A field of study can be defined as a group of theorems and axioms that explain certain phenomenon related to a particular part of the universe. For instance, chemistry is the study of all the chemicals that are present in the universe and physics is the study of the physical objects of the universe and mathematics is the study of all the numbers and ways that these numbers can be manipulated using various theorems and proofs. But these fields are not mutually exclusive. It can be safely maintained that one field study needs knowledge from another field of study in order to advance significantly. From deriving new theories in theoretical physics to the finding a balance between environmental conditions to discover new elements in chemistry; mathematics is needed. On the other hand, mathematics is a very vast topic with a raft of theorems and postulates which might give the impression that mathematics is advancing significantly without the use of other subjects. But on a more close inspection, one can see that mathematics is composed of a vast variety of subjects. Thus, no field can advance without incorporating knowledge and experience from outside that field.
Starting with physics, one can always look at various theories that have been put forward in order to unravel mysteries of the universe. All these theories had to be proved in order to make them believable. And in order to prove these theories mathematics has been used extensively in physics. For instance, the theory of relativity put forward by Albert Einstien states that energy of a moving particle is equal to mass of the particle multiplied by the speed of light squared. To prove this, he used mathematical postulated and axioms put forward by Euler and the final result itself is a mathematical formula. It is elusive to think that physics would have advanced without mathematics.
Also, chemistry, which is the study of chemicals uses mathematics extensively in order to discover new elements. Mathematics is used to find equilibrium environmental conditions in which new elements can be discovered and have stability. Finding the correct values to be used for titrations is yet another scintillating example that shows chemistry depends heavily on mathematics as well.
While this passage propounds that subjects require other subjects like mathematics to make significant strides, someone might argue that since both these fields depend on mathematics, thus, mathematics can be called out as a field of study that doesn't need another field to make advances and that the opinion that all fields of study need another field of study is debunked. But, on close inspection, one can see that mathematics itself is composed of a plethora of other subjects like coordinate geometry, complex numbers, prime number conjectures etc. And all these fields complement each other and one helps in the advancement of the other. For instance, problems that cannot be solved purely using complex numbers can be solved easily by thinking of complex numbers as points in a plane and points in a plane is part of the coordinate geometry.
In order to recapitulate, every field of study needs knowledge and experience from another field of study in order to make significant advances. Physics and chemistry depend heavily on mathematics and mathematics itself is a very broad topic and is composed of various mutually supporting fields of study.
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