What is Mathematics 2

Basic ideas of math developed from observing patterns in the environment. But math seems to be breaking away from physical reality. Developments in math have been focusing increasingly on its abstract aspects.

The representation of numbers & operations with symbols, eliminated all references to any physical context.

For example, once a problem from real life was written down symbolically as 5 – 3 = 2, a radical mathematician could ask what then is 3 – 5? Mathematicians solved the problem by inventing integers!

Another such related question was what is (-3) X (-3)?

The above questions show an important difference between mathematics and physical sciences. In physical sciences questions asked are about phenomena observed in the environment. They try to explain these phenomena by conjectures, theorising and experimentation. The ultimate validity of the answer is whether they explain the observations.

But in math, as we have seen above, questions can be asked without any reference to “physical reality”.

Such questions were raised from time to time and were responsible for “pushing the limits” of math.

What is Math?

There are 4 views about ideas in math, take for example the number three.

1. Physical - Three is a common name for sets of “three” things. But math also has objects outside the physical reality.

2. Psychological -Three is an idea in our mind. Obviously math ideas are more unlimited than our minds.

3. Anti-Realistic - There is no such things as three. Math is invented.

4. Platonic -Three is an abstract idea which exists (Platonic) outside, time, space, senses etc. This math can be discovered. Math is completely embedded in physics, chemistry and all sciences & life itself. If physics is true, the math embedded in it also should be true.

There are many structures in math, one of them being the structure of natural numbers. Euclidean geometry, non-Euclidean spaces etc.

Development of an Internal Logic

To answer such questions math developed an internal logic of its own. Every new idea would be accepted by fellow practitioners who will check if it is logically derived from existing knowledge and is consistent with it.

A new idea does not replace an older idea. It is just extends the breadth & depth of that idea and the discipline of math. In that sense no previous concept in math is ever proved wrong by a new concept.

Math develops like a spider’s web, with new strands being attached to older strands as well as new points.

But what kind of questions can be asked in math? For this, mathematicians collectively evolved certain laws which had to be adhered to. Fundamental Laws of Arithmetic is one such law. This specifies the way numbers & operations can be manipulated.

But such laws are very few. There is a lot of freedom in mathematics to ask questions. Because of this freedom, numerous branches of mathematics have developed over the centuries.

In the last 2 centuries the pace of development in math has accelerated. Collaboration and exchange of ideas increased dramatically with invention of communication technology.

Today there are so many new areas in math that it is difficult for even great mathematicians to keep track of all the developments in the field. The days when the field of math was dominated by a Newton or Leibnitz or Gauss, who worked simultaneously in many branches of math, are over.

Math is the Science of Patterns

Today, math is seen as a tool for understanding, quantifying and explaining patterns. It is defined as a “science of patterns”.

The world around us is constantly changing in all aspects – physical, social, economic etc. The changes also seem to follow discernible patterns.

These patterns can be of various types — numerical, shape, motion, behaviour, population, voting, patterns, and repetition of events. They can be imagined or real, visual or mental, static or dynamic, qualitative or quantitative.

Math is used to study patterns in any complex field and understand the nature of a problem. If a problem is understood correctly, then the solution is easy.

Due to digitisation and electronic communication, the amount of data which is available in any field is humongous. The latest fields of math – Artificial Intelligence, Machine Learning & Data Science are being used for locating patterns in such data. It has even acquired a new name; data mining.

There are instances where AI seems to suggest ways of approaching problems which have eluded solutions for centuries.

Math is no more just about computations. In the last 50 years, computer software has taken over all computations.

Math is about understanding complex and abstract structures in any field and using mathematical tools to unravel the complexity.