I believe that mathematics is best-described as a language. Both mathematics and conversational language are rooted in logical structures. The foundational axioms of mathematics are, in fact, expressed in ordinary language. How can one define a limit without invoking ordinary language to describe the concepts of neighborhoods and delta-epsilon analysis? However, mathematics is far more concise than ordinary language. Mathematics is the compression of many ideas into compact equations.
Every mathematical equation could be completely expressed in essay form. The beauty of mathematics is that it describes concepts that occur in several different contexts. Every derivative in every textbook could be replaced by an essay describing what is meant by the word, but clearly this is inefficient. Why reinvent the wheel every time? While it takes some effort upfront to understand a derivative, the effort is rewarded by easier understanding of a vast range of ideas in the future. The term derivative is, in a sense, an incredibly useful vocabulary word with a definition of essay length.
Attempts to explain in plain English ideas concisely written in mathematical equations invariably lead to superficial or incorrect understanding. Take, for example, the Heisenberg Uncertainty Principle. This principle is a succinct mathematical inequality. However, the node is filled with sometimes incorrect, often long-winded explanations. Part of this is due to confusion inherent in quantum mechanics. Much of quantum mechanics cannot yet be written mathematically. For instance, we must resort to plain-English definitions of measurement like the Copenhagen Interpretation. The main reason for the length of the explanations is that most people have poor understanding of probability.
As an example of how plain-English translations of mathematical equations remove most of their meaning, consider one of Maxwell's Equations for electromagnetism. Maxwell's Equations describe most of what there is to know about electromagnetics. They account for the propagation of light and radio waves, the transmission of electricity in microprocessors and cables, the force that binds electrons to nuclei and one atom to another, why you can't push your hand through a brick wall, etc. The following equation is known as Faraday's Law.
Curl of E = -∂B/∂t
E represents the electric field and B represents the magnetic field. The ∂s indicate partial derivatives.
The plain-English translation is usually A changing magnetic field creates an electric field. If one were seeking an encylopedic but useless knowledge of science, this plain-English version might be suitable. However, in less space, the mathematical equation says a huge amount more.
First the boldfaced E and B imply that the electric field and magnetic field are vector quantities. This means that they have both magnitude and direction, which makes sense. Since the Lorentz Force Law shows that E and B are proportional to force, they must indicate both in which direction the force accelerates an object, as well as how much the force accelerates the object. The partial derivative is the limit as an increment of time shrinks to zero of the change in magnitude and direction of B during that increment of time, divided by that increment. The concept of a limit is very precisely defined in calculus so that there is never any confusion as to the meaning of a derivative. The curl of the vector E would take many paragraphs of ordinary language to explain, but it basically describes the directions of the electric field produced around a point by the changing magnetic field at the point.
My purpose is not so much to explain the equation, as it is to draw attention to the huge amount of information that is lost when a physical law is stated in plain English rather than mathematics. Eventually I could come up with a tedious essay that completely describes that Maxwell equation. The concepts I would need to express, however, reoccur in several other places. The remaining Maxwell's equations, the equation that governs diffusion of dopant atoms in a crystal, the dynamics of matter in the universe, etc. use the concepts of vector calculus as well. If people would spend some time thinking about a few fundamental mathematical principles instead of asking for mathematics in ordinary language, they would have a greater appreciation for and a better understanding of mathematical fields like physics and economics.