions. By hypothesis, someone who tacitly assigns a cosmic significance to the conventions of the grammar of her/his native tongue, or who unawarely projects the structure of its grammar, does not explicitly acknowledge doing so. Moreover, s/he doesn't so much as mention related topics such as *assigning a cosmic significance* or *projecting*, either.

Likewise, by hypothesis, the notion of *privilege*, as used in the phrase *privileged position*, entails subscribing to one of those "implicit and unstated" agreements shared by the members of a local speech community which Whorf refers to: It amounts to an unspoken agreement not to question, to disclose or otherwise to tamper with the hitherto unstated presuppositions encoded in the local frame of reference in question; and if possible, to prevent others from disclosing, discussing and criticizing those presuppositions. Moreover, a person who grants such a privileged position to her/his grammar does not acknowledge doing so; and s/he does not discuss or even mention the related topic of *the presuppositions encoded in a grammar.*

In our second assertion, on the other hand, we explicitly speak of each of these "forbidden" possibilities. By discussing these taboo topics - pointing out that we speakers/writers of WIE languages do appear to treat them as taboo - we not only invoke *not assigning a cosmic significance*, *not projecting*, *and not granting a privileged position* as possible ways for humans to operate, we also claim first-hand experience with them - enough experience to allow us to discriminate, to tell the difference between the two modes of functioning, and judge whether a given human group (one which appears not to use both modes) uses one rather than the other of them. If we had lacked such experience, we insist, we could not have made the diagnosis which we express in our second assertion.

Obviously, nobody can, by reading or listening, receive the non-verbal EXPERIENCE of operating in a manner contrary to the tenets of a shared language and/or foreign to her/his behavioral repertory. At best, s/he can take the writer or speaker as a role model, and **emulate** her/him: "If I intelligently imitate the way that person functions, I believe, I will find the outcomes personally satisfying, in the domain of our shared interests."[28]

In particular, as role models, we point to our willingness to take seriously some "outrageous" suggestions - notably, Korzybski's suggestion that we humans reject identity as valid. We invite our readers to reflect on the fact that, at one time (as we insist), every member of our research group subscribed to roughly the same body of agreements and presuppositions that (we say) the rest of the members of the WIE speech communities do; and that we did so about as blindly as (we maintain) everyone else does. Daily, in our speech and our behavior - indeed, from minute to minute - we repeatedly ratified these presuppositions. We ascribed a cosmic significance to the conventions of our grammar, and projected its structure and so awarded it a privileged position, along with the best of our fellow humans. Then - as suggested by the evidence of this paper, and our two assertions - we somehow managed to alter the presuppositions from which we live.

To us, the process we went through to alter our own presuppositions looks transactional - a stepwise reiteration, not in principle different from the way scientists often operate: By linguistic means - verbal or notational - generate a new "outrageous suggestion" consistent with the "outrage" of relying on non-identity - and pursue it till it leads to trouble. Then handle the trouble. Then press on to the next "outrageous suggestion."

When we followed the logical consequences of non-identity reasoning, that led to the "trouble" entailed by disclosing the usage of the postulate of tacit identity concealed in the WIE grammar, discussed above. When we dislodged and rejected it - disallowing identity as valid, even here - that left us with no basis for distinguishing between noun and verb. And without the (by postulate, SPURIOUS) distinction between these two most important "parts of speech," we could not generate a complete sentence in English or a well-formed formula in set theory. In other words, when one rejects identity as valid, the WIE grammar collapses, taking with it the current forms of our WIE logics, mathematics, sciences, philosophies, jurisprudences, religions, etc.[29]

To recover from this collapse, we found a way to derive a grammar from the non-aristotelian premises.[30, 31] On that derived grammar, we have already built up an alternative "Let's Keep Track of What We Say" notation, which occupies a role more or less analogous to that of a WIE symbolic logic/set theory. We have developed the beginnings of a non-aristotelian substitute for WIE number theory, algebra and analysis, and have begun outlining a substitute for WIE geometry.

Before we humans can create a full-fledged, functional non-aristotelian version of "quantum theory" - one which leads to experiments - we (or someone) will have to complete these edifices, and provide an articulated, working version of non-aristotelian "arithmetic", "algebra", "analysis", "geometry" and "physics."

2. PROOF OF GENERALITY

A bridge between the traditional WIE grammar and the frames of reference erected on it and our non-standard, derived grammar and the frame of reference erected on it already exists: We have published a **proof of generality** [32], which displays a restricted and restrictive assumption (the postulate of tacit identity), and shows how to inter-convert between the less general system of the mathematical theory of sets, and the more general system of our non-aristotelian notation, by eliminating or introducing this restrictive assumption.

In the absence of such a bridge, for over half a century, workers within WIE mathematical logic have striven to represent the peculiarities of the phase space of quantum theory by generating a specialized notation to model its characteristics and constraints. R. I. G. Hughes summarizes these efforts much as we would:

… One arrives at quantum logic by considering the mathematical structure of the formalism of quantum mechanics. That mathematical structure, however, is based on the deductive patterns of classical logic. Hence classical logic is presupposed in the development of quantum logic.[33:213]

Our frame of reference builds on a derived grammar rather than on the traditional grammar of the WIE languages. In principle, we know of only one way to spell out the relations between this traditionalistic quantum logic and our non-standard frame of reference: make use of the bridge provided by our proof of generality.

3. ORDERING

To anticipate something we say below, the construct of *order* (or better, *ordering*) plays an intrinsic role in our non-aristotelian frame of reference, as one of the three undefined terms: Every map generated by an-organism-as-a-whole-transacting-with-its-environment-at-a-date will specify *ordering* in some sense. So far, we have found a need for five special cases, two pairs of which come through as polar term-pairs. (a) In **spatio-temporal** ordering, the terms, Gestalten, etc., from some grouping differ from one another along a string of "when-wheres". In its polar opposite, **synchronous** ordering, the terms, Gestalten, etc., of this grouping do NOT differ in terms of "when-where" (or otherwise stated, they share a single "where-when" – perhaps they appear simultaneous). (b) As a prerequisite for **hierarchical** ordering, we posit a process which generates a number of adjacent "logical levels", which we designate as an **ordering on abstracting**, made up of a number of adjacent **positionings**. Then in *hierarchical* ordering, the terms, Gestalten, etc., of some grouping differ in "logical level" – we assign them to different (usually adjacent) *positionings *in this *ordering on abstracting*. In its polar opposite, **co-ordering**, the terms, Gestalten, etc., do NOT differ in "logical level" (they share a single "logical level"). (c) In **polar** ordering, the terms, Gestalten, etc., define the ends of some kind of scale or the extremes of some opposition. To date, we have not figured out how to frame a polar opposite of the construct of __polar ordering__.

In contrast, the presuppositions encoded in the WIE grammar do not require its speakers/writers to utilize any notion corresponding to what we call *ordering* at all. WIE languages, both discursive and mathematical, do have vocabulary items or constructs corresponding to the English noun-form __order__, but their use occurs at the level of "content," as a kind of "afterthought", grammatically speaking. In the proof of generality, one of the steps in interconverting between the two frames of reference consists of introducing, or eliminating, *ordering* as an intrinsic structural aspect.

4. GESTALT AND NON-IDENTITY

In our alternative frame of reference, no matter what explicit topic we may discuss, we find ourselves constrained to do so from within the setting of *transacting* - or stated in other words, from within a **dramatic situation** which we can indicate by means of run-on phrases such as **"one-particular-organism-as-a-whole-dealing-with-its-environment-at-a-date, as viewed by a specified observer."**

Then the non-aristotelian premises **partition** this setting, in various ways and on various "logical levels".

a. The undefined terms function as postulates of a particular kind: namely, ones for which no one can state in words exactly what they "say" or "mean" or "presuppose".[34:153] However, even though, by definition, we have nothing "familiar" to substitute for these "unfamiliar" terms and so cannot **define** our undefined terms, we can at least tell others how to USE them (which comes down to spelling out how WE use them). We use the undefined terms to point to particular EXAMPLES of "happenings" in the domain of interest, each one a special case of "an organism-as-a-whole-dealing-with-its-environment-at-a-date, as viewed by a specified observer." We can engage in such pointing-to before we have defined any other terms at all, and even before we have stated our explicit postulates. Thus, in the most general sense, we use our undefined terms to bridge between "logical levels", e.g. between the **non-verbal** and the **verbal**.

b. The non-aristotelian postulates (called *Non-identity*,* Non-allness* and *Self-reflexiveness* - see ms p. 5 and Note 7), which we can state in terms of the undefined terms, describe the processes by which (in the opinion of a specified observer) an organism-as-a-whole deals with her/his/its environment. In particular, these postulates spell out limits to the precision, accuracy, scope, usefulness, etc., of the maps, the knowledge, etc., which these processes engender.

We use our run-on phrases to encompass at least two perspectives: that of

i) the **behaving** of our organism, in context (as viewed by others), and that of

ii) her/his/its **experiencing**, in context (as viewed by the organism her/him/itself).

In so doing, we use these phrases to deny the relevance and the validity of dualistic divisions such as "physical" vs. "mental", "body" vs. "mind", "intellect" vs. "emotions", "head" vs. "heart", etc.

c. The three postulates appear to have different foci:

__Non-identity__, we hold, deals with what our organism knows, and can know, about what goes on in and around herself/himself/ itself.

__Non-allness__, we hold, deals with what passes back and forth across the boundary - the relations between organism and environment.

__Self-reflexiveness__, we hold, deals with the relations of organism with itself (the ways our-organism-abstracting transacts with our-organism-abstracting).

In order to discuss these matters, form within the setting of *transacting*, we need a circle of five inter-defined terms, none of which "makes sense" in the absence of the other four - namely,

Organism

Environment

Abstracting (or map-making)

Representing (or abstraction or map)

An ordering on abstracting (or "logical levels")

Since in the present paper we do not make the finer details of abstracting our main focus, we shall not take the trouble here to define and inter-define these terms explicitly.

As noted above (ms pp. 2), the postulate of Self-reflexiveness discloses the "internal structure" of the process of abstracting, and of any map (any representing generated by abstracting) - it shows abstracting as intrinsically **selective**, and any map therefore as partitioned in (at least) two different ways:

i) into an **attended to** component (*figure*) and a **more or less ignored** component((back)*ground*),

ii) where* the attended to* aspect consists of both a **focal** component and a **subsidiary** component.

EXAMPLE

An example might serve to make clear this array of distinctions: Say that you play the violin, fairly well, and that you have a master violin teacher. Say that, at your violin lesson, you perform from the latest piece you have under study: say, Beethoven's Violin Concerto in D. For your teacher, you play a particular lyric passage.

This situation, we point out, serves as a suitable example of the setting of* transacting*, and provides an opportunity to use one of the undefined terms (e.g. *structuring*, as in you-structuring-yourself-and-your-environment, where the term *environment* includes the *violin* and* bow*, the* musical score*, your* teacher*, etc.).

In playing this passage, I assert, you develop two kinds of abstractions or maps, two kinds of *attended-to*/*ignored* awareness: To play anything, you MUST feel (generate and maintain a mainly-proprioceptive awareness of) what you do with the fingers of your left hand on the fingerboard, with your right hand as you hold and move the bow, etc., against the background of the feel of other bodily activities. While you perform the solo melody, however, your awareness of these details of your functioning must remain subsidiary - to play the melody, you MUST focus on (generate and maintain a focal mainly-auditory awareness of) the passage, the lyric theme and your production of it, against the background of other sounds, and against the subsidiary mainly-proprioceptive awareness of bodily activities involving fingers, hands, etc.

These distinctions give a language for making sense of the particular example of behaving-and-experiencing under discussion. We point out that these distinctions accommodate a wide body of research findings: those of the transactional psychologists and the neurophysiologists concerning how organisms "perceive"; the findings of Gestalt psychologists, as re-interpreted by Polanyi[35:55-65]; etc. Moreover, we can express these distinctions in a set theory notation - e.g. as "(x/(E - x)i" (including the quotation marks) to signify "the focal awareness of the environmental object x Î E , against the background (E - x), via sensory modality i ," and as "Sfi" , including the reversed quotation marks, to mean the subsidiary awareness of the processes of the Self, by which the organism puts together its focal awareness. These set theory conventions enable us to state the Postulate of Self-reflexiveness, by asserting the mutual necessity of these two kinds of awareness:

"(x/(E-x)i" Û
" Sf_{i} " Note 7, Premise 5

By making these interconnections we show the terminology of Gestalt psychology and Gestalt therapy as consistent with the non-aristotelian premises, and make it possible to use this lexicon with full mathematical rigor. But the Postulate of Self-reflexiveness goes even deeper, for it entails a non-aristotelian version of the logic of opposites (see ms pp. 28-9 and Notes 7 and 9) - as we can show by returning to our example.

Say that now your teacher stops you: "Go back to measure 90. When you come to the E-flat, why don't you switch over to the G-string, staying in third position? I believe you'll find that much easier - and more graceful."

So you **negate** your overall Gestalt - you interchange *figure *and (back)*ground*, or interchange the focal and the subsidiary kinds of awareness, so as to pay focal attention to the fingering (against the ground of other bodily activities), and subsidiary attention to the melody (against the ground of other sounds).

