C. A. Hilgartner

Ronald V. Harrington

Martha A. Bartter


We hold that, for every human, life itself depends, completely and irrevocably, on the actions of humans who have preceded us -- the kinds of actions which result in knowledge, in the form of tested guesses.1 Without necessarily saying so out loud, virtually every human strives to increase human knowledge of what-goes-on-around-and-in-us. They -- we -- do this first in order to improve the accuracy of our short-run predicting, and so to become more able to satisfy our basic needs; and in an ultimate sense, in order to increase the likelihood that individual, group and species will survive.2 Our very lives depend on the body of accumulated knowledge gained through those actions -- a heritage which the dead freely bequeath to the living.


Every human born into a culture somewhere, somewhen, assimilates some fraction of the inherited knowledge available within that culture at that date; contributes to it during her/his lifetime; and passes on the augmented heritage to her/his progeny or other successors and to the generations yet unborn.3

Most of the time, what we humans say-and-do -- trivial everyday things as well as major survival-oriented actions -- fits smoothly and easily within the framework which we share with our speech community and uphold in our various social groups. What we receive from our heritage and what we contribute to it dovetail neatly within its traditional, expected patterns. Our "doings" follow from our usual shared assumptions, and reaffirm our accustomed relationships with ourselves, each other, and the rest of our surroundings.


The practical and logical consistency of our "doings" works to our advantage in many ways -- it underlies our demonstrated ability to develop cultures that last many generations, to maintain customs and lifestyles, to adjust them to fit changing conditions, etc. But it does not lead to the kind of new knowledge which rests on new premises, inconsistent with the old ones. A cursory glance at our surroundings, however, suffices to show that we humans have at times abruptly altered our ways of talking-and-acting; and we may well do so again.


How do we go about changing our current, received, assimilated, perhaps treasured ways of doing things, and their underlying frames of reference? With Kuhn, the authors hold that the methods by which such changes come about entail "doings" which alter the view of what-goes-on-around-and-in-the-person(s)-involved; and these "doings" may well unsettle the person(s) involved, and perhaps others around them, in fundamental ways. In one breath, these "doings" both alter the environment4 of the person(s) involved, and also alter the person(s) themselves. Then, in accordance with this altered view, the person(s) proceed(s) to transact5 in a new way with this altered environment. But that brings us back to the question: How do we manage it? What do we DO when we do this?



In this paper, the authors designate these unsettling "doings" as innovating. We discuss this as a pattern, a process with a definite and definable structure. In principle, innovating can carry over and affect humans other than the innovator(s), and can even alter the shared viewpoint of the whole speech-community-in-its-environment.6


Using the history of geometry to provide a worked example, we show that that pattern which we call innovating depends upon a kind of reasoning pervasively used by humans but not often recognized as such nor consciously applied. We designate it as non-identity reasoning. Frequently seized upon with glee when available, the RESULTS of this kind of reasoning can lead to important advances that, in multitudes of ways, affect the lives of millions or billions of us humans. But the METHODS used to obtain these tradition-altering results remain shrouded in mystery.


Part of this mystery follows immediately from the current dogma held by exponents of the Western Indo-European (WIE) tradition. Regardless of whatever they may say to the contrary, the logicians, mathematicians, scientists, philosophers, and other workers within the WIE tradition still rely on a dualistic frame of reference -- which includes spurious distinctions such as 'mind' vs. 'matter' and 'self' vs. 'outside world'. In holding that terra incognita exists independent of any observer -- "out there," in the "outside world" -- awaiting discovery, our philosophers and logicians of science (among others) subscribe to such dualisms. From within such a dualistic framework, it appears that no one can describe how an innovator transcends the standard pattern. As Root-Bernstein remarks, the standard view concerning how humans "discover" previously unknown aspects of the "outside world"


... suggests, paradoxically, that illogical processes have led to the most logical constructs known to man -- mathematics and science. And it fails to explain where problems come from, what scientists do from day to day, and how they actually think.7


One good reason for questioning our dualistic frame of reference shows up when we examine the philosophy of science, as expounded by Peirce and his successors. They discuss science as a "thing" which exists "out there" rather than as a special case of the transacting of humans. This leads philosophers of science to deal with scientific advances in the same way that a religious person must discuss a miracle -- as an occurrence outside of nature, observable only after the fact, and explainable only through faith. For a scientist, such a "miracle" occurs as the unexpected "breakthrough", the sudden gift of insight, the accident that works. Root-Bernstein's careful study of the topics of "discovery" and "proof" shows that philosophers of science, including Beveridge, Braithwaite, Hempel, Hull, and Popper, hold innovations as equivalent to miracles -- something that just happens "out there." Then, since language for discussing "someone doing something, somewhere, somewhen, and producing an innovation" remains out of their reach, those who subscribe to our dualistic tradition posit an unbridgeable gap between "the process of discovery" and "the process of proof", and maintain that 'logic' and 'reason' apply only to proof.



Fig 1. about here (Sydney Harris "Miracle" Cartoon)



Unlike discovery, scientific validation is thought to consist of two distinct mental activities: induction (deriving general rules from particular instances) and deduction (making specific predictions based on general rules). The objective of science, according to this philosophy, is simply to validate or invalidate inexplicable insights.8


The acid test for any of the current versions of the logic of science comes when we consider the logic of science itself as a self-reflexive enterprise, a behavioral theory, in which, for an audience of humans, some human(s) endeavor(s) to account for certain aspects of what humans DO. The multiordinal term to account (for) bears special weight in that phrase.9


In general, in order to survive, any organism accounts, transactionally, for -- makes guesses (in the form of Gestalten) about -- what goes on around and in itself; and it guides its further activities by this transactional accounting, these guesses. In so doing, it subjects the guesses to (informal) experimental test. (For many organisms, these accountings or guesses remain entirely non-verbal, and the results of testing them also show up non-verbally, in how well the organisms fare, in survival terms. With language users, the guesses exist on both non-verbal and verbal levels, and so do the results of testing them.) Scientists, as the main point of their job description, take on the job of accountingn for human accountingm in various domains. Physical scientists SAY they study the physical (non-living) "happenings" that take place around them, as if these "happenings" had no effect on THEM. But from our point of view, physicists appear transactionally to study THEIR TRANSACTING with their so-called "physical" environments; and this not only leaves their environments (e.g. experimental systems) altered, but also alters the physicists themselves. Thus the study of physics already entails at least two layers: accounting2 for human accounting1. In a similar fashion, biological and human psycho-social scientists transactionally study the "happenings" which make up the transactional accounting of non-human, and human, organisms. In particular, the study of any of the psycho-social sciences entails at least three layers: accounting3 for how humans account2 for human accounting1. 10


Then, in these terms, the job of logicians of science goes at least one layer deeper, and consists of accounting4 for how humans, e.g. scientists, account3 for how humans account2 for human accounting1.


Dualistic frames of reference have an irreflexive structure. Instead of accountingn for accountingm (a reflexive relation, with many logical levels, such as m,n,etc.), they rely on causal reasoning. This presupposes a dualism, with one and only one logical level, which by definition has an irreconcilable split down the middle. It further entails two schematic chains, one in each of the two opposing domains of this dualism: e.g., that 'matter' versus this 'mind'. In the domain of 'matter', we have the 'facts' that A causes B, and B causes C, and C causes D, and so on, regardless of any observer (where A,B,C,D,etc., designate static-and-unchanging things). In the realm of 'mind', we have the proposition, " a causes b , and b causes c , and c causes d , and so on," (where a,b,c,d,etc., designate self-identical noun-phrases). Causal reasoning provides an explanatory principle by assuming that a designates A, b designates B, etc., with nothing left out and nothing left over. But this poses an epistemological question, in the arena of "How do we know?": In light of the irreconcilable split of the dualism, and in the absence of some kind of "pre-formed harmony" between its two sides, it remains a mystery how a particular "mind" becomes apprised of what happens in or to any particular "matter," and generates a proposition (of this form, or of any other). Yet our logicians of science presuppose precisely that mystery when they say that "discovery is a product not of particular methods of logical inquiry but of being in the right place at the right time." Since this causal reasoning, the spurious separation of "mind" from "matter," etc., prevent accounting for one's own accounting, the only act available to scientists, "according to this philosophy, is simply to validate or invalidate inexplicable insights." Their dualistic frame of reference makes it impossible for our WIE philosophers of science to do their job (as we define it).


