What is Mathematics 1

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As we near the end of the book, we once again try to understand what mathematics is in the light of how we have developed the theme through various chapters. It is said that math is the study of patterns &amp; relationships. What does this mean? We will try and explore this statement.

Math is in our Brain

Neuroscience research indicates that there is part of our brain where ideas related to math are processed. This part of the brain tries to process questions like &ldquo;how many&rdquo; and &ldquo;how much&rdquo;. Math grew from such rudimentary skills which evolution has endowed us with.

Natural Numbers &amp; Operations

It is very obvious that we developed the idea of numbers one to ten &amp; the four basic operations from our observations of patterns in the parts of our body and the environment in which we live and the daily transactions that we have. But the real breakthrough was the representation of numbers with numerals (which are abstract representations) and operations with symbols. This act freed humans from being tied to physical reality and encouraged them to think out-of-the-box!

Freedom from Physical Reality

For example, 5 – 3 = 2 represented in summary, an infinite variety of transactions which happen in the world. But it also allowed mathematicians to ask questions like &ldquo;what is 5 – 5?&rdquo; and even &ldquo;what is 3-5?&rdquo;. These questions may have never beeb asked (3 - 5) &amp; even considered trivial (5 – 5) if we had not invented numerals &amp; were working only with physical materials!

When mathematicians asked such questions, most other mathematicians never said that these questions were meaningless or irrelevant. They were only interested in the consequences of following from these questions. So the question &ldquo;what happens if?&rdquo; is an important question for mathematicians.

Internal Logic of Mathematics

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

Bur math is different. It has no direct relation to the physical world, though the basic ideas of math came from observing patterns in the environment. But math develops an internal logic of its own. So every new idea must be accepted by fellow practitioners who will check if it is logically derived from existing knowledge.

Questions in mathematics may not be related to events in the environment. They develop from an internal logic of the discipline itself. For example, the question &ldquo;what is 3 -5?&rdquo; arose from the statement &ldquo;5 – 3 = 2&rdquo;. It is such questions that extend the discipline of mathematics.

In that sense, every new idea extends the field of mathematics. It does not replace an older idea. In that sense no previous concept in math is proved wrong by a new concept. It just extends it. Math spreads like a spider&rsquo;s web, with new strands 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 &amp; operations can be manipulated. But such laws are very few. There is a lot of freedom in mathematics to ask questions. Because of these innumerable branches of mathematics have developed over millennia.

In the last 2 centuries the pace of development has accelerated. This is because 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.

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