Once you have gone through the altered fingering several times, you get excited - yes, it really does seem more graceful (and easier)!

Then you re-negate your overall Gestalt, once more paying focal attention to the melody, and subsidiary attention to the fingering, etc.

Anyone LISTENING could immediately tell the difference between the three situations I have described - the first sounded would sound like a good student playing that lyric theme, the second sounded would sound like someone practicing something, and the third sounded would sound like a BETTER student playing the theme.

In WIE logic, when we negate a proposition (or a term), and then re-negate it, that brings us back "identically" to our starting-place. In this non-aristotelian frame of reference, negating and then re-negating a Gestalt does NOT bring us back to our starting-place. (And in human behaving-and-experiencing, the act of negating one’s Gestalt and negating it again in order to study a situation, in a manner similar to that described here, does not bring one back to one's starting-place. Here, it brings us into the presence of a BETTER student.)

This way of stating-and-understanding the Postulate of Self-reflexiveness gives us the means to formalize the version of the **logic of opposites** [36] (the relations which hold between a **term** and its negation or contradictory or opposite or **complement**) which operates in a non-aristotelian system. In the following listing of points, we take care to say where the map or representing or abstraction which we negate comes from, as well as how to negate it.

FIGURE 1 (STEP DIAGRAM) about here

d. NEGATING:

1. Our incompletely-informed and inaccurately-informed and self-referentially informed (symbolic) *organism*

2. Consists of spatio-temporally-ordered "doings" or "happenings" (aspects of an *ordering on abstracting*)` which occur within a (specific delimited) overall setting known as *transacting*.

3. By her/his* abstracting*, our organism elaborates a Gestalt (*abstraction*) composed of

a) a figure with an aspect which focally interests the organism, coupled with an aspect of subsidiary interest to the organism

b) specified against a (back)ground which does not (at present) much interest her/him.

4. Any Gestalt further consists of two components

a) one of which tells about one aspect of the *territory*, namely, the external environment, and

b) the other tells about another aspect of the territory, namely, the organism which elaborates the Gestalt.

5. In negating a Gestalt, our organism interchanges the figure and the (back)ground (or sometimes that means interchanging the focal and the subsidiary aspects), and alters none of the other conditions listed here.[37]

To re-state a point made on ms p. 17, since this frame of reference shows an intrinsic spatio-temporal ordering, when you negate and then re-negate a Gestalt, you do not come back to your original starting-place. For example, even with so "simple" a situation as the drawing presented as Figure 2 (ms p. 28), someone who sets out to negate and re-negate her/his visual Gestalt of that drawing 100 or more times in succession will grow bored, and discover for her/himself that what we call "boredom" functions as a kind of pain – clearly a change from her/his starting situation.

5. INCREASES OF KNOWLEDGE AS NON-IDENTITIES

Discussions of the development of physics, e.g. from Newton to relativity and quantum theory, hinge on how the discussants view the topic of how humans generate new knowledge. And that in turn depends on how they view the "old" knowledge, which gets modified or replaced by the "new" knowledge.

In accounting for new knowledge, exponents of the WIE tradition up till now have had no choice but to invoke their dualistic frame of reference – they found themselves required to hold that, independent of any observer, "things" "exist", and that identities and similarities among these "things" also "exist". An increase of knowledge results, they say, when someone **discovers** a pre-existing but hitherto overlooked "similarity" (e.g. within the domain of *matter*), and **describes **it (in terms which belong to the domain of *mind*). As Poincaré puts it,

In sum, ALL THE SCIENTIST CREATES IN A FACT IS THE LANGUAGE IN WHICH HE ENUNCIATES IT.[38; emphasis in original]

That assumes that "the fact" exists independent of "the scientist". The "old" knowledge, therefore, got itself discovered in a similar fashion – as a kind of fortunate accident which proved not only useful but "true, until displaced by the "newer" "truth".

In contrast, we hold that new knowledge arises by a process by which an individual (or a small group) makes new distinctions, which entails replacing at least one map-territory identity with a map-territory non-identity. This may sound facile and simple, but anyone who works it out – replaces a map-territory identity with a new distinction – has had to engage in a rigorous and exact sequence of "doings", many of which then seem to disappear from view, absorbed between the lines of "the language in which [s/he] enunciates it."

Below, we discuss the topic of the *setting* for our frame of reference in some detail. Here it suffices to say that it makes more sense – it sets forth experience in ways more nearly similar in structuring to our observations – to postulate a **specific delimited setting** for our frame of reference, which we designate by terms such as *transacting*, or by run-on phrases such as "*one-particular-organism-as-a-whole-dealing-with-its-environment-at-a-date, as viewed by a designated observer*."

In the domain of "how we hold "old" knowledge" – including historic frames of reference – our stance rules out awarely relying on map-territory identity, and with it, rules out relying on dualistic frames of reference. It disallows traditional constructs such as "mind" vs. "matter", etc., as manifested in the notion of static "things" that "exist" "out there", independent of any "observer". In particular, our frame of reference disallows explaining the process by which we humans gain new knowledge by postulating the disembodied "dis-covery" of some "already-existing" "similarity" by some "mind" that "thinks" about "things". Instead, we hold that terms such as *observer* and *observed* (e.g. *physicist* and *elementary particle*) signify INFERENTIAL entities, (rather than "the really real"). Further, we hold that what we perceive as, and call, (non-verbal) *observer* and *observed* arise together, in the course of the kind of (mutually-altering) transacting which we call *observing.*

In our setting, then, the traditional notions of "*similarities*" and "*differences*" come to refer to constructs which our organism or our observer might generate, by a process of abstracting. These constructs occur on verbal levels, and refer to verbal/symbolic levels (specifically, to the products of abstracting). In general, we regard "similarities" as something our organism generates, for her/his own purposes (e.g. a need to form generalizations and make predictions), and that s/he generates these "similarities" by a process of ignoring certain specific "differences".

To focus again on the topic of increases of knowledge as non-identities, as we see it from this frame of reference, any person, or any social group of persons, already has a viewpoint, a body of received knowledge. And, as the Gestalt psychologists have taught us, any viewpoint seems somehow "complete" at any given moment - it shows **closure**. As one consequence, personal or shared viewpoints contain few openly acknowledged blank spaces, few "I don't know"s. Mostly, where we don't know, where we have no first-hand experience, we have already found some way to cover over or otherwise conceal our ignorance - from ourselves as well as from others. To use a typesetting image, instead of acknowledging a hiatus in our knowledge/experience, we "delete and close up," and so conceal it.

Furthermore, few human groups (we believe) teach their members to hold the views put forth by the group itself in an "as if", tentative fashion. Instead, most hold the shared views in the rigid mode of "that's the way things really are" - which implicates tacit identity.

We see these matters illustrated in the debate over quantum theory. As we show over and over again in this paper, even the most accomplished physicists find it difficult if not impossible comprehensively to abandon the assumptions they grew up into, such as the assumptions encoded in the WIE languages – even though these fit poorly with the findings and viewpoint of quantum theory.

To generate new knowledge, then, someone (or some group) has to find some way of setting aside the rigidly-held, shared views on the topic of interest, and see or otherwise experience for her/himself how things appear to work in that domain. Having done so, s/he must devise some linguistic way of describing what s/he has found-and-created, so s/he can remember it her/himself and so s/he can pass it on to peers and progeny or other successors.

We call this process** innovating**, and focus on the ways an innovator has to make new distinctions, and so replace a map-territory identity with a non-identity. To illustrate these points, we offer six examples, the last two of which discuss aspects of quantum theory relevant to the present study:

EXAMPLES:

5. INCREASES OF KNOWLEDGE AS NON-IDENTITIES

Discussions of the development of physics, e.g. from Newton to relativity and quantum theory, hinge on how the discussants view the topic of how humans generate new knowledge. And that in turn depends on how they view "old" knowledge, that gets modified or replaced by "new" knowledge.

In accounting for new knowledge, exponents of the WIE tradition up till now have had no choice but to invoke their dualistic frame of reference - they hold that, independent of any observer, "things" exist, and identities and similarities among these "things" also "exist". Although no one explicitly says so, this formulation requires its exponents to display both sides of any given dualism (e.g. mind/matter). An increase of knowledge results, they say, when someone **discovers** a pre-existing but hitherto overlooked "similarity" (e.g. within the domain of __matter__), and describes it (in terms which belong to the domain of __mind__). As Poincaré puts it,

In sum, all the scientist creates in a fact is the language in which he enunciates it.[37]

In contrast, we hold that new knowledge arises by a process of making new distinctions, in which someone replaces a map-territory identity with a map-territory non-identity.

Below, we treat the topic of the *setting* for our frame of reference in some detail. Here it suffices to say that we envision a context for our frame of reference, which we designate by terms such as *transacting*, or by run-on phrases such as"one-particular-organism-as-a-whole-transacting-with-its-environment-at-a-date, as viewed by a designated observer."

This rules out dualisms, such as "mind" and "matter", etc., with static "things" that "exist" "out there", independent of any "observer". In particular, this disallows the disembodied "dis-covery" of some already-existing "similarity" by some "mind" that "thinks about" "things". Instead, we hold that terms such as __observer__ and __observed__ (e.g. __physicist__ and __elementary particle__) signify INFERENTIAL entities, not "what really exists, independent of any observer". Further, we hold that what we call __observer__ and __observed__ arise together, in the kind of transacting which we call __observing__. Here the notions of "similarities" and "differences" refer to constructs which our organism or our observer might generate, by a process of abstracting. They occur on verbal levels, and refer to verbal/symbolic levels (the products of abstracting). In general, we regard "similarities" as something our organism generates, for her/his own purposes (e.g. a need to form generalizations and make predictions), and that s/he does so by a process of ignoring certain specific "differences".

In this frame of reference, any person, or social group of persons, already has a viewpoint, a body of received knowledge. And, as the Gestalt psychologists have taught us, any viewpoint seems somehow "complete" at any given moment - it shows **closure**. As one consequence, personal or shared viewpoints contain few openly acknowledged blank spaces, few "I don't know"s. Mostly, where we don't know, where we have no first-hand experience, we have already found some way to cover over or otherwise conceal our ignorance - from ourselves as well as from others. To use a typesetting image, instead of acknowledging a hiatus in our knowledge/experience, we "delete and close up," and so conceal it.

Furthermore, few human groups (we believe) teach their members to hold the views put forth by the group itself in an "as if", tentative fashion. Instead, most hold the shared views in the rigid mode of "that's the way things really are" - which implicates tacit identity.

We see these matters illustrated in the debate over quantum theory. As we show over and over again in this paper, even the most accomplished physicists find it difficult if not impossible comprehensively to abandon the assumptions they grew up into, such as the assumptions encoded in the WIE languages, even though these fit poorly with the findings and viewpoint of quantum theory.

To generate new knowledge, then, someone (or some group) has to find some way of setting aside the rigidly-held, shared views on the topic of interest, and see or otherwise experience for her/himself how things appear to work in that domain. Having done so, s/he must devise some linguistic way of describing what s/he has found-and-created, so s/he can remember it her/himself and so s/he can pass it on to peers and progeny or other successors.

We call this process __innovating__, and focus on the ways an innovator has to make new distinctions, and so replace a map-territory identity with a non-identity. To illustrate these points, we offer five examples, the last one of which entails a discussion of aspects of quantum theory relevant to the present study:

EXAMPLES:

a. Declining to question the received construct(s)

Apropos of what we call **the is of identity** (locutions consisting of

All pines are trees, but not all trees are pines.

We claim no competence in ancient Greek, but at least in English, we find no overt linguistic markers to differentiate the relation expressed by the copula in that sentence from the overtly symmetric sense of the copula in sentences such as:

A pine is a pine.

For over two thousand years, commentators reflecting on this problem have made excuses for *is*, rather than regarding this "difficulty" as grounds for holding the construct of *identity*, including its manifestations in *is of identity* constructions, as intrinsically ambiguous, logically unsatisfactory and/or empirically disconfirmed. Aristotle's discussion, for example, amounts to a metacomment, after the fact, which does not modify the grammar, or the assumptions it encodes. Instead, it defends Aristotle’s accustomed languaging and his (mainly unstated) presuppositions.

b. Piecemeal questioning of the received construct(s)

In devising his construct of universal gravitation, Newton comes to utilize non-identity reasoning. Prior to 1665, the commonly-held, unquestioned view or map held that "Things like balls or apples just naturally fall, while the Moon naturally stays up there -- that's the way things really are." To call this view *unquestioned* means that those who held the view did not treat it as a provisional construct, but rather, tacitly treated this map as somehow identical with the territory. Galileo and others had done earlier work dealing with topics in mechanics, including problems involving falling objects. But in 1665-6, we know of no one besides Newton who had considered that the moon could fall, much less asked how come it didn't.

