What is Maths? What is it not?

Mathematics, which we cannot fully understand because it is not taught correctly, is not only an unpopular field for most of us, but for some of us it has even turned into a nightmare full of fear. The many mathematical formulas and rules that we are forced to memorise, which we are not told exactly where they come from, which for some reason do not even need to be questioned whether they are correct or not, and which unfortunately we are forced to memorise, as if they had descended from the sky, have been enough to both hinder our creativity and to make us tired of our lives during our student years.

However, when we really start to understand mathematics, that is, when we learn where its formulas and rules are derived from, how they were discovered, and when we really grasp the mathematical proofs that show the correctness of these formulas and rules, mathematics can turn into a very valuable occupation that we can enjoy and that can add meaning and colour to our lives. It should not be forgotten that human beings love what they can understand.

Real mathematics is not a soulless mass of formulae and rules. Real mathematics is a thrilling adventure of discovery in order to understand the mysterious world of numbers and shapes. A real mathematician is a person who somehow finds himself/herself in the middle of this adventure, who feels the excitement of this adventure in the depths of his/her heart and cannot stop enjoying it. Anyone can do mathematics, regardless of their profession, level of education or age; the German mathematician Carl Friedrich Gauss, called the ‘Prince of Mathematics’, was only a primary school student when he made his first discovery. The French mathematician Pierre de Fermat, on the other hand, was a lawyer who studied law and served in the Supreme Court.

Maths is not done for the sake of doing something. The main mobilising force behind it is, in a word, curiosity. However, if we still want to see what mathematics is good for, it will be enough to look around us, at nature and at the wide variety of advanced tools and equipment we use in almost every field; it is thanks to mathematics that we have been able to understand natural phenomena and to make tools and equipment that make our lives easier. Mathematics is the common language of all basic sciences such as physics, chemistry, biology, astronomy and their sub-branches, which we have developed to understand nature, the universe we live in and moreover ourselves.

As you see the interesting coincidences, aesthetic beauties and still unsolved mathematical problems it contains and start to really understand them, you can personally witness that a love for mathematics, or at least an interest or curiosity, begins to form in your heart. In an attempt to mediate between you and mathematics and to end the ancient resentment, or perhaps hatred, between you and mathematics, sometimes due to lack of understanding, sometimes due to misunderstanding, I would like to offer you two small examples, one from the mysterious worlds of numbers and the other from the mysterious worlds of shapes.

As a first example, let us look carefully at the following three equations or equations about numbers;

Furthermore, the 0th power of a number a is always defined as equal to 1. That is, a°=1. 

In other words, if we multiply zero a's by each other, we get zero! I will not go into how to give an acceptable logical explanation for this absurdity in this article in order not to deviate from our main topic.

Let's make one more reminder; the factorial of any number a is indicated by placing an exclamation mark (!) right next to the number a, and the product of the numbers starting from the number 1 and up to the number a (including the numbers 1 and a), i.e. a! = 1.2.3....a. In addition, the factorial of the number zero is also taken to be 1, which may seem absurd at first glance.

I will not discuss in this article why this should be so, so that we do not stray from our main topic once again. By the way, when we use the word ‘number’ to define the factorial of a number, we mean the number 0 and positive integers such as 1, 2, 3.

In the light of these reminders, why don't you take a piece of paper, a pencil and an electronic calculator and ask yourself whether the above three equations are indeed true? Who knows, maybe they are not true. How do you know I am not fooling you! 

Joking aside, don't you find the surprising coincidences based on symmetry in the above equations strange and exciting?

Our second example will be from the world of shapes. But first, a few definitions will be useful. A circle is defined as a geometric shape formed by all points on the same plane that are equidistant from any point on the plane (this point is called the centre of the circle). In other words, every point on the circle is equidistant from the centre of the circle according to this definition. This distance is called the radius of the circle, and 2 times the radius is called the diameter of the circle. The diameter of the circle is also equal to the distance between two points located in opposite positions on that circle. The figure below will help to better understand these definitions.

Here we encounter an interesting situation; no matter which point we choose on the circle, the angle from that point to the diameter of the circle (or the angle formed by the diameter of the circle at that point) is always 90 degrees. 90-degree angles are also called right angles, and they are shown by placing a small square at the corners of the angle, as can be seen in the figure above.

This feature, which constitutes only one of the many astonishing properties of circles, was named Thales' Theorem in memory of the ancient Greek philosopher named Thales who first realised it. As it can be seen with the help of the figure above, the angle APB, which sees the line segment AB, the diameter of the circle, from the point P on the circle, is also a right angle, and the angle ATP, which sees the diameter AB from the point T on the circle, is also a right angle. For some reason, this is always the case for every point on the circle.

To test the truth of Thales' theorem, you can take a compass and a protractor, draw a similar figure to the one above on a piece of paper, then choose any arbitrary point on the circle, measure the angle from this point to the diameter of the circle and check whether this angle is 90 degrees. Suppose you find that this angle is 90 degrees. However, this result is not sufficient for you to prove Thales' theorem. Who knows, maybe this angle is not 90 degrees for another point on the circle that you did not measure. Finding even one such point would mean that Thales' theorem is false. Therefore, if you want to prove Thales' theorem, unfortunately, you have to make angle measurements for “all” points on the circle. No one will have enough time to do this. How many points on the circle do you think there are in total? So many that you can never count them. In fact, there are infinitely many points on a circle. Then, there must be another way to prove the truth of Thales' theorem other than the above method, which we call trial and error. I think it would be better to leave the mathematical proof of Thales' theorem for another article.

The above two short and simple examples about numbers and shapes, and many other interesting examples we can give, are proofs that mathematics is too perfect, flawless, ideal, precise, strange and beautiful to be true.

Moreover, mathematics is the only reliable basis that shows us how we can use our minds correctly in order to find the truth and the right, and not to be mistaken or misled. Perhaps this is why the famous Ancient Greek philosopher and mathematician Plato's Academy had the inscription ‘No one who does not know mathematics is allowed to enter’ on the entrance door. Without knowing mathematics, we can neither understand nature nor emerge victorious from the war against ignorance. Both history and today's world are full of examples of the pathetic situations in which societies that move away from mathematics and science fall into.