The topic I wish to talk in this post is about **Mathematical maturity**. It is connected to the abstract nature of mathematics and humans’ endeavor to cope with it.

Let me start by asking you a question. Have you ever seen a circle in real-life?. You might say yes because we all draw many in our school days using a compass and pencil.

But is that a real circle?. What if we zoomed enough using a magnifier or high-resolution camera?. We might notice the imperfections, sloppy edges, distorted by carbon powder all over it. In theory, it is only circle-like. The real circle only exists in the abstract mathematical world. Same with the case of numbers. If I say number 4, it could mean four books, four people, four sheep, etc. But the real number 4 is abstract.

In these two examples, we were lucky because at least we could relate the abstract concept to some physical reality.** What about an n-dimensional world? or infinity?** **can you related it to any physical reality even vaguely?**.

I remember myself struggling to figure out what negative frequency in Fourier transforms means during my masters.

Some mathematical concepts defy our logic and intuition. And we are not the first ones to struggle with this.

In 19th century, mathematicians were puzzled at a problem called **Banach–Tarski paradox**. It essentially says in theory you can take a sphere and cut it into pieces and reassemble so that you would get two identical spheres of the same size as the original one !.

This result caused a major shift in the approach towards mathematics. It became more abstract. To trust such counter-intuitive results we will have to be 100% sure that our usage of mathematics is right. Now we know mathematicians are so rigorous about its usage as a language :-).

*All this points to a skill called mathematical maturity. It is the quality of being comfortable with and handle abstract mathematic ideas. As mathematically matured people, we will have to be precise about the mathematical language and trust the mathematical results over our intuition even if it is counter-intuitive.*

Understanding mathematical maturity itself gives us a new perception while learning hard concepts. At some point in our professional life, we will reach such a situation. So the next time you come across a strange mathematical concept, don’t freak out!.

I end with the quote of the famous mathematician John von Neumann, **“Young man, in mathematics, you don’t understand things. You just get used to them”**.