The Mathematics - What is Mathematics?

 

What is Mathematics?

The mathematics that we study in school doesn’t really do justice to the field of mathematics. We only capture a brief look at it, though mathematics, as combined, is a vast and beautifully diverse subject. My point with this article is to give you an idea of what mathematics actually offers.

A Brief History of Mathematics

By looking at the start, we find mathematics in counting. We know that human started counting by witnessing checkmarks made in bones in ancient times. This lead to the foundations of counting numbers. There then happened progress in the subject. Egyptians managed to have the first equation and Greeks were the one doing a lot of work in areas like geometry and numerology Before-Christ. Chinese invented the negative numbers, and zero was used for the first time as a number in India. Then in the Golden Age of Islam. Persian mathematicians made further strides and the first algebra book was written. After this, mathematics gained its actual pace in the Renaissance along with other sciences.

Surely there is a lot more to the history of mathematics but we now simply jump to the modern form of mathematics, to which we usually regard.

Areas in Mathematics

Modern mathematics can mainly be distributed in two halves: Pure Mathematics, which is everything you study in mathematics for the sake of its own, and Applied Mathematics which comes into the picture when you use maths to help solve some real-world problem.

Both are different but overlap each other very frequently. Many times in history, someone created a whole bunch of new mathematics just out of curiosity which was interesting to him but doesn’t solve any problem. Yet, after years, someone sitting on some problem discovers that the bunch of mathematics is exactly what he needs to solve his problem.

Although pure maths does not directly involve in any practical application, yet pure maths itself has value too. It has a beauty that more often than not makes it an art.

Let's dive further into pure maths.

What is Pure Mathematics - Subjects in Pure Maths

'Number Systems' is the study of numbers and starts with 'Natural Numbers'. Having natural numbers for counting will let you do simple arithmetic. Then we involve other numbers such as 'Integers' with negative numbers and 'Rational Numbers' like one-half and one-quarter. All these numbers together with numbers like the exponential ‘e’ or pi ‘π’, make up the set of 'Real Numbers'. And finally the set of 'Complex Numbers' with every number inside it.

These collections of numbers also have some interesting properties, for example, there are infinitely many real numbers and infinitely many natural numbers, but there are more real numbers than natural numbers. Though infinity is regarded as something immeasurable and not defined, some infinities are bigger than others.

These numbers take us to 'Structures' which studies numbers by putting them in equations in either form of a variable or a constant.

'Algebra' defines rules for how you handle these equations. In the study of structures, you will also find multi-dimensional numbers called Matrices and Vectors, and the rules of how vectors and matrices relate to each other are found in 'Linear Algebra'.

The studies that feature number and everything like to do with numbers is 'Number Theory'. 

'Combinatorics' deals with the properties of certain structures like shapes, graphs, and other things that are made of discreet units that can be counted.

Objects that are similar to each other are investigated in 'Group Theory'. A famous example of this is a Rubik’s cube which expresses a permutation group.

'Order Theory' studies the arrangement of objects in some definite rules, like how something is a larger quantity than something else. Anything with a two-way relationship can be ordered. The natural numbers is an example of an ordered set.

Pure maths also concerns with shapes and their behaviour in spaces. This starts with 'Geometry', which includes Pythagorean mathematics and goes all the way to 'Trigonometry'.

'Fractal Geometry' is another fun thing where we find scale-invariant patterns, means you can zoom into them forever and they will always repeat themself, appearing the same.

'Topology' studies the properties of shapes where you continuously deform: extend, wind, fold and twist shapes but not join or tear them. For example, a Möbius strip has only one surface and one edge, no matter what you do with it. Topologically speaking - a cup and a swing needle are the same things (both have one hole).

The method of assigning values to sets and spaces and tying together numbers and spaces, intuitively interpreted as to its size is called 'Measure Theory'.

'Differential Geometry' studies the characteristics of shapes on curved surfaces. 

This brings us to another portion which is ‘change’.

The mathematics of change is 'Calculus', which includes functions and the properties and behaviour of their slopes including techniques like integration and differentiation on those functions. Also, 'Vector Calculus' is there which is the same thing for vectors.

This also comes with some additional areas like 'Dynamical Systems' which investigates systems that go through the time and change from one state to another, for example, the swing of a pendulum or the flow of fluids. And the strange-sounding 'Chaos Theory', that looks at dynamical systems with irregularities that are governed by laws which are sensitive to initial conditions.

At last, 'Complex Analysis' is the study of the properties of complex functions also regarded as the theory of complex functions. With this, we step into applied mathematics.

 

Applied Mathematics - Subjects in Applied Maths

We start with 'Physics', which uses most of the mathematics that we talked about earlier. Theoretical physics and mathematics, especially pure maths, has a close connection. It is said that the language of Physics is Mathematics.

After Physics, mathematics largely rule 'Engineering' with building things, since Egyptian and Babylonian times, highly complicated electrical systems such as power grids or aeroplanes use techniques of Dynamical Systems called Control Theory.

Other natural sciences also require mathematics such as Chemistry and Biomathematics with modelling molecules or evolutionary biology etc.

'Numerical Analysis' contains mathematical techniques and tactics used in places where analytic solutions fail to exist or become too complicated to solve completely. Instead of solving traditionally, numerical methods use simple approximations and combine those approximations to get a nice approximate answer, which can be obtained by using computer programming.

'Game Theory' investigates the most suitable choices within a given a set of rules among competing players and is used in economics and other areas like Psychology, and Biology.

'Probability' studies random events like cards shuffling or dice rolling.

'Statistics' studies methods of gathering and analyzing data. This relates to 'Optimisation', in which to determine the best choice from a set of different options or constraints.

Another subject that is classified in applied mathematics but deeply related to pure mathematics is 'Computer Science'. The rules of computer science were worked out in pure mathematics and were derived a lot before programmed computers were built.

'Computation' takes the theory of Cryptography and uses pure mathematics such as Combinatorics and Number Theory to study what can and can not be computerized in mathematics.

This is pretty much everything that pure and applied mathematics offers. But we haven't talked about the fundamentals of mathematics yet.

 

Foundations of Mathematics

This section works at the properties of mathematics and the ground of all the rules in mathematics.

The question of a complete collection of fundamental rules, called axioms, from which the entire mathematics is originated, and can we prove that this set of axioms is all enough for justifying mathematics?

'Set Theory' and 'Category Theory', combined known as Mathematical Logics, tries to answer questions like these and a famous result is Gödel's Incompleteness Theorems (two theorems) which implies that mathematics does not have a complete set of axioms, which also means that all the mathematics is kind of incomplete and made by us humans and doesn't have any natural existence. Which is quite surprising.

Seeing a manmade thing explaining everything in our universe such beautifully is actually very weird and a mystery in its own.

 

The love thing about learning mathematics is the feeling you get when something seeming terribly confusing suddenly clicks in your mind and in a epiphany moment everything starts making sense. In fact, some of the most satisfying intellectual moments ever been would be uncovering some part of mathematics.

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