Intuitive theory, definition

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. 

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 . 

Owing to the very simple and intuitively clear definition of the equation of state, the topology of the vapor-liquid-phase transition and critical point is examined easily using the methods of dynamic system and bifurcation theory. 

Bifurcation theory studies the changes in the phase space as we vary the parameters of the system. In essence, this is the authentic notion of bifurcation theory proposed originally by Henry Poincare when he studied Hamiltonian systems with one degree of freedom. We must, however, note that this intuitively evident definition is not always sufficient at the contemporary stage of the development of the theory. One needs, in fact, to have an appropriate mathematical foundation to define the notions of the structure of the phase space and the changes in the structure. 

We have already mentioned that real-world data have drawbacks which must be detected and removed. We have also mentioned outliers and redundancy. So far, only intuitive definitions have been given. Now, aimed with information theory, we are going firom the verbal model to an algebraic one. 

The study of chemical reactions requires the definition of simple concepts associated with the properties ofthe system. Topological approaches of bonding, based on the analysis of the gradient field of well-defined local functions, evaluated from any quantum mechanical method are close to chemists intuition and experience and provide method-independent techniques [4-7]. In this work, we have used the concepts developed in the Bonding Evolution Theory [8] (BET, see Appendix B), applied to the Electron Localization Function (ELF, see Appendix A) [9]. This method has been applied successfully to proton transfer mechanism [10,11] as well as isomerization reaction [12]. The latter approach focuses on the evolution of chemical properties by assuming an isomorphism between chemical structures and the molecular graph defined in Appendix C. 

Many theories developed in this book are expressed by equations or results involving continuous functions for example, the spatially variable concentration c(r). Materials systems are fundamentally discrete and do not have an inherent continuous structure from which continuous functions can be constructed. Whereas the composition at a particular point can be understood both intuitively and as an abstract quantity, a rigorous mathematical definition of a suitable composition function is not straightforward. Moreover, using a continuous position vector f in conjunction with a crystalline system having discrete atomic positions may lead to confusion. 

The HSAB principle can be considered as a condensed statement of a very large amount of experimental information, but cannot be labelled a law, since a quantitative definition of the intuitive concepts of chemical hardness (T ) and softness (S) was lacking. This problem was solved when the hardness found an exact, and also an operational, definition in the framework of the Density Functional Theory (DFT) by Parr and co-workers [2], In this context, the hardness is defined as the second order derivative of energy with respect to the number of electrons and has the meaning of resistance to change in the number of electrons. The softness is the inverse of the hardness [3]. Moreover, these quantities are defined in their local version [4, 5] as response functions [6] and have found a wide application in the chemical reactivity theory [7],... 

Each of these aspects of the field of chemistry is connected through the basic principle of chemical structure, which is a profound physical feature of the molecular world where we live. At its most fundamental, stereoelectronic structure is a quantum-mechanical reality of all molecules, with the intrinsic uncertainty that this reality implies. Thus, perfectly accurate structural descriptions of molecules are both elusive and potentially cumbersome. Instead, chemists have devised an exceptional model of molecular structure by inference. This model has been built over decades between evolving theory and experiments that measure various molecular properties that derive from structure itself. Closely aligned with our intuitive definition of structure , of course, are methods that provide direct information about... 

Another relevant concept within information theory, in some cases strongly related to the aforementioned measures, is the so-called complexity of a given system or process. There is not a unique and universal definition of complexity for arbitrary distributions, but it could be roughly understood as an indicator of pattern, structure, and correlation associated to the system the distribution describes. Nevertheless, many different mathematical quantifications exist under such an intuitive description. This is the case of the algorithmic [19, 20], Lempel-Ziv [21] and Grassberger... 

The basis of chemistry as it grew up in the nineteenth century is Dalton s atomic theory. In this the intuitive idea of ultimate particles is applied to explain definite quantitative laws of chemical composition, those, namely, of constant, multiple, and reciprocal proportions. These rules could only have emerged after a long empirical study... 

A general and, at the same time, intuitive definition of open-shell molecules is not that straightforward to give, but it can be provided in the context of MO theory, where a molecule is considered to be open shell if its valence shell contains orbitals that are partially occupied. Open-shell systems can be encountered when the number of a-electrons differs from the number of p-electrons, and, as a consequence, nonzero eigenvalues of the total squared spin operator are obtained (in atomic units),... 

The basic steps to a formal treatment of mereology were taken by Stanislaw Lesniewski s mereology (Srzednicki and Rickey 1984) is a basic system of rules for valid reasoning about wholes and their parts. David Lewis carried through the mapping of intuitive mereological rules onto the formal system of set theory. To do so he began with two definitions (1991 73). 

To say the least, the above description of the theory leading to the definition of the Hausdorff dimension does not suggest a simple and intuitive way of evaluating this dimension in practical cases. Indeed, the calculation of Hausdorff measures and dimensions is, in general, more than a little involved, even for simple sets [e.g. 26... 

Chemists are more used to the operational definition of symmetry, which crystaUo-graphers have been using long before the advent of quantum chemistry. Their ball-and-stick models of molecules naturally exhibit the symmetry properties of macroscopic objects they pass into congruent forms upon application of bodily rotations about proper and improper axes of symmetry. Needless to say, the practitioner of quantum chemistry and molecular modeling is not concerned with balls and sticks, but with subatomic particles, nuclei, and electrons. It is hard to see how bodily rotations, which leave all interparticle distances unaltered, could affect in any way the study of molecular phenomena that only depend on these internal distances. Hence, the purpose of the book will be to come to terms with the subtle metaphors that relate our macroscopic intuitive ideas about symmetry to the molecular world. In the end the reader should have acquired the skills to make use of the mathematical tools of group theory for whatever chemical problems he/she will be confronted with in the course of his or her own research. 

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