The shared viewpoint of the authors, which a) admits no dualism, and b) has an explicitly self-reflexive structure, shows few of these disadvantages. We deny that any 'outside world' or 'self', 'mind' or 'matter', etc., "exist," or can "exist", independent of each other.11 Instead, empirically speaking, we find our actual experiencing occurring solely and exclusively within "worlds" which INCLUDE human observers. We human observers transact only within "worlds" which transact-back with us. As Perls, Hefferline and Goodman put it, "We speak of the organism contacting the environment, but it is the contact which is the first and simplest reality."12 In accordance with our own shared viewpoint, the authors each assert:


"The environment" forms the other side of my skin; "I" form the other side of the environment's skin."13


Hence, in the present paper, rather than discussing the discovery of what lies 'outside' of 'us', we discuss transactional INNOVATING.


Looking at the process of innovating -- analyzing it, with examples, from a point of view which shows both its pervasiveness and its importance -- may make this way of operating more available for conscious use.





Non-euclidean geometries play an important role in current science. Further, from within his general-semantics viewpoint, Korzybski claims that the work that culminated in the publication of non-euclidean geometries represents the first break with the traditional way of doing things -- the first step in freeing ourselves from the identity-based framework of Aristotle's, Euclid's and Newton's interlocking systems.14 Korzybski's claim suggested to one of us (CAH) that the non-euclidean geometers relied in some fundamental sense on the logical relation of non-identity rather than on identity, and that by doing so, they somehow opened the way for other theoretical systems, which in some sense depend even further on non-identity rather than on identity: non-newtonian physics, and non-aristotelian systems (of which Korzybski laid the first explicit foundations.)15


But the premises of the non-euclidean geometries differ from those of Euclidean geometry -- as we shall see in detail below -- only in that they include "non-standard" variants of Euclid's fifth postulate, the parallel postulate. They do not, on the surface, include any version of the Postulate of Non-Identity. In what sense do they depend on non- identity? In the present paper, we answer that question.


To begin this inquiry, consider first where our emphasis on the term-pair non-identity vs. identity comes from. As one of his greatest innovations, Korzybski questions the legitimacy of the logical construct of identity. In effect, he makes a quick tour of the universe, asking, "In a cosmos which has human observers in it, when and where may we LEGITIMATELY use the logical construct of identity? Where and when does the notion of entire and absolute agreement or negation of difference prove valid? Under what circumstances may we apply it?" His considered opinion: We may not. Under no circumstances does identity survive scrutiny.16 And then Korzybski makes an outrageous suggestion: Since identity never holds, let's not rely on it.


Korzybski proposes that we reject identity -- disallow it as a valid "relation." In effect, he suggests that we reserve it to provide a way of talking about situations in which someone takes B as if it "were" A, and so (by the definition of the term mistake) makes a mistake.


To state the matter in more general terms, Korzybski discloses that there exist not one but two ways to handle the paired terms identity and non-identity: a) One can "like" identity as a foundation, and "dislike" non-identity (as our linguistic forbears did); or else b) one can "like" non-identity and "dislike" identity. Once a choice becomes possible, any human who uses this term-pair at all must choose which way to use it. Korzybski himself takes a stand, declaring himself one of the company -- the first of the company -- of those who explicitly prefer non-identity.


Korzybski thus creates a choice with consequences -- a fork in the road, or a turning-point, for the human race -- where no choice had previously existed.


To reiterate, then, we maintain that the non-euclidean geometers came to prefer, or at least rely on, non-identity rather than identity. In this paper, we examine that supposition.


To understand a deviation, one must understand what has changed -- truly comprehend the "starting point." Consider a few details from the history of geometry. Apart from some minor contributions of the early inhabitants of the Fertile Crescent and Asia Minor, geometry had its beginnings in Egypt prior to 1700 BC. There, where the annual flooding of the Nile deposited fertile soil and erased man-made boundaries regularly, the emphasis lay on practical methods of surveying. The arts, and pyramid-building, provided further incentives toward accumulating geometrical knowledge. Extant manuscripts indicate that the Egyptians interested themselves mainly in the practical rather than the strictly logical aspects of their geometry, and that they relied almost entirely on intuition, experiment and approximation in accumulating geometrical relationships.


Around 600 BC, Greek travelers, notably Thales (ca. 640-ca. 546 BC), introduced the study of geometry into Greece. Thales developed the method of demonstrating a geometrical relationship (theorem) by showing that it follows logically from certain universally-accepted statements ("self-evident truths"), called axioms or postulates. In the Elements, Euclid (ca. 330-ca. 275 BC) gathered most of the mathematical knowledge of his day into thirteen books, of which seven dealt with geometry. He re-worked geometry in order to present it as a coherent body, the first formalized deductive system, built up so as to depend on what he regarded as the simplest of premises.


Euclid's premises consist of four parts. First, he utilizes a number of undefined terms (terms for which he provides no explicit definition, and which later workers have not succeeded in defining in a satisfactory manner): e.g. point, line, plane, parallel, etc. The further premises tell how to use the undefined terms. Second, he utilizes what we would now call logical axioms, which specify the construct of equality in a mathematical or geometric setting: e.g. how equal quantities behave under the various operations of addition, subtraction, multiplication, division, powers, roots, etc. Third, he specifies four geometric axioms which a) hold that through two given points one and only one straight line can pass, b) hold that in size and form geometric figures remain constant when moved in space,    c) define the construct of congruent (geometrically identical) in terms of the intuitive construct of "can be made to coincide," d) explicate the undefined construct of line in terms of the equally undefined construct of distance: "A straight line is the shortest distance between two points." Fourth, he states the "constructions which are admitted to be true without proof" -- the five postulates.


1. A straight line may be drawn through two given points.


2. A straight line may be drawn indefinitely or limited at any point.


3. A circle may be described about any given point as center, and with any radius.


4. All right angles are equal.


5. Through a point not in a straight line one and only one straight line may be drawn parallel to the given line.


For the next two millennia and thirteen decades, we Westerners thought of that mathematical entity which we call a line -- "the shortest distance between two points" -- as straight.


Meanwhile, from the first appearance of Elements, mathematicians had regarded the fifth postulate of Euclid as troublesome and perplexing. Intuitively, it seemed somehow "different" from the other four. Workers attempted in vain to derive or prove the fifth postulate. Finally, in the nineteenth century, several people demonstrated its independence of the other four by inventing geometries which utilize, as written, all of the other axioms and postulates of Euclid, but alter the fifth postulate. As we now know, Gauss (1777-1855) posited more than one parallel, and derived a modified geometry. He expected this procedure to yield a contradiction; but it did not. He did not publish his results (but then, he did not publish a large proportion of his investigations. His feat came to light only from later studies of his private papers.) About 1830, Bolyai (1802-1860) and Lobachevsky (1793- 1856) independently postulated that through a given point can pass two (or more) lines parallel to a given line; and each built up a consistent geometry. In 1854, Riemann (1826-1866) constructed a consistent geometry using the postulate that no line parallel to a given line can pass through such a point.


