|HOW GENERAL SEMANTICS CAN RESCUE BIOLOGY FROM ITSELF|
A Biology with Biologists In It
C. A. Hilgartner
In this paper, I intend to describe briefly a major flaw that has plagued biology for at least the past half century, and to show how progress in general semantics has begun the process of removing the fundamental logical-and-empirical error that underlies that flaw.
The flaw consists of the by now nearly universal consensus among biologists that they cannot define key terms such as life or living at all, with the unstated corollaries that no one else can now or ever will manage to do so. Back in the 1930’s and 1940’s, when the leading biologists of the day proposed rejecting these key terms, they appeared unaware of the way any theory-based discipline depends upon the ability to draw a boundary around the domain of discourse. Any scientific discipline includes a theory or theories which purport(s) to model the "doings" or "happenings" that take place within this domain, and a boundary which excludes those "doings" or "happenings" which occur within neighboring but incompatible domains. When the biologists rejected their key delimiting terms, they thereby rejected the possibility of drawing such boundaries. That not only eliminates the ability to exclude inappropriate topics from consideration – it also eliminates the possibility of testing, and perhaps disconfirming and rejecting, one’s own suppositions; for without a boundary, you have no delimited domain to reject them from. Thus by entering into this consensus, biologists destroyed –or rather, further delayed – biology as a science.
Starting about 165 years ago, biologists began the process of rejecting teleological explanations (in my opinion, appropriately). They replaced these with mechanistic ones. As a tactic for bringing experimental method to bear within biology, this substitution has turned out wildly successful. But as a method of study within the domain of biology, the program of the mechanistic/reductionist biologists unavoidably fragments biology. It focuses on bits and pieces of organisms (rather than on organisms-as-wholes-in-their-environments); and as with Humpty Dumpty, it cannot put these pieces back together again. After a century of research based on grinding up or otherwise fragmenting living organisms, biologists came to reject not only the teleological way of explaining biological observations, but also some of the key observations that teleology had purported to explain. The language and tactics they adopted from physics actively disallow the topic of apparently-purposive "doings" or "happenings," and so fail to account for important aspects of the domain it purports to account for. In the process, more or less unawarely, they destroyed biology as a theory-based discipline.
Ever adaptable, biologists became sub-specialist chemists and physicists. In their newly devised fields, they encountered so-called "borderline cases" – filterable viruses, blood platelets, and other systems, which function in ways that seemed odd, escaping or violating the older categories and generalizations. In the 1930’s and 1940’s, after a century of remarkable successes in their empirical investigations and perhaps half a century of study of various unsettling "borderline cases", biologists rejected the central defining terms of their discipline: life and living.
Exponents of general semantics can, as I will show below, offer new explanations, with new underpinnings, to replace both older views. Once solidly grounded, biology and biologists may do more than they can today to improve their discipline, and the prospects for the survival of our grandchildren.
(See Figure 0)
Alfred Korzybski (1879-1950), on the page facing the title page of the first edition of ^ , lists some "Volumes in Preparation," and also some other titles for which, he promises, "The Names of the Authors … To Be Announced Later." On that page alone, Korzybski outlines an ambitious program for his new discipline of general semantics.
I and several associates have, over the past thirty-plus years, generated an alternative frame of reference which synthesizes Korzybski’s non-aristotelian premises, other aspects of general semantics, and contributions from a number of other workers, including the anatomist and cyberneticist Gerd Sommerhoff. In a number of domains, including that of biology, we have considerably advanced Korzybski’s program.
In this frame of reference, I treat theories and theorizing as forms of human behavior –human behaving-and-experiencing – and I treat human behaving-and-experiencing as intrinsically transactional. Thus in this frame of reference, any theory, e.g. a theory of biology, must describe and deal with some kind of interchange which produces fundamental, irreversible alterations. To represent this peculiar requirement in a Western Indo-European (WIE) language such as English or the mathematical theory of sets, I invoke some pair of related terms, e.g. organism and environment, and treat them as a mutually-defining polar term-pair, where the transacting in question occurs as some kind of two-way interchange – "contacting" or "abstracting" – between the members of this term-pair. As Perls, Hefferline & Goodman put it,
We speak of the organism contacting the environment, but it is the contact[ing] that is the first and simplest reality. (Perls, Hefferline & Goodman, 1951, p. 227)
Thus I treat the construct of the environment as designating the other side of the organism’s skin, and treat the organism as designating the other side of the environment’s skin.
