To many people, there’s a certain four letter word that strikes great fear into their hearts: math. Mathematics has a reputation for being a subject of the elite–a terrible, confusing, jumbled mess of illogical expressions and rules, which many people just give up trying to decipher at some point. Nevertheless, many students of mathematics (formal and informal), persevere through years of algebra and arithmetic to find themselves facing a very different beast – Calculus.
In truth, mathematics IS complicated and advanced, and it took hundreds of years to develop this language–the language that can accurately describe the universe we live in. Initially, math arose to solve problems and predict outcomes in everyday life, and as humans became more interested in how the world worked, they were faced with limitations of their current mathematical theories – which is why many people throughout history worked to create new and better models of nature, leading to advanced mathematics – here is how Newton (among others) created some of the most dreaded mathematical equations that we know today.
To describe the need Newton felt for more precise mathematics, you first need a brief understanding of what math was before Newton came along and changed everything. In the ancient times (around 580-212 BC) three notable philosophers – Pythagoras, Euclid, and Archimedes – created what we know now as algebra and geometry, but still struggled to unify them. Until this point, algebra was done without the tools necessary, like a numbering system that was easy to work with (imagine doing algebra with Roman Numerals) as well as many of the symbols like the plus, minus, and multiplication/division symbols. In the late 1500s, Rene Descartes unified algebra (which was used as an analytical tool) along with the geometric shapes. He found that, a point on a plane can be described using two numbers, and from that, equations of geometric figures were born.
Around the 1670’s, two great men discovered and developed calculus independently from each other. Sir Isaac Newton of England, and Gottfried Wilhelm Leibniz of Germany, both did quite a lot of work forming a language of numbers that could accurately describe nature. It is worth noting, however, that Newton developed calculus 8 years before Gottfried, but Gottfried is known for developing modern European mathematics by introducing carefully drawn symbols and rules (many people say the equals sign ‘=’ was the work of Gottfried) – Both men will claim that the other plagiarized them for the rest of their lives, a conflict known as the “great sulk”.
So why was this new and complicated form of mathematics invented? And how does one manage to come up with such an abstract idea? Newton was foremost a physicist, and in his day, he tackled many difficult issues in physics, probably the most famous of these issues is gravity. Newton is known for developing the laws of motion and gravitation, which undoubtedly led to his work in calculus. When trying to describe how an object falls, Newton found that the speed of the object increases every split second and that no mathematics currently used could describe the object at any moment in time.
Another tale of Newton’s reasoning is that a colleague of his asked “Why are the orbits of the planets in an ellipse?”, to which, Newton took some time to ponder, and came back with the fact that the ellipses are, in fact, sections of cones, and armed with calculus, he could describe exactly how these sections behaved. These were the types of questions (in conjunction with the fact that Cambridge University -where Newton studied – was closed due to numerous outbreaks of the plague) that drove Newton to expand on mathematics and develop the concepts of differential and integral calculus.
One may ask “how does one invent a new form of mathematics” this answer is quite simple; Isaac Newton forced relationships between physical phenomena and the mathematics of the day. Through trial and error (and quite a bit of ingenuity), Newton saw the need for a whole new math, and this came from his conceptual understanding of physics. More simply, Newton already knew the concepts of calculus because he was describing gravity and planets; it was a matter of writing it down and showing proof that it works.
Nobody would claim that Newton wasn’t a brilliant man, but calculus has been refined for over 300 years with many features added (like limits, which make everything easier), meaning that no longer does one have to be a genius to understand it. I want to make a point that, a student who has a concept of algebra already knows more than Newton did when he invented calculus, and in fact, calculus was born out of what we know as geometry and algebra. Math students of the present day should not fear, calculus is not just for the elite, not just for the genius, but for anyone who has the time and discipline to learn it. I fight for equal treatment of all mathematics, let’s take away the stigma of calculus!