The modified postulate that many lines parallel to a given line can pass through a given external point turns out to yield a geometry defined on a pseudo-sphere (a surface of constant "negative" curvature) rather than on a plane. Here the construct of line (or "straight line") comes to signify a segment with a "negative" curvature equal in value to the curvature of the pseudo-sphere. On a pseudo-sphere, more than one line parallel to a given line can pass through a given external point. The modified postulate that no line parallel to a given line can pass through a given point turns out to yield a geometry defined on the surface of a sphere rather than on a plane. Here the construct of line (or "straight line") comes to signify a segment with a "positive" curvature equal in value to the curvature of the sphere, namely, a segment of a great circle. On a sphere, all great circles intersect: parallel lines cannot exist.


The achievements of the non-euclidean geometers leave a curiously diversified situation. Their predecessors had one and only one mathematical construct of "the shortest distance between two points," and could regard as valid the identity-statement (special case of Aristotle's Law of Identity):


A line is a line. [1]


Their peers and successors, however, have at least three mathematical constructs for "the shortest distance between two points." Further, they have a totally unexpected new construct -- namely, the "space" (planar, spherical or pseudo-spherical), or more generally, the setting -- to specify before they can posit a line or other geometrical figure. In the newer, diversified situation, the construct of line qualifies as incomplete until after the setting for it gets specified. In other words, the generally valid statement about the GENERAL construct of line negates Identity, holding that:


A line IS NOT a line. [2]


After all, if nobody has previously specified one "space" or another as the setting for this proposition, nothing prevents the substitution of one "kind" of line on one side of the negated copula and another on the other, converting [2] into the following, "obviously" correct proposition:


A (planar) line IS NOT a (spherical) line. [2a]


Moreover, we no longer regard even a "straight line" quite as we used to: we now describe it as "a segment having a radius of curvature infinitely great." While few of us may actually use such terms, many of us use the results, for instance, in Great Circle airline routes, etc.




The outre' expression [2] -- a statement of Non-identity rather than Identity -- does provide a way of summarizing these revolutionary ideas so as to make them appear as deviations from earlier viewpoints. But more important, it also opens up a way of specifying, both logically and psychologically, just HOW an innovator arrives at an innovation. To finish this job, we need four fundamental constructs to use in our analysis: namely, the logic of opposites; the Sapir-Whorf-Korzybski principle; a logically and empirically satisfactory way of handling the apparently 'purposive' activities of living systems; and a way of characterizing the environment unique to humans. After presenting these constructs, we shall use them to discuss four aspects of the process of innovating: the perceptual, the feelingful, the logical and the linguistic.


a. The logic of opposites


The question of how an innovator innovates depends on what we call the logic of opposites 17: the relations which exist between a construct and its opposite or contradictory or complement -- in particular, between Identity (or the binary relation of identical with) and Non-identity (or the binary relation of distinguishable from).


Humans through history and around the planet have devised a number of versions of the logic of opposites. For one example, speakers/writers of certain languages, such as English, generate a term to indicate the opposite of a given term (usually, a noun or noun-phrase) by the "simple" expedient of affixing a particle (e.g. not-). The two opposing terms which they produce in this fashion appear discrete, separable, with nothing in common but their opposition -- the procedure does not require them to posit a single setting to encompass both. We can display this kind of relationship of opposing terms ( C and not-C ) by drawing a single circle on an undelimited plane. (See Figure 2.) The points within the circle exemplify C , and those outside the circle exemplify not-C , which by default amounts to "everything else." In such languages, even when the vocabulary contains common terms for both sides of a polarity (e.g. black vs. white, night vs. day, etc.), its speakers treat these explicit opposites as discrete, separable, as if no relationship existed between the two beyond their opposition.


For a contrasting example, other languages, such as Chinese or the modern symbolic logics and set theories developed after the 1902 publication of Russell's Paradox18, constrain the process of forming opposites by requiring their users from the very beginning to postulate a single setting on which to define both the term and its opposite. That gives quite a different configuration -- like the construct of "order" and that of "chaos," defined as opposites on a setting of "the political condition," or the construct of "buy" and the opposing one of "sell," defined on a setting of "trade". We can display this kind of relationship -- an opposing term-pair composed of terms C and not-C defined on a specified setting -- by drawing two concentric circles on a plane. (See Figure 3.) The outer circle encloses a delimited domain. The points within the inner circle exemplify C , and those between the inner and the outer circles exemplify not-C . Here, no one will mistake not-C for a representation of "everything else, period" -- instead, it looks like "everything else within the delimited domain." Procedurally, this way of forming an opposing term-pair makes it impossible to TREAT the opposing terms as discrete and separable.


b. The principle of linguistic relativity


The question of how an innovator innovates hinges on the principle of linguistic relativity.19 Whorf expresses this principle as follows:


We dissect nature along lines laid down by our native languages. The categories and types that we isolate from the world of phenomena we do not find there because they stare every observer in the face; on the contrary, the world is presented in a kaleidoscopic flux of impressions which has to be organized by our minds -- and this means largely by the linguistic systems in our minds. We 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 and is codified in the patterns of our language. 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 the organization and classification of data which the agreement decrees.20


The terms of the agreement may seem obligatory, but languages do not remain fixed. An innovator, of logical necessity, alters the terms of this "agreement" in ways that support the new vision.


c. The apparently 'purposive' activities of living systems


The "doings" by which an innovator innovates appear 'purposive', in the biological sense of the term. So let us look more closely at the apparently 'purposive' "doings" of living systems.


Whenever we look in detail at what living systems DO -- at their behavior, ecology, morphology, anatomy, physiology, biochemistry, etc. -- we find that whatever particular details we examine seem "goal- directed." They fit into biological activities on the next higher level as if "on purpose," so as make the higher level functions work. Furthermore, underlying whatever details we may examine, we find subsidiary details, occurring on the next lower level of activities (down to atomic and quantum-mechanical levels), which fit in as if "on purpose," so as to make the details under examination also work.


Accounting for these apparently 'purposive' activities has long posed severe problems. Elsewhere we have discussed various of the attempts to model them, and the ways that most of these attempts have proved unsatisfactory.21 To repeat those comments here would exceed the scope of the present paper. The model proposed by Gerd Sommerhoff22, which he calls directive correlation, has thus far survived logical and empirical testing.23 To express this model, we define four constructs, which occur in an ordered sequence that spans the interval t0 to t2 . To express the first of these, we utilize two terms ("initial conditions" and "goal"), and posit a strict logical relation between the two.


i) Initial conditions and goal: The construct of initial conditions signifies a grouping of existing or possible "disturbances". When something disturbs an-organism-and-its-environment, that initiates a 'purposive' sequence.


The logically related construct of goal, a subsidiary portion (or "subset") of the construct of "outcome", gives the criterion for an outcome which appears 'favorable' from the point of view of the organism.


Each of these constructs stands as polar to the other: neither could "exist" or "occur" without the real or imagined presence of the other. (If, for example, the "initial conditions" consisted of a "need," e.g. a relative lack of nutrients -- which a human might express by saying, "I feel hungry!" -- then the "goal" would consist of the criterion for whatever conditions it would take to satisfy this "need," e.g. to leave this human well-fed and feeling replete.)


ii) Effects of the "initial conditions" on the organism: This construct expresses "what the organism does" when the "disturbances" impinge on it.


iii) Effects of the "initial conditions" on the environment: This construct expresses "what the environment does" when the "disturbances" impinge on it.


iv) Outcome: The interplay between (ii) and (iii) leads eventually to an "outcome" of the sequence. This may (or in the negative case, may not) satisfy the "goal," the criterion for "'favorable' from the point of view of the organism."


A successful directively correlated sequence, then, over the interval t0 to t2 , involves the following doings or happenings:


At t0: the "initial conditions" function;


At t1: "what the organism does" and "what the environment does" in


At t2: the outcome, produced by the interplay between "what the organism does" and "what the environment does," occurs; and it satisfies the "goal."