Transacting affects both participants, both poles. This kind of setting allows no "unmoved movers."
Within the available space of this paper, I cannot state my presuppositions in notation. But I can give relevant citations. Furthermore, for the frame of reference I use here, I can, and do, verbally summarize the setting, the undefined terms, a cluster of inter-defined terms, and the postulates that underlie it.
Notation: For a summary of the conventions of set theory notation which I use here, see Ashby (1962). A briefer, more sharply selected account appears in Hilgartner & Randolph (1969a), pp. 325-336.
Setting: As various twentieth century workers have pointed out, whenever we humans produce, or otherwise attend to, a locution – spoken, written, signed, etc. – we treat it as if it "occurs" or "exists" within some specified or specifiable set of circumstances. In English, we designate such circumstances as the context or setting for the locution. For my alternative frame of reference, I posit a definite and distinctive setting, which I designate as specific and delimited. I can express this setting by means of a run-on phrase such as "One particular organism-as-a-whole-dealing-with-its-environment-at-a-date, as viewed by a designated observer." To express the role of the designated observer, I hold that our observer observes the observed, and writes down, mainly in notation, expressions intended to describe what she has observed, to infer about it, etc., in accord with the presuppositions of the language she chooses to use. Any time you, as a reader of this document, see a notational expression, I invite and request you to take it initially as indicating "what our designated observer says (writes) to express what she observes." Furthermore, out of sheer necessity, I appropriate certain grammatical forms for my own purposes: To point to this designated observer, I utilize feminine pronouns (she, her, hers, etc.), without implying "female", and, to designate our organism, utilize masculine pronouns (he, him, his, etc.), without implying "male" (as I just did).
^ – In general: Workers at the end of the nineteenth century (e.g. David Hilbert (1862-1943), 1898-9) proposed that the premises of a formalized axiomatic system must include some terms for which we offer no verbal definitions. These workers had noticed that the process of defining amounts to replacing a more or less unfamiliar term with a somewhat more familiar one – and if you continue the process, defining some term A and then in turn defining each of the defining terms B, C, D,…, the process sooner or later becomes circular: the person producing the string of defined terms eventually comes to repeat her/himself, using some term N to define a term M and then using that term M to define the term N. Besides, at the beginning of the process of framing a formalized axiomatic system, one has no "already-familiar" terms to work with. Hence, the process of providing rigorous definitions for one’s terms eventually turns into the process of explaining what one has left undefined.
As Korzybski (1933, p. 154) points out, undefined terms represent
"blind creeds which cannot be elucidated further at a given moment…."
I maintain that the undefined terms provide a bridge between those verbal constructs that make up the theory and those non-verbal "doings" or "happenings" which the theory purports to describe or model.
The process of mastering an axiomatic system starts, then, with the process of learning how to USE the undefined terms.
^ – Specific: As her undefined terms, our designated observer follows the lead of Korzybsky, and chooses structure, order, and relation. To remind her readers that she does not subscribe to the traditional presuppositions of WIE languages, she treats these not as noun-forms but as verb-forms: To structure, to order, to relation.
Term-cluster: Within this specific delimited setting of transacting, our specified observer needs vocabulary items with which to describe the "doings" or "happenings" which she observes so she can state her chosen premises. To that end, she lists a grouping of five inter-dependent (or mutually-defining) terms: the polar "term-n-tuple," or term-cluster, which consists of
Abstraction (map) and
An ordering on abstracting (or the notion of "logical levels").
Postulates in notation: The earliest set theoretic statement of the premises (undefined terms and non-aristotelian postulates) of this frame of reference appears in Hilgartner & Randolph (1969a, pp. 295-7). Also, please examine the step-diagram in Figure 11 and my interpretation for it (Appendix).
Postulates in words:
Presume that no structuring, ordering or relationing stands as identical with any structuring, ordering or relationing (including itself).
