Axiomatic theories

It has been possible to develop axiomatic theories for complete and partial ignorance that, by studying the sets of acts and states, yield a better understanding of the choice of criteria, and are in harmony with logical formalization of intuitively acceptable notions. We have treated in a rudimentary fashion a subject that is deep, novel, and promising. 

Shamanin, V. V Bases of the Axiomatic Theory of Addition Polymerization. VoL U2,pp. 135-180. 

The purpose of this research was to explore the representation, manipulation, and utilization of strategic knowledge in organic synthesis planning. The method we decided to explore was to create an axiomatic theory to replace our intuitive theory about chemical synthesis. This formal method of reasoning is very powerful in that it completely eliminates any questions about the method used to reach a conclusion. Since any conclusions reached would be theorems of the axiomatic theory, the acceptability of the conclusions rests completely on the acceptability of the postulates and not upon the method of reasoning. We are then free to focus on the chemical principles which are provided as postulates. 

Definition of Axiomatic Theories. An axiomatic theory is an attempt to formalize an intuitive theory. Geometry was intuitive before Euclid wrote "The Elements . An intuitive theory is defined as a body of knowledge which attempts to express relationships and causality between objects, but is not formal. Most modern science is still intuitive, even though it may represent many of it s findings in exact mathematical formulae. As long as the entire corpus of knowledge is not expressed in a single formal system, it will remain intuitive. 

The following list of the steps is necessary to create an Axiomatic Theory ... 

Provide a set of axioms (postulates). If one is attempting to create an axiomatic theory which mirrors experimental reality, then these axioms should express some fundamental properties of the system you are trying to model. 

In modern axiomatic theory, postulates and axioms are defined simply as given statements. By the definition of an axiomatic theory the concept of truth is not considered relevant to its construction. If we can derive a theory which seems to mirror reality as reported by our current experimental knowledge, then we consider the postulates to be successful in some sense of the word. If the theory derived from the postulates clash drastically with our observations, the postulates can be thrown away as non-relevant . If the differences are slight, or if the theory predicts new experiments which should show differences from what the intuitive theory would predict, we can even call the axiomatic theory interesting . 

One of the most important requirements for an axiomatic theory is to determine the validity-round of the laws, and to verify of the self-consistency in the theory. The M-M experiment proved that the prediction of the classical physics was not valid for light propagation, or rather, for Maxwell s theory of electromagnetism. This is an applicability limit of Newtonian physics. Beyond this limit, Newtonian physics becomes incomplete. 

General Remarks. In this section, we will briefly outline an axiomatic theory that encompasses within a single quantum structure both mechanics and thermodynamics. A single quantum structure is necessary because the theory includes quantum mechanics. It is also sufficient because we consider only one type of uncertainty, namely, that inherent in quantum mechanics, and not two types of uncertainty, namely, one for quantum effects and another for incomplete information. 

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