In these terms, the practice of science functions as a biologically 'purposive' activity of the human species. Root-Bernstein, who does not explicitly utilize the construct of directively correlated as a basis for his formulations, puts it this way:


The passion of any real scientist is to expand our knowledge of the world, not merely to confirm it.24


We repeat: In our more general setting, we maintain that we humans practice science with the proximate 'goal' of increasing human knowledge of what-goes-on-around-and-in-us, and thus, our ability to predict accurately in this domain, with the ultimate 'goal' of increasing the likelihood that individual, group and species will survive.


d. The uniquely human environment


An innovator innovates in contact with an environment. We need to spell out just what particular environment proves unique to humans, and the range of the ways a human can transact with it.


In order to characterize this uniquely human environment with acceptable precision, we first need to specify two biologically 'purposive' terms. The first of these we call organic integration. The second depends on the observable fact that certain kinds of organisms associate with others of their kind. This associating may also entail a mechanical connection, as in colonial hydrozoa, or the organisms may remain spatially separate and individually mobile, as with the members of a wolf-pack or a band of humans. We need a criterion by which to determine whether or not to classify an apparent associating as social.


i) Sommerhoff defines a system of organic activities as integrated in the biological sense provided that


the activities are directively correlated and [that] these correlations themselves are again directively correlated inter se (e.g. if their respective [goals] may in turn be regarded as a set of directively correlated variables).25


For example, someone eating at table in a Western, e.g. American, setting, uses fingers, hand, arm, etc., to manipulate a fork so as to move a piece of meat from the plate into her/his mouth -- an apparently 'purposive' procedure, which we can represent in directively correlated terms. But before s/he can manipulate the fork and meat, s/he must engage in the subsidiary activities of maintaining a sitting posture -- a procedure which we can also represent in directively correlated terms. Furthermore, these two 'purposive' procedures appear directively correlated inter se, with the achieved "goal" of maintaining a sitting posture figuring among the "initial conditions" of the directively correlated procedure of manipulating fork and meat.


ii) Further, Sommerhoff designates an associating as social provided that



there exists a system of integrated directive correlations between the states or activities of the members which has the continued existence of the [associating] as an ultimate [goal].26


Individually and collectively, we human organisms show both the behavioral capacity, and a fundamental need, to form (an) associating(s) with other humans, and, once having done so, to continue the associating. Elsewhere, we have shown that the "states and activities" involved in associating satisfy the criteria as both directively correlated and integrated. 27 Thus we humans function as a social species. The functioning of some non-human organisms, such as wolves, termites, colonial hydroids, etc., also satisfies these criteria -- so they too qualify for classification as social species.


To proceed in a useful manner, now we need to distinguish the social transacting of humans from that of other species. Korzybski pointed the way in 1921, by re-framing the traditional philosophers' question ("What 'IS' Man?"). He asks instead, "What do we humans DO that distinguishes us from other living creatures?" 28 He asserts that we humans accumulate human knowledge, at exponential rates. This process, which he labels time-binding, both forms the basis for how we humans gain our living in the biosphere, and also provides the defining mark of the species. We humans cooperate to apply what we know, in the process coming to know more. The rate at which we gain new knowledge depends on how much we already know.


We humans, then, like other living organisms, transact with the aspects of our environments which we call physical, biological, etc. Unlike other organisms, we also inhabit a unique environment or matrix, which includes symbolic aspects, such as what we commonly call "language" and/or "culture".


Every human who can talk and/or sign with her/his fellows, and so observably functions as a member of a "speech-community" and a "culture", thus demonstrably lives in a primary, direct relationship with that body of accumulated knowledge available within her/his speech-community/culture. This relationship has three main facets:


1. Each human inherits this body of knowledge, unconditionally (and, in the process of growing to maturity, assimilates some fraction of the time-binding heritage, making it her/his own); after having assimilated it,


2. Each of us administers and cares for it, and contributes to it so as to augment it; and


3. Each of us passes on the resulting body of knowledge to peers and progeny and to the generations yet unborn.


This means that any human social transacting -- any associating -- hinges on a body of shared knowledge. This time-binding heritage consists of at least partially explicit lived theory and "received wisdom," and remains in principle accessible to the members who associate. Anyone who participates in social transacting within that social setting has already assimilated this heritage to some degree. Consider for example a young couple -- a sexually mature male and female, not previously married -- who declare their intention to live together within the social forms of the institution of marriage, as it exists within their culture(s) of origin. In making that declaration, they show that they have apprenticed themselves to the elders of their social environment, and have assimilated enough of their tradition to seek to enter this next developmental stage of its social forms. They thereby continue to apprentice themselves to their tradition, and in the process take on a further body of lore, a vast collection of tested guesses, which they must assimilate (to some degree) in order to carry out their stated intentions. These include the guesses which underlie how the two of them transact together, given their respective "personalities," backgrounds, etc.; those which underlie the sexual practices of their culture(s) and how this couple enacts them; those which underlie how they engage themselves in the customary division of labor, responsibilities, obligations and activities between spouses; and so on. Other areas of this body of lore include the change of roles for each involved in going from "unmarried" to "married"; the "proper" behavior towards kin and towards non-kin; etc., etc., etc.


Our view of the uniquely human environment includes fine details concerning the structure of human knowledge. In our view, any transacting, any guess, any knowledge has two components. One of these "tells about" some aspect of what goes on around and in the human organism who generates it; the other component gives some kind of representation of that human.29


Thus we view a culture as consisting of tested guesses which, on its "other" side, gives its exponents guidance as to what aspects of the environment to consider important; and on its "self" side, encodes a more or less tacitly held, normative view of Man. Or otherwise stated, we hold that the shared expectations of the members of a given culture, sub-culture, group, etc., include rules and standards for how "WE PEOPLE" behave -- what "we" must, and must not, do, etc.


This view of the environment unique to humans contrasts with traditional views, as they bear on this topic. So far as the authors know, the members of no traditional culture currently on record explicitly hold that their shared view of "we-people-and-our-surroundings" arises self-reflexively, from the symbolizing-activities of "the people" involved. Moreover, the members of no culture on record explicitly base their shared expectations, their rules and standards, on evidence. Finally, the members of no culture on record frame their shared expectations so as to make these disconfirmable.30


These remarks give some sense of the existence of a uniquely human environment, and of its pervasiveness throughout human living. As for the ways a human can deal with it -- transact with her/his time-binding heritage -- let us now consider two polar constructs, which we call self-eliminating and self-affirming.


i. Self-eliminating


A person who takes a self-eliminating attitude toward her/his heritage tends to regard the heritage as FIXED -- perhaps, the product of a Golden Age of REAL knowledge and wisdom, which has gotten imperfectly transmitted to the degraded present. S/He tends to disparage her/himself in comparison with authority-figures, considering the Authorities who created the heritage as somehow "better" than her/him, and the heritage itself not for the likes of her/him to tamper with. Further, s/he holds an attitude of more or less covert FEAR: To tamper with the heritage might prove dangerous. First, at the level of "logical consequences," s/he says in effect, "If I deviate from 'the right way to do things' and so 'AM' wrong , some THING I don't know about might hurt me." Second, at the level of social consequence, "If I deviate and so 'AM' wrong, people might LAUGH at me or otherwise harm me." S/He holds that her/his heritage exists in a context of scarcity, of "not enough to go around" and of "win/lose" solutions to conflict situations. In short, s/he holds her/himself as subject to authoritarian authority, and as subservient to "the weight of tradition".


ii. Self-affirming


At the other extreme, a person who takes a self-affirming attitude toward her/his time-binding heritage tends to regard the heritage as a free gift of the dead to the living, and to consider her/himself free to receive it, assimilate it, augment it and pass it on to peers and progeny. In other words, s/he sees the heritage as growing and expanding, a living tradition. Her/his general attitude looks like confidence and faith: "If I take a step forward, or take a stand, the universe will support me." S/He acts on the expectation that what s/he figures out will prove useful to others as well as to her/himself; and that the contributions of others will support her/him in her/his own quest. S/He holds that the heritage exists in a context of sufficiency, of "win/win" solutions to conflict situations. In short, s/he holds her/himself as empowered with personal authority, and as both worthy and able to contribute to the heritage. Then the big question remains, how to do so.