In effect, in developing the construct of universal gravitation, Newton changes what he assumes. As we picture the situation, Hhis first Gestalt in the relevant sequence has as its background the traditional view, which tacitly assumes map/territory identity; and has as its figure a questioning of the traditional view, so that in effect this Gestalt distinguishes between map and territory: *What if the commonly held view amounts to only a surmise, instead of expressing "the way things really are"*? This Gestalt then becomes the background of the next Gestalt encoded in the insight we attribute to Newton. In expressing its figure, Newton devises an alternative map: *What if the ball (or apple) and the moon both fall -- and the apple reaches the surface of the earth whereas the Moon does not.* * How would I account for that?*

Newton completes this sequence by using his newly-developed mathematics of fluxions (calculus) to compute the centripetal and centrifugal forces on the moon, and to compute the value for the period of the moon's orbit which would make these forces equal. His first rough calculation gave a value that matches the true value of about 27.25 days "near enough." In so doing, he TREATS his surmises concerning gravitation as disconfirmable, and so relies on a tacit version of the Postulate of Non-identity. This stands as an important step in the development of the practice of science, but does not overtly modify the WIE grammar or the assumptions encoded therein. (At other places in his theory, e.g. in positing "absolute space", "absolute time", or "absolute simultaneity," Newton clearly projects the assumptions of his grammar and so relies on tacit identity.)

c. Non-identities (distinctions) in logic and mathematics

Exponents of nineteenth-century logic and set theory, including Gottlob Frege, introduce distinctions such as that between *Name* and *Thing Named*, and that between the *use* and the *mention* of a term. These distinctions - of the form of explicit non-identities, although no one else has called them that, to the best of our knowing - got incorporated into the practice of WIE logic; but again, they do not modify the WIE grammar or the assumptions encoded therein. However, the growing number of "exceptions", of practices that lead seasoned exponents to a kind of linguistic wariness, leaves an increasing sense of "cognitive dissonance" within science.

d. Non-identities in the premises of set theory

Following the paradox concerning "sets which do not belong to themselves" he put forth in 1902, Bertrand Russell (in his theory of types), and Zermelo, Fraenkel and others (in what we now call the modern mathematical theory of sets), introduce *ad hoc* provisions to prevent or otherwise avoid the kinds of trouble posed by the paradox. Zermelo's version includes a specific axiom designed to forbid treating the undefined relation of **belongs to** as reflexive:

No set may belong to itself.

Such provisions DO modify the assumptions encoded in the notation – in particular, provides an altered version of the logic of opposites (see Note 9 and ms p. 28-9 for details). But the presuppositions encoded in the WIE grammar operate at a more basic level than do those which make up the premises of the notational system: Set theory still treats of self-identical *sets* (noun-surrogates) which enter into more or less transient *relations* (verb-surrogates).

e. Non-identities in the theory of relativity

Bernard d'Espagnat discusses a structure of assumptions widely held by speakers of WIE languages, including many scientists - which quantum theory calls into question. He writes,

… Even if quantum mechanics is considered to be no more than a set of rules, it is still in conflict with a view of the world many people would consider obvious or natural. This world view is based on three assumptions, or premises that must be accepted without proof. One is realism, the doctrine that regularities in observed phenomena are caused by some physical reality whose existence is independent of human observers. The second premise holds that inductive inference is a valid mode of reasoning and can be applied freely, so that legitimate conclusions can be drawn from consistent observations. The third premise is called Einstein separability or Einstein locality, and it states that no influence of any kind can propagate faster than the speed of light. The three premises, which are often assumed to have the status of well-established truths, form the basis of what I shall call local realistic theories of nature. … [and] are essential to a common-sense interpretation of the world. [39, p.158b-c]

He further maintains [39, pp. 174-8] that these assumptions form the fundamental basis for WIE science, and that many scientists cannot see how to abandon even one of them, and recoil from doing so.

We have already discussed the first of these assumptions, and shown that the construct of "some physical reality whose existence is independent of human observers" requires human observers to hold it - observers who on some fundamental level (such as that of grammar) fail to distinguish between* map* and* territory*.

The second of these premises (as we see it) amounts to the claim that the grammar of the WIE mathematical languages has a cosmic validity - that "the world" or "reality" really does consist of static-and-unchanging things and more or less evanescent relations, exactly suitable for us to represent by means of self-identical *quantities* or other noun-cognates and *operations* or other verb-cognates, with nothing "left over" - so that any symbolic transformation which proves logically valid by the rules of WIE mathematics and logic will also prove empirically valid within WIE physics. (But this amounts to a map-territory identity, as forbidden in our frame of reference.)

The third of these premises has no place within the traditional WIE grammar. As its name suggests, Einstein, early in the development of special relativity, introduces one version of this assumption as an antidote to a hidden assumption he attributes to the received physical theories of his day (and so eventually to Newton and his successors).

Newton discusses the occurrence of two events as if they either really do occur **simultaneously**, or else they really do not. To express a twentieth-century concern, he tacitly assumes that whatever signals or stimuli make it possible for observers to judge when events occur will reach them, as Alice in Wonderland puts it, in "no time at all." Likewise, where he posits action-at-a-distance, as with the force of gravity, he discusses the manifestations of such *forces* or such *actions* as taking place **instantaneously**. "Observers" or the process of "observing" (abstracting) make no difference – any observer, or any number of observers, observing competently, would see the two events, or the manifestations of such forces, the way they really happened, and would readily agree that they saw them that way.

Where it becomes necessary to convert mathematically from one frame of reference to another of the same kind, Newton appropriates Galileo's transformation rules, which codify the view that for physics, there exists a single underlying framework for the entire Cosmos. Within that framework, the difference between the observations made within one set of, say, Cartesian coordinates and those made within another depends on the distance between their axes and the time it would take to get from one to the other (the velocity of travel). Newton sees no need to distinguish between "what happened" and "MY VIEW of what happened" (or in our terminology, between *territory* and *map*). Although at times he seemed uncomfortable with the construct of *instantaneous-action-at-a-distance*, his theory requires this construct, and indeed, with it in place, his theoretical system continued to survive experimental testing for 200+ years. Newton seems to have expressed no sense of impending difficulties concerning the notion of *simultaneity. *

During the last two decades of the nineteenth century, reputable physicists, utilizing acceptable methods of study, made a series of perhaps several hundred anomalous observations, each of which appeared to disconfirm tenets of Newtonian physics.

The innovations in physics which Einstein proposes turn out to account for these anomalies, by abandoning the notion of a single framework for the entire cosmos. In its place, he introduces the alternative supposition that each observer has a different frame of reference. But this requires him to begin to take into account matters centering about the notions of *observers* and *observing* (*abstracting*). This Einstein does by discussing how observers might use signals, including light signals, to compare their observations and reconcile their findings. Einstein expresses his insights by a synecdochal usage of the notion of *the velocity of light*. The grammar of his mathematics does not allow direct discussion of the topics of *observers* or *abstracting,* any more than the grammar of Newton's mathematics does. Einstein gets around this limitation by substituting the construct of *a coordinate system* (usually Cartesian) for the notion of* observer*.[40] Einstein posits that (a) light propagates at a finite velocity (b) which remains constant (has an upper bound) for all "observers" (coordinate systems), regardless of their motion relative to one another or to the light source. To convert mathematically from one observer's frame of reference to another's, he utilizes the **Lorentz-Einstein transformations**, which amount to the Galileo transformations with a correction factor - in which the square of the velocity of light appears under a square-root sign, as the denominator of a fraction. From our frame of reference, we would speak of this relativisitic viewpoint as showing one variant of spatio-temporal ordering.

Einstein's innovations not only provide an acceptable mechanics and end up handling the anomalous findings, but also provide insights concerning how to bridge between Newtonian and relativistic mechanics. Physicists noticed that, if they treated the notion that light propagates at an "infinite velocity" as a restricted and restrictive assumption, and substituted that value into the Lorentz-Einstein transformations, that converts the value of the correction factor to 1, which "reduces" the Lorentz-Einstein transformations to a symbolic structure equivalent to that of the Galileo transformations. Conversely, if they eliminated this restrictive assumption from the Galileo transformations, that "expands" them into a symbolic structure equivalent to that of the Lorentz-Einstein transformations. This, of course, amounts to an early variant of what, above, we call a *proof of generality* (ms p. 12-13). Here, the restrictive assumption that *light propagates at an infinite velocity* (or its velocity has no upper bound) appears so restrictive as to hold under no circumstances whatsoever. In other words, it stands in for what, in our frame of reference, we call *a map-territory identity* or *a usage of the Postulate of Tacit Identity*.

To see what we mean by that, let us scrutinize the notion that light propagages at an "infinite velocity." Exponents of WIE physics may regard that as a familiar, and acceptable, notion. Let us take another look. In WIE physics, we define *velocity* **v** as *distance* **s** divided by *time* **t** . To obtain the value of an INFINITE velocity, someone has to evaluate the fraction s/0 - and the rules of arithmetic forbid dividing by zero. Hence, at least within our non-standard frame of reference, the notion of an "infinite velocity" fulfills the role of a **meaningless noise**. In this instance, to rely on THIS meaningless noise as if it qualified as a legitimate construct has the effect of making "observers" or the process of abstracting irrelevant - or, to coin a phrase, has the effect of eliminating the observer from consideration. And from our point of view, that appears tantamount to relying on a usage of map-territory identity.

(Some other workers have made a diagnosis remarkably similar to this one, but framed in other terms and using other grounds.)[41]

Up to this point, we have inter-translated between our frame of reference and several variants of the traditional WIE frame of reference. In particular, we have inter-translated between "a view of the world" which d'Espagnat asserts that "many people would consider obvious or natural" and our own frame of reference, and have related the three assumptions which d'Espagnat holds as underlying that "world view" to our own views.

As a further point, d'Espagnat asserts that quantum theory "is still in conflict with" that traditional world view, even when we take quantum theory as "no more than a set of rules."[42] We propose to use the resources developed here to examine that conflict.

f) The measurement problem

A little reflection suggests that our non-identity-based frame of reference may provide insight into the so-called** measurement problem** in quantum theory. Ohanian phrases this as follows:

The weirdest feature of the Copenhagen interpretation is that it requires that the wave function suffer a discontinuous, unpredictable change during the measurement. Consider, for instance, the impact of an electron on the fluorescent screen in the electron diffraction experiment. The impact and the flash of light released in it constitute an (approximate) measurement of the position of the electron. Just before this measurement, the wave function was spread out all over the screen; immediately after the measurement, the electron position is known to lie within some small spot on the screen, and the wave function must therefore have an extent no larger than this spot. Thus, during the measurement, the wave function suffers an unpredictable collapse or reduction. The collapse is unpredictable, since we have no way of knowing onto what part of the screen the wave function will collapse – we know only the probability distribution of the spots on which the wave function collapses, that is, the probability distribution of positions for the electron on the screen.[43]

For today’s exponents of the dualistic WIE versions of quantum theory, this discontinuous collapse of the wave equation still occurs, it still remains unpredictable, and workers still cannot explain it in terms of the dynamics of the theory

As we point out above (ms pp. 6-10), those who rely on traditional, intrinsically dualistic WIE frames of reference systematically eliminate the observer from consideration. Any physicist who uses WIE-based mathematical and/or discursive languaging in framing her/his quantum theory may encounter difficulty in avoiding joining the ranks of those who eliminate the observer. We suggest that exponents of WIE quantum theory may even generate the measurement problem for themselves in the course of the way(s) that they eliminate the observer; and in contrast, we suggest that exponents of a frame of reference which systematically takes the observer into account might encounter no measurement problem.

Ohanian, for example, in his description of the measurement problem, manages to avoid describing in detail what a physicist DOES in the course of performing an electron diffraction experiment. Indeed, he avoids even mentioning any observer at all, except as concealed in his usages of the passive voice ("the electron position is known to lie… ") or in generalized usages of the first-person-plural pronoun ("we have no way of knowing … – we know only the probability distribution …"). Many scientific writers, we believe, follow similar editorial conventions in writing about quantum theory (among other topics).

To begin the process of explicitly taking the observer into account here, let us start with an obvious factor that determines the design of experiments, which probably does not get listed in most experimental protocols: We human organisms find ourselves limited in what, with our unaided sensory receptors, we can sense. When we design and perform experiments, most of the details we focus on come from the necessity to make the ‘happenings’, the ‘changes’ of interest, detectable by the experimenter(s). Quantum theory tells us about continuously-evolving ‘happenings,’ of the curious form of superpositions of states, etc. However, when we make a measurement, we ask for something else: we ask for enduring traces of these transient ‘happenings’, of a type that we humans can detect. An instrument such as a cloud chamber, bubble chamber, fluorescent screen, photographic plate, etc., can give us these enduring traces – at a price. Present evidence suggests that this price includes momentarily registering our altered demand, as a momentary interruption of the continuous development of the superpositions of states – precisely what, in the present WIE versions of quantum theory which (inconsistently) eliminate the observer from consideration, we regard as "the measurement problem."

As everyone already recognizes, verbal expressions such as "I see an electron!" or "I detect an electron!", emitted upon seeing (for instance) what looks like a flash of light at some point on a fluorescent screen, stand as hyperbole – a metaphor. We may SEE what we accept as an "enduring trace", but at best we only infer the electron.

