The title alludes to a diagram in Brian Cantwell Smith's paper The Limits of Correctness : the diagram shows a pair of relationships computer ↔ model ↔ world. The model, Smith writes, is the "glasses through which the computer sees the world." Model theory, he adds, studies the relationship on the left between computer and model, but the right-hand side relationship remains problematic: for the right-hand side "we have no theory." A historic example showing how much this matters. In 1960, in the Ballistic Missile Early Warning System (BMEWS), a defective model misinterpreted radar reflections from the moon as a launch of Soviet missiles against the USA: fortunately, the threatened outbreak of war was averted. Might a good theory have prevented this defect?
The model is the "glasses through which the computer sees the world." Yes: but the computer also sees the world directly, through the interface of sensors and actuators. These are physical things—domains—that effectuate causal links by which the world and the machine constrain each other. Further causal links are effectuated within and between domains contained in the world—and similarly in the machine (although we rarely think of the machine like this). The governed world model maps the domains of the world and interface, and their causal links. This map shows what effects the machine can evoke—directly and indirectly—at each actuator, and what information about the governed world can be inferred—directly and indirectly—at each sensor during a behaviour enactment.
We can liken the map of causal links in the world to a map of one-way roads over the domain infrastructure, in which road junctions correspond to state and event phenomena. To design a behaviour is to specify a set of possible complex journeys between and within the machine and the governed world. The designer relies on the map to show which routes, passing through which physical phenomena, are possible in the world. The model—that is, the map—can be perfectly formalised, allowing perfectly reliable inferences. Why, then, is the right-hand side relationship problematic?
Einstein stated the problem succinctly: "... as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” The map cannot be perfectly reliable: the map is fixed, but the roads change; minor roads are omitted; access may be subject to unstated conditions; a road may be blocked; a path may be cut off by an accident. Contingencies like these have their counterparts in physical failures that may vitiate the designed behaviour.
Should we then, aim to solve this problem by a theory of the right-hand side relationship? No. To achieve its purpose, the theory itself must be formal; but relating a formal theory to the non-formal physical world would embody the original problem again—in an infinite regression. Instead we should seek a sound practical discipline of developing models that—although imperfect—are fit for purpose. This is not a disappointment. Fit for purpose is exactly what an engineering product must be.
 Brian Cantwell Smith; The Limits of Correctness; ACM SIGCAS Computers and Society, Volume 14,15, Issue 1,2,3,4, pages 18-26, January 1985.
 Albert Einstein; Geometry and Experience; Methuen, 1922.
Links to other posts:
← Models: Types and purposes of models of the physical world
← Physical Bipartite System: The nature of a bipartite system
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