We hold that innovating initially takes place as the developing of a Gestalt -- a figure of focal interest to our organism, against a (back)ground relatively empty of interest31 -- within the personal version of the World-View which some human (or some small group of humans) share(s) with her/his/their speech community. In order to account for innovating, we discuss this developmental process as a series of nested Gestalten, where the figure of Gestalt1 becomes the background of Gestalt2; and the figure2 of that Gestalt2 in turn becomes the ground of Gestalt3, which itself has a new figure3. Within this general configuration, we treat the details relevant to the stage of innovating of interest at the moment as a) a Gestalt with several components (e.g., logical as well as the perceptual, feelingful and motor ones), and as b) involving figure/background relations in each component. Altering one portion of such a Gestalt perforce produces compensatory alterations in at least one other portion.


a. Overall setting


Any innovating arises against the background of the innovator's "pre-innovation" view of the topic. Even within one cultural and linguistic tradition, people differ from one another -- appear unique -- including in their approaches to living and to dealing with the uniquely human environment. Historically speaking, relatively few members of the community of scholars and scientists have done significant innovating. Root-Bernstein suggests that it might prove useful to examine the process of innovating (or, as he terms it, discovery) empirically.


Why not admit that discoveries derive from the ways in which particular scientists logically go about their work? ...

... Again and again, the record reveals that the discovery is not a fluke but the inevitable, if unforeseen, consequence of a rational and carefully planned line of inquiry initiated by a scientist. It follows that, contrary to philosophical orthodoxy, the tests of an incorrect hypothesis often result in surprises that lead to discovery, and that discoverers are not just beneficiaries of fate. They seem to have ways of courting the unexpected, which improve their chances of making novel observations. So there must be a logic, or at least a set of strategies, in discovery. The question is, Why are discoveries made by certain scientists rather than others? Can their strategies be learned? ...

It should be clear by now that scientific discovery is never entirely accidental. It holds an element of surprise, to be sure, the effective surprise that changes a person's perception of nature. But the best scientists know how to surprise themselves purposely. They master the widest range of mental tools (including, but certainly not limited to, game playing, universal thinking, identification with subject matter, intuition, and pattern recognition) and identify deficiencies or inconsistencies in their understanding of the world. Finally, they are clever enough to interpret their observations in such a way as to change the perceptions of other scientists, as well. As Albert Szent-Gyo"rgyi put it, "Discovery consists of seeing what everybody has seen and thinking what nobody has thought."32


In our own terminology, someone whose dealings with her/his heritage follows mainly a self-eliminating pattern seems less likely to do much innovating than does someone who follows mainly a self-affirming pattern.

In the course of its directively correlated transacting with its surroundings, an organism generalizes, and then uses these generalizings to generate higher-ordered generalizings, etc. In a frame of reference in which we treat differences -- non-identities -- as fundamental (rather than so treating similarities, or their limiting case, identities), we hold that organisms create generalizings (similarities) for their own needs -- and that they do this by a process of ignoring differences.


Here, by ignoring some of the differences between self-affirming patterns, we create several features common to such patterns. For example, we hold that, instead of regarding her/his surroundings as more or less threatening and trying to hold them off at a distance, a mainly self-affirming person finds her/himself "at home," intimately involved with and participating in her/his surroundings. Her/his experiencing shows the marks of a "strong Gestalt" -- whatever s/he takes part in appears in general bright, clear, unified, fascinating, graceful, vigorous, releasing, etc., depending on whether it occurs primarily in a perceptual, feelingful or motor context.33 S/He finds her/his surroundings fascinating in some way or ways. In effect, s/he "falls in love" with "It," with her/his favorite arena of transacting; and in so doing, s/he dissolves the traditional disjunction between "Observer" and "Observed," or "I" and "It." This kind of participatory involvement stands as one of the first steps in what Polanyi calls "choosing a problem."


In choosing a problem the investigator takes a decision fraught with risks. The task may be insoluble or just too difficult. In that case his effort will be wasted and with it the effort of his collaborators, as well as the money spent on the whole project. But to play safe may be equally wasteful. Mediocre results are no adequate return for the employment of high gifts, and may not even repay the money spent on achieving them. So the choice of a problem must not only anticipate something that is hidden and yet not inaccessible, but also assess the investigator's own ability (and those of his collaborators) against the anticipated hardness of the task, and make a reasonable guess as to whether the hoped for solution will be worth its price in terms of talent, labor and money.34


b. Perceptual aspect


The perceptual alteration central to innovating -- Gestalt psychologists call it the "Aha!" experience -- hinges on the construct of distinguishable from (or not-identical with): In a situation which fascinates (or at least focally interests) her/him, our person notices something that in some sense s/he did not expect, something that alerts and excites her/him. S/He attends to this unexpected situation long enough to complete the Gestalt, making in it a differentiation which her/his peers and predecessors have not made. For example, s/he discerns two or more distinguishable aspects where s/he, and everyone else, previously had found only one aspect, and where the traditional linguistic system contains one single term to describe this "single" aspect of the situation. Often, an innovator reports that s/he had great difficulty in seeing the newly differentiated aspects for the first time.


c. Feelingful aspect


While in the directively correlated process of developing a Gestalt, the person(s) involved do(es) not know its "contents," and perhaps not even the nature of its "goal". To guide her/his/their transacting, s/he/they keep(s) track of how the "happenings" develop on both sides of the organism/environment boundary. Hilgartner & Randolph35 describe the guiding process in terms of excitement.



Excitement. No organism could survive at all if it did not have the capacity to monitor and keep track of the progress of its own activities toward the achieving of its ["goals"]; but since the exact details of this "achieving" are unknown a priori, being worked out as the organism proceeds, and since perhaps the entire nature of the ["goal"] may be unknown to the organism, by what mechanism might the organism monitor its progress? ("The coming but as yet unknown solution."36) In a situation in which the organism is vitally involved, in which its survival is in some way or other at risk, the organism is to some degree or other excited; and each alternative it faces promises to increase or to decrease the excitement. Furthermore, its own state of excitement is apparent to the organism, by proprioception. And finally, each choice it makes which brings it nearer to the achievement of its ["goal"] serves to increase the level of excitement, up to the climax of the experience. Excitement, then, comprises a feedback-process without which directively correlated activities in principle could not achieve the ["goal"], [i.e.] could not exist.37


Excitement, and the proprioceptive monitoring of excitement, play central roles in Gestalt-formation in general and in innovating in particular. In innovating, this excitement occurs in the context of another aspect of feeling, namely, the felt difference between seeing (or in general, sensing/feeling/moving) in a traditional and self-eliminating manner, on the basis of authoritarian authority, and seeing (or in general, sensing/feeling/moving) for oneself in a self-affirming manner, on the basis of personal authority.


At this point, a mainly self-eliminating person, seeing a novel situation in a rigidly traditional manner, displays an unspoken (and perhaps entirely unaware) background of fear. Grounds for fear do of course exist, in the midst of other possibilities: in her/his social transacting, a human faces a spread of possible outcomes ranging from the most favorable (the extremes of support, respect, esteem, reward, etc.) to the most unfavorable (the extremes of obstruction, disapproval, disesteem, punishment, coercion, rejection, etc.). In following a mainly self-eliminating pattern, this human appears to expect an unfavorable outcome. Any Gestalt which s/he develops will reaffirm the traditional position.


