Here we are going to discuss about some basic ideas of Mathematical Reasoning. All of us know that human beings evolved from the lower species over many millennia. The main asset that made humans “superior” to other species was the ability to reason. How well this ability can be used depends on each person’s power of reasoning. 

How to develop this power? 

Here, we shall discuss the process of reasoning especially in the context of mathematics. 

In mathematical language, there are two kinds of reasoning – inductive and deductive.

Mathematical Reasoning

The basic unit involved in mathematical reasoning is a mathematical statement.

Let us start with two sentences:

• In 2017, the president of India was a woman.
• Human being weighs more than a Sheep.

When we read these sentences, we immediately decide that the first sentence is false and the second is correct. There is no confusion regarding these. 

In mathematics such sentences are called statements.

On the other hand, consider the sentence:

Women are more intelligent than men.

Some people may think it is true while others may disagree.

Regarding this sentence we cannot say whether it is always true or false . 

That means this sentence is ambiguous. 

• Such a sentence is not acceptable as a statement in mathematics.
• A sentence is called a mathematically acceptable statement if it is either true or false but not both. 
• Whenever we mention a statement here, it is a “mathematically acceptable” statement.

While studying mathematics, we come across many such sentences. Some examples are:

• six plus two equals eight.
• The sum of two positive numbers is positive.
• All prime numbers are odd numbers.

Of these sentences, the first two are true and the third one is false. There is no ambiguity regarding these sentences. Therefore, they are statements.

An example of a sentence which is vague or ambiguous:

The sum of x and y is greater than 0

Here, we are not in a position to determine whether it is true or false, unless we know what x and y are. 

For example, it is false where x = 1, y = –3 and true when x = 1 and y = 0. 

Therefore, this sentence is not a statement.