THE MATHEMATICS OF RADICAL UNCERTAINTY

 

C. A. Hilgartner

R. V. Harrington

M. A. Bartter

 

INTRODUCTION

 

In this paper, the authors introduce readers to a new formalized language of the "Let's Keep Track Of What We Say" type. In writing it, we have created and utilized a frame of reference fundamentally different from that underlying the comfortable symbolic logics, set theories, etc., of the Western mathematical tradition. Readers should expect to find it unfamiliar in unfamiliar ways.

 

In setting up our notation, we have deliberately avoided using the two-term grammar of Western Indo-European (WIE) discursive or formalized languages. In this notation, you cannot resolve a "complete sentence" or "well-formed expression" into at least one noun-phrase next to at least one verb-phrase. It uses nothing that resembles substantives and verbs, or subject and predicate, or actor and action, or things and relations between things, or objects and their attributes, or quantities and operations, etc. (cf. Whorf, 1956, p. 241@). The notation does consist of strings of markings, but none of its groupings of markings will have a form similar to, or analogous to, or derivable from, familiar phrases such as

 

a + b = c

 

or

 

P(x,y) v Q(y,z)

 

or

 

x (element of) A .

 

 

The notation itself forms a part of a larger theoretical system, a general theory of human behavior which also amounts to a general theory of social systems. This theoretical system asks and answers the following fundamental question:

 

How does a dynamically changing organism, which guides its transacting with its dynamically changing environment by means of sensory information that remains in principle inaccurate, incomplete and self-referential, manage to survive, to keep itself intact-and-growing from one moment to the next throughout an entire lifetime?@

 

 

@Whorf, 1956, p. 241

 

@Refs for fundamental question

##

Traditional WIE logical, mathematical, scientific, philosophical, etc., viewpoints simply do not address this question or the existential situation it presupposes. Instead, insofar as they discuss organism and environment at all, usually they tacitly treat them as isolated from one another in some primary sense; or, at best, they secondarily "recombine" them. And when they do that, they tacitly presume that the organism operates from "perfect" information, as if possessed of "absolute certainty."

 

As we develop our notation, we shall generate a frame of reference capable of handling the key construct of inaccurately, incompletely and self-referentially informed in a rigorous manner. In this Introduction, before we actually start setting forth the notation, we shall show what it takes to do that. We will consider some things which may sound like "philosophy," and others like "the history of mathematics," and others which may come through as "kinds of material which no self-respecting mathematician would pretend to command, or should feel required to consider". But those who persist to the end of this Introduction will not only come to perceive why we use first-person grammatical forms in discussing a mathematical notation but also will find themselves oriented to what we do/say in building up the notation.

 

Let us start with two of the strengths of the theory. Briefly stated, our alternative notation makes it possible:

 

1. Systematically to take the observer (theorist, mathematician) into account; and

2. To show what one has to assume in order to take the observer into account, as opposed to what one has to assume in order to eliminate the observer from consideration.

 

THE STARTING POINT: Time-binding

 

These two self-reflexive topics bring up the starting-point, both historical and logical, for our alternative notation: Namely, the problem of defining the species-term Man.

Any human born and raised among other humans cannot help having some kind of a definition, at least tacit and informal, for his term for humans. And this informal definition -- which he builds up through transacting with his fellows, and so probably shares with them -- occupies the role of a self-fulfilling prophecy: he will enact in his own life, and will expect from others, the kinds of behavior spelled out in his definition for his term for human.

 

In the various Western cultures, our informal definitions have centered about the view of humans as a species of depraved animals. In other locations, the inhabitants have held other views, some of them thoroughly incompatible with the generically Western ones. (We discuss the logic of this Western pattern below, under the heading of "The construct of Setting." (Ms p. 11))

 

Few humans have taken care to make their definition for the species-term Mam explicit and testable, so that it can conform to evidence. Doing so requires setting aside the tacit, informal definition one shares with one's fellows.

 

A. The first explicit, testable definition for this species- term came from Korzybski (1921)@. Instead of asking, "What IS Man," as Western philosophers had done back to the era of Aristotle, Plato and Socrates, Korzybski asked:

 

"What do we humans DO that distinguishes us from other living

systems?"

 

 

@Korzybski, 1921

##

 

Let us express the answer he gave in several different ways:

 

1. As the defining mark of the species, humans accumulate human knowledge (in the form of guesses which have survived testing), at exponential rates.

 

a. Any human gets born at a particular time-and-place, into some specific culture, which has some body of human knowledge available within it. In the course of growing to maturity among these humans, (s)he will assimilate some fraction of this body of knowledge; and having assimilated, (s)he may then contribute to it, and pass her (his) contributions on to peers and successors.

 

b. In terms of what (s)he can do by way of contributing to the body of human knowledge, it matters WHEN a human gets born. For someone born in 1900, considering becoming the first human to visit the moon and return remained only a pipedream; for someone born in 1932, it stood as a real possibility; for someone born in 1964, it passed beyond reach, into history, during her/his childhood.

 

2. As a species, we gain our living in the biosphere by cooperating to apply what we know, in the process coming to KNOW more.

 

a. Humans generate and use knowledge cooperatively. The generating, testing, judging and possibly the revising of guesses or hypotheses occurs within the human community, as a joint creation, rather than in the setting of private possession or property. Even within Western societies, with our ethos of "private advantage" and our well-developed constructs of "private property" and "trade secrets" and "security clearance," the currently-available human knowledge affects everyone in the community. For example, "the lone inventor" takes out patents on the knowledge he generates, and manufactures or licenses the manufacture of his widget; even secret weapons research takes place within specially designated groups, with the sanction of the community; the technological devices developed from new knowledge get deployed in one way or another throughout the larger community, affecting the lives of everyone.

 

b. New theories also arise cooperatively, even if the theorist lives as a recluse. No theorist can possibly generate ALL the knowledge which (s)he uses to theorize with -- (s)he cannot avoid making use of the accumulated body of explicit human knowledge. Although in the Western Indo-European cultures we have the custom of naming a new theory after an individual or a small group, we would do better to regard it as a species accomplishment.