In contrast, a mainly self-affirming person, seeing a novel situation as interesting to her/him above and beyond its traditional valuation, displays an unspoken confidence, appearing to expect a favorable outcome. Any Gestalt which s/he develops will include the possibility of deviating from the traditional position.


d. Logical aspect


Neither an innovator nor her/his critic seems likely to engage in logical analysis of an innovation until after it gets completed. Thus we restrict the discussion of the logical aspect of innovating to after-the-fact rather than in-process inferences.


The logical aspect of innovating, in the present viewpoint, centers on two topics: an analogy between living systems and formal deductive systems,38 and the criterion of logical generality.


The present viewpoint compares human behaving-and-experiencing (behaving as viewed from the "outside," experiencing as viewed from the "inside") to a formal deductive system. In line with the above comments about geometry, a formal deductive system has an array of premises: undefined terms, which we take on faith and do not explicitly define; axioms or postulates stated in terms of the undefined terms; an array of defined terms; and rules of inference, which tell us how we may, and may not, combine or otherwise manipulate these pieces. And it has its resulting conclusions, along with standards of proof, so we can know when we have proved a conclusion consistent with, or not consistent with, the premises we started with. The comparison between human behaving-and-experiencing and a formal deductive system posits that every human assumes (has or holds premises); and that what s/he DOES functions at the level of conclusion. This amounts to as if reasoning, in which we INFER premises: We account for the "doings" of a human logically, AS IF her/his "doings" follow strictly from her/his assumings -- AS IF under no circumstances can a human NOT assume.


The logical aspect of innovating also hinges on the criterion of generality. This criterion has several parts. It posits a) a specifiable special assumption, restricted and restrictive, which may or may not form a portion of the premises of a given theory. It also posits b) a coherent body of observations, and c) two or more theories which purport to account for these observations. Specifically, it expresses the logical relation between two theories, one of which accounts pretty well for a restricted portion of this body of observations but cannot account for the rest, and the other of which accounts (equally well) for the portion of the body of observations accounted for by the first theory, and also accounts for a portion of the body of observations which the first theory cannot account for. One can establish that the second theory qualifies as more general than the first if and only if one can demonstrate that two conditions obtain:


i) That the premises of the first theory include the special, restrictive assumption, whereas those of the second do not; and


ii) That by systematically eliminating the restrictive assumption from the premises of the first theory, one can "expand" it into a logical structure equivalent to the second; whereas, by systematically introducing this restrictive assumption into the premises of the second theory, one can "collapse" it into a logical structure equivalent to the first.


By this criterion, the theory of relativity qualifies as more general than does Newtonian physics.39 And by the end of the present paper, we will have demonstrated that these two conditions apply to the non-euclidean geometries also, and that these therefore qualify as more general than Euclidean geometry.


The present viewpoint accounts for innovating in these logical terms, including that done without benefit of the present theory: It holds that the traditional shared view of the problem-situation follows from shared premises -- namely, those which underlie the agreements Whorf speaks of, and codified in the patterns of the language of that speech community. The altered view of the problem-situation generated by the innovator also follows from premises -- perforce, premises which differ from those shared throughout the speech community in question. Specifically, the altered premises differ from the shared premises in accordance with the criterion of generality: The innovator generates her/his innovation by the process of eliminating from her/his own premises some specifiable, restrictive assumption which s/he previously shared with the speech community.


To complete this argument we must specify that restrictive assumption which, in general, underlies the traditional views that the innovative ones have replaced. Innovating, as noted above, hinges on the opposition between a) the construct of distinguishable from (or not-identical with), e.g. A =_/ B , and b) the opposite of distinguishable from (namely, identical with), taken in a special, tacit sense. Those who non-verbally failed to distinguish (e.g. between the aspects of the problem-situation which our innovator comes to discern) showed undiscriminating perceiving of that problem-situation. In logical terms, this undiscriminating perceiving appears based on a special, restricted and restrictive assumption -- specifically, the unstated assumption that those now demonstrably different aspects of the problem- situation (territory) qualify as IN-distinguishable (identical), just as the traditional symbolic map says they do. We call that special restrictive assumption tacit Identity. It qualifies as so restrictive as to hold under no circumstances whatsoever: By the definition of the term mistake, given an A which qualifies as distinguishably different from B , non-verbally to TAKE A for B constitutes a mistake. (Such a mistake may, or may not, have immediate survival-consequences.) Let us regard tacit Identity, then, as a disqualifying assumption: Any viewpoint which includes tacit Identity among its premises thereby qualifies as ultimately unsatisfactory.


In these logical terms, the perceptual alteration central to innovating (the "Aha!" experience) depends on distinguishing -- on eliminating a specific usage of the disqualifying assumption of tacit Identity.


Tacit Identity also forms the core of the construct of authoritarian authority, mentioned above under the rubric of the feelingful aspect of innovating. As noted above, the traditional map -- the sanctioned manner of seeing the problem-situation -- involves tacitly failing-to-distinguish-between those discernible aspects of the territory which our innovator actually comes to discern. As another aspect of this undiscriminating perceiving, the fact that the speech community has an "official" way of perceiving the problem-situation, one represented in the vocabulary (in the WIE family of languages, probably as a noun-form) and encoded in the structural patterns of the shared language, has consequences: It means that the members of the speech community share another usage of tacit Identity, one which holds that the group's map (the traditional way of perceiving the problem-situation) qualifies as identical with the territory (the "doings" or "happenings" signified by the phrase the problem-situation itself): "That's the way it REALLY IS!"


One can imagine a human group which systematically encourages its members to examine situations for themselves, and supports perceiving on the basis of personal authority. But many groups do otherwise -- appear tacitly to identify their shared beliefs or views with their individual or group survival, and so greet any deviation from the group-sanctioned views, with their encoded usages of tacit Identity, with reprisals or threats of reprisals of various sorts. The term authoritarian authority, then, designates that constellation of threatening attitudes.


e. Symbolic aspect


Innovating requires stubbornness verging on monomania, and a symbolic invention. After generating her/his initial multi-faceted insight, the innovator persists in her/his investigations as long as necessary to test the null hypothesis -- the guess that the insight qualifies as entirely mistaken. (In so doing, s/he sharpens still further her/his non-verbal perceiving of the problem-situation, as well as providing opportunities for her/his peers and/or successors to sharpen their own.) In the process, s/he creates symbolic means to describe what s/he finds -- s/he has to do that, in order to "remember" her/his own insight -- and develops these so as to present the extensions s/he makes to her/his fellows and to the young. Often, in order to describe/explain the initial insight, s/he generates a metaphor by juxtaposing two dissimilar constructs so as to make them one. Then the new metaphor has implications of its own, and so suggests new observations, further innovations and further creative descriptions/explanations. In developing these, s/he applies and extends the original insight as far as seems useful; and s/he presents her/his innovations to her/his fellow-humans.





To see an example of how this model of innovating works, let us return to the geometries. Evidence in the form of mathematical presentations indicates that geometry-innovating actually took place roughly between 1830 and 1854. From this evidence we infer that the perceptual, feelingful, logical, etc., components of innovating did occur -- perhaps along the following lines:40


a & b. Perceptual-feelingful aspects


Like other arenas of human knowledge, geometry emerged out of the less differentiated traditional past, and became progressively more and more articulated and articulate. As we said, it had arisen out of practical concerns, e.g. surveying, pyramid-building, etc. The Greeks had transformed it into an abstract theory, and Euclid made geometry systematic and formal. But he presented his five postulates as "self-evident truths." His successors may soon have abandoned that way of presenting the postulates, but in a sense the expression summarized how geometers actually HELD their discipline. For over two thousand years, exponents of geometry automatically and unreflectingly TREATED the postulates as well as the theorems of Geometry as "TRUE" empirically -- monolithically true, with the weight of authority behind them. Prior to about 1830, the idea that geometry might comprise a 'map', and worse still, one which represents its 'territory' imperfectly at best, appears not to have arisen.