We find similar grounds for holding that we also only infer the observer who claims to "see" an electron, - but these seem much less widely recognized. As an initial way to explore that possibility, let us look "between the lines" of the way Ohanian presents the example in his discussion of the measurement problem: He seems to us to posit a physicist functioning in at least three different roles, during at least three different moments of her/his life and on at least three different "logical levels" (positionings in an ordering on abstracting):

a) As an **experiment-designer**, manipulating some portion of the formalism of quantum theory in the context of some class of ‘observables’. The Gestalt (abstraction) s/he generates has as its (back)ground the formalism of quantum theory available to the worker; and as its figure, has the transacting with the design for the experiment which the worker comes up with. This amounts to transacting with symbols – written, spoken, signed, etc.

b) As an **experiment-performer**, actually making a measurement of some ‘observable’. To skip lightly over the elaborate procedures of acquiring the necessary apparatus, assembling it into the requisite experimental system, the process of following the written protocol for the experiment, the process of taking notes on the results, etc., the Gestalt (abstraction) s/he generates has as its (back)ground the experimental design from (a), and as its figure, has the crucial transacting with the actual measurement – e.g. with the flash of light at a particular location on the fluorescent screen. This amounts to transacting with non-verbal ‘happenings’.

c) As an **experiment-interpreter**, contemplating the discontinuity between her/his transactings with the symbol-level predictions concerning the evolution of her/his experimental system which s/he generated by manipulating the relevant formalism vs. her/his transacting with the non-verbal ‘happenings’ s/he obtained when s/he made her/his measurement. The abstraction s/he generates has as its (back)ground the details of the figures of both (a) and (b) – the experimental design, the apparatus, the notes, etc. – and as its figure, has her/his transacting with, as Ohanian puts it, "the probability distribution of the electron on the screen" vs. "the flash of light released …[as] an (approximate) measurement of the position of the electron."

At this point, we expect that we can only annoy our readers by elaborating on the truism that

verbal-level symbol non-verbal ‘happenings’,

and the finding that by posing the measurement problem in a frame of reference which includes the observer, we have changed the "weird" discontinuity from a "how come it does that???" into an acknowledgment of the alteration of our own transactional demands on the experimental system.

III. QUANTUM THEORY AND NON-IDENTITY

A. THE DOMAINS OF CLASSICAL AND OF QUANTUM-THEORETIC 'HAPPENINGS'

Physicists claim to provide a description of "reality" - specifically, of "physical" reality or "external" reality.

In our opinion, the term *reality*, with or without these modifiers, invites its users to beg some key questions. In its place, we propose to use the term *'happenings'*, including the single-quote marks. Please regard it as a "pointer", chosen so as to beg as few questions as possible.

Classical Newtonian physicists prior to 1905 discuss physical 'happenings' **quantitatively**, in terms of "the state of the system in phase space". They utilize the calculus - in principle made up of arithmetic or algebraic transforms of the **complex numbers** - to do so. They speak of these 'happenings' as if they occurred independent of any observer. Otherwise stated, in the absence of symbolic resources that would allow them to operate by any other pattern, they have no choice but to invoke the "matter" side of their dualistic world-view, and posit a "physical" domain P; and in the same breath, to invoke the "mind" side, and posit the "language" domain of mathematical analysis in which to describe and discuss this domain P. However, their symbol-systems include no provision that requires them explicitly to distinguish between those supposedly independent (physical) 'happenings' and the (mental) numerical representations of them. In other words, they silently rely on the Postulate of Tacit Identity, operating as if a symbolic structure framed in terms of (self-identical) quantities and (not-self-identical) operations matches perfectly the 'happenings' they study. By their own example, they pronounce the branch of mathematics called *analysis* - including *arithmetic*, *algebra*, *calculus*, etc. - as capable of entering into an exhaustively complete, entirely accurate one-to-one relation with "what really happens, independent of any observer."

The quantum theorists, in their turn, posit a domain Q of quantum theoretic 'happenings'; and these seem in some elusive way non-classical. These workers have reached no general agreement as to just how these quantum 'happenings' differ from classical 'happenings'. This bothers physicists trained in representing "real" happenings in "real" notation.

Werner Heisenberg,[44] dissatisfied with the epistemological hodgepodge of the quantum theory of his day, sets out to axiomatize the quantum approach and produce a self-consistent physical theory. As he laboriously worked out the needed relationships, someone looking over his shoulder Born & Jordan [45] pointed out that Heisenberg seemed to them to have employed the new field of matrix algebra. When Heisenberg looked into that work, he found He finds that matrix algebra makesmade it possible for him successfully to represent the 'happenings' of concern from the quantum domain Q . A matrix comprises, not a self-identical number, but rather an intricate relation (arrived at by a linear transformation rule from self-identical numbers). For example, the matrix multiplication rule specifies that square matrices need not commute.

mn ¹ nm (1)

Heisenberg's matrix elements exhibit relations that impose limitations on the precision of our measurements of what we can observe in the region where quantum effects dominate. He surmises that these restrictions or uncertainties in measurements cause us to abandon, to some extent, our reliance on the classical notions of *particles* and *waves* because of our inability to measure them with arbitrary precision in the quantum region.

Niels Bohr opposes this view, insisting that the only contact we have with the real world comes through classical terms such as wave and particle. The Complementarity Principle becomes manifest in statements such as the following:

The unambiguous interpretation of any measurement must be essentially framed in terms of classical physical theories, and we may say that in this sense the language of Newton and Maxwell will remain the language of physicists for all time.[46]

To Bohr the process of measurement, always performed in the classical sense, necessitates a choice between complementary outcomes. Therefore, for example, we either observe wave phenomena or particle phenomena, never the two simultaneously.

B. COMPLEMENTARITY AS A "REVERSIBLE" GESTALT

Now let us shift to a frame of reference with a setting which we can represent in terms of "one particular organism-as-a-whole-dealing-with-its-environment-at-a-date, as viewed by a specified observer." Then this description of two aspects of complementary effects resembles the description of a certain kind of Gestalt (defined as "a figure of focal interest to the organism, against a background relatively empty of interest", where the focal aspect subsumes the operations of setting up and performing the experiment, and the subsidiary aspect subsumes the question of which of the available experiments the experimenter performs).[47] In one experiment, one term of the complementary term-pair stands sharply in focus (we know it precisely) while the other stays out of view (we know it imprecisely). Then we may consider the precisely-known term as occupying the role of the figure of the Gestalt, and the imprecisely-known term as the background of the Gestalt. Alternatively, in a different experiment, the formerly out-of-view term now stands sharply in focus, and the formerly in-focus term now stays out of view. Consider this as a different Gestalt, with figure and background from the former Gestalt now interchanged in the new one.

This situation resembles a famous Gestalt picture (Figure 1) - when you look at it one way, you see a white chalice against a black background; seen another way, it looks like two shadowed faces in profile against a lighted window.

FIG. 2 ABOUT HERE (Faces/vase)

Consider as an example of this kind of "reversible" Gestalt the often discussed double--slit experiment. A stream of electrons from a single source passes through a barrier that has two small openings, and from there to a distant fluorescent screen which can detect where they hit. When we look at the flashes on the screen, we see a pattern equivalent to a two-slit interference pattern produced with monochromatic light - a distribution of intensity, marked by successive minimum and maximum points. Quantum wave mechanics asserts that wave packets representing electrons going through one hole interfere on the screen with the respective wave packets of electrons going through the other hole to produce the interference pattern. We would still see the construction of this interference pattern even if we could slow the stream of electrons enough so it consisted of one electron at a time. In this case, discrete detections build up, dot by dot, into a characteristic interference pattern over a period of time. Quantum theory argues that the evolution of the wave packet represents the probabilities of the future positions of the electron. For a single electron such probabilities come from superimposing the wave packet representing the electron passing through hole 1 on the wave packet representing the electron passing through hole 2.

FIG. 3 ABOUT HERE (Double-slit)

When we place detectors at or near each hole, in hopes of determining which one the electron actually passed through, we get a different pattern. The interference pattern on the distant screen disappears. In its place we see a pattern, on the nearby screens, which indicates that the electron either passed through one hole or the other. We get no indication of a probability distribution like that which the interference effect suggests. Having gained positive knowledge as to which hole the electron passed through, we find we must set equal to zero the wavefunction corresponding to the probability of passing through the other hole. Therefore, two wave packets no longer exist; hence no superposition effect; hence no interference pattern. Now the difficulty arises in that if we take as "real" the one wave packet actually followed by the electron, we must regard the other as "having no physical reality" - we must regard it as retained only as long as we did not know which wave packet the electron would follow. If we have reasoned this out correctly, we must face the question of how the wave can interfere. How can a "real" wave interfere with an "illusion" to produce an observable interference pattern?

Heisenberg's Principle of Uncertainty rescues physics from this dilemma. The Principle asserts that in the physical act of making the observation, in this case finding out which hole the particle actually passed through, we have disturbed the event enough to alter its future history and bring its particle nature into existence. The interference effect endures as long as the particle remains unobserved. If we don't directly observe it, we can view the particle as existing in both wave packets at the same time. This representation clearly conflicts with the traditional understanding of the construct of *an electron*, or of any particle, as a discrete entity occupying a well-defined position in space at any given instant. In this interpretation we see the kernel of Bohr's Principle of Complementarity. We must view the particle aspect and the wave aspect as complementary and as exhibiting different aspects of the same underlying reality. When we look to the distant screen we detect wave patterns (against the remembrance of particles), but when we look at nearby screens close to the holes we detect particles (against the remembrance of wave patterns). These complementary findings appear strikingly similar to figure/background 'happenings'. In fact, we say precisely that: the Principle of Complementarity functions like a "reversiblereversible" Gestalt.

In another context, let us look at quantum field theory. Here we can picture any particle interaction(s) in space-time diagrams, and associate each diagram with a mathematical expression which allows one to calculate the probability that the corresponding process will occur. In 1949, the late Richard Feynman established the exact correspondence between the diagrams and the mathematical expressions, and consequently physicists know these representations as Feynman diagrams.[48] As its central idea, quantum field theory holds that when particles interact, they exchange energy and momentum in quantized packets known as virtual particles.[49] To picture this viewpoint as a Gestalt, we say that the force of interaction (electric, gravitational, nuclear) causes the exchange, and treat that as the figure of the Gestalt; we relegate the production of virtual particles to the background. With Feynman diagrams, we bring the virtual particle production into the foreground and treat the force as the corresponding background.

FIG 4 ABOUT HERE (Feynman diagrams)

C. THE SETTING FOR THE QUANTUM-THEORETIC GESTALT

These insights bring up an interesting question: In what kind of world (or world-view) would the ELEMENTARY PARTICLES behave like a Gestalt?

In order to answer this question, we will have to consider, from our more general frame of reference, the setting underlying each of several world-views, including the setting for any frame of reference based on the non-aristotelian premises of Korzybski. (To do so requires attending to the structuring of world-views other than our own - probably an unfamiliar exercise for most of us.)

1. EASTERN

Exponents of Eastern world-views (Ea) hold that "Everything is Self or Soul or Atman" - that "All else is *maya*, illusion." As a favorite metaphor puts it, "The Atman, the One, plays at being the Many, and does such a good job of play-acting that it fools Itself." Such exponents do not do "logic" precisely as WIE exponents do; but they appear to take the views pointed to by such remarks as in some sense fundamental for their frame of reference and then work out the details within such frames of reference.

We generalize by saying that the primary relations dealt with in this world-view take place between Atman and Atman. If we wanted to draw a graph of such primary relationships, we would have to give both axes the same name - say, call the X-axis "organism" and the Y-axis "organism" as well. Or as a Cartesian product space, designate it as O ´ O , "organism-cross-organism."

Ea = {(o,o) | (o,o) Î O ´ O}

From the standpoint of our theory, we say that the Eastern world-view eliminates from consideration the Environment.

2. WESTERN

Exponents of Western world-views (Ws) say in effect, "Well, I don't feel so sure about the Atman or the self or the soul, but the Outside World - that's Really Real!" Again, such exponents appear to take the views pointed to by such remarks as somehow foundational.

We generalize by saying that the primary relations dealt with in this world-view take place between Outside World and Outside World. We even have a specialized science that studies these relations - we call it Physics. Again, if we wanted to draw a graph of such primary relationships, we would again have to give both axes the same name - say, call the X-axis "environment" and the Y-axis "environment" as well. Or as a Cartesian product space, designate it as E ´ E , "environment-cross-environment."

Ws = {(e,e) | (e,e) Î E ´ E}

We regard the Western world-view as manifesting the presuppositions encoded in the grammar common to the Western Indo-European (WIE) family of languages, and say that it systematically eliminates the Observer from consideration.

3. NON-ARISTOTELIAN

Exponents of a world-view based from the very beginning on the non-aristotelian premises (Non-A) envision a **specific delimited** setting, which we can point to in English with words such as *contacting* or *transacting*, or by run-on phrases such as "one-particular-organism-as-a-whole-dealing-with-its-environment-at-a-date." Instead of regarding "the organism" and "the environment" as separate THINGS, which occasionally come into contact with each other, they (we) regard *my*__ __*environment* as "The other side of my skin," and *me* as "The other side of the environment's skin."[50] With Perls, Hefferline & Goodman, they (we) say,

We speak of the organism contacting the environment, but it is the contact[ing] that is the simplest and first reality.[51]

And from this "simplest and first reality," this setting, we INFER *organism* and *environment*. (Here, as usual, we maintain the map-territory distinction: We regard the territory - Thing(s) Named - as inferred, no matter what Name(s), terms or maps we may use to name or designate or refer to it.)