 

3. Every human lives in a primary relation with the body of accumulated knowledge. In other words, every human lives concurrently as

 

a. Heir of the entire cumulative body of human knowledge: the trial-and-error, trial-and-success of all past generations.

b. Steward for, administrator of, and contributor to, the present store of human knowledge; and

c. Trustee for all future generations.

 

Korzybski summarized these considerations by characterizing the human species as a time-binding class of life, occupying a different dimension than do other living systems. (For the contrasting constructs of space-binding ("animal") and chemistry-binding ("plant"), cf. Korzybski (1921)@.)

 

@Manhood of Humanity. New York: E. P. Dutton; second edition, Lakeville CT: Institute of General Semantics, 1950).

##

 

THE ROLE OF LANGUAGING IN TIME-BINDING

 

When we ask how time-binding works, we arrive immediately at the construct of languaging, both signed or spoken, and written.

 

Consider even the earliest of humans, entirely pre-literate and, perhaps, still doing languaging of a structure only slightly more discriminating than the non-linguistic symbolizing that makes up the call systems common among social land mammals@. But this slight advantage suffices to support the beginnings of time-binding. In other words, it allows the humans (in Whorf's (1956, p. 213-4) metaphor) to "slice up the world" into practically-useful symbolic chunks, which they use to predict what will happen. When they proceed to guide themselves by these predictions and so test them, they generate human knowledge. Even early languaging, then, proves more powerful than does the symbolizing based on call-systems done by their merely space-binding predators, prey, or ecological rivals. For example, assume that a member of the tribe under consideration has, by trial-and-error, trial-and-success, come to distinguish reliably between a poisonous species of insect (such as the Monarch butterfly, Danaus plexipus) and an edible species whose markings mimic those of the poisonous species (such as the Viceroy, Limenitis archippus), so that he can reliably predict whether the possible morsel of food at hand will taste awful and make him feel ill or will taste good and nourish him. The ability to sign or speak allows him to encode his useful distinction so that he can re-use it himself, and also can pass it on to his peers and progeny. And the increase in encoded knowledge changes the environment of the tribe by increasing its food base.

 

In principle, the growth of knowledge AMOUNTS TO self-reflexively changing the environment of the humans involved.

 

 

@Ascher & Hockett on call systems

##

 

Consider also the symbol-system of the Natufian hunter-gatherers of the Fertile Crescent, some 12,000 years ago, which apparently worked very well in the human environment of the time -- the Natufians seem to have flourished there. But imagine a Natufian who found himself somehow transplanted to the "same" region 7000 years later: His symbol-system probably would not give adequate guidance for living -- would not enable him to predict accurately -- in the strange new environment made up of the first Mesopotamian city-states, supported by settled, irrigated agriculture, which had arisen there in the meanwhile. For example, he might "hunt" an animal from someone's domesticated herd, and end up in real danger.

 

Meanwhile, the current tribe-members also change the languaging system progressively and irreversibly, by changing the patterns of pronunciation, by incorporating new terms, by shifting the "meaning" of older ones, etc. But in principle, language change occurs at some more or less constant (or at least non-accelerating) rate, whereas the increases in explicit knowledge follow an exponential curve. Hence, when current tribe-members contribute to and so increase the explicit time-binding environment of human knowledge, they produce a potential and potentially increasing mis-match between the tacit knowledge encoded in their culture and their languaging-system and the environmental conditions they have produced by use of the exponentially-increasing body of explicit human knowledge. This mis-match plunges everyone in the cultural and linguistic community into an increasingly strange environment. Eventually, faced with this increasing discrepancy, the humans will have to revise their linguistic and cultural symbol-systems, or lose their ability to predict, and so perish.

A NEW PATTERN

A. The present situation of the human species:

 

Today, we humans find ourselves in precisely that danger. We have not consciously and intentionally revised the knowledge encoded in our linguistic and cultural symbol-systems since pre-historic and early historic times. The inherited, traditional world-views which we absorb from infancy, as we learn to function with the humans we got born among and to speak-and-listen to them, do not prepare us for current conditions. As the anthropological linguist Edward Sapir puts it,

 

It is almost as though at some period in the past the unconscious mind of the race had made a hasty inventory of experience that allowed of no revision, and saddled the inheritors of its language with a science that they no longer quite believed in nor had the strength to overthrow. Dogma, rigidly prescribed by tradition, stiffens into formalism. Linguistic categories make up a system of surviving dogma -- dogma of the unconscious.

(Sapir, 1921/1949, p. 100)

 

We manifest this "dogma of the unconscious" by learning to operate divisively, as if from "absolute certainties," at a point when our accumulated human knowledge enables us to use cosmic forces (such as nuclear fission and fusion -- A-bombs and H-bombs) to defend these supposed "absolute certainties." Thus in the environment of our current knowledge, the linguistic and cultural traditions that lead us to pretend to "absolute certainties" have outlived their usefulness, and now cannot guide us toward survival, but only toward species suicide and extinction.

 

However, so far we persist in following these inherited patterns, and so we keep arranging, in ever more numerous and more ingenious ways, for ultimate catastrophe in general and for our own self-destruction in particular.

 

No attempt at a full-scale revision of our symbol-systems has yet achieved general acceptance, and in fact, no other attempt at revision has met with visible success.

 

B. How the present frame of reference modifies this situation

 

1. The authors and our colleagues have created a new linguistic pattern, a derived grammar. On it we have built up an alternative mathematics, which appears more general than older ones. (It includes them as special cases.)

 

2. Potentially: The new pattern frees the human species, so we humans (in principle) no longer need find ourselves BOUND to the older patterns.

 

a. Since the new pattern qualifies as more general than old ones -- includes the older patterns as a special case -- we can still use the older patterns whenever they might prove useful.

 

b. The advantages (if any) of the new pattern should in principle extend to personal, professional, etc., levels as well as to the species- as-a-whole.

 

3. In the remainder of this Introduction, we take it as our immediate job to contrast the structure of the old and the new symbol-systems, and to display this contrast.

 

STRUCTURAL FEATURES OF THE WIE LANGUAGES: The Search for Certainty

 

Western cultures have a long tradition of striving for certainty. As we shall show, the tacit assumptions encoded in the grammar of the WIE languages presumes a situation of absolute certainty. These hidden assumptions, perhaps, dictated this search over the centuries and millennia.