Nonetheless, skilled geometers felt the fifth postulate as somehow disturbing. That fact, too, had soon become tradition; and for two thousand years, all efforts to show the fifth postulate as redundant by deriving it from the other four postulates, or proving it as a consequence of the other four, had failed.


The perceptual differentiation made by the geometry-innovators centered on the fifth postulate considered apart from the rest of the premises, and some "What if -- ?" reasoning, along the following lines:


"Perhaps the fifth postulate does not qualify as true in quite the same way that the rest of the premises do;... perhaps it qualifies as somehow restrictive.... Aha! What would happen if one expressed it differently, e.g. changed its wording? What if one posited not one and only one, but rather two or more lines passing through a given point and parallel to a given line?"


The results of these (inferred) perceptual-feelingful insights we know.


c. Logical aspects


The above manner of expressing the perceptual-feelingful aspects of geometry-innovating already imply the logical aspects, namely, the criterion of generality. It implies that traditional Geometry includes among its premises some specifiable assumption, stated or unstated, which qualifies as restricted and restrictive; and that the non- euclidean geometries do not include that special assumption among their premises.


As a candidate for the role of special restrictive assumption, consider the unquestioning attitude of the earlier geometers, that Geometry accurately and adequately represents the sidereal universe. In other words, we hold that the earlier geometers FAILED to assume that the premises and theorems of Geometry qualify as distinguishable from (not-identical with) the non-verbal world -- the sidereal universe -- which the geometers inhabit. And that failure amounts to a special case of the disqualifying assumption of tacit Identity.


The non-euclidean geometers, in contrast, specifically eliminated that special restrictive assumption. As far as we can now tell, Gauss went only part-way. He devised a procedure which tests precisely that UNSTATED assumption: He altered the fifth postulate and generated a geometry, expecting this procedure to yield a contradiction. It did not do so. But the fact that he expected it to yield a contradiction means a) that he held the traditional unquestioned and unquestioning view, that Geometry identically represents the sidereal universe, and b) that he had not made explicit the hypothesis which his procedure tests. Consequently, when the results disconfirmed the hypothesis, he may not have recognized the significance of that outcome: that Euclidean geometry does NOT identically represent Space. At any rate, the insights contained in the relevant sections of his personal papers got published only posthumously.


Lobachevsky and Bolyai did not explicitly state a hypothesis questioning whether or not Geometry uniquely represents Space, either. Instead, each one implicitly raised that question by changing the wording of the fifth postulate and generating a geometry that questioned the accepted view, and publishing. Judging from what they DID, they must have presumed that "A different geometry represents a different space."


Riemann appears to have assumed that, if changing the fifth postulate one way produces one alternative geometry, perhaps changing it in another way might produce yet another alternative geometry; and so it turned out.




As promised, we have accounted for how the non-euclidean geometers (tacitly) utilize non-identity in generating their geometries: From their formalized systems, we show, these innovators specifically eliminate that restrictive tacit assumption encoded in the older system, to the effect that Geometry accurately and adequately models Space. Instead, they distinguish the premises and theorems of Geometry -- both Euclid's and their own geometries -- from the non-verbal world, the sidereal universe which they inhabit. Thus, in a piecemeal fashion, they eliminate one reliance on what we call the disqualifying assumption of tacit identity.

In answering this question, we have proposed a full-fledged theory, an alternative to the standard WIE logic of science. Furthermore, our version of the logic of science has the advantage of generality over the standard version.


a) Our version accounts for many of the observations concerning how scientists actually do science, including most of those presented in Root-Bernstein's paper. We tested our proposals by using them to re-construct how the non-euclidean geometers generated their innovative geometries. (As noted above, the standard WIE version does not account well for observations.)

b) We display a restricted and restrictive assumption, which we call the assuming of tacit identity.


c) We demonstrate that this restrictive assumption forms a part of the premises of Euclidean geometry (as the unstated assumption that Geometry accurately and adequately model Space); and of the standard WIE version of the logic of science (as the unstated basis for its irreflexive and dualistic structure).


d) We demonstrate that the non-euclidean geometers obtained their "revisionist" geometries by eliminating this restrictive assumption. Similarly, we obtained our own theoretical framework by eliminating this restrictive assumption, in several manifestations, from the premises of our own developing theoretical framework.



e) We leave it as an exercise for the reader to convince her/himself that to re-introduce that restrictive assumption into each of the non-euclidean geometries in turn would, in each instance, lead to a system logically equivalent to Euclidean geometry. Elsewhere, we have published a notational proof which demonstrates that our own non-standard theoretical framework qualifies as more general than the WIE mathematical theory of sets (which, in the 1990's, WIE logicians and mathematicians hold as most general formalized language of the WIE tradition -- the paradigm and exemplar of "a mathematical language of known structure").41


In principle, we have framed our theory so as to make it testable.

We show that any logic of science which cannot account for innovating cannot account for the DOING of science -- but rather, only for the post hoc effects of doing science.



1Hilgartner, C. A. (1968). "General Semantics, Psychotherapy, and the Logic of Science." Unpublished manuscript, 1963; revised 1967. Abbreviated version, ETC.: A Review of General Semantics 25:315-324.


2Hilgartner, C. A. (1965). "Feelings, Orientation, and Survival: The Psychological Dimension of the Current Human Crisis." Presented at the Ninth International Conference on General Semantics, San Francisco State College, August 1965.


3To convince yourself of this, consider a human who (like the authors) lives in a "temperate" climate -- one where in the winter, it snows. How well would this person fare in the absence of the technologies and the underlying knowledge represented by buttons, cloth, tailored garments, houses, a winter food supply, etc. Yet no one alive today created that knowledge de novo, nor first developed those technologies.


4For humans, environment includes 'physical', 'biological', 'psycho- social', and 'symbolic' components. Consequently, even what we traditionally refer to as a "purely cognitive" change betokens an altered environment for the humans involved.


5Dewey, John & Arthur F. Bentley (1949). Knowing and the Known. Boston: Beacon Press. Paperback edition, 1960. These authors contrast mechanical interacting, that may alter the physical state, the shape, etc., of the relevant components, but otherwise leaves them fundamentally unchanged -- a construct which seems well-suited for describing the billiard- ball "behavior" of the macroscopic non-living, against the back-and-forth transacting characteristic of living systems, which leaves both parties fundamentally transformed.


6Polanyi, Michael (1964). Personal Knowledge: Toward a Post-Critical Philosophy. Chicago: University of Chicago Press, 1958. Torchbook edition, New York: Harper & Row, 1964, p. 171.


7Root-Bernstein, Robert S. (1988). "Setting the Stage for Discovery." The Sciences 28(3):26-34, May/June 1988, p. 29.


8Root-Bernstein, 1988, p. 27-8.


9Korzybski, Alfred (1933). Science and Sanity: An Introduction to Non- aristotelian Systems and General Semantics.. Chicago: Non-Aristotelian Library Publishing Co. 4th edition, Institute of General Semantics, Lakeville CT 06030, pp. 433-442.


10The use of the indexed terms here suffices to show that we distinguish between these hierarchically-ordered, multiordinal usages of the term to account -- one mark of non-identity reasoning -- instead of treating them as equivalent or indistinguishable (the mark of identity-based reasoning).


11But in practical terms, whether any separate 'outside world', or 'self', "actually" exists or not seems irrelevant. Even if such 'entities' did 'exist', we humans could not in any way transact with them, experience them, and hence know them or know about them -- could not find our viewpoints altered by our dealings with them.


12Perls, Frederick M., Ralph Hefferline & Paul Goodman (1951). Gestalt Therapy: Excitement and Growth in the Human Personality. Julian Press, New York, p. 227.


