The Importance of Mathematics to Address Physics Situations , Is introduced by understanding that mathematics is the language to formulate empirical laws of nature. Physics was created to explain the relationships between measurements of physical objects.
A large portion of mathematics - not to say the whole branch - is determined by the understanding and definition of the relationships between objects. Consequently, physics is a specific example of mathematics.
Link between mathematics and physics
Generally considered a relationship of great intimacy, some mathematicians have described this science as an"essential tool for physics,"and physics has been described as"a rich source of inspiration and knowledge in mathematics."
The considerations that mathematics is the language of nature can be found in the ideas of Pythagoras: the conviction that"numbers dominate the world"and that"everything is number". These ideas were also expressed by Galileo Galilei:"The book of nature is written in mathematical language."
It took a long time in the history of mankind before this to discover that mathematics are useful and even vital in the understanding of nature. The Eastern thought was dominated from the antiquity, until the Renaissance, turn in turn, by Plato and Aristotle.
Aristotle He thought that the depths of nature could never be described by the abstract simplicity of mathematics. Galileo Recognized and used the power of mathematics in the study of nature, which allowed its discoveries to initiate the birth of modern science.
The physicist, in his study of natural phenomena, has two methods of progress:
- The method of experiment and observation, and
- The method of mathematical reasoning.
One could describe the mathematical quality in nature by saying that the Universe is constituted of mathematics from its conception.
However, recent discoveries in the physical sciences show that this statement is rather trivial. The connection between mathematics and the description of the Universe goes much further than this concept.
Mathematics in the Mechanical Scheme
The mechanical scheme considers the entire Universe as a dynamic system (an extremely complicated dynamic system), subject to the laws of motion that are essentially of the Newtonian type.
The role of mathematics in this scheme is to represent the laws of motion through equations, and to obtain solutions of the equations that refer to the observed conditions.
The dominant idea in this application of mathematics to physics is that the equations representing the laws of motion must be made in a simple manner. The success of the scheme is due to the fact that the equations in a simple way seem to work.
This method of simplicity is very restricted, but this principle of simplicity applies only fundamentally to the laws of motion, not to all natural phenomena in general.
The discovery of the theory of relativity made it necessary to modify the principle of simplicity. Presumably one of the fundamental laws of motion is the law of gravity which, according to Newton, is represented by a very simple equation, but according to Einstein, the development of an elaborate technique is necessary before this equation can be written.
Quantum mechanics
Quantum mechanics requires the introduction into the physical theory of a vast domain of pure mathematics, the complete domain connected with non-commutative multiplication.
It might be hoped in the future that the mastery of pure mathematics will be involved with fundamental advances in physics, whose tendency is pointed to the introduction of new geometries from the theory of relativity.
Role of mathematics in physical theories
It is true that physics chooses certain mathematical concepts for the formulation of the laws of nature, and surely only a fraction of all mathematical concepts is used in physics.
It is true, furthermore, that the concepts that were chosen were not arbitrarily selected from a list of mathematical terms, but were developed in many cases independently by physicists and recognized after being conceived and approved by a mathematical.
Now it begins to appear that not only complex numbers (also called analytic functions) are destined to play a decisive role in the formulation of quantum theories. These functions refer to the rapid development of the theory of dispersion relations.
Mathematics as mechanized thinking
Once an idea is expressed in mathematical form, you can use the rules (axioms, theorems, etc.) of mathematics to change it to another statement. If the original statement is correct, and you follow the rules, the final statement will be correct. This is what you do when solving a mathematical problem.
Static Mechanics, Dynamic Systems and Ergodic Theory
A more advanced example that demonstrates the deep and fruitful relationship between physics and mathematics is that physics can eventually develop new mathematical concepts, methods, and theories. This has been demonstrated by the historical development of static mechanics and Ergodic theory .
For example, the stability of the solar system was an old problem investigated by great mathematicians since the eighteenth century.
It was one of the main motivations for the study of periodic movements in body systems, and more generally in dynamical systems especially through the work of Poincaré in celestial mechanics and the investigations of Birkhoff in general dynamic systems. These paved the way for the development of the modern theory of dynamic systems.
Differential equations, complex numbers and quantum mechanics
It is well known that from the time of Newton, differential equations have been one of the main links between mathematics and physics, taking both important developments in analysis and in the consistency and fruitful formulation of physical theories.
It is perhaps less well known that much of the important concepts of functional analysis originated in the study of quantum theory.
References
- Klein F., 1928/1979, Development of Mathematics in the 19th Century, Brookline MA: Mathematics and Science Press.
- Boniolo, Giovanni; Budinich, Paolo; Trobok, Majda, eds. (2005). The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects. Dordrecht: Springer. ISBN 9781402031069.
- Proceedings of the Royal Society (Edinburgh) Vol. 59, 1938-39, Part II pp. 122-129.
Mehra J., 1973"Einstein, Hilbert and the theory of gravitation", in The physicist concept of nature, J. Mehra (ed.), Dordrecht: D. Reidel. - Feynman, Richard P. (1992). "The Relation of Mathematics to Physics". The Character of Physical Law (Reprint ed.). London: Penguin Books. Pp. 35-58. ISBN 978-0140175059.
Arnold, V.I., Avez, A., 1967, Problems Ergodiques de la Mécanique Classique, Paris: Gauthier Villars.