 

Aristotle of Stagira (384-322 BC) sought to refine the process of inferring from premises so as to make the conclusions partake of absolute certainty (even if the premises did not). He reduced the previously ill-defined process of deductive reasoning to fourteen rules and a few canons.

 

The canons included the "Laws of Thought":

 

The Law of Identity: Everything is identical with itself.

The Law of Contradiction: Nothing can both be and not be.

The Law of Excluded Middle: Everything must either be or not be.

 

Aristotle apparently developed the rules from his meticulous study of syllogisms, whereas he intended the canons to express verities that many of us would call common sense. (Guillen, 1983@)

 

From the present point of view, the "Laws of Thought" seem both deeper and more superficial than whatever philosophers may mean by terms such as common sense or thought -- they appear to codify the rules for naming or "nouning" in WIE languages such as ancient Greek or modern English. These rules function within the following overall structure:

 

A. Rudimentary Grammar

 

The vocabulary of WIE discursive languages, and of the specialized technical sub-languages, discursive or formalized, derived from them, consist mainly of two kinds of terms. Traditionally, we label these two kinds of terms by means of paired substantives such as noun/verb, subject/predicate, actor/action, thing/relation, object/attribute, quantity/operation,form/substance, etc. (Whorf, 1956, p. 241)

 

By far most of the entries in a large dictionary of English, for example, bear the label of noun or of verb.

Then to form what we call a complete sentence (in formalized languages,a well-formed formula or WFF), a speaker/writer places at least one noun or noun-phrase, e.g. the cat, next to at least one verb or verb-phrase, e.g. grinned:

 

The cat grinned,

 

or

 

2 + 2 = 4

 

Furthermore, we distinguish between these two kinds of term by means of the construct of identical with and Aristotle's "Law of Identity": Those terms which we classify as nouns (or subjects, quantities, things, etc.) we implicitly regard as self-identical. One might say, for instance,

 

"A cat is a cat."

 

But we regard those terms which we classify as verbs (or predicates, operations, relations, etc.) as not-self-identical. One would not say,

 

"A grinned is a grinned."

 

In this setting, the term self-identical appears to signify persisting, really existing, or (although this seems to embarrass us) static-and-unchanging. (Hilgartner, 1977/78@)

 

WE INFER: These structural features yield a language-system composed of (posited) static-and-unchanging THINGS which enter into (posited) more-or-less-transient RELATIONS.

 

The resulting languages turn out particularly adept at representing situations involving "one-way causality": "A causes B, and B causes C, and C causes D, and D causes E, and ..." Contrariwise, these languages show marked difficulty in handling recursive relations, two-way causality, transacting, self-reflexive relations, etc.

 

 

@Michael Gullen, "Logic and Proof: A Certain Treasure," in Bridges to Infinity. (1983)

 

@Cf. C. A. Hilgartner, "Some Traditional Assumings Underlying Western Indo-European Languages: Unstated, Unexamined, and Untenable." General Semantics Bulletin Nos. 44/45, 1977-78, pp. 132-153.]

##

 

B. Dualism

Native speakers of the various WIE languages have utilized paired terms (noun-forms), often called dualisms, in building up their frames of reference. In a dualism, one of the terms suggests "some-thing" permanent in some sense, and the other suggests "some-thing" in some sense transient -- evanescent, immaterial, perhaps even mysterious.

 

Let us give some examples: Matter/soul, matter/mind, body/mind, intellect/emotions, cognition/conation, objective/subjective, language/thought, language/world, etc.

 

For the moment, let us focus on the last of these: We Westerners posit a "World" separate from "Language," which we use Language to describe.

 

As my crucial point, in developing a dualism we project the noun-phrase/verb-phrase structure of our Language onto The World:

 

Without even saying so in words, we ACT as if we hold that The World, like our Language, consists of "static and unchanging THINGS, (exactly right for designation by noun-forms or noun-phrases), that enter into transient RELATIONS, (exactly right for designation by verb-forms or verb-phrases)."

 

Thus certain philosphers have postulated a "PREFORMED HARMONY" between Language and World.

 

Or in other words, we give the grammar of our native languages a Cosmic significance. For us, it occupies a Privileged Position. We secretly hold that what we describe in language is REAL, that "things are REALLY LIKE THAT."

 

Suppose that to say an organism lives means that it generates some kind of sensory "map" of that "territory" composed of what goes on in and around itself: Then in giving a privileged position to the WIE grammar, we tacitly and unquestioningly presume that OUR OWN "maps" give a point-for-point perfect representation of this "territory" (exhaustively complete and entirely accurate) -- we presume that we operate from some kind of MAP-TERRITORY IDENTITY, which confers "absolute certainty."

 

So the discussion to this point brings us very naturally to the topic of the logical construct of identity.

 

 

WHAT DO WE MEAN BY THE PHRASE, THE LOGICAL CONSTRUCT OF IDENTITY

When we speak of identity, we do NOT use the term in the sense of that by which a thing is definitively recognized or known, nor of inter-personal role in the sense of "Who You ARE," nor of the identity in question when you experience an "identity-crisis," etc. Instead, we take the term identity in its LOGICAL sense.

 

DEFINITION OF IDENTITY

We define the logical construct of Identity as signifying "Absolute sameness in all respects, or negation of difference." (Cf. Webster's@)

 

 

 

@Webster's

##

 

Furthermore, we discern two ways of using identity:

 

1. Traditional

 

In the traditional usage, we treat Identity as necessarily valid: ("That's How Things Really Are!")

a. At least since the time of Aristotle of Stagira (384-322 BC), the construct of identity has served as the avowed foundation for our logics, mathematics, sciences, philosophies, religions, jurisprudences, etc.

 

b. The construct of Non-identity gets used only for contrast.

 

For example, this pattern shows clearly in a quote from Benson Mates's book Introductory Logic:

 

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

 

x is different from x .

