Axioms & Common Notions

In mathematics, an axiom (also known as a postulate) is a statement that is accepted as true without proof. Axioms are the foundational building blocks of a mathematical system. They are considered “self-evident truths” that are used to derive other statements (theorems) through logical deduction. Common notions are similar to axioms and often refer to general principles that apply across various areas of mathematics.

Think of axioms as the rules of the game. You have to accept them as true to play the game (do math)! They are the basic starting points that everyone agrees on.

Examples

Here are some examples of common axioms and common notions:

Example-1: Euclid’s Common Notions (from geometry)

• Things which are equal to the same thing are also equal to one another.
• If equals are added to equals, then the wholes are equal.
• If equals are subtracted from equals, then the remainders are equal.
• Things which coincide with one another are equal to one another.
• The whole is greater than the part.

Example-2: Axiom of Completeness (from real number system)

• Every non-empty set of real numbers that is bounded above has a least upper bound (supremum) in the real numbers.

Common mistakes by students

Students sometimes struggle with the concept of axioms because:

• They try to prove them. Axioms are not proven; they are accepted as the starting point.
• They confuse axioms with theorems. Theorems are statements that *can* be proven based on axioms and previously proven theorems.
• They fail to recognize the importance of axioms in building a logical system. Without axioms, there is no foundation for mathematical proofs.

Real Life Application

While axioms themselves are abstract, the principles of axiomatic systems are crucial for building logical and consistent frameworks in many areas. Consider these examples:

• Computer Science: Axiomatic systems are used in logic programming and the design of programming languages to ensure consistent and reliable behavior.
• Legal Systems: Laws and constitutional principles can be seen as axioms or foundational rules on which the legal system is built. Changes (like amending a constitution) can be thought of as altering the set of axioms.
• Engineering: Engineers uses some axioms to determine structural integrity of the projects.

Fun Fact

The development of axiomatic systems allowed mathematicians to formalize their work and make it more rigorous. The idea of starting with a few basic truths and building up an entire mathematical world from them is a truly powerful concept! Before the axiomatization of geometry (e.g., by Euclid), mathematical knowledge was less organized and more prone to contradictions.

Recommended YouTube Videos for Deeper Understanding

Q.1 If equals are added to equals, the wholes are equal. Which of the following scenarios BEST illustrates this axiom?

Check Solution

Ans: C

C

Q.2 “Things which are equal to the same thing are equal to one another.” Which of the following is a direct application of this common notion?

Check Solution

Ans: A

A

Q.3 Which of the following statements embodies the axiom that “The whole is greater than the part”?

Check Solution

Ans: B

B

Q.4 What common notion is demonstrated by the following statement: “If the cost of a shirt is the same as the cost of a pair of pants, and the cost of the pants is the same as the cost of a jacket, then the cost of the shirt is the same as the cost of the jacket.”

Check Solution

Ans: A

A

Q.5 Consider the following scenario: John and Mary have the same amount of money initially. John earns \$10 and Mary earns \$10. Based on which common notion can we conclude that John and Mary still have the same amount of money?

Check Solution

Ans: B

B

Next Topic: Postulates: Geometric Assumptions