| A NON-ARISTOTELIAN VIEW OF QUANTUM THEORY |
C. A. Hilgartner
Hilgartner & Associates
College of Notre Dame of Maryland
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.
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.
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.
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. 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) ^ . 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. 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 (which includes (i) a formal, notational, definition for the term living, (ii) a novel accounting for the origin of living organisms, (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) ^ . When in 1971 we turned to the topic of logical foundations, however, we encountered a major obstacle: a new kind of self-contradiction.  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. 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.
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. 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.
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.)
We shall discuss quantum theory in terms of assumptions.
Our assumptions concerning the term assumptions (a multiordinal inquiry – see ) 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. 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.
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 .
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.