### Encyclopedia of Mathematical Physics. Contributors

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In present-day terminology, however, a distinction is made between the two. Whereas most of theoretical physics uses a large amount of mathematics as a tool and as a language, mathematical physics places greater emphasis on mathematical rigor, and devotes attention to the development of areas of mathematics that are, or show promise to be, useful to physics. The results obtained by pure mathematicians, with no thought to applications, are almost always found to be both useful and effective in formulating physical theories. Mathematical physics forms the bridge between physics as the description of nature and its structure on the one hand, and mathematics as a construction of pure logical thought on the other.

This bridge between the two disciplines benefits and strengthens both fields enormously. See Physics , Theoretical physics. The methods employed in mathematical physics range over most of mathematics, the areas of analysis and algebra being the most commonly used. Partial differential equations and differential geometry, with heavy use of vector and tensor methods, are of particular importance in the formulation of field theories, and functional analysis as well as operator theory in quantum mechanics.

Group theory has become an especially valuable tool in the construction of quantum field theories and in elementary-particle physics. There has also been an increase in the use of general geometrical approaches and of topology. For solution methods and the calculation of quantities that are amenable to experimental tests, of particular prominence are Fourier analysis, complex analysis, variational methods, the theory of integral equations, and perturbation theory.

See Variational methods physics , Vector methods physics. Mathematical physics is closely connected with physics inasmuch as it deals with the construction of mathematical models; at the same time it is a branch of mathematics inasmuch as the methods used to investigate the models are mathematical.

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The concept of mathematical physics also includes those mathematical methods that are used to set up and study mathematical models that describe large classes of physical phenomena. The methods of mathematical physics as the theories of mathematical models in physics were first developed intensively in I.

The further development of the methods of mathematical physics and their successful application to a wide range of physical phenomena are associated with J. Lagrange, L.

Euler, P. Laplace, J. Fourier, K.

Gauss, B. Riemann, and M. Ostrogradskii, among others.

Liapunov and V. Steklov made major contributions to the development of the methods of mathematical physics. Beginning in the second half of the 19th century, the methods of mathematical physics were used successfully in studying mathematical models of physical phenomena related to various physical fields and wave functions in electrodynamics, acoustics, the theory of elasticity, hydrodynamics, aerodynamics, and other fields related to the study of physical phenomena in continuous media. The mathematical models of this class of phenomena are described most often by means of partial differential equations called the equations of mathematical physics.

In addition, integral and integrodifferential equations, variational and probability theory methods, potential theory, the theory of functions of a complex variable, and many other branches of mathematics are also used in describing the mathematical models of physics.

### Encyclopedia of Mathematical Physics

With the rapid development of computer mathematics direct numerical methods based on the use of computers and especially finite-difference methods of solving boundary value problems have acquired particular importance in the investigation of mathematical models in physics. Theoretical investigations in quantum electrodynamics, the axiomatic theory of fields, and many other branches of modern physics have led to the creation of a new class including the theory of generalized functions and the theory of continuous-spectra operators of mathematical models constituting an important branch of mathematical physics.

The formulation of the problems of mathematical physics consists in the setting up of mathematical models describing the basic regularities shown by the class of physical phenomena being studied. Such a formulation involves the derivation of equations differential, integral, integrodifferential, or algebraic that are satisfied by the quantities characterizing the particular physical process. In so doing we proceed from the fundamental physical laws that take account of only the most significant features of the phenomenon, disregarding its secondary characteristics.

Its efficient cause is the sculptor, insofar has he forces the bronze into shape. The formal cause is the idea of the completed statue. The final cause tends to be the same as the formal cause, and both of these can be subsumed by the efficient cause. Of the four, it is the formal and final which is the most important, and which most truly gives the explanation of an object.

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The final end purpose, or teleology of a thing is realized in the full perfection of the object itself, not in our conception of it. Final cause is thus internal to the nature of the object itself, and not something we subjectively impose on it. To Aristotle, God is the first of all substances, the necessary first source of movement who is himself unmoved.

God is a being with everlasting life, and perfect blessedness, engaged in never-ending contemplation. Aristotle sees the universe as a scale lying between the two extremes: form without matter is on one end, and matter without form is on the other end. The passage of matter into form must be shown in its various stages in the world of nature. To do this is the object of Aristotle's physics, or philosophy of nature.

It is important to keep in mind that the passage from form to matter within nature is a movement towards ends or purposes. Everything in nature has its end and function, and nothing is without its purpose. Everywhere we find evidences of design and rational plan. No doctrine of physics can ignore the fundamental notions of motion, space, and time. Motion is the passage of matter into form, and it is of four kinds: 1 motion which affects the substance of a thing, particularly its beginning and its ending; 2 motion which brings about changes in quality; 3 motion which brings about changes in quantity, by increasing it and decreasing it; and 4 motion which brings about locomotion, or change of place.

Of these the last is the most fundamental and important. Aristotle rejects the definition of space as the void. Empty space is an impossibility. Hence, too, he disagrees with the view of Plato and the Pythagoreans that the elements are composed of geometrical figures. Space is defined as the limit of the surrounding body towards what is surrounded. Time is defined as the measure of motion in regard to what is earlier and later. It thus depends for its existence upon motion.

If there where no change in the universe, there would be no time. Since it is the measuring or counting of motion, it also depends for its existence on a counting mind. If there were no mind to count, there could be no time. After these preliminaries, Aristotle passes to the main subject of physics, the scale of being.

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The first thing to notice about this scale is that it is a scale of values. What is higher on the scale of being is of more worth, because the principle of form is more advanced in it. Species on this scale are eternally fixed in their place, and cannot evolve over time. The higher items on the scale are also more organized. Further, the lower items are inorganic and the higher are organic. The principle which gives internal organization to the higher or organic items on the scale of being is life, or what he calls the soul of the organism.

Even the human soul is nothing but the organization of the body. Plants are the lowest forms of life on the scale, and their souls contain a nutritive element by which it preserves itself. Animals are above plants on the scale, and their souls contain an appetitive feature which allows them to have sensations, desires, and thus gives them the ability to move.

The scale of being proceeds from animals to humans. The human soul shares the nutritive element with plants, and the appetitive element with animals, but also has a rational element which is distinctively our own.

The details of the appetitive and rational aspects of the soul are described in the following two sections. For a fuller discussion of these topics, see the article Aristotle: Motion and its Place in Nature. Soul is defined by Aristotle as the perfect expression or realization of a natural body.

From this definition it follows that there is a close connection between psychological states, and physiological processes.