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Notes in Mathematics and Computer Science - Mathematical Maturity - Issue #1

Fares AlHarbi
Fares AlHarbi
Welcome to Notes in Mathematics and Computer Science! I’m aiming to let this newsletter as sharing notes from my ongoing research. I’m planning to write in Mathematics and its relation with Computer Science.
Usually, I will start with a certain subject. Showing its current state, and what’s there right now – Literature review. After, I will discuss every part in detail as much as I can. Why? Trying to go deeper on the subject and developing a new understanding of it.
Nice to mention, I would be happy if someone share his thought with me, and started a long discussion about the subject.
Since it’s the first issue, I will not discuss a technical subject, rather I’m opening the playhouse with “Mathematical Maturity” – What every mathematician seeking.

Mathematical Maturity - A Personal Aspect
I remember, well and clear, the first sign of it. The first time I feel it. One day, I opened the book and trying to explain hyperbola equations and properties. It was obvious, clear, something you can see in your head, imagine it, and enjoy by reading it. That was my starting point in the world of Mathematics.
At that time, I didn’t know what it is, why I see it, I feel it, but its beauties were enough to make a tough decision: That’s my road. There I belong. Later, I learn it’s called “Abstract Thinking”.
The following years were critical. It started with a feeling but ended by a lot of research papers and books around, different visions, approaches, and aspects, re-and-re analyze almost everything.
Reduction, Freedom
The core of Mathematics is Reduction. Which in its heart: Language. A language of lazy humans who likes to simplify complicated things, and accomplish goals with the least possible of work. One of my professors told me a joke:“The mathematicians are smart because they are lazy.” I couldn’t agree more.
The upper element is “Freedom” – Literally, I mean it – you have to be free in interpreting concepts in different terms. When you have that sense, your mathematical maturity will develop over time. By the end, you will see the “empty space” as “Product of Identity”.
Knowledge, Derivation
Sets, Differential Calculus, Abstract Algebra, Analysis, Geometry, Numbers, Matrices, and lastly but not last Topology. How these things at first thought looks separated, but deeply, is connected with elegance.
To answer a mathematical question, you need two things: knowledge, and derivation. Started as a beginner, you don’t know enough about the subject. You need to study it in-depth, looking for sources of derivation. For example, if you need to show that the summation of two even numbers is even, you cannot show it without knowing the definition of “even number”. Does this enough? Absolutely not! Even you know the definition, you have to know how to derive the conclusion from it. Someone could write:
Let x=2k, y=2u, then x+y=2k+2u ; where k,u are natural numbers.
But unfortunately, he stopped before getting the diamond. Since he didn’t derive the even number definition in x+y case, he didn’t write x+y as 2p for a natural number p . The process of derivation needed to be accurate, precise, and completed like surgery.
Developing these twos:
  • Deep understanding of the subject,
  • And derivation concept.
is important to get the mathematical maturity.
Backing to the mathematical subjects, when the mathematician arrives at a point where he stucks in a problem, his mathematical maturity will be the mentor, the guidance, and the soul mate. It will guide him to which subject he would get rich results, useful tools, and proper intuition.
Intuition, Convincing
Any mathematical work started with intuition - The Internal Logic. Something you can see, you know it, feel it, but you cannot prove. Over time, it will lead you to creative and extraordinary ideas. I guess you ask: How can I improve my intuition? The answer was in the previous section: Deep understanding. Deep knowledge. By far, it will simplify your work to tiny lines hard to get from elementary people.
You get the intuition and worked hard to investigate its correctness, eventually, you succeeded. Now, it’s the time for convincing other people - The External Logic. Here, knowing the level of the audience is very important, with a beginner you need to start from almost scratch to convince him. With a professional, his background is the major player in the process, you have to find a common ground, then build on it. Your Mathematical Maturity is measured by how you’re doing in this phase.
Connecting the dots
Each of the three headlines above contains two words, ordered on purpose. I assume the reader by now gets it, and he connected the dots:
  • Reduction, Knowledge, Intuition: Simplicity, Deepness, Believe - Relationship with Internal.
  • Freedom, Derivation, Convincing: Playing with the origins, Recombination, Showing with trust - Relationship with External.
This is the mathematical maturity in general terms. Becoming an adult in the world of mathematics.
See you in the next issue!
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Fares AlHarbi
Fares AlHarbi

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