"Our designated observer holds that the map IS NOT the territory it stands for."
Presume that no structuring, ordering or relationing can represent all the aspects of any structuring, ordering or relationing.
"Our designated observer holds that no map includes a representation of ALL aspects of the territory."
Presume that no structure, order or relation exists free of aspects which refer to itself and/or to the organism which elaborates it.
"Our designated observer holds that any map includes a representation of the map-maker."
(see Figure 1)
Let me start by characterizing the current condition of biological theory, as I see it.
My desk dictionary defines the term biology as follows:
biology n. 1. The science of life and life processes, including the study of structure, functioning, growth, origin, evolution and distribution of living organisms.
However, today’s biologists assert that terms such as life or living lack rigor, so that no one can offer satisfactory definitions for them. What difference in "the study of … living organisms" would an exponent of general semantics find, when the practitioners abandon the key terms used to define – set the boundaries of – their discipline?
This brings up a major issue of theory: I do believe that vitalism merited rejection and replacement. But so does mechanism/reductionism. Starting in about 1830, increasing numbers of biologists adopted mechanist/reductionist dogma to replace teleology. In so doing, they declared organisms "nothing but" mechanisms, each in effect "equal to the sum of its parts" and so fully describable from within the terminology and linguistic habits of nineteenth-century physicists and chemists. They thereby threw out the baby with the bath-water: They rejected not only the unsupportable dogma of teleology and vitalism, but also threw out the OBSERVATIONS which earlier biologists had EXPLAINED by means of that dogma. For logically speaking, teleological/vitalist explanations invoke a hierarchical structure, in which, for example, what happens on one "level of organization" makes possible what happens on another, higher "level". As an explanatory principle, of course, such views cannot further account for what they invoke. In contrast, logically speaking, a mechanism or collection of mechanisms remains "flat" – it has no way of generating a hierarchy of "logical levels", nor logical structures which can occupy these various "logical levels" (such as those invoked by teleological/vitalist dogma). In other words, the dogma of mechanist/reductionist biologists disallows multiordinal relations.
My approach to biological theory, which I frame in mathematical languages of known structure (such as the mathematical theory of sets), can and does define these terms, in a manner that makes them consistent with my chosen premises. Furthermore, the theory-based line which it draws between the constructs of living and non-living runs about where most people (including biologists) would want it to run.
(See Figure 2)
In Figures 2 and 3, I spell out my view of how we humans generate new knowledge, such as the findings about filterable viruses and other "borderline cases" that so excited and troubled biologists, and cast such doubt on the generalizations and terminology of biological theory.
In Figures 4, 5 and 6, I outline some of the presuppositions which I find encoded in the grammar of WIE languages, which have gotten in the way of generating logically-and-empirically satisfactory theory in biology.
Readers will, I expect, find that these figures express their points without need for further explanation.
(See Figure 3)
WESTERN INDO-EUROPEAN (WIE) EXPLAINING
(See Figure 4)
DUALISM as UNAWARE PROJECTING
(See Figure 5)
(See Figure 6)
COLLAPSE OF TERMINOLOGY
(See Figure 7)
By the 1940’s, biologists had concluded that the definitions for terms such as life or living had collapsed. For example, in 1937, the Cambridge University biochemist N. W. Pirie, published a paper entitled "The Meaninglessness of the Terms Life and Living." Eleven years later, the Nobel Prize laureate virologist Wendell Stanley (the first person to crystallize a virus) justified this stance as follows:
With the realization that there is no definite boundary between the living and the non-living, it becomes possible to blend the atomic theory, the germ theory, and the cell theory into a unified philosophy, the essence of which is structure or architecture. The chemical, biological and physical properties of matter, whether atoms, molecules, germs or cells are directly dependent upon the chemical structure of matter, and the results of the work with viruses have permitted the conclusion that this structure is fundamentally the same regardless of its occurrence. (Stanley, 1948; emphasis mine)
Thus, Stanley jubilantly proclaims the ultimate success of the mechanist /reductionist program.
(See Figure 8)
We humans have had half a century to get used to that conclusion, which has become widely accepted.