We generalize by saying that the primary relations dealt with in this non-aristotelian world-view take place between (the inferred) Organism and Environment. If we wanted to draw a graph of such primary relationships we would have to call one axis, say, the X-axis, "organism," and call the other "environment." Or as a Cartesian product space, designate it as O ´ E , "organism-cross-environment."

Non-A = {(o,e) | (o,e) Î O ´ E}

This world-view systematically takes both *organism* and *environment* into account.

D. EXPERIENCING THIS SETTING

Perhaps the construct of an alternative world-view with a specific delimited setting, based on the non-aristotelian premises, seems very abstract, unfamiliar and unreal to you. A simple experiment will serve to bring it within your own immediate firsthand experience.

**Instructions:** Here-now, please reach out and touch something - the arm of your chair, a friend's hand, the frame of your glasses, or whatever. Continue touching it for ten seconds or more, while letting yourself experience doing so. But do your best not to SAY anything, aloud or to yourself, about this experiencing.

(If you do find yourself speaking aloud, or sub-vocally "to yourself", pay no attention to the speaking voice, and just keep noticing your non-verbal experiencing, while continuing to touch this object in your environment.)

Take a look at your own experiencing. We imagine that you had no difficulty distinguishing what we might call "the feel of what you touched" from what we might call "the feel of your hand touching it."

Consider that experiencing as a case in point, an example: The setting on which we build up this non-aristotelian world-view consists precisely of transacting(s) (as we say it in English) like what you just produced in your own experiencing by touching something.

From such transacting or contacting, then, you INFER constructs such as "I"-and-"it",

"I"-and-"you", "organism"-and- "environment," etc. As a quantum physicist, you might infer constructs such as "observer"-and-"elementary-particle". But the two go together, like the "heads" and "tails" of a single coin. The setting of* transacting* requires us to consider, at a minimum, "an observer observing the observed, as viewed by a designated onlooker."

In quantum physics, this would mean something like, "a physicist-observer observing an elementary particle, as viewed by a designated onlooker." Notice that this specific delimited setting disallows constructs such "an 'it' that occurs or exists independent of any observer". Instead, once you have grounded your constructs on this setting of *transacting*, not only can you account for the finding that your "elementary particles" show Gestalt behavior, but it becomes intuitively necessary to regard them this way. From certain experiencing, you infer the "elementary particle" in question. But you cannot do that without also inferring the "physicist-observer" who observes this "elementary particle." And any *observing* which (s)he does entails this physicist-observer generating a Gestalt of the "elementary particle" in question.

IV. A SET-THEORY PROOF

Heisenberg’s axiomatization of quantum theory, and his Uncertainty Principle, antedate Korzybski’s Postulate of Non-identity, so he could not possibly have relied explicitly on that postulate. Above (ms p. 9), we offer the hypothesis (A) that there exists an innovative strand within quantum theory which presupposes a tacit version of the Postulate of Non-identity.

To test this logical surmise, we perform a set theory proof. The fact that we utilize a non-identity-based frame of reference, and that this framework enjoys the advantage of generality over the identity-based mathematical theory of sets (cf. ms p. 21) gives us a certain freedom here. Our readers will find that we do not merely bend the rules of set theory, we break them. Systematically. Perhaps a main point of the demonstration lies in its power to convince us Western scientists that we cannot deal adequately with the domain of quantum theory within identity-based restrictions – in order to play on that court, we MUST, in a disciplined fashion, violate the rules of set theory.

But that means that our proof (or "proof") bridges between the familiar, identity-based WIE frame of reference and a newer, more unfamiliar, non-identity-based frame of reference which we (clumsily) designate as *non-aristotelian*. To many readers, our "proof" may seem "analogical" rather than "strict."

PART 1:

First, within the domain of **human-created ‘symbols’** in the most general sense, we characterize the structure of the domain of classical physics, and contrast that to the structure of the domain of quantum theory. In order to do this,

1) We point out that from Galileo, Kepler and Newton on, classical physicists framed the domain of classical physics within the field of analysis. Thus they express the **phase space** of Newtonian physics in terms of complex numbers and transforms on complex numbers. To summarize this domain in our set theory notation, we adduce the domain of self-identical WIE notational languages, which includes the set of complex numbers, the logical axiom of identity, the equivalence relation of identity, the relations of associativity, commutativity, distributivity, etc., and the construct of the empty set.

WIE mathematicians do not usually make a point of distinguishing between the relation of identity as used in the modern logical axiom of identity and identity as an equivalence relation. We do, in order to partition the domain of ‘symbols’, including complex numbers and the **‘observables’** (or ‘operators’ which extract ‘information’) of quantum theory, into at least two parts.

i) One part appears quite traditional: e.g., ‘symbols’, like some x,y,z , for which the reflexive, symmetric and transitive relations hold. According to the rhetoric generated by making this distinction, the fact that these ‘symbols’ satisfy the equivalence relation of identity means that they therefore satisfy the logical axiom of identity and so belong to the domain of self-identical WIE languages. When we go to perform arithmetic or other analytic operations on them, they will "behave" like proper, self-identical "things" (e.g. complex numbers) and yield up the right (classical, Newtonian) answers, as we expected.

ii) In contrast, at least one other part of this domain appears non-traditional, as if such ‘symbols’ (e.g. ‘observables’) rest on some basis for science different from that of the WIE tradition: e.g., a collection of ‘symbols’, like some a,b,c , for which at least one of the key relations (symmetric, transitive, reflexive) does NOT hold. According to the rhetoric generated by making this distinction, if, for a given grouping of ‘symbols’, one of the three relations characteristic of an equivalence relation does not hold, it doesn’t matter whether the other two do or not – from our non-traditional, non-WIE point of view, such ‘symbols’ do NOT satisfy the equivalence relation of identity, and so do NOT satisfy the logical axiom of identity, and therefore do NOT belong to the domain of self-identical WIE languages. When we go to perform the non-aristotelian analog of "arithmetic" or other "analytic" operations on them, they need NOT behave like proper, self-identical "things" – they might not and yield up classical Newtonian answers. They might, however, yield up correct quantum-theoretic answers.

2) With reference to arithmetic or algebraic operations, e.g. multiplication, we specify the constructs of **commutative** and **non-commutative**.

Like Einstein faced with the Lorentz contracture and the Lorentz transformations, we reason that the procedure Heisenberg used did not merely deliver a "right answer" - instead, we hold, he had disclosed something fundamental. The historical accounts tell us that as Heisenberg got started on his project of axiomatizing quantum theory, he built up a notation in which, when he performed operations (e.g. multiplication) on the ‘symbols’ he used to represent the ‘happenings’, the ‘observables’, of quantum theory, these operations did not commute. We take that as an insight concerning something fundamental to the quantum domain. Meanwhile, as noted above, someone else, looking over Heisenberg’s shoulder,Born & Jordan pointed out the similarity between what he had already written and the new branch of mathematics known as matrix algebra. That may have eased his Heisenberg’s path, giving him an acceptable notation within which to pursue the further development of his insights – but at the price of constraining him to remain within the assumption-structure of WIE languages as he did so. We take his use of matrix algebra as providing "a way to get the right answers", which spared him the necessity to question, disclose and replace other presuppositions, other linguistic habits.

3) In order to pursue our own inference here, we show that an "if and only if" relation holds between the construct of *commutative* and that of *symmetric*.

We already know that, in order suitably to represent the ‘happenings’ which make up the quantum domain, we require a language in which ‘observables’, such as complementary coordinates, may not commute under multiplication. By reference to the above distinction (by which we partition the domain of ‘symbols’ into at least two parts), that non-commutativity implicates the subset of ‘symbols’ which make up the second part of our partition. In other words, in order to satisfy this condition and so suitably represent those quantum ‘happenings’, we require a language of ‘observables’, where we may perform a given string of operations in two different orders: and, and we use one term‘symbol’ from the second subset to represent the results of performing these operations in one order and another to represent the results of performing them in another order. Then we find that these terms

i) appear non-commutative under multiplication;

ii) and therefore appear non-symmetric;

iii) and thus the relation between such terms, and the relations between such terms and "themselves", fail(s) to satisfy the equivalence relation of identity;

iv) and therefore fail(s) to satisfy the logical axiom of identity;

v) and from these logical relations, in turn, we infer that therefore these quantum-theoretic ‘observables’ do not belong to the domain of self-identical WIE languages. Instead, we suggest, they satisfy the Postulate of Non-identity and so belong to a non-aristotelian domain, e.g. that of *transacting*.

1. The domain of classical physics:

__a. The complex numbers and the logical axiom of Identity__

Consider the set S of complex numbers. Modern mathematicians have characterized the complex numbers in terms of the conventions of set theory. Or in other words, every complex number (every element of the set S , which itself belongs to the blank delimited domain D ) satisfies all of the axioms of set theory, including the modern Logical Axiom of Identity:

" x Î S: x º x . (2)

__b. Identity and the properties of equivalence relations__

The construct of Identity (º
) comprises an **equivalence relation** W (the strictest one known), where one reads y W x as " x stands in the equivalence relation W to y ," or " x is identical with y ."

__c. The properties of equivalence relations __

As an equivalence relation, Identity shows three properties:

" x Î S: x º x

Û
|**Reflexive**: x W x . (3)

|**Symmetric**: y W x Þ
x W y (4)

**|Transitive**: [y W x and z W y] Þ
z W x (5)

__ d. The arithmetic operations, e.g. multiplication__

On the members of the set of complex numbers, one can define various **operations**, e.g. *multiplication*. To represent these operations, define a set A whose elements consists of ordered pairs (e.g. (m,p)). In each of these ordered pairs, let the first element (e.g. m) itself consists of an ordered pair (e.g. (n,o)). Thus any element of A has the form ((n,o),p), where n * o = p. In other words,

A = {((a,b),c) | a * b = c} (6)

The operation of multiplication, then, requires a relation R such that,

Given element ai, element bj, and element c, each of which belongs to S , then

R(c) = {(ai,bj) | ((ai,bj),c) Î A} . (7)

EXAMPLES:

((2,1),2)

((2,2),4)

((2,3),6)

2. Some relations between sets (such as the complex numbers), etc.:

__a. Associativity, commutativity, etc__.

According to the theorems of set theory, on any sets (e.g. sample complex numbers) A , B , and C Î S one can perform certain operations; or otherwise stated, sets (such as the complex numbers) show the relations of:

**Associativity**: e.g. A È
(B È
C) = (A È
B) È
C . (8)

**Commutativity** : e.g. A È
B = B È
A (9)

**Distributivity**: e.g. A È
(B Ç
C) = (A È
B) Ç
(A È
C) . (10)

The **principle of duality**, with respect to the **empty set** Æ
and the **universal set** U:

e.g. A È Æ = A , and (11)

A È
= *U* (12)

(The law of contradiction)

__b. The empty set__

Set theorists sometimes define the empty set as follows:

There is a set having as members all objects that satisfy the sentence-form

x is different from x .

[Or, in notation, x x (13)]

Obviously this set has no members; it is called the empty set.[52]

However, the empty set itself satisfies the logical axiom of identity.

There exists at least one Æ Î S: Æ º Æ (14)

**3. Some connections between these constructs:**

__Relations between commutative and symmetric __

Given: Sentence (7). Then

Multiplication on the set A qualifies as commutative iff

For all c Î S: R(c) is symmetric (15)

Or in notation,

[((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Î
A)]

Û
|[u_{i }W a_{i} Þ
a_{i} W u_{i}] |

|[v_{j} W b_{j} Þ
b_{j} W v_{j}]| (16)

The above remarks and sentences summarize a key portion of the rules governing the WIE mathematics of complex numbers. For the purposes of this proof, we will not need to characterize the classical framework more extensively or in greater detail.

4. The domain of quantum theory:

In contrast to the classical framework, let us now turn and consider the domain of quantum theory. As noted above, to workers in the field the ‘happenings’ in this domain seem in some elusive way *non-classical*. Furthermore, quantum theorists have reached no general agreement as to just how these quantum ‘happenings’ differ from classical ‘happenings’.

We begin by pointing out that Heisenberg in particular (and, perhaps, quantum theorists in general) do(es) not deal with complex numbers, but rather with ‘observables’ (‘operations’ which extract ‘information’) – and these ‘observables’ do not satisfy the commutative relation. Stated in terms of Sentence (7), and expressed in terms of standard, self-identical WIE numbers, e.g. a , b , and c , the construct of *non-commutative under multiplication* means

a * b = c_{1} and

c_{1} ¹
c_{2 }

b * a = c_{2} , where

under the explicit presupposition that

c_{1 } º
c_{1}

and

c_{2} º
c_{2 }.

But here, in the quantum domain (in order to take Heisenberg’s initial method seriously, as if it indicates something fundamental about the quantum domain) perhaps we must abandon the familiar, comfortable ‘symbols’ of the identity-based WIE frame of reference, and try out some other class of ‘symbols’ – ones, perhaps, for which* non-commutative* might mean:

a * b = c , and

b * a = c , and

c ¹ c .