 

Obviously this set has no members; it is called the empty set. ... Correspondingly the universal set ... is the set of all objects satisfying the sentence-form

 

x is identical with x . ... (Mates, 1965@, p. 29)

 

Similar passages appear in Tarski (1965@, pp. 72-3), Halmos (1960@, p. 8), etc.

 

 

@Mates (1965)

@Tarski (1965)

@Halmos (1960)

##

 

In this traditional way of dealing with the construct of Identity, then, every "thing" (noun) within this universal set or universe of discourse "obviously" qualifies as identical with itself.

As long as we go along with the traditional ways of doing and saying, this manner of handling the construct of Identity and the logic of opposites seems, as Mates puts it, "obviously" correct.

 

2. Non-traditional

 

We discern also a non-traditional way of using Identity, suggested by the review of the findings of modern science done by Alfred Korzybski (1879-1950): When looking around in a world which includes time-binding humans, Korzybski finds NO situations in which he can appropriately utilize a construct of "absolute sameness in all respects." The findings of our science, even though framed in the static WIE discursive and mathematical languages, told him only of a universe of ceaseless changing, in which the construct of Identity doesn't apply.

So Korzybski judges the construct of Identity as "Never Valid," and suggests that we categorically REJECT it.@ The rejection of identity, then, forms the central premise of our alternative frame of reference (theory, model).

 

@In this non-WIE frame of reference, Identity serves one and only one functin: To provide a way to discuss situations in which "Somebody Goofed!" by TAKING A as if it "WERE" B. (By the definition of the term mistake, to take A as if it "WERE" B qualifies as a mistake.)

##

This DISRUPTS the WIE grammar. Taking the construct of  Non-identity as valid, and disallowing that of  Identity, blocks us WIE provincials from distinguishing the nouns from the verbs. That leaves us unable to name "things", or to form a complete sentence.

 

a. This tactic does not leave us mute and helpless. Instead, it opens a way for us to DERIVE a grammar, and to build up a "Let's Keep Track of What We Say" notational language on this DERIVED grammar. This tactic leaves us with a symbolic system not derived from the structure of the WIE family of languages, or from that of any other traditional language or family of languages.

 

b. When we build up our theory of  human on a DERIVED grammar, wethereby eliminate the privileged position of the WIE grammar (or of any other traditional grammar).

 

THE CONSTRUCT OF SETTING

 

The unexpected fact that we humans may choose between at least two ways of holding the opposing pair of terms, Identity and Non-identity, turns out closely connected to the construct of setting ("context," "domain of discourse"), and to the topic of the relations between a term and its opposite (or contradictory or complement) -- sometimes called "the logic of opposites." It turns out that we humans have also devised several versions of the logic of opposites.

 

A. No setting

Exponents of traditional WIE viewpoints posit no setting (domain, context) for a term (noun-form).

 

1. In a frame of reference which posits no setting for a term, the opposite of the term A , namely not-A , consists of "everything else."

 

For example, let us consider the term "Friend" as opposed to "everything else."

____________

| |

| A | not-A

| (friend) | (not-friend)

------------------

 

 

2. We point out that the construct (the undelimited setting) of "everything else" includes "white," but it also includes "tomatoes," "male sexual functioning," "Cantor's trans-finite cardinal numbers," etc.

 

When we do that, we thereby show ourselves as heir to a newer tradition that knows how to make trouble for that older traditional way of framing the logic of opposites. But instead of depending on our skills, let's consider some evidence.

 

This traditional WIE version of the logic of opposites leads to difficulties in the search for certainty in inference:

 

Around 300 BC, Euclid utilized Aristotle's syllogistic methods to prove the theorems of geometry. Mathematicians hailed his work as the model of certainty for the next 2200 years.

In the latter part of the nineteenth century, mathematicians set out to extend the realm of "Absolute Certainty" so as to include arithmetic as well as geometry. Gottlob Frege, one of the front-runners in this effort, spent many years on the task of deriving the mathematical structure of arithmetic from Cantor's mathematical theory of sets. Volume 1 of his Grundgesetzge der Arithmetik had come out in 1893, and he already had received the galleys of Volume 2 in 1902, when he received a letter from Russell. Russell proposes a paradox concerning "Sets which do not belong to themselves."@ This paradox discloses that the premises of Cantor's "intuitive" set theory (namely, the Laws of Thought) lead to a contradiction -- it shows Aristotle's rules for achieving absolute certainty of inference as flawed.

 

(For me, Frege occupies the role of a culture hero. On page 1 of his book, Frege inserted the footnote, "It is hardly possible to imagine a more unpleasant circumstance than to have the foundation crumble just as the edifice is completed. It is in just that position that I was placed by a letter from young Bertrand Russell," and he published Russell's letter as an appendix to his book.)

 

 

 

@Cf. Cohen & Hersh (1967)

##

 

B. The abstract (or blank) setting

 

Ernest Zermelo (1908), seeking to get around the logical difficulties of Russell's paradox, proposed one way to re-establish mathematical certainty. He set forth axioms for set theory designed not to fall prey to Russell's paradox. (Fraenkel (1925) provided an additional axiom.)

 

1. Zermelo's axioms posit an abstract and delimited domain for any given set, so that its complement consists of "everything else within the domain" rather than of "everything else (period)."

__________________

| | Delimited

| _____ | Domain

| | | not-A | D

| | A | |

| |_____| |

----------------------------

 

His axioms provide a protocol for building up sets, as needed, on this delimited domain.

 

2. They also contain a specific axiom which forbids taking the undefined term belongs to in a reflexive sense. The axiom states, "No set may belong to itself."

 

3. In this frame of reference, Aristotle's "Law of Identity" gets paraphrased into the modern Logical Axiom of Identity:

 

For all x which belong to the delimited domain D,

x =_ x .

 

 

Between the 1920's and 1950-60's, all forms of Western mathematics got placed on a set theoretic basis. Today, the mathematical theory of sets stands as the paradigm and exemplar of "a mathematical language of known structure": THE GENERAL logical language (of the WIE tradition).

 

Meanwhile, in 1931@, Kurt Go"del's published a paper which dashed once and for all the hopes of developing a formal deductive system which can deliver absolute certainty in the realm of inference: His "undecidability" theorem provided a "negative proof" which demonstrated that "Any system as complicated as arithmetic will contain theorems the truth or falsity of which one cannot establish from within the system."

