Question

What is the need for introducing axioms?

Answer 1

Hint: Define what an axiom is, related to mathematics. Find how axioms are used in two related by distinguishable senses with examples and thus find the need of axioms.

Complete step-by-step answer:
An axiom can also be called a postulate Axiom is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. As used in mathematics, the term axiom is used in 2 related by distinguishable senses: “logical axioms” and “non-logical axioms”.

Logical axioms are usually statements that are taken to be true within the system of logic.

For e.g.:- (A and B) implies A.

While non-logical axioms are given by e.g.:- a + b = b + a.

They are substantive assertions about the elements of the domination of a specific mathematical theory.

The need for introducing axioms is that axioms depend upon a certain primitive notion like points, straight lines, planes and space. But this was not enough to deduce everything. They had to be set up for certain statements, whose validity was accepted unquestionably. Thus there was a need to introduce axioms.


Note:
Non-logical axiom is simply a formal logical expression. It might or might not be self-evident in nature. For an axiom to be “true”, it is a subject of debate in the philosophy of mathematics, whether it is meaningful or not.