But please look again at what (as I see it) Stanley asserts:
"With the realization that there is [within the non-verbal territory] no definite [verbal-level construct of a] boundary between [the verbal-level constructs of] the living and the non-living,…
I regard the constructs of the living, the non-living, and the boundary between them as TERMS, belonging to the domain of map – and further, constructs central to how we define the term biology.
Stanley, however, expresses himself as if he expects to find these verbal-level TERMS within the non-verbal TERRITORY that biological science purports to describe. He declares the territory DEFICIENT because he can’t find the map already pre-formed within it.
(See Figure 9)
This stands as a fundamental error – a logical-and-empirical error, which contaminates (poisons) many of the broad conclusions drawn from otherwise useful findings.
Logically speaking, mathematicians (e.g. Frege) criticize this kind of logical maneuver as confusing name with thing named. Logicians (e.g. Turbayne) call this a sort-crossing error. General semanticists (e.g. Korzybski) call this confusing levels of abstraction. I call it map-territory identity. We concur in calling it A MISTAKE.
Scientifically speaking, Stanley holds his own map (mechanist/reductionist view of biology) as "True", and declares the territory DEFICIENT because it "fails" to match his map.
Stanley’s dictum offers an excuse: His conclusion PARDONS himself and his colleagues for failing to provide theory which accounts for what biologists set out to account for.
Further, it contains an implied dogma: Since (in his view) the territory ‘IS’ deficient, he implies that ‘Nobody else will EVER manage to offer logically and empirically satisfactory definitions for terms such as life or living.’
(See Figure 10)
I regard Stanley’s quoted conclusion as INCOMPETENT – logically and scientifically – and I REJECT IT.
I also regard it as HAZARDOUS TO LIFE ON EARTH.
In my opinion, a scientist may NOT allow her/himself to explain any mismatch between theory (map) and observations (territory, or evidence for a territory) by declaring the TERRITORY deficient.
Nevertheless, the conclusion which Stanley expresses got widely accepted. After about ten years, it passed out of the realm of discussion and into that of dogma, becoming part of the non-verbal practice of what now passes for "biology". Most of today’s biologists unquestioningly, even unawarely, regard the topic of definitions for terms like life or living as a dead issue. Today’s biologists cannot draw a theory-based line between the living and the non-living. Hence their "science" has nothing to say about living organisms, about biology.
(See Figure 11)
I observe a particular situation, which includes a North American lawn by daylight, with air containing about 20% oxygen at roughly one atmosphere of pressure, moist soil, grass, a population of earthworms (^ ), and a male robin (Turdus migratorius). Robins belong to the order of passerine birds, whose primary gait for non-airborne locomotion consists of hopping. However, a hunting robin often moves by placing one foot before the other – he strides or stalks, without much head-bobbing. Here-now, our robin stalks about through the grass; he freezes, turns his head this way and that, and then very suddenly thrusts his beak into the ground (into a worm-hole). After a vigorous display of tugging behavior, the bird straightens up with an earthworm grasped in his beak; he shakes it vigorously, mashes it with his beak and otherwise abuses it, and then swallows it.
Here I regard the robin as fulfilling the role of the organism which I observe, and the other aspects of this situation as making up the robin’s environment. That uses two members of my ring of five terms.
Let me focus particularly on the moment, very shortly before he pecks, when our robin-organism cocks his head as if "listening". (I have heard that robins find their prey mainly by auditory cues.) I take this postural display as the outward and visible sign that the robin has engaged in abstracting so as to generate an abstraction – which I could express in English as the assertion, "I HAVE LOCATED A WORM". That uses two more of my ring of five terms.
By acting on his abstraction – thrusting his (partly-opened) beak into the ground – our robin-organism treats his abstraction as a guess or behavioral hypothesis, and puts it to test. Then when he captures a worm, he must render it unable to crawl back out of his crop and escape, and must swallow it, or else he will derive no biological advantage from his capture. If he had come up with no worm, he would have to discard that behavioral hypothesis, and set about to generate another – which means, resume stalking earthworms.
