Math is a fascinating language. We use the terms such as addition, subtraction, and other mathematical operations as if we fully understand what these words mean. In school, mathematics is often taught as a serious & sombre subject. It is very common to run into someone who is afraid of math purely because that person often got confused and was never really able to visualise the ideas presented. Meanwhile, there would be a group of other kids in class who seemed to understand just about everything ever taught in the class. Dear reader, this writer was never in the group that understood math. In my pursuit of trying to understand things, I have painfully been made aware that a lot of people, even my teachers, never fully understood the words they used. No one really stops to ask what these terms fundamentally mean.
We must begin by agreeing on the idea that written language was invented by humankind to save knowledge from one generation and share it with the ones that came after. This changed human development as we were not like other animals whose successive generations start at nearly the same point of intellect, save for the evolutionary changes. Numbers were not invented by someone who wanted to do complicated math. What seems to be likely is that when human civilisations started to form, specialisations came about. Each human had a different specialisation or interest in being able to contribute to their society. This is not a creation of civilisation in itself. It is in nature that these inequalities exist. If we were to take a sodium ion (Na+) and a chlorine ion (Cl–), the difference is easy to establish. Both of these ions are unstable. One has a deficit, while the other has a surplus, as we can see with the ‘+’ & ‘–’ signs, which we commonly use. However, what does this stability mean? Stability is when nothing happens over a long period of time. In other words, the overall sum of forces acting on the object sum up to zero. I know I am still using these terms that we think we understand, but I will dive deeper into what they mean as I go.
What does it mean to say that the forces sum up to zero? What is a force? I am not a purist for definitions, so I would articulate a force as being anything that causes a change in an object. A balance of these forces would mean that there are multiple forces acting on a certain object of observation. The change that can be caused by one force is equal to what another force can cause on the same object, but these two forces add up to nothing. What does this addition mean? What does sum mean? Linguistic etymology aside, it boils down to something absolutely axiomatic. If we were to take two objects, and pile them one over another and tie them together so that we now have a bigger object comprising of the two previous objects, what it means is that these two objects have been subjected to the operation of addition. Now, if we were to take a single object and break it into two parts, the object has been subjected to the operation subtraction. Where addition does something close to joining & conjoining, subtraction means the object has been broken down into two parts. If we were to take object A and object B, and add them to one another to get an object C, we can safely say the sum of A & B is C. If object B is then broken from object C, we can say that when object B is subtracted from object C, we get object A. In other words, the difference between object C & object B is object A. An empty bottle is filled with water, and then the water is emptied.
- Object A – water
- Object B – bottle
- Object C – bottle filled with water.
Now, what does it mean to say that the forces sum up to nothing? Let us consider the forces to act subsequently, instead of simultaneously. The change added by one is subtracted by the immediate one that follows. In doing so, we are talking about a system to describe the object to be able to say this. Let us say this force moves the object a little, while the other force moves the object back to where we started observing the object. This would mean that if the two forces acted at the same time, the forces would not move the body. And when an observed object cannot be described as having changed, in any manner at all, over a long time, it is stable. So, what does it mean to say the ions are unstable? Sodium has one less electron needed for achieving the state where nothing happens in the atom over a long period of time, while Chlorine has an extra electron that stops it from achieving the similar stability. When these two ions are in proximity, the additional electron from chlorine passes on to Sodium and both the ions achieve stability. This exists in nature, right from the elements that we are made of.
It is only natural that humans have similar inequalities that become specialisations. It is very easy to perceive this deficit & surplus, as the existence of that inequality is the very reason why trading exists. Someone with surplus cotton and a deficit of wheat runs into someone with a surplus of wheat and a deficit of cotton. These two individuals want to trade. The animalistic instinct kicks in. neither of the two wants to give up more than they need to. The question is how to ensure both end up with quantities of wheat and cotton without surplus or deficit. This is one transaction. Now, in a society such transactions could happen on a regular basis. We are dealing with two items, one is used to make fabric while the other is consumed as food. Like how language was invented, a method was needed to address this concern as well.
The greatest achievement of written language and the thing we had learned from its invention was that sounds could be written down as combinations of a finite number of elements – alphabets. English has only 26 letters, yet we add new words every year to the lexicon. The beauty lies in the aspect that this language was created with 26 characters, each denoting a sound to utter. When combined differently, all the letters have a composite sound which becomes the word we utter. These 26 letters/characters are the fundamental elements of language. All of them share the similar burden – denoting a sound to be made. Each represents a different sound. The existing alphabet system cannot solve the problem of articulating the surplus or deficit. This is where the natural sequence of events directs someone to break something down into uniform elements to compare. Remember, the sodium ion had just as much deficit as the chlorine ion had a surplus of – a single electron. We know this now, but it isn’t that obvious if we begin to ask what things mean.
The existing physical object, let’s say cotton for now, needs to be compared between both the individuals. How much more than one of the two does the other have? Let us say these two individuals meet somewhere and place their respective bales of cotton. At the end, both want to end up with equal amounts of cotton. Both agree to take what appears to be the excess in one and place it in the bale with lesser cotton in it. They wish to repeat the same for wheat, but one of the two has this feeling that he is giving up more cotton than how much wheat he is getting in return. At the end, both have equal quantities of cotton and wheat. Yet, that feeling of uneasiness persists in one of the two. He is unable to articulate why. So, the natural instinct to count kicks in. Let’s call these two A & B, to keep things simple. The next time they meet, A tells B to form a cup out of his hands and A wants to see how much cotton he is giving up for the wheat he’s getting. It seems natural that A begins counting without knowing he’s counting. Of course, humans even back then had 10 fingers. Every time B took some wheat or cotton in his hand and moved from the surplus to the deficit, A put up a finger. He noted that a few fingers less were put up for wheat than cotton. Inadvertently, A & B together came up with the concept of counting. What they counted became numbers. These numbers would become the elements of a new language that would articulate things not to extol their beauty like a line of prose. This new language would end up developing a set of rules that would solve much bigger problems, yet making the way these problems were solved understandable through symbolic representations, like the alphabet did with languages like Greek, Latin, English & so on, to name a few. As it becomes obvious, civilisation brought forward such advancements merely through the features of naturally occurring inequalities & differences.
Fundamental Review 