 

Findlay's (1952)@ verbal version of Go"del's theorem:

"We cannot prove the statement which is arrived at by substituting for the variable in the statement form 'We cannot prove the statement which is arrived at by substituting for the variable in the statement form Y the name of the statement form in question' the name of the statement form in question."

 

 

 

@Go"del (1931)

 

@J. Findlay, "Go"delian Sentences, a non-numerical approach, Mind, 51 (1952), pp. 259-65, 262.)

##

 

According to Guellen, many contemporary mathematicians

... coexist so uneasily with uncertainty that they go about their day- to-day business as though the events of this century had never happened. Or perhaps it is because, as Kline suggests, "they find it hard to believe that there can be any serious concern ... about their own mathematical activity" -- each mathematician behaves as if Go"del's uncertainty is something that affects the next person, but not him. (Guellen, 1983@, (p 473))

 

In our view, Go"del's proof hinges on self-reflexive considerations, and to accomodate to it fully requires taking the observer (mathematician) into account systematically. Since the grammar on which they build up the WIE mathematics systematically eliminates the observer from consideration, Western mathematicians have difficulty taking Go"del's uncertainty as affecting their own work.

 

 

@Michael Guellen, Bridges to Infinity. (1983. Textbook, p. 473)

##

 

Meanwhile, contemporary critics of set theory make it sound as if the foundations of the mathematical theory of sets stand in disarray. For example, Benson Mates, (1965@, pp. 148-50) points out that

 

The identity relation for a given domain is a relation that holds only between each element of the domain and itself....... [I]t must be pointed out that our own intuitive account of the identity relation is not free of objectionable features. For instance, we have no right to speak of THE identity relation; by our analysis the identity relation among the elements of one domain will be different from that among the elements of another. Also, we have explicated the term 'relation' in such a way that whatever cannot be a member of a set cannot be related by any relation. Thus insofar as identity is a relation in this sense, such a thing cannot even stand in this relation to itself. This would hold not only of the set of all objects that are not members of themselves, but also of sets described by phrases that give no hint of impending difficulties. This problem is closely related to Russell's [Paradox], and once again every way out seems unintuitive. (Mates, 1965@, pp. 148-50)

@Mates (1965)

##

 

No way out of these difficulties has yet gained general assent.

 

C. A specific setting

 

The present language posits a specific (rather than an abstract) setting. I can present it in simple language, provided that you will DO something I ask you to, so we will have the right kind of experience in common.

 

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

 

[UNTIL YOU HAVE PERFORMED THIS EXPERIMENT, DO NOT READ FURTHER!]

In English, we have separate vocabulary items for you and it, for organism and environment. Our linguistic pattern suggests that what we call organism and environment exist SEPARATELY and PRIOR TO that "touching" -- that you come into contact with it only during the period in which you follow our instructions. Or, stated in slightly different terms, it suggests that organism and environment exist as separate, static-and-unchanging "things," which enter into transient "relations."

 

Our present non-WIE mathematical language utilizes a pattern different from the one just described. In it, one can discuss only one topic: an observer observing the observed; or, where we have before us written (instead of spoken or signed) text, the topic of a writer observing an observer observing the observed. We call the character (the observer) who observes the observed "our organissing," and call the character who writes out the text "our logicking." Where we use a pronoun instead of the role-name, we distinguish between these two by discarding the usual notion of gender and appropriating the existing third-person forms: -- we refer to our logicking with she or her (without implying "female"), and refer to our organissing with he or him (without implying "male"). Here, we say that our logicking posits a specific setting on which she builds up the entire notation; and that the setting which she posits consists solely and exclusively of transactings (as we say it in English) like what you just produced in your own experiencing by touching something.

 

As Perls, Hefferline & Goodman (1951) put it,

 

We speak of the organism contacting the environment, but it is the contact that is the simplest and first reality. (PH&G, 1951@, p. 227)

 

 

@PH&G (1951), p. 227.

##

 

From such contacting (transacting), our logicking INFERS constructs such as "I" and "you," "I" and "it," "organism" and "environment," etc.

 

Furthermore, to specify the pattern of inferring which our theory allows, she uses a circle of 5 inter-defined terms:

 

Abstracting Organissing

Representing Ordering on abstracting

Environing

 

At the appropriate point in the presentation of the notation, she will develop these terms explicitly.

 

SOME CONSEQUENCES OF THESE FORMULATIONS:

 

The novel setting posited for the present non-WIE frame of reference raises anew, and answers anew, one of the oldest questions posed by the Greek philosophers of the Sixth Century BC. As Koestler (1959) sets the scene,

 

Thales of Miletos [640-546 BC], who brought abstract geometry to Greece, and predicted an eclipse of the sun, believed, like Homer, that the earth was a circular disc floating on water, but he did not stop there; discarding the explanations of mythology, he asked the revolutionary question out of what basic raw material, and by what process of nature, the universe was formed. His answer was, that the basic stuff or element must be water, because all things are born from moisture, including air, which is water evaporated. Others taught that the prime material was not water, but air or fire [or earth]; however, their answers were less important than the fact that they were learning to ask a new type of question, which was addressed not to an oracle, but to dumb nature. (Koestler, 1959, p. 22)

 

To designate "the basic raw material" (or "the first principle or cause of all things"), the ancient Greeks used the word ____ (arche).

 

Up until now, we humans have given only two main answers to questions about the nature of the arche, namely, the generically Eastern and the generically Western.

 

According to the Eastern view, "Only the Atman (soul, self) exists; all else is Maya, illusion. The One (the Atman, the Self) plays at being the Many, and does so with such skill that it fools itself."

 

Exponents of Eastern views use no construct analogous to arche; and they do not do "formal logic," specifying undefined terms, postulates, rules of inference, etc., the way we Westerners now do. But speaking from the present alternative viewpoint that insists on attributing assumptions, (including tacit ones), we infer that, formally speaking, underlying their constructs of the One and the Many and so fulfilling the role of the Arche, they at least tacitly place Atman or Self.