By using these terms to describe what I observe, I indicate that I attribute to the robin (or to any other living organism) the ability (non-verbally) to abstract (to generate a ‘map’ of "what goes on in and around this organism"), and the ability (non-verbally) to distinguish between (non-verbal) map and (non-verbal) territory. I can state this point in more abstract terms as (i) the ability to generate a map or abstraction, and (ii) the ability to generate an ordering on abstracting. That, then, completes the circle of five terms I started with.
In analyzing any other detail of the performance of our robin-organism transacting with his environment, I would again use all five members of this polar term-cluster. Then in order to present my proposals for biology in a comprehensive fashion, still within the set-theoretic approximation to the non-standard frame of reference, I would have to treat the other topics listed in Figure 11 in a similarly detailed and loving fashion.
STEP-DIAGRAM – Displaying the Logic of Opposites
The step-diagram in Figure 11 gives a graphical representation of these "doings" or "happenings". I discuss it in detail in the Appendix.
(See Figure 12)
In 1950, the British anatomist and cyberneticist Gerd Sommerhoff performed a logical analysis of the traditional biological construct of the apparently-purposive activities of living organisms. He generated a mathematically-defined model of it, which he calls directive correlation. He used it to analyze the major teleological terms, and to provide logically and empirically satisfactory definitions for them.
In 1962, W. Ross Ashby provided a summary of a Bourbaki algebraic set theory notation, and translated Sommerhoff’s construct into that notation. Further, he provided a theorem – an element-free expression – which generalizes Sommerhoff’s definition. In 1965, using Ashby’s notation, I incorporated Sommerhoff’s construct into my developing theoretical system. In order to do so, I had to reinterpret it in light of Korzybski’s premises in general, with special attention to the Postulate of Non-identity.
Let me return to the example of the robin and worm, discussed above – this predator/prey encounter which I observed. To describe this biological occurrence in terms of the modified construct of directively correlated, I must partition my picture of these "doings" or "happenings", and assign the various aspects of this setting to the relevant parts of the construct. (I began that process when I designated robin as organism and worm, etc., as environment.) Now, considering the various aspects separately and/or together in accord with this interpreted model, I must show what they do.
In order to do this, I'll resort to a mathematical vocabulary, but in a way that will allow persons with little mathematical background to follow the argument. In particular, I'll need two main mathematical constructs: variable and function.
a) The term variable refers to a kind of mathematical "blank check," to which you can assign various values.
b) The term function designates a kind of "logical machine" which, when you give it one value, returns you another value, according to some "rule."
Then to spell out the construct of directively correlated, I require at least two variables and three functions.
Coenetic variables (CV) (Sommerhoff, 1950. Coenetic from Greek, coenos, "They vary together."): A grouping of initial conditions which, at the beginning of the ‘purposive’ sequence’, affect both organism and environment (as the designated observer sees it). I can express this construct in terms of a disturbance d , where d CV , or in words, the disturbance d belongs to (stands as an element of) the set CV .
In this instance, these coenetic variables resemble a Gestalt in that they include background factors (the lawn-worm-robin complex), and foreground factors (for the robin: whatever interoceptive and/or sensory cues that led the robin to start hunting ["hungry", "stalking"], and whatever sensory cues that led him to peck into the ground at that spot ["I detect a worm"]; for the robin’s environment: harder to specify but still in principle specifiable).
^ (FC), the criterion for a favorable outcome, from the point of view of the organism (as the specified observer sees it).
In this instance, the specified observer would regard this condition as satisfied if and only if the robin obtains a worm to eat, and eats it.
The construct of focal condition implies-and-assumes a broader construct, Outcomes (Oc), of which FC forms a part (FC Oc , the focal condition forms a subset of outcomes). Let oc signify a particular value of Oc , e.g. "ending up obtaining a worm", or "ending up not-obtaining a worm." Then oc might or might not satisfy this criterion, and so might or might not belong to FC .
^ , which we can agree to express as the function f .
As a set ("collection of the effects of various values of CV (e.g. effects of various sensory cues, etc.) on the organism"), f signifies "what the organism does" – which includes the making of maps (abstractions) and the process of guiding itself by these maps, or in other words, the activities which I call abstracting.
