Importance of Mathematics to Address Physics Situations

The importance of mathematics to address physical situations, is introduced by understanding that mathematics is the language to formulate empirical laws of nature. 

A large portion of mathematics is determined by understanding and defining 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".

Considerations that mathematics is the language of nature can be found in the ideas of Pythagoras: the conviction that "numbers rule 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 human history before someone discovered that mathematics is useful and even vital in understanding nature..

Aristotle 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, allowing his discoveries to usher in the birth of modern science.

The physicist, in his study of natural phenomena, has two methods of progressing:

• the method of experiment and observation
• the method of mathematical reasoning.

Mathematics in the Mechanical Scheme

The mechanical scheme considers the Universe as a whole as a 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.

The dominant idea in this application of mathematics to physics is that the equations representing the laws of motion must be done in a simple way..

This method of simplicity is very restricted; applies 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.

Quantum mechanics

Quantum mechanics requires the introduction into physical theory of a vast domain of pure mathematics, the entire domain connected with non-commutative multiplication.

One might expect in the future that the domain of pure mathematics will be engulfed with fundamental advances in physics..

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 may 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 18th century..

It was one of the main motivations for the study of periodic movements in body systems, and more generally in dynamic systems especially through Poincaré's work in celestial mechanics and Birkhoff's investigations in general dynamic systems..

Differential equations, complex numbers, and quantum mechanics

It is well known that since Newton's time, differential equations have been one of the main links between mathematics and physics, both leading to important developments in analysis and in the consistency and fruitful formulation of physical theories..

It is perhaps less well known that many of the important concepts of functional analysis originated from the study of quantum theory..

References

1. Klein F., 1928/1979, Development of Mathematics in the 19th century, Brookline MA: Mathematics and Science Press.
2. Boniolo, Giovanni; Budinich, Paolo; Trobok, Majda, eds. (2005). The Role of Mathematics in Physical Sciences: Interdisciplinary and Philosophical Aspects. Dordrecht: Springer. ISBN 9781402031069.
3. 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.
4. 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, Problèmes Ergodiques de la Mécanique Classique, Paris: Gauthier Villars.