How to prove Math is cool

It is sad to say that Math, most of the times, is depicted as a kind of scary monster; and simply that's not fair… usually the aversion is not due to math actually being bad but due to bad representation it gets.
By writing this post, my purpose is to get rid of some misconceptions about Mathematics, to show its beauties and to prove it isn't so bad; the article is also intended to make you learn a little of mathematical reasoning.
In order to do that, I'll use a simple language with some technical reference, I'll show one of the most elegant examples of what math actually is and by a pseudo-formal reasoning I'll prove Math is cool.

Mandelbrot Set: the "heart" of Mathematics 💖

Before you start reading, keep in mind that...

You're a Mathematician! You're proving something!
You have to use all your intuition and imagination! And then demonstrate!
When you try to crack the problem, you should write down everything you think (everything!), then you judge if the idea was brilliant or silly.
It's good practice to have fun along with struggling (poor mathematicians… 😔)

Declare our purpose

As soon as you feel comfortable with your new identity of mathematician, you should start to state what you want to demonstrate; that's our conjecture.

"Mathematics is cool"
Th. (Thesis)

Translate Th. in mathematical language

Let's define the following function:

C(T_n) = coolness of T_n
P.1 (Property 1)

Where T_n represents the n^th topic of Mathematics

By P.1, we can rewrite Th. as follows:

C(T_n) > 0
∀ n ∈ ℕ

What did we just do?

We defined the property P.1 that represents the coolness of the n^th topic of Mathematics, we also defined the n^th topic cooler than the (n - 1)^th topic. So the grater is n, the greater is the coolness of T_n; it is like saying "the more you go in dept in math and the more you discover cool things".

The Proof

In proof theory, a demonstration needs, to be valid, a correct mathematical proof; and a method to get it.
In this case, we'll take advantage of Mathematical induction principle^[1], because it is almost always used to demonstrate that a property holds for every natural number.
To keep it simple, you need to check two things (cases):

If the property is satisfied by a reasonable number, which is usually 0.
If the property is satisfied by a general natural number "n" and "n+1"

If the two conditions are met, then the property is satisfied by all the natural numbers greater than or equal to the starting number.

Base Case: The Mandelbrot Set

We need to find the natural number for the first case, this case has got a name, the "base case". And our base case will be a super cool topic: the Mandelbrot Set.

The Mandelbrot Set is mostly know due to its representation on the complex plane, because it is a particularly suggestive fractal. A fractal is geometrical shape that repeats a part of itself infinite times, and the repetition gets gradually smaller, infinitesimal; this property is called "self similarity".

What's inside Mandelbrot?

Another Mandelbrot! As I told you, fractals auto-replicate in a very similar form, but Mandelbrot is special because it has got super fascinating spirals!

Zoom in Mandelbrot Fractal^[2]

Relations with other sets and fractals

One of the most unexpected things about Mandelbrot Fractal is that you can find it in other famous fractals!
As an example, the Collatz Fractal is related to one of the most difficult open problems of our time: the Collatz Conjecture. Open problems are generally unsolved demonstrations of conjectures or hypotheses, they haven't to be necessarily true, it is just needed to demonstrated if they're always valid or not.

The Collatz Fractal: do you see our Mandelbrot?

There is another wonderful fractal containing Mandelbrot Set, it is the Julia Set Fractal. This set has got a complementary set: the Fatou set. Julia is a "chaotic" set, whereas Fatou is "regular"; but neither of them is unique there exist infinte Julia and Fatou sets.

An animation of similar Julia Sets Fractals, of Maxter315

In the end?

I think it is unnecessary to say that such a beauty is also tremendously cool, we just demonstrated our base case.

Are you sure...?

Now we know that Mandelbrot is cool, but we actually don't know the starting number n₀!
Do you remember the first definition of Mandelbrot I gave you?
I told you it is the "heart" of Mathematics: the "heart" of a set of topics has necessarily to be the first topic, hence n = n₀ = 0.
By our reasoning we found out that:

C(T₀) > 0

Induction Step

Next, we need to verify the second condition for the second case, it is called the "induction step":

In order to apply the Mathematical Induction, it is crucial to think about a relation between T_n and T_(n+1); in other words it is the relation between a topic and the next.
As we know that mathematical topics are strongly bonded together and similar to each other, it's easy to conclude that the next topic, which is the same as the previous but it goes even further, has to be even cooler! (e.g. the Julia and Collatz Sets)

Let's write it in mathematics:

C(T_n) > C(T_{n - 1})
∀ n ∈ ℕ

Conclusion

This is what we got:

C(T₀) > 0
C(T_n) > C(T_{n - 1})
∀ n ∈ ℕ

Do you see it?
We know that the first topic is cool and any topic is cooler than the preceding, thus we know that all topics are cool in Mathematics and they became way cooler the more you go forward, by Mathematical Induction we demonstrated Th; now it is a Theorem.

To summarize…

We declared what we wanted to demonstrate
We defined a property in order to write the thesis in formal mathematical language
We got some intuitions
We demonstrated it by Mathematical induction
We got a theorem

Final Notes

So, ~~maybe~~ probably someone is arguing that my theorem isn't demonstrated and so on… And they're right, my theorem isn't demonstrated at all, but it wasn't intended to. I wanted to make you see mathematics in a different way, to put to your attention some of its beauties, to give you a model of mathematical thinking and problem solving, in a new, fancy manner.
Obviously that math is cool isn't a matter of fact but it's absolutely real for somebody.

Let me know in the comments if you love math now! So my theorem is demonstrated!