Axiomatic method

In view of the success of the axiomatic method for teaching NM, MT, STR and GTR, it should be no surprise that QM is also taught this way. The works of Dirac(4), Von Neumann(5) and Pauli(6) all were written by masters in the field who were completely familiar with all the phenomena relating to QM. To them and their colleagues these works are a clean axiomatic summary of what they knew. 

The need for writing this book was dictated by the peculiar situation which had become established in mathematics in the middle of the twentieth century. The deductive-axiomatic style, which emerged at the end of the nineteenth century in works on the foundations of mathematics, found many adherents. (As Bertrand Russell said in promoting the axiomatic method, the latter has many advantages similar to those of stealing versus doing honest work.) By the mid-twentieth century use of this method had become widespread in the teaching of mathematics as well. 

Anderson was able to move from chemical facts to geological phenomena because he was using the same axiomatic method that allowed him to extend the rules of chemistry into agriculture. In his discussion of stalactites and limestone strata, he differentiated between the Operations of Art and the Operations of Nature. The former were, in this case, the principles of chemistry.74 He used these to obtain the facts of composition and then he noted other properties like shape and mass so that he could then inductively determine the process by which the stalactite had formed. Once this was done, he could then address larger Operations of Nature like limestone strata ... 

The principles formulated below and coinciding in their meaning with selection rules are essentially postulates because they do not follow from anything. They should not be understood as something quite fixed and invariable. This is only the first and possibly not the best attempt to formulate tentatively the physical bases of catalytic polymerization on the basis of the axiomatic method. 

It may not be entirely vain, however, to search for common properties among diverse kinds of complex systems. The ideas that go by the name of cybernetics constitute, if not a theory, at least a point of view that has been proving fruitful over a wide range of applications. It has been useful to look at the behavior of adaptive systems in terms of the concepts of feedback and homeostasis, and to analyze adaptiveness in terms of the theory of selective information. The ideas of feedback and information provide a frame of reference for viewing a wide range of situations, just as do the ideas of evolution, of relativism, of axiomatic method, and of operational-ism. ... 

Acceptance rates 231 Acrylamide 124 Activity inversicm effect 172 Additicxi polymerization, elementary act 135, 146, 153 Adhesicxi 286 Aging, effect 61, 68, 79, 85 AIBN 124 Alkyl vinyl ether 4 Amplitudes 221 Asymmetry 239, 266 Axiomatic method 340... 

NoU, W. The foundations of classical mechanics in the light of recent advances in continuum mechanics. The Axiomatic Method, with Special Reference to Geometry and Physics (Symposium, Berkeley, 1958). North Holland, Amsterdam (1959). See Ref. [41]... 

Euclid (c. 330-c. 260 bc) Greek mathematician, who worked at the academy in Alexandria. He is best known for his 13-volume work the Elements, which deals mainly with geometry and established the axiomatic method of reasoning. 

The axiomatic way of teaching quantum mechanics (QM) is anaiyzed in the light of its effectiveness in making students ready to understand and use QM. A more intuitive method of teaching QM is proposed. An outline of how a course implementing that method could be structured is presented. 

In this chapter we discuss techniques for program verification and their mathematical justification. The basic idea behind these methods was originally presented by Floyd mathematical formulations and logical justifications were developed by Cooper and Manna, and others, and continued in King s Ph.D. thesis in which he presented the development of a partial implementation for these techniques. A sanewhat different axiomatic approach has been pursued by Hoare et al. The reader who has never made acquaintance with the formalism of the first order predicate calculus should at this point turn to Appendix A for a brief and unrigorous exposition of the material relevant to this chapter. 

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. 

Parallel with the phenomenological development, an alternative point of view has developed toward thermodynamics, a statistical-mechanical approach. Its philosophy is more axiomatic and deductive than phenomenological. The kinetic theory of gases naturally led to attempts to derive equations describing the behavior of matter in bulk from the laws of mechanics (first classic, then quanmm) applied to molecular particles. As the number of molecules is so great, a detailed treatment of the mechanical problem presents insurmountable mathematical difficulties, and statistical methods are used to derive average properties of the assembly of molecules and of the system as a whole. 

The synthetic methods that have been employed for the preparation of the cycloproparenes fall into two distinct categories, namely those that commence with a preformed aromatic moiety to which is appended a three-membered ring, and those that build the molecular framework prior to aromatization. The sections that follow have been composed to provide a continuity of purpose rather than a strict division between these two axiomatic methodologies. 

The burden of rigorous justification forced authors to throw out of textbooks on analysis any nontrivial and truly interesting applications. A comparison of typical purged modern textbooks with the brilliant courses written fifty years before (for example, by the German mathematicians R. Courant and D. Hilbert, and others) clearly shows how much of importance was lost with the introduction of axiomatic and set-theoretical methods in the teach-... 

Sterilization processes are discussed in detail in Chapter 20. However, it is axiomatic that whatever method is chosen, the process should not cause damage to the product. By reference mostly to moist heat sterilization processes (the reader should remember that there are parallel approaches to other methods of sterilization) this section illustrates the factors that must be considered in the design of a sterilization process. 

A formal similarity arises because both the scattering matrix in quantum electrodynamics and the wave operator in a full coupled-cluster method [cf. Equations (1) and (4)] are exponential operators. In Bogo-liubov s axiomatic formulation of the scattering matrix,29... 

A major improvement in the precision of the more complex methods has been achieved by the now almost standard practice of using an internal standard of radioactively labeled steroid. Some features of this practice need discussion. It is axiomatic that the mass of added tracer must be small compared with the mass to be estimated in the sample, and that the precision of addition and measurement of the internal standard be as great as possible. Also, the tracer affords a correction for losses only in the stages of the procedure following its addition. Where... 

In contrast to HDT, when using the Utility Function, each (normalized to the scale [0,1] and then called instead of q now p,) descriptor is given a weight indicating the relative importance of that particular descriptor. More complex assumption, how a descriptor can be transformed into an "individual" preference function can be found by Schneeweiss (1991), where also the axiomatic foundation of this method is discussed. Instead of the simultaneous consideration of the descriptors the function... 

The limitations of modem experimental physico-chemical methods in the direct and detailed study of the monomer insertion in the polymer chain on the one hand, and the large amount of information about the dependence of the macromolecules primary structure on the reaction conditions on the other hand should be taken into account. Consequently, it seems that the only real path of the search for relationships between the electronic aspects of the reaction and the observed microstructural statistics of polymers is the heuristic method of formulation of definite rules. It is most suitable for carrying out an adequate formulation of these imaginary discoveries in the framework of an axiomatic approach. 

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