 

According to the Western view, the arche consists of "some-thing" outside of the Self. In keeping with what we say about the structure of WIE languages, the Western view projects the noun-verb structure of WIE languages onto the Cosmos: it contrasts the Self with the Other, regarding the Self as verb-like -- somehow transient, evanescent, non-specifiable -- and regarding the Other as noun-like -- persisting, really existing, specifiable, e.g. as "objective reality," or in other words, the arche.

 

Pythagoras of Samos (582-c. 500 BC), who founded science (as we understand the word today), nominated the construct of number as fulfilling the role of the arche. As Koestler (1959) elucidates the Pythagorean view,

 

... the Pythagorean tenets [include the dictum] that "philosophy is the highest music", and that the highest form of philosophy is concerned with nubers: for ultimately "all things are numbers". The meaning of this oft-quoted saying may perhaps be paraphrased thus: "all things have form, all things ARE form; and all forms can be defined by numbers." (Koestler, p. 30)

 

Modern scientific answers to the question of the nature of the arche, starting with the neo-atomic theory of John Dalton (1766-1844) and proceeding down to the contemporary, mathematically-defined constructs of modern physics, have built upon the Pythagorean version of the Western view.

 

So far as I know, no one previously has succeeded in reconciling the generically Eastern and Western views. Nor has anyone found grounds, logical or empirical, for showing that one of these views has any discernable advantage over the other.

 

The present non-standard frame of reference changes that. Perhaps the following mathematical analogy will suffice to show the main relationships here. Let us start with the dictum of Perls, Hefferline & Goodman (1951), and our development on it:

 

We speak of the organism contacting the environment, but it is the contact that is the simplest and first reality. (PH&G)

 

From such "contact" (transacting), our logicking INFERS constructs such as "I" and "you," "I" and "it," "organism" and "environment," etc.

 

Or otherwise stated, this dictum and our elaboration on it gives a new viewpoint (World-View), a new version of the arche -- one which includes both the Eastern and the Western versions as special cases.

 

To show this, we shall a) develop an analogy which uses the familiar WIE mathematical theory of sets to represent our novel version of the arche, and b) show how to "reduce" our version of the arche so as to yield a representation first of the Western version of the arche, and then of the Eastern version.

 

In Western languages such as English or the mathematical theory of sets, we have no way directly to represent the setting of transacting, from which to infer "you" and "it," "organism" and "environment," etc.

 

But to approximate this setting, we can utilize the construct of Cartesian product space (a generalization of Cartesian coordinates (graph)), and the associated construct of identity mapping. I can indicate a Cartesian product space by the notation X X Y, where the ordered pair (x,y) belongs to X X Y ; whereas the ordered pair (y,x) does not. (However, (y,x) does belong to Y X X .)

 

An identity mapping, then, constitutes a mathematical trick by which to eliminate one of the axes (sets) from X X Y , yielding either the Cartesian product space X X X or Y X Y (depending on which way you do the trick).

 

To approximate the required setting of transacting, as required for the present frame of reference, we posit a Cartesian product space O X E , where we expressly insist that the sets O ("organism") and E ("environment") do not "pre-exist," as they would in the standard Western frames of reference; but rather, these constructs (sets) get inferred from the "referent," the non-verbal experiencing, for which O X E stands. In this approximation, every point within the domain of discourse in question consists of an ordered pair (o,e), which we can take as an approximately adequate representation of an example of contacting or transacting.

 

To obtain a representation of the generically Western frame of reference, then, I perform an identity mapping on the setting O X E so as to eliminate the axis representing "the Self," namely, O . That leaves me with the Cartesian product space E X E , which stands as a fair metaphor for "objective reality," as posited in the Western frame of reference.

 

To obtain a representation of the generically Eastern frame of reference, I perform the converse identity mapping on the setting O X E so as to eliminate the axis representing "the Other" (or "the Many"), namely, E . That leaves us with the Cartesian product space O X O , which stands as a fair metaphor for "the Self" or "Atman," as posited in the Eastern frame of reference.

 

QED: The (analogical) novel version of the arche includes both the Western and Eastern (analogical) versions as special cases.

 

TAKING THE OBSERVER INTO

 

What does one have to assume in order to take the observer into account, as compared to what one has to assume to eliminate him from consideration?

 

The authors know only one route to taking the observer into account. First, one must assume that humans assume, that they cannot not-assume. Then what a human DOES follows from what he ASSUMES -- in roughly the way that, in a formal deductive theory such as Euclidean geometry, a theorem follows from the premises of the system. And second, one must assume that a human can change what he/she assumes only by assuming something else instead; he can NOT assume nothing-at-all.

 

The constructs of assuming and of changing what one assumes hinge on an analogy which compares the construct of living to the process of map-making: To say that an organism lives means that it generates maps of that territory composed of what goes on in and around the organism, and then it guides its doings and choosings by these maps.

 

When one uses a map to guide one's behavior, one can in principle neglect the distinction between map and what the map refers to. We refer to this form of indiscriminateness as postulating map-territory identity. This neglect manifests itself in treating the map as if it yielded some kind of "absolute certainty," and in holding oneself unwilling even to consider questioning, testing or revising it. At the other extreme, one can remain conscious of the distinction between map and what the map refers to. We refer to this form of discriminating as postulating map-territory non-identity. This discriminating manifests itself in remembering to treat the map as more or less tentative or approximate, as incomplete, and as created from one's own point of view for one's own purposes; in remaining willing to test it for accuracy; and in holding oneself in readiness to revise it at need.

 

Using this terminology, we can make explicit what one has to assume in order to take the observer into account -- namely, on the specific setting of transacting, the three postulates set forth by Korzybski:

 

Non-identity: Presume that the map IS NOT the territory for which it stands.

("The word is not the thing it stands for.")

 

Non-allness: Presume that no map includes representations of ALL the characteristics of the territory.

 

Self-reflexiveness: Presume that no map exists free of some kind of representation of the map-maker.

