Mathematics and physics are two branches of science that share a lot of similarities. Some people often view math as a theoretical subject while physics is applied mathematics. This is a novice way of looking at it but there is some truth in it. There are major differences between these two sciences but they also have a symbiotic relationship that leads to their development. This paper will discuss the relationship between them by highlighting the nature of these sciences and how they interact.
The Nature of Maths and Physics
Maths has often been described as a conceptual and theoretical science. There are several philosophies that seek to describe it such as formalism and mathematical platonism. One is that maths is a play of numbers and shapes. This philosophy describes maths as a formal manipulation of symbols as an intellectual pursuit. This places a sense of naturalism in maths in that it exists as a creation of human endeavor rather than existing in a supernatural realm. Mathematical platonism in turn describes maths as an abstract field that exists independent of the physical world. The mathematical objects that we see in the subject do not exist in the physical world and we only use them via mathematical intuition. This could be reconciled with naturalism in that the mathematical intuition we use to develop this abstract realm is as a result of general intelligence which leads to humans reasoning about the physical world using mathematical objects. Mathematicians are therefore interested in the understanding of abstract topics where they use proofs and mathematical reasoning to support these ideas (Vinitsky-Pinsky & Galili, 2014). The approach used by mathematicians to approach a topic are different from physics but these mathematical models and objects are often the language of physics and indeed all science.
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Physics is a natural science which studies the laws of nature. It is empirical in nature and can be proven by experiments. In physics computation and calculation is done with the ultimate goal of getting an answer to an hypothesis (Vinitsky-Pinsky & Galili, 2014, Penn, n.d.). Physicists are therefore not interested in the understanding of abstract theories but rather the empirical evidence as to the existence or absence of certain aspects of matter such as gravity in blackholes, development of subatomic particles, gravity in space, Penrose tiles and others.
The development of modern physics has led to the adoption of sophisticated abstract mathematics in physics theories. This relationship could be attributed to the theory that maths is a form of natural science as well (Pospiech et al., 2019). Mathematics is derived from natural law, human reasoning, and tells the story of our universe. Abstract mathematical ideas are a formalistic manipulation of maths which do not directly refer to anything that exists in the real world but maintains a tether to empiricism and the law of nature. This would explain why some math in those abstract models are later evidenced in natural science.
Relationship Between Mathematical and Physical Theories
Maths is viewed as a tool for physics whereas physics has been described as a source of new mathematical tools and models. Mathematical branches such as geometry and calculus were formulated as a result of development in physics while branches such as particle physics, string theory and quantum mechanics have been developed by existing mathematical tools.
The development of theoretical concepts in physics necessitates the development of new mathematical tools to verify those concepts. Isaac Newton for example developed calculus in his quest to develop the laws of motion (Penn, n.d.). Calculus was used to study the rates of change and assisted in his discovery of gravity. The geometric theories were important in the construction of Albert Einstein’s theory of general relativity and later in the superstring theory (Marianne, 2018).). He used non-Euclidean geometry developed by Bernhard Riemann to show that the curvatures of time and space could be attributed to gravity. Non-Euclidean geometry was able to accomodate a space made primarily of curves and where all lines eventually meet at a common point. The greatest point of relationship between the two however was in the development of quantum mechanics and the superstring theory.
Quantum Mechanics
This theory was developed by Niels Bohr and Erwin Schodinger to describe the structure of the microscopic atoms. Atoms are treated as waves and particles. The subject of study in this theory were so small or so large that actual experimentation was impossible. This therefore required the use of maths to compute these structures in empirical calculation. This theory combined with others such as thermodynamics, general relativity, and mathematics were used to study blackholes. Roger Penrose and Stephen Hawking showed that the centers of blackholes (singularities) where space and time are so distorted that normal physical laws cease in significance existed and could be formed under certain conditions. Hawking was also able to show that they do not completely violate the physical laws of nature since they radiated a small amount of heat.
String and Superstring Theories
The treatment of subatomic particles as dimensionless points proved a challenge for physicists due to complexities encountered in studying their characteristics. The idea was born to treat the particles as loops or strings whose vibration determined their characteristics and how they developed into larger atomic particles and generated forces such as electromagnetism and gravity. This theory would help study gravity in the quantum mechanic field. It was referred to as the string theory which faced several challenges. In the 1980s the theory was reborn as the superstring theory. This theory relied on the mathematical concepts of symmetry. The strings were placed in a supersymmetric 10-dimensional universe which helped study these particles better. This universe contains the three dimensions of space we have, one dimension of time, and six dimensions which curl up so tightly they cannot be seen (Marianne, 2018). The particles are so small that there is no hope they can ever be detected or observed and, therefore, sophisticated mathematical representation remains the only way of exploring this theory.
Eugenio Calabi formulated the existence of a 6-dimension manifold in the 1950s. This topological manifold would allow complex structures such as string theory to be understood easilier (Penn, n.d.). The manifold was proved by Shing-Tung Yau in 1978 and would be referred to as the Calabi-Yau manifold. These manifolds were difficult for physicists to incorporate in the string theory. A physicist called Ed Witten who was well versed in mathematics and the mathematician Michael Atiyah were able to translate the manifolds in a way physicists could apply them to the super string theory. This application of new mathematical models into developing theories in physics is a critical element in the advancement of both sciences (Vinitsky-Pinsky & Galili, 2014). An interdisciplinary approach among professionals and learners from both sciences is a key step in understanding the theories from both fields
Conclusion
Mathematics and physics are sciences that share an intimate and symbiotic relationship. Mathematical tools are the language used in physics especially when investigating subjects which cannot be physically detected and investigated. Physics on the other hand offers mathematicians the chance to create new tools and also gives mathematicians a chance to apply abstract and sophisticated mathematical tools into useful and verifiable theories of the universe. Learning institutions are encouraged to note the relationship between these sciences and teach their students to think or approach studies from a perspective that combines both sciences.
References
Marianne. (2018, June 21). The unreasonable relationship between mathematics and physics. plus.maths.org. https://plus.maths.org/content/unreasonable-relationship-between-mathematics-and-physics#:~:text=The%20unreasonable%20relationship%20between%20mathematics%20and%20physics,-Submitted%20by%20Marianne&text=People%20often%20talk%20about%20the,a%20mere%20human%2Dmade%20tool
Penn. (n.d.). Where math meets physics. Penn Today. https://penntoday.upenn.edu/news/where-math-meets-physics#:~:text=For%20physicists%2C%20math%20is%20a,mathematicians%20to%20develop%20new%20tools
Pospiech, G., Eylon, B., Bagno, E., & Lehavi, Y. (2019). Role of teachers as facilitators of the interplay physics and mathematics. Mathematics in Physics Education, 269-291. https://doi.org/10.1007/978-3-030-04627-9_12
Vinitsky-Pinsky, L., & Galili, I. (2014). The need to clarify the relationship between Physics and Mathematics in science curriculum: Cultural knowledge as possible framework. Procedia-Social and Behavioral Sciences, 116(2014), 611-616. https://doi.org/10.1016/j.sbspro.2014.01.266