Then f(d) mathematically signifies "What the organism does with the disturbance d " – e.g. the organism generates some particular maps of or abstractions about what in particular goes on in and around itself, at the moment(s) in question, e.g. "hungry", "stalking", and "detecting a worm".
^ which we can agree to express as the function g .
As a set ("collection of the effects of these factors on the environment"), g signifies "what the environment does."
Earthworms form a part of our robin’s environment, so "what the environment does" includes "what earthworms do."
Then g(d) signifies "What the environment does with the disturbance d " – here, it includes what this particular earthworm does. Among other things, when an earthworm in its burrow feels something grab it, it extrudes a set of thick bristles built into its hide into the surrounding burrow, so that it sticks there rather like Velcro.
^ , which we can agree to write as the function .
As a set, signifies "How what the organism does and what the environment does articulate."
Then (g(d),f(d)) signifies "How what the robin does this time meshes with what the environment (including the earthworm) does this time" (so as to arrive at an outcome).
A directively correlated (apparently purposive) sequence, then, over the interval t0 to t2, involves the following "doings" or "happenings" (as our designated observer sees them):
At t0: d lawn-robin-worm-etc. system ...
At t1: f(d) and g(d) robin detects earthworm,..., pecks...
At t2: (g(d),f(d)) = oc FC struggle,..., robin eats earthworm
So far, so good – when presented in this verbal fashion and deployed in some detail, the construct of directively correlated gives a convincing enough accounting for this predator/prey encounter between robin and worm. From this point on, those with scant background in "Let’s Keep Track of What We Say" types of mathematics will have to take my word for it to an increasing degree, while I translate this verbal presentation into a set theoretic definition and three theorems.
(See Figure 13)
The construct of directively correlated (as our designated observer sees it), as defined in a Bourbaki algebraic set theory notation, goes as follows:
Given: Spaces O , E , CV O × E , and Oc O × E , together with an onto function g:CV E and a function :E × O Oc ; then, with FC a subset of Oc, we have
DEFINITION: A function f:CV O qualifies as directively correlated with respect to g , and FC if and only if
d CV : (g(d),f(d)) FC. Sentence (1)
The graph drawn in Figure 13 may help in visualizing this construct.
(See Figure 14)
The main point of describing a situation of interest in a mathematical language lies in the support for further developments which the mathematical language provides.
As presented, Sentence (1) refers to specific situations (as our designated observer sees them), such as a robin-and-worm – predator-and-prey – encounter. As displayed, it shows both sets (e.g. CV , f , g , , etc.) and some of the elements of these sets (e.g. d , f(d) , g(d) , the ordered pair (g(d),f(d)) , etc.).
The tools of set theory make it possible to rearrange a definition like Sentence 1 to bring any one of its constituent terms under scrutiny (in the process, concealing the elements). Here, I scrutinize the term f , "what the organism does (with d )". This entails generalizing Sentence 1 so as to describe not just the structure of specific predator/prey encounters, but rather, to specify the part played by the chosen key component of the construct of directively correlated in relation with the other components.
THEOREM: In the notation of sentence (1), f qualifies as directively correlated with respect to g , , and FC if and only if
f -1(FC) o g . Sentence (2)
PROOF. A proof appears in the Appendix, and also in Hilgartner & Randolph (1969a), p. 336.
Biologically Integrated Relations between
(See Figure 15)
Sommerhoff defines the construct of integrated in the biological sense as follows:
A set of organic activities is integrated in the biological sense if the activities are directively correlated and if these correlations are again directively correlated inter se (e.g., if their respective focal conditions may in turn be regarded as a set of directively correlated variables). (Sommerhoff, 1950, p.195)
I display two systems, P and Q , each of which (as viewed by our designated observer) qualifies as directively correlated in this notation. I define P O × E so that (1) and (2) hold, while Q O × E involves a space J O × E , an onto function n:J E , a function m:J O , and a function :E × O Z , with H Z . Then m qualifies as directively correlated with respect to n , H and if and only if
j J : (n(j),m(j)) H . Sentence (3)
Then by reasoning like that displayed for sentence (2),
m -1(H) o n . Sentence (4)
Simplest case: Let us suppose that the set of ‘favorable outcomes’ (focal conditions) for P consists of the same elements as does the set of ‘disturbances’ (coenetic variables) for Q , or in other words, FC = J .
