While proving any statement logically, you have to depend on theorems, postulates and axioms for reasoning. It is impossible to establish logical proofs without using the mathematical axioms and geometrical postulates.
Do you know the difference between axiom, postulate and theorem? Or, when a particular statement is an axiom or not a theorem or a postulate?
To understand these terms, let’s first look into the definitions of these three terms.
What is an Axiom?
An axiom is a statement or proposition which is regarded as being established, accepted, or self-evidently true on which an abstractly defined structure is based.
The word ‘Axiom’ is derived from the Greek word ‘Axioma’ meaning ‘true without needing a proof’. A mathematical statement which we assume to be true without a proof is called an axiom. Therefore, they are statements that are standalone and indisputable in their origins.
In simpler words, these are truths that form the basis for all other derivations and have been derived from the basis of everyday experiences. In addition to this, there is no evidence opposing them.
Examples of axioms are:
- 2 + 2 = 4
- 1 x 5 = 5
- A straight line can extend upto infinity
What is a Postulate?
Now, let’s look what is postulate meaning.
Postulate means a fact, or truth of (something) as a basis for reasoning, discussion, or belief. These are the statements assumed to be true without any requirement of proof. They are built upon the knowledge that satisfies the reader (or listener) in terms of veracity. Postulates are the basic structure from which lemmas and theorems are derived.
For example, knowing that a stick A is 5 inches long and that a stick B is longer than the stick A, one can create a postulate that states that the stick B is at least 5 inches long.
Difference Between an Axiom and a Postulate
The difference between the terms axiom and postulates is not in its definition but in the perception and interpretation. An axiom is a statement, which is common and general, and has a lower significance and weight. A postulate is a statement with higher significance and relates to a specific field. Since an axiom has more generality, it is often used across many scientific and related fields.
Axiom is an older term while postulate is a new term in mathematics.
What is a Theorem?
A theorem is a statement proven based on axioms, other theorems, and some set of logical connectives. Theorems are often proven through rigorous mathematical and logical reasoning, and the process towards the proof will, of course, involve one or more axioms and other statements which are already accepted to be true.
Theorems are often expressed to be derived, and these derivations are considered to be the proof of the expression. The two components of the theorem’s proof are called the hypothesis and the conclusion. It should be noted that theorems are more often challenged than axioms, because they are subject to more interpretations, and various derivation methods.
- An axiom is a statement that is assumed to be true without any proof, while a theory is subject to be proven before it is considered to be true or false.
- An axiom is often self-evident, while a theory will often need other statements, such as other theories and axioms, to become valid.
- Theorems are naturally challenged more than axioms.
- Basically, theorems are derived from axioms and a set of logical connectives.
- Axioms are the basic building blocks of logical or mathematical statements, as they serve as the starting points of theorems.
- Axioms can be categorized as logical or non-logical.
- The two components of the theorem’s proof are called the hypothesis and the conclusion.
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