Reflection on Equations of Science
At the very onset, let me make it categorically clear to the readers that this write-up has nothing to do with the nature of equations as to whether they are invented or discovered or to undermine the role of inequalities that far outweigh equations in terms of their significance and importance in mathematics. This I will leave for philosophers of mathematics to answer. On this point sufficient enough will be to quote Robert Lanza from his acclaimed book ‘Beyond Biocentrism: Rethinking Time, Space, Consciousness, and the Illusion of Death’:
“One might see that the physical world is not the same as the abstract mathematics or even simple logic we might use to describe it. Logic demands symbolic thinking, whereas the actual world doesn’t have to play by those semantic rules.”
Coming back to my point, this write-up, as the title suggests, is all about equations and the role they have played in shaping the modern world. Before I delve deep into answering this question I would love every one of us to ponder over the point as to whether our world has always been the same as it is today. If it has changed, which in reality is a fact, what we owe that change to? Without any doubt do I think that we owe the change to equations of Physics and Mathematics formulated over a time extending over hundreds of years.
So what are equations? Why do we need them? What is the role they have played in making the world as it is today? Ian Stewart, the British mathematics popularizer, while answering why we need equations, writes in his book ‘In Pursuit of the Unknown: 17 Equations that Changed the World’:
“Equations are the lifeblood of mathematics, science and technology. Without them, our world would not exist in its present form.”
He also writes:
“The course of human history has been redirected, time and again, by an equation. Equations have hidden powers. They reveal the innermost secrets of nature.”
“Equations model deep patterns in the world.” The power of equations lies in the philosophically difficult correspondence between mathematics, a collected creation of human minds, and an external physical reality.”
However, equations can be scary as well. Prof Hawking in his masterpiece ‘A Brief History of Time’ writes that his publishers told him that every equation included in his book may halve his sales but despite that, he could not do away without including Einstein’s famous mass-energy equivalence relationship.
As young students, the first and foremost equation that we are introduced to is Pythagoras’ theorem that connects the three sides of a right-angled triangle. The theorem and its consequences had a gigantic impact on human history. It literally opened up our world. The power of this theorem lies in the fact that it has given birth to a new branch of mathematics called Trigonometry. This equation relates Algebra and Geometry. In terms of its application, it continues to play a key role in modern-day constructions, surveying, navigation besides having played a key role in the formulation of Einstein’s special and general relativity – the two most beautiful theories on space, time and matter. Pythagoras’ theorem also played a pivotal role in the invention of Coordinate Geometry. Its extension to triangles without right angles and the triangles on a sphere allows us to map our continents and measure our planet.
The second most beautiful equation that a student comes across is Newton’s 2nd Law of Motion expressed generally in mathematical form as F = d(mv)/dt. This law, according to many physicists like Kirchhoff, occupies a central position in Newton’s laws of motion. The discovery of this Second Law of Motion, elegantly expressed in the above form, was the dramatic moment in the history of science. This equation is the prescription for formulating the dynamical equations of motion in inertial frames. The second law gives us a specific way of determining how the velocity changes under different influences called forces. An important thing to realize is that this equation or relationship involves not only changes in the magnitude of momentum or of the velocity but also in their direction. The motion of pendulums, oscillators with springs and weights in them etc. could all be analyzed completely after Newton’s law was enunciated. It beautifully explained the motion of planets which were a complete mystery before Newton. This equation has in it a beauty that it can be derived from Schrodinger’s equation, provided the quantities it relates are understood to be averages rather than precise values.
The third class of equations that have shaped the modern world is Maxwell’s set of four equations that unified electricity and magnetism. Maxwell through these equations is considered to have made one of the great unifications in physics. The beauty of Maxwells equations is paraphrased in these words of acclaimed Nobel Prize winning American theoretical physicist Richard Feynman:
“God said let there be light and it was. Maxwell could say, Let there be electricity and magnetism, and there is light.”
The applications of Maxwell’s equations are far too many to count. From MRI scanners in hospitals to the creation of computers, from generation of electricity to magnetic tapes; it has a sole role in all of them. One of the pioneering tasks that Maxwell’s equations did was the prediction that electromagnetic waves do exist, travelling at the speed of light. The light itself is such a wave which in turn motivated the invention of the radio, radar and wireless communication. It won’t be exaggerated to mention here that Maxwell’s equations did not change the world rather they opened a new world; the world we live in.
So equations do have a profound role in our lives. Coming up with new equations at earliest will not only lead to better understanding of the universe we live in or reveal the new secrets of nature but they can also improve our technology and can expose us to new horizons of life as well. Other equations will be taken in a separate write-up.
The writer is pursuing his masters from Department of Physics at University of Kashmir and can be reached at email@example.com.
Disclaimer: Views expressed are exclusively personal and do not necessarily reflect the position or editorial policy of Oracle Opinions.