Then, given a mapping, the projection onto the O axis PjO: E O O , then the relation between P and Q qualifies as ‘integrated’ (a ‘directively correlated’ relation between two ‘directively correlated systems’) if and only if
f -1(m-1 o PjO o -1(H)) o g . Sentence (5)
PROOF: A proof appears in the Appendix; and first got published in ^ (Hilgartner, 1971).
(See Figure 16)
Sommerhoff defines the term living organism (for which I substitute living system, (LS)) in terms of integrated relations between directively correlated systems.
A living organism may be described as a compact physical system of mechanically connected parts whose states and activities are related by an integrated set of directive correlations which, over and above any proximate focal condition, have the continued existence of the system as an ultimate focal condition. Death may be described as the breakdown of these directive correlations. (Sommerhoff (1950), pp. 195-6; cf. also 161 ff.)
Here, besides the multiordinal usages of directively correlated (as our designated observer sees it), Sommerhoff also specifies multiordinal usages of focal condition (and thus, by polar relations, multiordinal usages of coenetic variable as well). In my non-standard frame of reference, I take Sommerhoff’s definition as spelling out the conditions someone must satisfy in order to classify something as living.
In the notation of directively correlated systems and of integrated relations between directively correlated systems, I can express the sense of Sommerhoff’s formulation by using Sentence (5a) to define the set of living systems, (LS). I make this explicit in Sentence 6 below.
GIVEN: a designated observer who regards the following situation as given:
A system which I shall call our (supposed) organism O , which contains a finite set of parts (sub-systems which involve the states or activities of our organism), namely,
P1, P2, P3,..., Pn , for each of which sentences (1) and (2) hold, and where FC1 FC2 FC3 ... FCn Z ; and given that our supposed organism contains another sub-system Q for which sentences (3) and (4) hold and for which J = FC1 × FC2 × FC3 × ... × FCn ; and finally, given that the focal condition H which figures in Q refers to what Sommerhoff calls "the continued existence of the system (as an ultimate focal condition)", which I have rendered as the ultimate focal condition of all organisms ... the preservation-and-growth of the organism (Pr) throughout some finite time-interval (Hilgartner & Randolph, 1969a, p. 312):
Then, in notation.,
O LS [fi -1(m-1 o PjO o -1(Pr)) o gi]i Sentence (6)
Proof. A logical proof of (6) would closely follow the lines of the proof of (5).
That completes the description of the mathematical construct of directively correlated.
(See Figure 17A)
As a test case, I propose to use my frame of reference to reexamine enzymic reactions.
A bit of background: Biologists who study enzymic reactions traditionally start by grinding up living things. In 1897, for the first such reaction ever performed in a laboratory, the Büchner brothers mixed a dollop of yeast with some sand, and ground up the mixture with a mortar and pestle. When, under the microscope, they no longer could see any intact yeast cells, they filtered out the sand, collecting the liquid. Then they put a sample of their liquid extract into some kind of flask or test-tube, poured in some sugar solution, and swirled the mixture. While they watched, bubbles (of carbon dioxide gas) formed. They inferred that in vitro (in their glassware), and in the absence of intact yeast cells, the extract had fermented the sugar – in violation of then-current dogma, to the effect that fermentation can take place only by the agency of the vitalists’ life-force, as embodied in intact cells.
Later workers both greatly improved the methods of obtaining and purifying enzymes (from formerly living cells); and increased their understanding of the chemistry of enzymic reactions (as these occur in vitro).
These workers also made key inferences, to the effect that (a) enzymic reactions also occur in vivo, inside intact cells (which remains an inference, no matter how much supporting evidence we may adduce). They further inferred that (b) these reactions in vivo differ in NO particular from the enzymic reactions as they occur in vitro – a dictum which expresses the mechanist/reductionist dogma that has supplanted vitalist dogma. And as a further inference, these workers (c) arranged the numerous enzymic reactions they discerned within cell extracts into (plausible) sequential metabolic pathways, which allegedly operate in vivo - without mentioning the teleological, or at least apparently-‘purposive’, construct of integrated (in the biological sense). Thus they disallow regarding enzymic reactions in an intact cell or intact organism as evidence for an apparently-‘purposive’ configuration – one which, in a hierarchically-ordered fashion, contributes to the survival of a cell and, perhaps, in turn, of an entire organism.