I can summarize these three expressions by means of the following conditionals:

(((a_{i},b_{j}),c) Î
A) Þ
;

or

(((a_{i},b_{j}),c) Î
A) Þ
(((b_{j},a_{i}),c) Ï
A) .

As "outlandish" as this may appear on first encounter, a little reflection may bring forth numerous supporting examples, where we might find it necessary to write

B ¹ B .

To examine an example which appears, at best, tangential to physics, consider the following scene: a house with a broken window, with an irate-looking man standing in the front doorway, holding a baseball and looking out at two figures on the doorstep – a small boy who wears a baseball cap and holds a bat and a fielder’s glove, and a man with hands lifted in an apologetic gesture, who says,

"Boys will be boys!"

Let us examine the man’s spoken apologia with some care. The usage of the word *boys* which appears on the left side of the copula appears to designate *the culprit*, whereas the usage which appears on the right side appears to refer to *a stereotype of "boy-children as noisy, vigorous, and often in trouble with their adults."* Clearly, at the level of referent, what we call *the *(non-verbal) *culprit* IS NOT what we call *a* (verbal or symbolic)* stereotype*. When one usage of a given term designates something non-verbal, and another designates something verbal or symbolic, we call the term *multiordinal* and say that the two usages of the term refer to different positionings in an ordering on abstracting (or different "logical levels").

After enduring this exegesis, our readers might well feel little sense of shock if we proceeded to replace the copula which connects these two noun-phrases with a mathematical symbol or discursive term for *not-equal to* or *not identical with*:

Boys boys .

Indeed, after this exegesis, our readers might find that the usage of the copula in the first locution (taken literally) appears inaccurate, mistaken, even dishonest. In fact, the blandishment offered by the man depends on this dishonesty.

To return to the topic of quantum theory and this set-theory proof, we offer as a "plausibility argument" the possibility that, in this non-classical domain and in the context of multiplication on the set A, when we perform our operations in one order, perhaps we generate a ‘symbol’ on one "logical level," and when we perform them in the other (another) order, we generate one on another, different "level". Then the *not equal to* appears "inevitable" rather than "outrageous."

At any rate, granted this unexpected sense of the construct of *non-commutative under multiplication*, we proceed to reverse the chain of reasoning built up for the classical domain.

To express the first step of this reversal in notation, we point out that

Multiplication on the set A does not qualify as commutative iff

$ c Î S: R(c) is not symmetric. (17)

Or otherwise stated,

[((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Ï
A)]

Û
|[u_{i} W a_{i} Þ
a_{i} /
W u_{i}]|

|[v_{j} W b_{j} Þ
b_{j} /W v_{j}]| [((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Ï
A)] (18)

Sentences (17) and (18) do not seem to pertain to WIE complex numbers. But as we have insisted, quantum theory involves ‘happenings’ which seem somehow non-classical, not representable in terms of ordinary WIE numbers. So, ignoring for the moment the difficulties which these sentences entail (e.g. with the definitions for sets S , A , R , etc.), let us allow the present line of reasoning to go wherever it will.

__Commutative and symmetric, their contraries, and Identity vs. Non-identity__

Operations or relations which appear non-commutative under multiplication on a set also appear non-symmetric.

By (18),

[((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Ï
A)]

Þ R(c) is not symmetric (19)

To say "R(c) is not symmetric" means that

For not-all c Î
S: b_{j }W a_{i} Þ
a_{i} W b_{j} (20)

Or,

$
c Î
S: b_{j} W a_{i} Þ
a_{i} b_{j } (21)

In turn, operations or relations which appear non-symmetric fail to satisfy the equivalence relation of identity, and so in turn fail to satisfy the Logical Axiom of Identity.

By Sentence ((4),^{ }

*:

[ $
c Î
S: b_{j} W a_{i} Þ
a_{i} b_{j}] Þ
[ $
c Î
S: c c] . (22)

Such operations or relations satisfy the defining condition of Sentence (13), namely

*:

c c

and so they qualify as members of the empty set.

c c Þ c Î Æ (23)

No WIE mathematics based on complex numbers has the means for performing operations on elements of the empty set.. By the presuppositions of WIE mathematical languages, the empty set "obviously" has no members.

Hence, since they qualify as not-identical with themselves, such ‘elements’ q Î
Q do not satisfy the WIE logical axiom of identity and so do not belong to the domain of self-identical WIE languages, but rather satisfy the Postulate of Non-identity and so belong to a non-aristotelian domain (e.g. to the specific delimited domain of *transacting*).

[ " q Î Q: q q]

Þ [Non-identity, Non-allness, and Self-reflexiveness] (24)

That completes the first part of our proof.

The above remarks and sentences summarize a key portion of the rules governing the WIE mathematics of complex numbers. Some branches of WIE mathematics do have non-commutative relations, but these relations do not deal directly with complex numbers. In the domain which we consider here, we must consider "numbers" which fail to show the commutative and symmetric properties.

ii) Multiplication on the set A does not qualify as commutative iff

$ c Î S: R(c) is not symmetric. (17)

Or otherwise stated,

|[u_{i} W a_{i} Þ
a_{i} /
W u_{i}]|

| | Û

|[v_{j} W b_{j} b_{j} /
W v_{j}]|

[((a_{i},b_{j}),c) Î
A)

((b_{j},a_{i}),c) Ï
A)] (18)

Sentences (17) and (18) do not seem to pertain to WIE complex numbers. But quantum theory involves 'happenings' which seem somehow non-classical, not representable in terms of ordinary WIE numbers. So, ignoring for the moment the difficulties these sentences entail (e.g., with the definitions for sets S , A , R , etc.), let us follow the present line of reasoning wherever it takes us.

b. Commutative and symmetric, their contraries, and Identity vs. Non-identity

Quantities which appear non-commutative under multiplication on a set also appear non-symmetric,

By (18),

[((ai,bj),c) Î A) Þ ((bj,ai),c) Ï A)] Þ

R(c) is not symmetric (19)

To say "R(c) is not symmetric" means that

For not-all c Î
S: b_{j }W a_{i} Þ
a_{i} W b_{j} (20)

Or,

$ c Î S: bj W ai Þ ai / W bj (21)

In turn, quantities which appear non-symmetric fail to satisfy the equivalence relation of Identity, and so in turn fail to satisfy the Logical Axiom of Identity.

By (4),^{ }

*:

[ $ c Î S: bj W ai Þ ai / W bj] Þ [ $ c Î S: c / º c] . (22)

Such quantities satisfy the defining condition of (13), namely,

*:

c / º c ,

and so they qualify as members of the empty set.

c / º c Þ c Î Æ (23)

But no WIE mathematics based on complex numbers has the means for performing operations on elements of the empty set. By the presuppositions of WIE mathematical languages, the empty set "obviously" has no members.

Hence, since they qualify as not-identical with themselves, such 'elements' q Q do not satisfy the WIE logical axiom of identity and so do not belong to the domain of self-identical WIE languages, but rather satisfy the Postulate of Non-identity and so belong to a non-aristotelian domain (e.g. to the specific delimited domain of transacting).

[ " q Î Q: q / º q] Þ [Non-identity, Non-allness, and

Self-reflexiveness] (24)

That completes the first part of our proof.

PART 2:

For the second part of our proof, we start with

i) the specific setting for a non-aristotelian frame of reference, which we can call *transacting*, or can designate by means of a run-on sentence such as *the dealings of one-particular-organism-as-a-whole-with-its-environment-at-a-date; *

ii) the map-territory analogy, which holds (i) that the term *living* signifies "making 'maps' of that 'territory' made up of 'what-goes-on-in-and-around-me'," and (ii) that our observer generates such 'maps'; and

iii) the Postulate of Non-identity, which holds that *no map qualifies as identical with the territory it (allegedly) represents. *

Let us posit a verbal/symbolic domain of "quantum happenings" (a map) Q , and a non-verbal territory T . We do not regard Q as identical with, or in one-to-one correspondence with, T . But the symbolic domain Q does somehow represent the non-verbal T ; and does so accurately enough so that at least some of the hypotheses based on its quantum theory (or rather, on the WIE analogue of its quantum theory) have survived testing.

If in fact one of the foundational relationships of quantum theory amounts to a special case of the Postulate of Non-identity, then the '‘happenings’' which make up this symbolic domain Q will both satisfy the Postulate of Non-identity and form a part of the basis for quantum theory.

In this part of our proof, we show that, by setting Planck's constant h = 0 , we can "reduce" these symbolic "quantum happenings" q Î Q so that they come to satisfy the logical axiom of Identity. We show that they then belong to the domain of the self-identical WIE languages. When we do this, we produce a theoretical system isomorphic with that of classical physics. But we intended to describe the non-verbal territory T ; and in the quantum domain, predictions about T based on the presuppositions of classical physical theory fail to survive experimental testing.

1. The Symbolic Domain:

We may not regard the symbolic 'happenings' of our domain Q as identical with the non-verbal territory T :

*:

Q T . (25)

Moreover, Q does not qualify as identical with itself.

*:

Q Q . (26)

Finally, the 'happenings' q that make up Q do not qualify as identical with themselves either,

*:

" q Î Q: q q , (27)

in accordance with the Postulate of Non-identity.

2. Setting Planck's constant h to zero:

Heisenberg developed the equation which quantifies the uncertainty principle on theoretical grounds, but it reflects (in standard WIE mathematical terms) the repeated finding that, when performing experiments in the domain of "quantum happenings", one may set up the experimental equipment in at least two complementary ways. Done one way, the experiment shows some q as made up of one or more "particles"; done the other, it shows this q as made up of "waves." Or, metaphorically speaking, what you see when you look at some q "from the left" and what you see when you look at that q "from the right" DON'T MATCH.

Heisenberg’s equation,

D P ´ D X ³ h/4 p , (28)

where D P and D X stand for the uncertainties in the precise values for the complementary coordinates of a given system (e.g. momentum and position) which a given measurement will yield, describes this situation. The non-commutativity for multiplication discussed above (sentence (1), ms p. 47), gives an equivalent expression of the uncertainty principle.

PQ - QP = hi/2 p (29)

The equivalency does not appear obvious on the surface, but physicists know the mathematics of how to get from one expression to the other.

A derivation[53] of Heisenberg's equation (29) utilizes the properties of linear operators in Hilbert space.

Let A and B represent observables in an arbitrary state and have the operators A, B defined by

D A = A - <A> D B = B - <B> ,

where <A> and <B> represent expectation values.

Define the Hermitian operator C by the relation

[A,B] = AB - BA = iC . (30)

Then the uncertainty relations state that

D A * D B ³ ½ <C> . (31)

The proof rests on

a) the Schwarz inequality

<a
| a
><b
| b
> ³
<a
| b
> ^{2} ;

b) the knowledge that the expectation value of a Hermitian operator is purely real;

c) the knowledge that the expectation value of an anti-Hermitian operator, A = -A^{+} is purely imaginary.

In performing these algebraic transforms, the quantum theorists did not fully recognize that the reasoning they employed follows from some analog of the Postulate of Non-identity. They still overtly operated within the identity-based WIE frame of reference, under the presupposition that the domain Q , and all the ‘happenings’ which make it up, satisfy the logical axiom of Identity.

The non-commutativity of complementary coordinates, and the difficulties in interpreting quantum theory, stand as the main markers of this (presumed) mistaken presupposition.

If we abandon this tacit reliance on analogs of the Postulate of Non-identity, and set h = 0 , that allows the possibility of simultaneously knowing the position and the momentum with arbitrary precision. That in turn makes the complementary coordinates operators (and any other arithmetic transforms on Q ) perfectly standard - commutative, and symmetric, - so that both Q and all its members satisfy the equivalence relation of identity and the logical axiom of identity, and so belong to the domain of self-identical WIE languages.

Let h = 0 . (32)

[h = 0 ] Þ [PX - XP = 0] . (33)

Having made that change, then, in the notation of Sentence (7) (ms. p. 30, which shows the operation of multiplication as a set A of ordered pairs in which the first element of each itself consists of an ordered pair), it follows that multiplication on the set A qualifies as commutative.

[PX - XP = 0] Þ
[((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Î
A)] (34)

If multiplication on the set A qualifies as commutative, then

" c Î S: R(c) is symmetric (35)

[((a_{i},b_{j}),c) Î
A) Þ
((b_{j},a_{i}),c) Î
A)]

Û
|[u_{i} W a_{i Þ
} a_{i }W u_{i}] |

|[v_{j }W b_{j Þ
} b_{j }W v_{j}] | (36)

If " c Î S: R(c) is symmetric, then " c Î S: c º c . (37)

|[u_{i }W a_{i} Þ
a_{i} W u_{i}] |

|[v_{j} W b_{j} Þ
b_{j} W v_{j}]|

Û [" q Î Q: q º q] (38)

That reduces quantum theory to classical Newtonian mechanics, in effect implying an alternative condition for the Correspondence Principle.

