Foerster H. von (1971) Computing in the Semantic Domain. Annals of the New York Academy of Science 184: 239–241. Available at http://cepa.info/1645

“Environment” appears in three distinct domains: in the domain of the “real world” (W) ; in the domain of “cognitive processes” (C), which provide an organism with an internal representation of his surroundings; and in the domain of an organisms “descriptions” (D) of his world. Environment is the triadic relationship E (W, C, D) between these domains.

From this, three dyadic relations can be derived. RD (W, C), which is determined by an organism’s perceptive potential and gives rise to complementary concepts as, e.g., “niche” and “instinct”, “reality” and “consciousness”; RC (W, D) , which is determined by an organism’s behavioral potential, and gives rise to complementary concepts as, e.g., “territory” and “control”, “objects” and “names”; and RW (C, D), which is determined by an organism’s cognitive potential, and gives rise to complementary concepts as, e.g., “volition” and “action”, “conceptions” and “propositions”.

While these dyadic relationships have only recently been discovered as initial but essential steps in comprehending the holistic concept E, it is only descriptions, D, of single domains in isolation, the “major disciplines,” as “physics” D(W), “psychology” D(C), “linguistics” D(D), and so on, to which the scientific community is accustomed to addressing itself, and for which powerful analytic formalisms have been developed.

Of the many causes that may have led to the contemplation of these domains in isolation, two appear to have exerted a major influence in this development. One is that in dyadic relations of the kind described above the observer always appears in his descriptions – i.e., is implicitly or explicitly making statements about himself.

However, “self-reference” in scientific discourse was always thought to be illegitimate, for it was generally believed that The Scientific Method rests on “objective” statements that are supposedly observer-independent, as if it were impossible to cope scientifically with self-reference, self-description, and self-explanation – that is, closed logical systems that include the referee in the reference, the observer in the description, and the axioms in the explanation.

This belief is unfounded, as has been shown by John von Neumann, (1951, 1966), Gotthard Günther (1967), Lars Löfgren (1962, 1968), and many others who addressed themselves to the question as to the degree of complexity a descriptive system must have in order to function like the objects described, and who answered this question successfully.

The other cause for the increasing compartmentalization of knowledge into “major disciplines” is that the syntactic relations in all logicomathematical formalisms are reasonably well understood, while the semantic relations that underlie formalisms as expressed, for instance, in natural languages are only recently being investigated and are as yet little understood.

On the other hand, it is clear that the tautological transformations per se within a logicomathematical formalism cannot lead us to new insights by providing “solutions” to a “problem” (As long as no errors in calculation occur, they are just equivalent paraphrases.) unless by such paraphrasing new semantic relations become uncovered (the solution), which were opaque in the original formulation of the state of affairs (the problem).

Consider, for instance, The Ecological Problem. It consists of identifying n variables xi (i = 1, 2, 3… n) and n + mn2 parameters Aiμ, and aij (μ = 0,1,2 … m), and “solving” the set of n simultaneous, ordinary, first-order differential equations with respect to time, t:

x_{i} = \displaystyle\sum_{\mu=0}^{m} A_{i\mu}\left(\displaystyle\sum_{1}^{n} a_{ij}x_{j}\right)

with the convention:

a^{\nu}_{ij} \equiv a_{ij\nu}

and, sometimes, with the popular constraint:

m = 2.

This means expressing Eq (1) in the form of:

x_i = F_i(t,[ x_{ok}, a_{kj\nu}, A_{k\mu} ])

(i,j,k=1,2,\dots n);(\mu,\nu=0,1,\dots m)

where the xok are the n initial conditions.

At this moment it is beside the point to ask whether Eq. 1 indeed constitutes the formulation of the ecological problem, that is, is there a valid model of the ecology underlying this expression in which the various parameters and the quantitative relations between the variables can be justified. (In fact there is none! This expression has to be taken as a “first principle.”) The question here is whether or not the insight gained by contemplating Eq. 2 is more valuable than the insight lost by formulating Eq. 1.

It has become abundantly clear today that, although for small values of n and m (n Save Selection

How to overcome this dilemma is now the question.

There appear to be two avenues that will allow these difficulties to be bypassed. One was taken by physicists when they contemplated the thermodynamics of aggregates of very many particles. The idea is to forsake voluntarily the detailed available knowledge of the equations of motion of all the particles involved, and to develop measures of certain aspects of the system as a whole as, e.g., its energy, entropy, enthalpy, etc. Similarly, the systems engineers are content when they derive such notions as, e.g., uncertainty, redundancy, transmission, etc., which characterize in quantitative terms some general aspects of the systems under consideration.

The other possible avenue has been opened recently by our beginning to understand the relations between semantic and syntactic structures, and by the development of computer systems of unprecedented power (Barnes et al. 1968) together with relational program structures of unprecedented sophistication (Weston 1970). The idea is to abandon the strategy of reformulating the problem into terms that smack of mathematical rigor but lack the contextual richness originally perceived, and to develop the algorithms that transform the descriptions of certain aspects of a system into paraphrases that uncover new semantic relations pertaining to the system as a whole. This is computation in the semantic domain, rather than in the syntactic domain of logicomathematical formalisms. Here the possibility is opened for inductive inference, a logical modality that is denied all formalisms that rest on the infallibility of the tautological transform.

References

VON NEUMANN, J. 1966. Theory of Self-Reproducing Automata. A. W. Burks, Ed. University of Illinois Press. Urbana, Ill.

VON NEUMANN, J. 1951. The general and logical theory of automata. In Cerebral Mechanisms of Behavior. L. A. Jeffrees, Ed. : 21. John Wiley & Sons. New York, N.Y.

GÜNTHER, G. 1967. Time, timeless logic and self-referential systems. In Interdisciplinary Perspective of Time. R. Fisher, Ed. Ann. N.Y. Acad. Sci. 132 (2): 396-406.

LÖFGREN, L. 1962. Kinematic and tesselation models of self-repair. In Biological Prototypes and Synthetic Systems. E. E. Bernard & M. R. Kare, Eds. : 342-369. Plenum Press. New York, N.Y.

LÖFGREN, L. 1968. An axiomatic explanation of complete self-reproduction. Bull. Math. Biophys. 30(3): 415.

BARNES, G. H. et al. 1968. The Illiac IV computer. IEEE Tr. Comp. C-17: 746757.

WESTON, P. 1970. Cylinders: a relational data strcture. Tech. Report No. 18, Biological Computer Laboratory, Univ. of Ill.

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