I reject mechanist/reductionist dogma. Taking the role of designated observer, I propose to treat the enzymic reactions which (we infer) occur in vivo, within intact organisms, as apparently ‘purposive’, and therefore as the kind of "doings" or "happenings" that I can represent by means of the construct of directively correlated. In order to do that, I must follow the procedure outlined in Figure 12: Using specifiable criteria in each step, partition my non-verbal picture of what happened, and assign the resulting "pieces of what happened" to the relevant parts of the construct, etc. In general, I must
(i) assign aspects of "what happened" to key terms of the model, e.g. the polar terms organism and environment,
(ii) specify the coenetic variables CV ("initial conditions which affect both organism and environment") and the focal conditions ("outcomes which appear favorable from the point of view of the organism"),
(iii) run this "interpreted model" and
(iv) compare what I obtain with my initial non-verbal picture of what happened.
As an example, consider one step in the ^ – the one catalyzed by the enzyme phosphohexose isomerase, which reversibly inter-converts between glucose-6-phosphate and fructose-6-phosphate. This reaction changes only the three-dimensional configuration ("shape") of the substrate molecule.
(See Figure 17B)
In vivo, e.g. within a cell, any aqueous enzymic reaction presumably takes place within a small volume of water, which contains molecules of the relevant enzyme E , the substrate ("starting material") S , various inorganic ions I+/I-, various non-ionic substances N , etc.
(I) On this "logical level," I assign the enzyme to the role of organism; and relegate the other parts of the system to the role of environment.
(II) The coenetic variables here include suitable concentrations of the enzyme, [E] , of the substrate [S] , of small-molecular-weight co-factors [CoF] (if the reaction requires any), and appropriate concentrations of the various ionic and non-ionic substances [I+/I-] and [N] ; and suitable values for physical-chemical variables such as temperature (T), acidity (pH) , ionic strength ( ), etc.
(III) The focal conditions include at least two items:
That the reaction proceed in the proper direction (measurable as the requirement that the reaction bring the ratio of product to substrate, [P]/[S], closer to the thermodynamic equilibrium constant for that reaction), and
That the rate of the reaction remain greater than Vl , some lower limit of velocity intrinsic to the cell (e.g. a rate great enough so that this reversible reaction will not form the rate-limiting step of the metabolic pathway(s) in which it plays a part).
As described in the literature of biochemistry, this reaction both can and does achieve these focal conditions. Therefore we may classify it as directively correlated.
At this level of analysis, if one enzymic reaction qualifies as directively correlated, so will any other which we may examine.
This drastic revision of enzyme chemistry brings that biological field of study within the domain of directively correlated in general, and of the three theorems presented above, in particular.
That leads, with full mathematical rigor, to various multiordinal instances of a key conclusion:
Any intra-cellular, in vivo enzymic reaction which satisfies the above focal conditions also satisfies a higher-ordered focal condition, namely, that it has "…the continued existence of the [cell] as an ultimate focal condition." (Sommerhoff, 1950, p. 195) Therefore I find that such a cell satisfies the criteria of Theorems 2 and 3 and I may classify it as living.
The construct of metabolic pathway entails a sequence of intracellular, in vivo enzymic reactions in which the product of one reaction serves as the starting material for the next. In my alternative frame of reference, those enzymic reactions which demonstrably make up a metabolic pathway satisfy the criteria of Theorems 2 and 3 and so qualify as integrated in the biological sense, and therefore once more allow me to classify the cell within which they occur as living (but now on a higher "logical level").
This multiordinal conclusion suffices to show that my alternative frame of reference provides acceptable definitions for terms such as living, on both logical and empirical levels, and so restores to biologists the ability to delimit their own field of inquiry.
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