In particular, that leads to REPRESENTING Q as if each q had "lost a dimension," so that, when you look at a q "from the right" and "from the left," you see THE SAME THING - namely, CLASSICAL 'happenings'.

But unfortunately, the predictions concerning quantum ‘happenings’ which we base on classical physics end up disconfirmed.

Only if we retain the non-identity basis for Q and q can we obtain hypotheses that survive testing.

From that we infer that when they framed the foundational constructs of quantum theory, the quantum physicists included some axiom tantamount to the Postulate of Non-identity.

[Non-identity, Non-allness, and Self-reflexiveness]

Þ [ " q Î Q: q q] (39)

That completes our proof

V. DISCUSSION

To focus our investigation, in the Introduction we propose two hypotheses, (A) and (B). We subject both of these hypotheses to test.

These two hypotheses occupy different logical levels; and taken on its own logical level, each expresses an aspect of a single generalization.

A. Our study discloses previously unsuspected premises for quantum theory - e.g. contradictory premises:

a) the Postulate of Tacit Identity, as concealed in the privileged WIE (generically Aristotelian) frame of reference, vs.

b) some analog of the Postulate of Non-identity, which underlies the non-aristotelian frame of reference.

B. Our study yields a new way of handling various aspects of quantum theory (such as complementary coordinates, the Uncertainty Principle, the double--slit experiment and Feynman diagrams) - e.g. viewing them as belonging to the domain of the Postulate of Non-identity. Moreover, our study shows how to visualize quantum theoretic constructs (in terms of Gestalten, framed in the setting of *transacting* - from which its user infers "the observer observing the observed").

We utilize a set theory proof to test hypothesis (A), which holds that upon investigation, the presuppositions which underlie quantum theory will turn out to include some tenet which amounts to a special case of the Postulate of Non-identity.

In the first part, our set theory proof shows that the constructs that make up the quantum domain, q Î
Q , appear non-commutative, and therefore non-symmetric, and therefore fail to satisfy the equivalence relation of identity, and therefore fail to satisfy the logical axiom of identity, and therefore, in turn, do not belong to the domain of self-identical WIE languages. Instead, they satisfy the definition of ""elements of the empty set,"" and therefore satisfy the Postulate of Non-identity, and so belong to a non-aristotelian domain, e.g. that of *transacting*.

In the second part, given the domain of the Postulate of Non-identity and given that the foundational relationships of quantum theory amount to a special case of the Postulate of Non--identity, then the 'happenings' that make up the quantum domain q Î Q will both satisfy the Postulate of Non-identity and also form the basis for quantum theory.

Then, by setting Planck's constant h equal to 0 , we "reduce" these symbolic quantum '‘happenings’' so that they come to satisfy the logical axiom of identity. The proof shows that these quantum '‘happenings’' then belong to the domain of the self-identical WIE languages, and form a theoretical system isomorphic with that of classical physics. (Thus, we produce an identity-based (generically Aristotelian) statement of the Correspondence Principle.)

But unfortunately, predictions concerning quantum '‘happenings’' but which we base on the presuppositions of classical physics end up disconfirmed.

Only if we retain the non-identity basis for Q and q can we obtain hypotheses that survive testing.

From both sections of the proof, then, we infer that when they framed the foundational constructs of quantum theory, the quantum physicists included some axiom tantamount to the Postulate of Non-identity, whether they noticed or not.

Our perspective on quantum theory suggests an explanation for Heisenberg's assertion that

When we get beyond this range of classical theory, we must realize that our words don't fit.[54]

If, as we suggest, the innovative strand of quantum theory implies-and-assumes some axiom tantamount to the Postulate of Non-identity, then efforts to ""intuit"" or reason about quantum theory in the identity-based WIE notational or discursive languages will not work well. The Complementarity Principle represents just such an effort. Using classical ideas and terms in the measurement process, we quickly run into situations not fully explainable in the quantum region. Holding on to our traditional notions we call quantum theory ""unphysical"" and ""counter-intuitive"".

For those who can reason in a frame of reference which does not award a privileged position to the viewpoints based on identity, this disconnection would not hold. For them the constructs of quantum theory (as re-formulated so as to make explicit the non-identity basis) reveal a world of constant interplay between ""phenomena"" and ""measurement"", or observer, observed and observation. In such a non-aristotelian viewpoint Complementarity evolves out of the non-identity based, specific delimited setting, rather than getting imposed from without as a consequence of some external demand. This alternative frame of reference gives us the language to visualize a seamless world in which quantum effects appear as intuitive realizations of the underlying structure.

VI. EXTENSION

The authors initiated this study to find out what a viewpoint based on the non-aristotelian premises might contribute to quantum physics. Can this alternative frame of reference lead to a better visualization of the foundational constructs of quantum theory? If so, then this approach has merit and warrants further investigation.

In order to frame ways of testing the new formulation, let us look at the fundamental problem of how to interpret quantum theory. In its history exponents have made several attempts to provide an interpretive framework and an intuitive understanding of quantum theory, but for the purposes of this discussion let us take another brief look at two, the statistical and Copenhagen interpretations.

The Copenhagen interpretation, closely associated with Bohr, treats indeterminacy as an inherent characteristic of nature and the wavefunction as the most complete description of a system (single or ensemble) before the performance of a measurement.

The statistical or ensemble interpretation, which we associate with Einstein, claims that the quantum wave function does not pertain to an individual particle or entity, but rather describes ensembles or aggregates of systems. The indeterminacy associated with using the wavefunction to compute probabilities results from this averaging or statistical approach. Consider a single system, which remains ""objectively real"" and whose properties we cannot determine with arbitrary accuracy by the methods of quantum theory: The wavefunction does not in any way describe a state of this system.

Instead, the ensemble interpretation treats the probability obtained as resulting not from an indeterminacy in the phenomena, but rather an inability to determine the actual state of the system. Because of this inability, Einstein regarded quantum theory as unable to provide a complete description of the physical reality of individual states.

These objections found their best expression in a 1935 paper now known as the Einstein-Podalski-Rosen paper.[55] The EPR paper tries to argue through a thought experiment that quantum theory does not provide a complete accounting for phenomena. The inclusion of additional information or hidden variables becomes necessary for a full understanding.

The physics community initially presumed that theories such as EPR that use hidden variables produce the same predictions as does quantum theory, thereby providing no experimental distinctions. However, in 1964 John S. Bell[56] disconfirmed this presumption. Imagining an experimental set--up that measures the correlations of spin components on two identical spin 1/2 particles in the single state, Bell derived an inequality relation that demonstrated that theories based on hidden variables do not always give the same results as different than those predicted by quantum mechanics. A series of actual experiments[57] have repeatedly borne out the predictions of quantum theory, leading to the rejection of certain hidden variable theories, and with them the ensemble interpretation of quantum theory.

Recently Greenberger, Horne, Shimony, and Zeilinger[58] have derived a stronger version of Bell's Inequality. It analyzes a system of three or more particles, and so violates quantum conditions for perfect correlations. In order to accomplish this they first look at the following premises of the EPR argument (adapted to David Bohm's thought experiment[59]).

(i) Reality: If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.

(ii) Completeness: Every element of the physical reality must have a counterpart in the (complete) physical theory.

(iii) Locality: Since at the time of the measurement the particles no longer interact, no real change can take place in the second system in consequence of anything that may be done to the first system.

Greenberger, Horne, Shimony, and Zeilinger proceed to demonstrate that in systems consisting of more than two particles these premises appear inconsistent with measurements in which knowledge of the result on one particle can with certainty predict the measurements on other particles of the system.

Recent discussions have centered around proofs that guarantee no local hidden variables[60] and around the conflict between local realism and quantum mechanics.[61]

If we accept these results, where does that leave us in terms of the assumptions on which the Bell inequality rests? Clearly, they seem to invalidate at least one of those assumptions which underlie local realistic theories. But it seems difficult to decide which one, for each holds a cherished place in the constructs of science. Western scientists find it virtually inconceivable to deny the first assumption, which implies an 'objective reality' apart from their 'minds'; they regard the second, which allows all the reasoning tools available in WIE mathematics, as essential to any formal study; the third, Einstein separability, although a cornerstone of the theory of relativity, seems currently the most suspect. Physicists currently do not regard the decision among them as particularly clear, however. No one has yet performed a direct experimental test of Einstein separability. Physicists may have cited it as the invalidating assumption in Bell's inequality mainly because they feel unwilling (or unable) to give up the other two assumptions, the notion of '‘objective reality’' and the unrestricted use of WIE mathematical inference.

But Bell's inequality exists within the standard WIE frame of reference, which (as we demonstrate above) grants a privileged position to the presuppositions encoded in the grammar of the WIE languages - including the tacit postulate of map-territory identity.[62] In particular, the first assumption listed for Bell's inequality accurately names that which remains left over from the transactional experiential world after we have eliminated the observer from consideration: a ""physical reality"" which ""exists"" independent of our ""minds,"" or independent of any observer. (Conversely, that assumption also implies that ""the mind,"" or ""the observer"" (as exponents of WIE frames of reference use these terms), ""exists"" independent of any ""physical reality."") And, as we point out above (ms p. 10), to rely on the postulate of tacit identity has the effect of systematically eliminating the observer from consideration, whereas to rely on map-territory non-identity, within the specific delimited setting of transacting, has the effect of systematically taking the observer into account. Hence, where we find the observer already eliminated, we infer a reliance on map-territory identity - in other words, that the first assumption of both d'Espagnat's and Bohm's versions of the premises of Bell's inequality amounts to (or at least, expresses an immediate consequence of) the postulate of (tacit) Identity.

Our viewpoint successfully abandons this assumption.

The second assumption posits a perfect one-to-one mapping of a mathematical model (in the physical theory) onto the physical world itself. This expresses a tacit claim that it remains in principle impossible to have '‘happenings’' of the physical world which we cannot describe in the terminology of the corresponding WIE physical theory. Thus, this assumption amounts to (or at least, expresses an immediate consequence of) the Postulate of Allness.

Again, our viewpoint successfully abandons this assumption.

As the main body of our text attests, the authors have already developed a frame of reference which systematically disallows tacit identity. We have discarded the WIE grammar, and along with it, the rest of the generically Aristotelian premises. To replace it, one of us (CAH) derived a new grammar from the non-aristotelian postulates, by a small number of explicit logical steps; and later, in a collaborative setting, generated a "Let's Keep Track of What We Say" notation, a non-aristotelian "logic" that rests on this derived grammar, and which systematically takes into account the observer.[63, 64] Thus (as we just said), we have already "given up" those two WIE-based assumptions, replacing them with tenets based on postulating the non-identity of map and territory. As for the difficulties surrounding the notion of Einstein separability or locality/ non-locality, these difficulties themselves appear related to the generically Aristotelian notions of *noun* and *verb*, or *things* and* relations*. Our non-aristotelian frame of reference does not utilize those constructs or the associated ways of segmenting experience and partitioning our linguistic constructs, so those difficulties do not transfer to our frame of reference. Meanwhile, our frame of reference seems to include constructs tantamount to Einstein separability, e.g. in the use of (*to*)* order* as an undefined term.[65] (Cf. ms p. 22)

As we insist, the work already completed falls short of providing a full-fledged, notational non-aristotelian ""physics."" As noted above (ms p. 21), in order to complete the task of writing a non-aristotelian ""physics"", someone will have to go through some four steps: a) derive a grammar based on non-identity (which then will systematically take the observer into account); b) build up an alternative, non-WIE ""logic,"" and c) a ""mathematics"" (including cognates of ""arithmetic,"" ""analysis,"" and an alternative, non-WIE ""geometry""); and, eventually, d) a ""physics"" - in other words, a complete frame of reference which does not award a privileged position to the WIE grammar (and to the assumptions encoded therein). As one criterion of success - one way we'll know that this innovator has done her/his job correctly - the alternative frame of reference will systematically take the observer into account throughout.

Our research group has already done a part of that job. As just noted, we have generated a grammar derived from the non-aristotelian premises (including the Postulate of Non-identity), and a ""logic"" built up on that grammar. Using that ""Let's Keep Track of What We Say"" notation, Hilgartner & Harrington[66] derived a ""numbering-theory,"" which covers ground cognate to what WIE mathematicians designate as ""the counting numbers."" To date, on those beginnings we have not yet completed the additional steps required to provide an "arithmetic," an "algebra" and an "analysis" which will support calculating. Hilgartner and Harrington did conduct some informal explorations into a ""personal geometry,"" but have not written up the results.

We can make the size of the leap required to get from WIE-based physics to a non-aristotelian ""physics"" immediately present for readers by pointing out that since a non-aristotelian ""physics"" disallows the construct of identity in any guise or form, explicit or tacit, and since the native speakers/writers of WIE discursive and formalized languages consider every noun-form or noun-phrase as self-identical, therefore we exponents of non-aristotelian frames of reference cannot import ANY terms from WIE physics into a non-aristotelian ""physics."" That prohibition includes the undefined terms of WIE physics: terms such as* mass*, *time,* *energy*, *space*, etc. Although in a ""physics"" we will perforce have to have ways of dealing with processes of measuring with balances, clocks, thermometers, rulers, etc., we cannot use the untransformed constructs (noun-forms) *gram,* *second*, *degree*,* meter*, etc.