 

The cautionary principles expressed by postulating map-territory non-identity and non-allness underlie the scientific method and form the basis for its power. Remember, the scientific method can accomplish one and only one thing: To provide the basis for selcting between guesses. In a fully specific setting (e.g., with reference to such and such kind of happenings, as tested by these specific methods, as judged by this criterion), it can either show one's hypotheses, assumptions or other guesses as in error; or else, THIS TIME, find nothing wrong with them. Whenever one violates the tenets of non-identity and non-allness, one thereby allows the possibility of starting from already-discovered error, and thereby predictably reduces the predictability of one's guesses (maps).

The cautionary principle expressed by postulating self-reflexiveness underlies taking the observer into account. To violate its tenets also predictably reduces the predictability of one's maps.

 

It becomes clearer that these postulates underlie taking the observer into account when we notice what it takes to eliminate the observer from consideration. For to do that (or, equivalently, to eliminate the construct of self-reflexiveness from consideration), on an undelimited or an abstract setting, one has to assume the converse of these three postulates -- although no one would state these counter-premises explicitly, nor even willingly admit to holding them:

 

Tacit Identity: (One may TAKE B as if it WERE A; one need not distinguish between map and territory.)

 

Allness: (One may TREAT one's map as if every point of the map represented one and only one point of the territory, and no point of the territory went un-represented.)

 

Linearity: (One may TREAT one's map as entirely and absolutely objective, with no taint of reflexiveness, no trace of contamination by the map-maker.)

 

If, indeed, one's map should qualify as identical with the territory which it represent, that would make it perfect, a matter of absolute certainty. Possessed of such a perfect instrument, one could dispel any misgivings about the territory merely by consulting one's map, and need not bother to survey the territory itself.

Also, it would qualify as completely "objective" (or metaphysically linear) -- it would have no room, and one would have no need, for any kind of representation of or contamination by the map-maker (observer) -- e.g. oneself.

 

Thus to assume (explicitly or tacitly) that one's map qualifies as identical with its territory profoundly violates the scientific method. It also suffices to elliminate from one's consideration the construct of the observer. In the same breath, it also eliminates the process of observing, and happenings to observe. That leaves one knowingly or unknowingly holding the attitude that one "just knows what really happened" -- the unsupportable claim to absolute certainty.

 

Conversely, we may take the finding that a map fails to take the observer into consideration as indicating that it includes among its premises the postulate of tacit identity.

 

RECONSTRUCTING THE OLD PATTERNS FROM WITHIN THE NEW

 

Above, I claimed that the new symbolic pattern which we have devised shows the advantage of generality -- it includes the older symbolic patterns, e.g., those of the WIE languages, as special cases.

 

To test that claim, I propose to derive several structural features common to the WIE languages from the analogous structural features of our symbolic pattern. In order to do that, I display a restricted and restrictive assumption, disallowed in our pattern; and I shall repeatedly introduce this restrictive assumption into these general features of our theory, and show that this then "reduces" them down to the analogous, restricted features of the WIE pattern.

 

A. Self-identity as the "degenerate" case of map-territory

non-identity.

 

The argument I shall present here does not depend solely on reasoning in discursive English. Instead, I have followed the lines of a "proof" or demonstration done in the new notation, which shows that the present theory qualifies as logically more general than do its WIE rivals.@

 

 

@(Hilgartner (1978), "The Method in the Madness of Western Man." Communication 3:143-242, pp. 223-232.)

##

 

In our theory, our logicking infers an organissing who generates a representing of his environing, and he then uses his representing to guide his further "doings" or "choosings." Our logicking takes it as axiomatic that the representing ("map") differs in fundamental ways from the environing ("territory") it represents. (E.g., the representing contains self-reflexive aspects, which represent the organissing rather than his environing; it occupies a higher positioning in an ordering on abstracting than does the environing; etc.)

 

In order to arrive at a construct corresponding to the WIE construct of self-identical, mobilize the standard WIE trick of failing to distinguish between distinguishably different "doings" or "happenings." In other words,

 

1. Presume that our "map" x DOES qualify as identical with the "territory" y it represents.

 

2. Then in effect, ask yourself: what sense does it make to give them different names? Instead, call them both x , and treat them as entirely interchangeable -- treat them as entirely "the same thing,"

 

x is identical with x .

 

3. Manage to forget about having ever distinguished them.

 

Under such circumstances, then, it seems perfectly appropriate to consider some expression which originally contained a distinguishable term for each -- "map" or x or representing, "territory" or y or environment -- as "REALLY" a locution about " x and itself." Replace each distinguishing term x or y with a designation for "x," and connect these symbols for "x" and "itself" with =_ , the sign for "absolute sameness in all respects or negation of difference."

 

x =_ x .

 

QED: Starting with a statement of the postulate of Non-identity, we introduce a restrictive assuming and arrive at a statement of the Law of Identity (or else of the modern Logical Axiom of Identity).

 

B. NO setting (an undelimited field)

 

In our theory, our logicking carefully spells out a specific, delimited setting on which to define any term A and its (carefully restricted) complement Not-A .

 

To arrive at the situation of positing NO setting, again mobilize the trick of failing to distinguish between distinguishably different "doings" or "happenings." This time, aim to eliminate the distinction between the complement of our given term (e.g., not-A ) and the setting on which we define term and complement. Formally speaking, one can accomplish this by non-verbally TREATING the two as identical.

 

1. Treat the setting D as identical with the complement (not-A) of the term A .

 

QED: The trick suffices: The setting disappears, leaving an undelimited field, in which the complement not-A comes to occupy the role of "everything else," defined on an undelimited field.

 

C. The general pattern of dualism

 

A dualism results from performing the act of sorting "things" (designated by nouns or noun-phrases) on an undelimited field.

 

One SORTS "Everything" into "A" and "Not-A". One then designates "A" by a self-identical noun-form or noun-phrase, regarding it as somehow concrete or material or real; and one designates "Not-A" by another noun-form or noun-phrase, regarding it as somehow verb-like: transient, un-specifiable or irrelevant.

 

As one example of a dualism, consider the one currently designated by terms like mind vs. matter. (In classical philosophy, workers would more likely use the term soul than mind, but this does not change the issues.)

 

This dualism underlies the classical division between what we now call physics and psychology.