13Hilgartner, C. A. & David L. Johnson (1989). "A Unifying Principle for Biology." Presented at the Annual Meeting of the Ohio Academy of Science, Cuyahoga Community College, 29 April 1989.


14Korzybski, 1933, p. 90-91.


15Korzybski, 1933, p. 91.


16Korzybski generated the innovations summarized in this paragraph during the most productive dozen years of his life (1921-1933).


17Hilgartner, C. A. (1974), in Coping with Increasing Complexity. D. E. Washburn & D. R. Smith, eds. New York: Gordon & Breach, pp. 31-67, pp. 33-4.


18If you want to know more about Russell's paradox and how it affected twentieth century WIE mathematics in general and set theory in particular, see Guillen, Michael (1983). "Logic and Proof -- A Certain Treasure." From Bridges to Infinity. Los Angeles: Jeremy P. Tarcher, Inc. For a succinct statement of Russell's Paradox itself, see Cohen, Paul J. & Reuben Hersh (1967). "Non-Cantorian Set Theory." Scientific American 217:104-116, December 1967, p. 105.


19Please note that we do not speak of "the Whorf hypothesis." Many who cite the "Whorf hypothesis" or "Sapir-Whorf hypothesis" or "Sapir-Whorf- Korzybski hypothesis" do so as a means of discrediting Whorf (and his allies). They claim that Whorf asserts, "Language determines thought" (the "strong form" of this "hypothesis"), or "Language influences thought" (its "weak form"). As Alford and others have pointed out, Whorf knew the difference between a hypothesis and a principle. He did not propose a hypothesis; he proposed "a new principle of linguistic relativity" (Whorf, 1956, p. 214), offering guidance towards the formation of possible hypotheses. We quote one statement of Whorf's principle of linguistic relativity in our text.


20Whorf, B. L. (1956). Language, Thought, and Reality: Selected Writings of Benjamin Lee Whorf. John B. Carroll (Ed.). MIT/Wiley, New York, pp. 213-4.


21---> Criticisms of "inner man" etc.


Hilgartner, C. A. & John F. Randolph (1969). "Psycho-Logics: An Axiomatic System Describing Human Behavior. A. A Logical Calculus of Behavior." Journal of Theoretical Biology 23:285-338, pp. xxx.

Also, ...


22Sommerhoff, G. (1950). Analytical Biology. London: Oxford University Press; also, Singer, E. A. (1946), "Mechanism, Vitalism, Naturalism." Philosophy of Science 13:81-99; also, Ashby, W. Ross (1962), "The Set Theory of Mechanism and Homeostasis." Technical Report No. 7, Electrical Engineering Research Laboratory, University of Illinois, Urbana, IL.


23We paraphrase Sommerhoff and designate this construct as directively correlated. See Hilgartner & Randolph (1969a), pp. xxx-x, yyy-y; also, ... .


24Root-Bernstein, 1988, p. 29.


25Sommerhoff, 1950, p. 195.


26Sommerhoff, 1950, p. 196.


27Hilgartner, C. A. & John F. Randolph (1969c). "C. The Structure of Empathy." Journal of Theoretical Biology 24:1-29, especially pp. 3-10.


28Korzybski, Alfred (1921). Manhood of Humanity: The Science and Art of Human Engineering. E. P. Dutton, 1921. 2nd Ed., (1950), M. Kendig, ed., Institute of General Semantics, Lakeville CT 06030.


29Hilgartner & Randolph, 1969a, pp. 296-7; Hilgartner, C. A. & John F. Randolph (1969b). "2. The Structure of 'Unimpaired' Human Behavior." Journal of Theoretical Biology 23:347-374, pp. 353-356, 357-363.


30The exponents of WIE science -- which perhaps we must regard as a sub-culture -- do aspire to base their shared views on evidence, and to make them disconfirmable. But scientists in general do not systematically and methodically take on the self-reflexive task of accounting for their own accounting. Instead, they restrict their theories to the irreflexive topic of how external 'things" interact with external 'things', independent of any human. In other words, at a level more fundamental than that of the "content" of what gets said -- specifically, at the level of the now-known fundamental assumptions encoded in the generalized grammar underlying their logics, mathematics, and sciences -- the exponents of WIE science systematically eliminate the transacting observer from consideration. Early exponents of relativity and quantum theory introduced the construct of taking the observer into account to the human community, and found limited ways of doing so within their theories. But they -- and their successors -- have persisted in framing their theories in terms of WIE mathematics, which means that they too, inconsistently, eliminate the observer from consideration. What they give with one hand, they take away with the other.


To express the matter in a slightly different way, up until now the exponents of WIE science have not "merely" and benignly subscribed to the dualistic WIE world-view. Instead, they have granted it a privileged position. Instead of treating their world-view and their most fundamental assumptions as a topic of inquiry -- as self-reflexive, based on evidence, and disconfirmable -- they have treated it as "the way things REALLY ARE," and defended them from scrutiny, from critical inquiry, and from testing. Even the best WIE scientists have not framed at least some of their shared expectations so as to make these disconfirmable.


31Polanyi, 1964, p. 55-62.


32Root-Bernstein, 1988, p. 30b, 34b.


33See Perls, Hefferline & Goodman, 1951, pp. 231-2.


34Polanyi, 1964, p. 124.


35Hilgartner & Randolph, 1969a, p. 320


36Perls, Hefferline & Goodman, 1951, p. 234.


37Hilgartner & Randolph 1969a, p. 320.


38Hilgartner, C. A., Ronald V. Harrington & Martha A. Bartter (1989). "Anomalies Generated by Contemporary Physics." Bulletin of Science, Technology & Society 9:129-143, p. 136a.


39The details go as follows: Restricted and restrictive assumption: that light propagates at an infinite velocity.


Theoretical structure whose premises include the restrictive assumption: Newtonian mechanics, especially the Galileo transformations.


Theoretical structure whose premises do not include the retrictive assumption: Relativistic mechanics, especially the Lorentz-Einstein transformations, which attribute to light a finite velocity which remains constant for all observers, independent of their relative motion.


The Lorentz-Einstein transformations differ mathematically from the Galileo transformations in that they contain a correction factor which has the speed of light in the denominator of a fraction. If we take the speed of light as infinitely great, that correction factor disappears, and the Lorentz-Einstein transformations "reduce" to an expression isomorphic with the Galileo transformations. Conversely, to eliminate the assumption that light has an infinitely great velocity, we introduce that correction factor, thereby "expanding" the Galileo transformations into an expression isomorphic with the Lorentz-Einstein transformations.


As a "restricted and restrictive assumption, the notion of an "infinite velocity" belongs to the class of assumptions so restrictive as to hold under no conditions whatsoever. In WIE physics, we define velocity as distance divided by time. In order to obtain a value "infinitely great" for that quotient, we divide by zero. But in WIE mathematics, we have a rule which forbids dividing by zero. Therefore, the notion that "light has an infinite velocity" amounts to a contradiction in terms -- logically forbidden.


40Someone might argue the futility of discussing geometry-innovating in as if terms, and might demand that we tell them "how it really WAS." In so doing, they miss a fundamental indeterminacy in human behaving-and- experiencing. For even the innovator can see her/his innovation only after (s)he completes it and assimilates it. And the act of completing and assimilating it changes her/his point of view.


From within a given point of view, one cannot see that which, if seen, would alter that point of view. Conversely, from within the altered, more general viewpoint, one cannot see with the eyes of the un-altered point of view. Nor can we establish with certainty how our eyes got changed.


Thus we humans have no tools, logical or linguistic, for discussing "how it really WAS" -- we have only after-the-fact, inferential, as if constructs.


41Hilgartner, C. A. (1978). "The Method in the Madness of Western Man." Communication 3:143-242. The proof of generality appears on pp. 215- 232.