As a milepost on the way toward a full-fledged non-aristotelian ""physics,"" we intend to derive a cognate of the Bell theorem, using the non-identity-based ""logic""; and then to use that to generate testable hypotheses or predictions, for use in a critical experiment which will also test hypotheses based on Bell's inequality and on WIE quantum theory.

Meanwhile, until we or someone else actually fill(s) in these developmental gaps in the alternative frame of reference, and derive(s) hypotheses from a (putative) fully-developed non-aristotelian ""quantum theory,"" no one can say with assurance what these hypotheses will say. Because the assumptions of such a ""quantum theory"" (as we foreshadow it) demonstrably contradict those of Bell's inequality, it seems very likely indeed that such hypotheses will differ from those derived from that inequality. Whether they will agree with those derived from the (inconsistent, but experimentally tested and non-disconfirmed) WIE versions of quantum theory remains for someone to demonstrate.

REFERENCES

[1] A. Korzybski, Manhood of Humanity: The Science and Art of Human Engineering. (E. P. Dutton, 1921). 2nd edition (M. Kendig, Ed.) (Institute of General Semantics, 1950).

[2] A. Korzybski, Science and Sanity: An Introduction to Non-Aristotelian Systems and General Semantics. (Non-Aristotelian Library Publishing Co., 1933). 4th edition (Institute of General Semantics, 1958).

[3] A. Korzybski, "General semantics, psychiatry, psychotherapy, and prevention." American Journal of Psychiatry **98,** 203-214(1941). Reprinted in Alfred Korzybski Collected Writings 1920-1950 M. Kendig, Ed. (Institute of General Semantics, 1990), pp. 297-308.

[4] C. A. Hilgartner & J. F. Randolph, J. Theor. Biol. **23**, 285-338 (1969a), pp. 295-7.

[5] C. A. Hilgartner, Rev. Gen. Sem. **25**, 315-324 (1968)

[6] C. A. Hilgartner, presented at the Ninth International Conference on General Semantics, San Francisco State College, August 1965; revised (1967). Copies available from author, at price of photocopying and postage.

[7] C. A. Hilgartner & J. F. Randolph, J. Theor. Biol **23**, 347-374 (1969b), pp. 353-7.

[8] C. A. Hilgartner & J. F. Randolph, J. Theor. Biol. **24,** 1-29 (1969c).

[9] C. A. Hilgartner & J. F. Randolph, unpublished ms (1969d).

[10] C. A. Hilgartner, Acta Biotheoretica **27,**19-43 (1978a).

[11] Hilgartner & Randolph (1969a), footnote p. 309.

[12] C. A. Hilgartner, R. V. Harrington, & M. A. Bartter, Bull. Sci. Tech. Soc. **9**, 129-43 (1989).

[13] J. DiRienzi & C. A. Hilgartner, Precis: The Journal of Liberal Studies **1,** 12-14 (1989)

[14] C. A. Hilgartner, General Semantics Bulletin **Nos. 44/45**, 132-154, (1977/78).

[15] Hilgartner (1977/78).

[16] C. A. Hilgartner, Communication **3**, 143-242 (1978b).

[17] C. A. Hilgartner, Eco-Logos: A Magazine of One-World Environmental Concepts, **Vol 24, No. 90**, Appendix V, pp, 45-78 (1978c).

[18] C. A. Hilgartner, R. V. Harrington & M. A. Bartter, Rev. Gen. Seman. **48**(2):172-97 (1991), p. 177.

[19] Korzybski (1933), pp. 433-443.

[20] Adelbert Ames, Jr., __The Morning Notes of Adelbert Ames, Including a Correspondence with John Dewey.__ H. Cantril, ed. (Rutgers, 1960). Also, W. H. Ittelson, __The Ames Demonstrations in Perception__, (Princeton, 1952).

[21] Korzybski (1933), pp. 644f, 655.

[22] Hilgartner (1977/78)

[23}Hilgartner, Harrington & Bartter (1991).

[24] B. L. Whorf, __Language, Thought, and Reality: Selected Writings of Benjamin Lee Whorf__, John B. Carroll, Ed. (MIT/Wiley, 1956), pp. 213-4.

[25] F. S. Perls, R. F. Hefferline & P. Goodman, __Gestalt Therapy: Excitement and Growth in the Human Personality__, (Julian Press, 1951), pp. 243-4.

[26] Hilgartner (1977/78).

[27] Hilgartner, Harrington & Bartter (1989).

[28] M. Polanyi, __Personal Knowledge: Toward a Post-Critical Philosophy__ (University of Chicago Press, 1958; revised edition, 1962. Paperback edition (Harper Torchbook, Harper & Row, 1964), pp. 207-9.

[29] C. A. Hilgartner, General Semantics Bulletin No. 47, 112-9 (1980), p. 114b.

[30] Hilgartner (1978b).

[31] Hilgartner (1978c).

[32] Hilgartner (1978b), pp. 221-32.

[33] R. I. G. Hughes, Scientific American **245** no. 10, 202-14 (Oct. 1981).

[34] Alfred Korzybski (1933), p. 153.

[35] Polanyi (1964), pp. 55-65.

[36] Hilgartner (1978b), pp. 196-9, 199-204, 218-20

[37] Hilgartner (1978b), pp. 217-9

[36] Hilgartner (1978b), pp. 217-9.

[38] H. Poincaré * The Foundations of Science *(Academie des Sciences de Paris, 1916); quoted in Korzybski (1933), p. 367.

[39] B. d'Espagnat (1979), "The Quantum Theory and Reality." Scientific American **241** no. 11, 158-81 (Nov. 1979), pp. 158b-c.

[40] M. Swanson, General Semantics Bulletin **Nos. 22 & 23, **35-40 (1958).

[41] A. d'Abro, __The Decline of Mechanism__ (D. van Nostrand, 1939). Second edition, retitled __The Rise of the New Physics: Its Mathematical and Physical Theories__ (Dover, 1951), p. 401.

[42] d'Espagnat (1979), pp. 158b,c.

[43] H. C. Ohanian, __Principles of Quantum Theory__ (Prentice Hall, 1990, p. 350.

[44`]W. Heisenberg, Zeitschrifft Physics, **33**, 879-893 (1925). Reprinted as Ch. 12 in __Sources of Quantum Physics__. (B. L. Van Der Waerden, Ed.). (Dover Publications, 1967).

[45]Max Born & Pascal Jordan, The. Physik, **34**, 858 (1925)

[46] A. Pais, __Niels Bohr's Times, In Physics, Philosophy, and Polity__. (Clarendon Press, Oxford, 1991), p 426.

[47]Hilgartner & Randolph (1969a), pp. 296-7, 303-8, 319-320.

[48] R. Feynman, Robert B. Leighton & Matthew Sands (1965). __The Feynman Lectures on Physics__, Vol III, Ch. 1. (Addison-Wesley Publishing, 1965).

[49] J. Gribbin, __In Search of Schroedinger's Cat__, (Bantam Books, 1984), Ch. 9, pp. 183-192. [50] C. A. Hilgartner & T. L. Miller, presented at the Ohio Academy of Science, 3 May 1992.

[51] Perls, Hefferline & Goodman (1951), p. 227.

[52] B. Mates, __Elementary Logic__, (Oxford, 1965), p. 29. See also Alfred Tarski, __Introduction to Logic and to the Methodology of Deductive Sciences__, (Oxford, 1965), pp. 72-3; Paul R. Halmos, __Naive Set Theory__, (Van Nostrand, 1960), p. 8; etc.

[53] J. J. Sakurai, __Modern Quantum Mechanics__. (Menlo Park, CA: Benjamin/Cummings, 1985), pp. 34-36.

[54] Pais (1991), p. 310.

[55] A. Einstein, B. Podolsky and N. Rosen, Physical Review, **47**, 777-780 (1935). Reprinted in __Physical Reality__, (S. Toulmin Ed.) (Harper & Row, 1970).

[56] J. S. Bell, __Physics__, **1**, no. 3, November/December (1964), 195-200.

[57] A. Shimony, Scientific American **258** no. 1, 46-53 (January 1988).

[58] D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, Am. Jl. of Phys. **58**, 1131-1143 (1990).

[59] D. Bohm, __Quantum Theory__. (Prentice Hall, 1951), 614-619.

[60] N. D. Mermin, Phys. Rev. Lett. **65**, 3373-3376 (1990).

[61] N. D. Mermin, Phys. Today, **43** (6), 9-11 (1990).

[62] Hilgartner & Randolph (1969b), pp. 353-7.

[63] Hilgartner (1978b).

[64] Hilgartner, (1978c).

[65] C. A. Hilgartner, R. V. Harrington & M. A. Bartter, Bull. Sci. Tech. Soc. **9,** 129-43 (1989).

[66]C. A. Hilgartner & R. V. Harrington (1977), "Non-aristotelian Numbering" (Unpublished).

[67] D. David Bourland, Jr. "A Linguistic Note: Writing in E-Prime," General Semantics Bulletin **Nos. 32/33**, pp. 111-4 (1965/66).

[68] E. W. Kellogg, III, "Speaking in E-Prime: An Experimental Method for Integrating General Semantics Into Daily Life," ETC.: A Review of General Semantics **44**,118-28 (1987).

[69] D. David Bourland, Jr., "To Be or Not To Be: E-Prime as a Tool for Critical Thinking," ETC.: A Review of General Semantics **46**,202-11 (1989)

[70] Polanyi (1964), pp. 55-65

[71] Hilgartner & Randolph (1969a), pp. 295-7.

[72] W. R. Ashby, Technical Report No. 7, Electrical Engineering Research Laboratory, University of Illinois (1962).

[73] Perls, Hefferline & Goodman (1951).

[74] Polanyi (1964), pp. 55-65.

[75] M. Black, __Language and Philosophy__ (Cornell University, 1949), p. 246.

[76] G. Frege, __Grundgesetze der Arithmetik: Begriffschriftlich Abgeleitet__, (second of two volumes), (H. Pohle, Jena, 1903).

[77] P. J. Cohen & R. Hersh, Scientific American **217,** no 12, 104-116 (Dec. 1967), p. 105.

[78] Hilgartner & Randolph (1969a,b,c,d).

[79] Hilgartner (1977/78)

[80] Hilgartner (1978b)

[81] Hilgartner (1978c)

[82] Hilgartner, Harrington & Bartter (1989)

[83] Hilgartner, Harrington & Bartter (1989)

[84] D. K. H. Alford, Berkeley Linguistics Society (Proceedings of), 1981, pp. 13-26.

[85] Whorf (1956), p. 214.

[86] Hilgartner, Harrington & Bartter (1991).

[87] Polanyi (1964), p. 59.

[88] Hilgartner & Randolph (1969b), pp. 353-6.

[89] Swanson (1958).

[90] Hilgartner, Harrington & Bartter (1989).

LEGENDS FOR FIGURES

FIGURE 1: ANY REPRESENTING

Every visible detail of this figure represents some **structuring, ordering,** or **relationing.** The "steps" signify a **hierarchical** ordering. The arrows labeled T1 and T2 indicate that adjacent positionings in this hierarchical ordering (adjacent "steps") do not take place "synchronously" or "simultaneously", but rather "occur" or "exist" in such a way that they show **spatio-temporal** ordering. Given any pair of adjacent "steps", I can designate the lower one as **'territory'** and the higher one as **'map'.** The irregular circle labeled H signifies that structuring (part of the 'territory') I call **our (human) 'organism'**; the one labeled Y signifies that part I call **her/his ('external') 'environment'**. The smaller interior circles signify that portion (of 'organism' or 'environment') **in principle detectable** by the 'organism'; the exterior rings, that portion in principle **not** detectable by the 'organism'. The arrow labeled r
signifies that relationing I call **self-referential abstracting**; the one labeled s
, the relationing I call **hetero-referential abstracting**. The structuring called a *‘map’ *or *abstraction* (irregular circle drawn on 'map' level, not labeled) consists of Self (Sf) and Other (Ot) "components." The inner half-circles signify those portions of the representing directly obtained by abstracting from the 'territory'; the outer half-rings signify those portions of the representing, unavoidably present, which do not in any sense derive from the 'territory', by abstracting, but rather, which have to do with the 'organism' who does the abstracting.

FIGURE 2: FACES/VASE

The "reversible" goblet, introduced by Edgar Rubin in 1915, still serves as a favorite demonstration of figure-ground reversal. Viewers see either a goblet or a pair of silhouetted faces (but not both at once).

FIGURE 3: DOUBLE SLIT EXPERIMENT

A. Pattern observed when observer does not keep track of which slit electrons pass through.

B. Pattern observed when observer does keep track of which slit electrons pass through.

FIGURE 4: FEYNMAN DIAGRAMS

A. A Feynman diagram provides a useful way of visualizing a subnuclear event. Two particles interact by exchanging a third particle. Each force in nature involves the exchange of specific kinds of particles.

B. Mutual repulsion of two electrons through the exchange of a photon.

C. Exchange of a pion (p ) between two protons (p) .

FIGURE 5: THE UNDELIMITED DOMAIN

FIGURE 6: THE BLANK DELIMITED DOMAIN