 

Kepler, Galileo and Newton and other early workers in physics do not question this dualistic framework. Instead, they solidly ally themselves with "the unspecifiable, or irrelevant" side of the dualism, assign their subject-matter (Matter) to the "specifiable" side, and study that.

 

In the twentieth century, the dualism in effect turned around and bit the physicists: Einstein found it necessary to "take the observer into account," and had some successes in doing so. Moreover, he advised his fellow humans likewise to take the observer into account.@

 

 

@Our own study of Einstein's results show that he succeeded in taking the observer into account within a limited domain of the practice of physics, namely, that domain covered in English by terms such as before vs. after and here vs. there, -- e.g., with respect to the process by which a physicist observes two lightning flashes, A and B, and finds that B occurred "after" A, and reconciles his findings with those of another physicist, standing "there" instead of "here," who observed flash B as having occurred "before" flash A.

 

As another illustration, our findings show that Einstein's results account for the processes by which one physicist transmits his findings to another physicist -- but do not account for other processes integral to the practice of physics, e.g., the sequential processes by which a physicist a) actually handles his instruments and makes his observations, b) translates them into findings, c) writes them up in a paper, d) sees the paper through the publication process, and e) handles the changed relationships with his peers which result from publication. Nor do his results account for the process by which one physicist takes in someone else's results, compares them with his own, and forms some generalizations from the results of the comparison.

##

 

By the time he got to the general theory of relativity, Einstein had come to show more interest in the structure of the theory than in further implications of his own injunction. And Einstein's successors have managed to re-interpret the old man's injunction so as to re-eliminate anything having to do with people. Instead of dealing with "a fellow-physicist's frame of reference", they have come to speak of "an inertial frame of reference," and to consider it "physical" rather than "psychological" ("Self").

 

"Observer," I understand, has a specialized meaning in quantum theory also -- again, one which has nothing to do with people.

The classical (nineteenth-century) psychologists did stick to their side of the dualistic bargain, and discussed "the soul" in the abstract, in proper, unspecifiable philosophical terminology.

 

Later, when psychologists started trying to deal with their field in scientific terms, they emulated the theoretical successes of the physicists, and started treating their discipline like a peculiar branch of physics. They too ally themselves with "the unspecifiable, or irrelevant" side of the dualism, and treat their subject-matter (the Greek stem psyche designates "the breath") as real, concrete, and material, and study that. But something of the older view still permeates the discipline -- although they can account for certain aspects of the field in physics-based theoretical constructs in psychology, most psychologists most of the time still regard human behavior or human psychology as unspecifiable, at least in terms of what we know today. (cf. Stent, (1975)@) But just give them another century of study, and -- !

 

 

@Stent (1975?)

##

 

QED: WIE scientific disciplines (psychology, physics) appear to follow the old dualistic patterns. Since the revolution in physics eighty years ago, this has become increasingly clumsy, but no fundamental revision of this pattern has yet gotten generally accepted.

7. SUMMATION

 

[RETURN TO THE TOPIC OF THE NEW NOTATION]

 

This has spelled out SOME of the background for the alternative formalized language.

 

 

REFERENCES

 

C. A. Hilgartner, "Some Traditional Assumings Underlying Western Indo-European Languages: Unstated, Unexamined, and Untenable." General Semantics Bulletin Nos. 44/45, 1977-78, pp. 132-153.

 

C. A. Hilgartner (1978), "The Method in the Madness of Western Man." Communication 3:143-242, pp. 223-232.

 

Ernest Nagel & James R. Newman (1958). Go"del's Proof. New York: New York University Press, 1958.

 

Paul J. Cohen & Reuben Hersh (1967). "Non-Cantorian Set Theory." Scientific American 217:104-116, December 1967, p. 105. These authors state Russell's Paradox as follows:

 

"There are two kinds of sets. First there are those, such as "the set of all objects describable in exactly 11 English words," having the peculiar property that they themselves satisfy their defining property; in other words, sets that contain themselves as elements. We call them R sets, the R standing for Russell. Then there are all other sets -- sets that do not belong to themselves. Call them the non-R sets. Now, said Russell, consider the collection of all non-R sets. (The word "collection" is introduced here simply as a convenient synonym for "set.") Call this set M . Then M is either an R set or a non-R set. But if M is a non-R set, then it belongs to M , by definition of M , so that it is an R set, by definition of R sets. This is a contradiction. On the other hand, if M is an R set, then by definition of M it does not belong to M . It does not belong to itself, that is, it is not an R set, which is again a contradiction."

 

Guellen (1983) points out that over the next thirty years, two main interpretations arose concerning the difficulties posed by Russell's paradox.

 

The Logicists, led by Russell himself, would have obviated Russell's paradox by modifying the rules for class membership. Specifically, they wanted to disallow a priori the possibility that a class could be a member of itself. To accompish this, they proposed that the principles of Aristotelian logic be supplemented hereafter vby what they called the "vicious circle principle," which states, "Whatever involves all of a collection must not be one of the collection." By the enforcement of this principle, the question of whether Russell's hypothetical [non-R] class is or is not a member of itself would be settled by fiat, and the vicious circle of Russell's paradox would thereby be avoided.

In contrast to the Logicists, the Formalists believed that the shortcomings revealed by the paradox were not in logic itself but in the semantic content of the language used to express logic. In particular, they traced the origin of many paradeoxes, including Russell's, to the ambiguous meaning of the world "all." Statements such as "All rules have exceptions" are either innocuous or paradoxical, depending on whether we interpret the word "all" to include or exclude the statement of which it is a part. Uncertainties such as these are semantic rather than logical, the Formalists maintained, and could be expurgated simply by bleaching logic of its semantic coloration. Thus, they set about re-expressing the logical arguments in mathematics in terms of strictly defined symbols that had no real meaniong, rather than in terms of words.

 

[Reflexivity argument, from our point of view.]

 

 

Whorf (1956)

 

CAH (Fundamental Question)

 

Korzybski (1921)

 

Guellen (1983) "The Search for Absolute Certainty"

 

The Laws of Thought

 

Mates (1965)

 

Tarski (1965)

 

Halmos (1960)

 

Perls, Hefferline & Goodman (1951)

 

Stent (1975)