Mathematical Thinking: What is Mathematics
The key feature of mathematical thinking is thinking “outside the box”.
Inspired by KEITH DEVLIN: Introduction to Mathematical Thinking (Fall 2013) BACKGROUND READING
All the time schools are trying to teach mathematics to apply various techniques and procedures to solve maths problems. They did not tell me what the subject is about. This is 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.
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. In other words, they both are related. If architecture wants to build a building in his first step he created a map and does some mathematics to create the 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, and computer science they do all kinds of school and college 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, and geometric diagrams. This represents the abstract nature of patterns but mathematical notations are the right way to represent the algebraic expressions and formulas.
For Example
Commutative law for addition in English 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. (4The 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 understand abstract concepts — a shift from performing calculations to understanding the nature of the problems.
KEITH DEVLIN: a British Mathematician and science writer wrote in his book:
Proving something was no longer a matter of transforming terms by rules, but a process of logical deduction from concepts.
For Example: In the nineteenth century:
y =3²+ 3x − 5
Function: produce a new number y by any given number x.
Then revolutionary Dirichlet came along and said, to 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 shifts their interests from computations to understanding underlying concepts. Computations and procedures are important but they are not enough.
We can see the implementation of maths everywhere in nature as galaxies spiral, seashells curve, flower petals, and even high tides in the sea are in the form of a spiral.
