Carl Fredrick Gauss once said and I quote, “Mathematics is the queen of the sciences,” Gauss is not being poetic here, his judgment carries a sound momentum. It is what he realized after working his whole life with something that has captured the attention of human possibly since the dawn of civilization and even before.
Now, Gauss was a very great mathematician, he contributed to almost all branches of mathematics from differential geometry to statistics; so probably he had explored every aspect of the beauty of mathematics before declaring it as a queen, as a matter of fact, he was a physicist and famous Gauss theorem of electrostatics is named after him. But this is not the way society treats mathematics and by society, I mean pretty much everybody but mathematicians.
If I had to ask you that what do you think of a mathematician, you would probably refer to the people whom analytical skills such as computation abilities are highly efficient or someone who talks only in equations. While it is true that most of the mathematicians have a higher sense of computation, it is not what makes them a mathematician, however, this notion is very vague among people and titles for different nature of mathematicians are not available; so, they all are tagged with mathematicians.
There should have been a classification to clarify the personalities related to the world of mathematics just like the real world does for people of other professions. For example, a doctor is a very general word to describe someone’s medical expertise, an ophthalmologist, however, is a suitable title to depict that referring person has expertise in ophthalmology. Same holds for mathematics but sadly it is not available, probably because most of the people are not introduced to the mathematical world. So, what is this mathematical world we’re talking about? Well, it is not exactly our world and it is neither pure imagination.
You get a mathematical world when you try to crystallize your imagination and represent them in the physical world. So we can say that the basic elements of mathematics are the product of imagination and reality. Once you have these elements, they have their own world which is different from the real world as well as that of pure imagination. Consider a line between two points, the inspiration of straight line is found in nature such as the shortest distance between two points or wooden stick or dimensions of simple shapes; A line in the mathematical world represents the set of points such that there is no difference between the nature of two points. Can we draw a line in the real world? You’d probably say that a wooden stick may represent a straight line, let’s try to analyze that.
Let’s assume that stick is made up of a single element and at any given point of time all other physical variables such as temperature, pressure, gravitational potential are kept constant. Now does it represent the set of points? To answer this we should first look at elements that we can call points. In a mathematical world, a point is a dimensionless entity, but no matter what we do, we can’t reduce our element into a dimensionless entity. We know for sure that once we try to go on the subatomic level, we’ll have a lot of dimensions instead of being reduced to a zero dimension.
A line in the real world only resembles a line in the mathematical world and only approximately follows the properties of a given line. For example, a line can be extended to infinity in both directions, as you can observe it is contrary to the existence of the real world. If you try to draw a line, let’s say on earth, soon you’ll find out the major limitations of being on a finite curvature, We can also try to do this by representing a ray of light as a line but as we know, gravity is known to affect the path of light. Similar holds true for other mathematical objects or forms. This mathematical world is often referred to as Platonic World.
So, the primary question one should ask is why then the platonic world is not pure imagination? We can always imagine a line of identical points growing infinitely in both directions, so why not it is a subset of our imagination? You probably know the answer, because as one can see mathematical objects doesn’t follow free imagination, they’re rather disciplinary. For example, when we state the fact that two parallel lines in a Euclidian plane never intersect, it is strictly held as a law in the platonic world. It also holds true for our strength of imagination; you can’t imagine two parallel lines intersecting anywhere. However, a third line can be introduced as a transverse line which intersects both of them, in that case, what is not possible to be done by imagination is to conclude that the sum of their interior angles is equal to the sum of two right angles.
This is where the platonic world goes beyond the imagination and begins the subject of mathematics! This property was missing in our imagination and could only be deduced by reasoning. Now is it clever to say that platonic world is the sum of deductions and imaginations of the human mind? No! First of all, because the deduction is not limited to a single mind and secondly because there are plenty of things that have been purely observed and exist without getting deduced from established facts. For example, the realization of the existence of prime numbers. Prime numbers don’t exist until you introduce the concept of division, you’ll naturally observe as soon as you’ve introduced the division, there is a whole class of numbers that can’t be divided. So, in the platonic world, new elements are formed as new processes or new forms are introduced. This is why one needs to discover the properties of mathematical objects.
Now, how does mathematics seem to help us when it is so different and mystical from our world and minds? It turns out that we can simulate our world to resemble the mathematical world. How do we do that? We create the idea of measurement! For example, a length which resembles a line segment in the mathematical world simulates the experience of distance between two positions in the real world. We also imagine virtual mathematical objects to be imposed upon real-world objects. For example, we imagine an angle between two objects as the angle between two lines representing the orientation of objects. This way we approximate our world to imitate the platonic world so that we can predict the behavior of real world in terms of simulations because due to their ideal nature, the platonic world is way easy to analyze, deduce and study.