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Archive for **February 16th, 2022**

### Mathematical Thinking? Thinking “outside the box”? Isn’t true for all scientific discoveries?

Posted February 16, 2022

on:**Inspired from KEITH DEVLIN: **

**Introduction to Mathematical Thinking (Fall 2013) BACKGROUND READING**

# We can see the implementation of maths everywhere in nature like galaxies spiral, seashells curves, flower petals, even high tides in the sea is in the form of a spiral.** **

**Note**: I discovered that all my university students were using specific equations and have No idea what a method is. I had to list for them a **taxonomy of methods** for them to comprehend the rich ways in perceiving thinking in solving problems

Why most pictures in mathematics and physics cram an entire board like this? Isn’t it a negative method that may alienate novices?

All the time schools Do **Not tell what the subject of math is about**.

Is it Not like teaching chess and various techniques but did not tell what the game is? The true nature of math is very beautiful and fascinating.

Everything around us is a number, in numbers…

Our solar system or living beings everything is created in a specific manner and functionality. They are created with special mathematical calculations and functions which exist in the whole universe.

These special mathematics and functions are called the **Fibonacci number and golden ratio**.

And they both are related.

If an architect wants to build a building, his first step is to create a map and do some mathematics to create an architecture of the building.

In today’s world, understanding the nature of mathematics is very important and valuable for any citizen. Some people graduated in **mathematically rich subjects **like engineering, computer science, physics, chemistry, biology…

Those graduates **do all kinds of Maths** without knowing the nature of the subject or without understanding what constitutes **modern mathematics.**

According to a new definition, mathematics is a “**science of patterns”. **

Mathematicians define patterns — numerical, shape, motion, behavior, population, voting, patterns, and repetition of events.

These patterns can be imaginative or real, visual or mental, static or dynamic, qualitative or quantitative. These patterns are the reason behind the **various new fields**.

In data science or Machine learning, algorithms find patterns in the data and solve the problems.

**Arithmetic and number theory**study the patterns of numbers and counting.**Geometry**studies the patterns of shape.**Calculus**allows us to handle patterns of motion.**Logic**studies patterns of reasoning.**Probability theory**deals with patterns of chance.**Topology**studies patterns of closeness and position.**Fractal geometry**studies the self-similarity found in the natural world.

Mathematicians use complex and complicated algebraic expressions and formulas, geometric diagrams. This represents the abstract nature of patterns. The mathematical **notations **are the right way to represent the algebraic expressions and formulas.

**For Example**

Commutative law for addition is:

*When two numbers are added, their order doesn’t matter*

In symbolic form:

*a+b=b+a*

KEITH DEVLIN: a British Mathematician and science writer wrote:A page of sheet music represents a piece of music; the music itself is what you get when the notes on the page are sung or performed on a musical instrument.

It is in its performance that the music comes alive and becomes part of our experience; the music exists not on the printed page but in our minds.

The same is true for mathematics; the symbols on a page are just a representation of mathematics. When read by a competent performer (in this case, someone trained in mathematics), the symbols on the printed page come alive — the mathematics lives and breathes in the mind of the reader like some abstract symphony.

For example, mathematics is essential to our understanding of the **invisible patterns of the universe**.

In 1623, Galileo wrote,

The great book of nature can be read-only by those who know the language in which it was written. And this language is mathematics.(The Assayer. This is an oft-repeated paraphrase of his actual words.)

The new and unique concept of mathematics is not to perform calculations or computations but to u**nderstand abstract concept**s — a shift from performing calculations to understanding the nature of the problems.

KEITH DEVLIN wrote:Proving something was no longer a matter of transforming terms by rules, but a process of logical deduction from concepts.

For Example of an equation from the the 19**th century**:

y =3²+ 3x − 5

This **Function: produce a new number y by any given number x.**

Then revolutionary **Dirichlet **came along and said, forget the formula and concentrate on what the function does in terms of input-output behavior. He said :

**The function is any rule that produces new numbers from old ones.**

There’s no reason to restrict your attention to just numbers. **A function can be any rule that takes objects of one kind and produces new objects from them.**

From this concept of function study of the behavior of functions started.

For example, does the function have the property that when you present it with different starting values it always produces different answers? (This property is called **injectivity**.)

There is an old saying; “If you give a man a fish you can keep him alive for a day, but if you teach him how to fish he can keep himself alive indefinitely”. This is the same for maths.

With the increasing complexity of new problems mathematician shift their interests from computations to understanding underlying concepts. Computations and procedures are important but they are Not enough.

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