An axiom is a foundational principle or statement accepted as true without proof. It serves as the basis for reasoning in fields like mathematics, logic, and philosophy. Unlike common sayings, which convey cultural wisdom or advice, axioms are universal and self-evident truths that underpin entire systems of thought.
Axiom in a nutshell
An axiom is a basic truth or principle that is accepted without proof and serves as a starting point for reasoning or arguments.
Definition: Axiom
An axiom is a statement or idea that is accepted as true without needing proof because it is considered obvious or self-evident. They are often used as the foundation for further reasoning, especially in math, logic, and philosophy.
Examples
- A thing cannot both exist and not exist at the same time.
- Things that are equal to the same thing are equal to each other.
- It is impossible for something to be both true and false at the same time.
These are statements people generally don’t question and used to build more complex ideas or arguments.
Note: It should not be confused with similar-sounding terms like axion (a theoretical particle in physics) or axon (a part of a nerve cell in biology). Each term belongs to an entirely different field and has a distinct meaning.
Etymology
The noun axiom originates from Ancient Greece and comes from the Greek word “axioma” (ἀξίωμα), meaning “that which is thought worthy” or “that which is assumed.” It is derived from “axios” (ἄξιος), meaning “worthy” or “deserving,” combined with the suffix “-ma”, which indicates the result or outcome of an action. This reflects the idea of a statement being self-evidently true or worthy of acceptance.
The concept of axioms was first used in philosophy and mathematics by Greek thinkers like Aristotle, who described them as foundational truths that require no proof and serve as the starting point for logical reasoning. Over time, the term became central to disciplines like geometry and logic, where such self-evident principles underpin entire systems of thought.
Note: The theory of evolution is not an axiom itself, but relies on axiomatic principles to build its structure and logical foundation.
Meanings in disciplines
The term “axiom” takes on distinct meanings across various disciplines, serving as the foundation for different systems of thought and practice. Below is an exploration of its application in key fields.
In philosophy, it’s a self-evident truth that requires no proof and forms the foundation for logical arguments or ethical principles.
Examples
- Nothing can both be and not be at the same time.
- The whole is greater than the part.
- All men are mortal.
In logic, an axiom is a fundamental statement or proposition assumed true within a system of reasoning, used to derive further conclusions.
Examples
- If A and B are true, then A is true. (Conjunction elimination)
- It is either true or false that A is the case. (Law of excluded middle)
- A proposition cannot be both true and false. (Law of non-contradiction)
In mathematics, it’s a basic assumption or rule accepted without proof, creating the foundation for a mathematical system or theory. There are logical (universally true statements based on logic) and non-logical axioms (specific to a mathematical system).
Examples
- Through any two points, there is exactly one straight line. (Euclidean geometry)
- For every number x, x + 0 = x. (Identity property)
- If a = b and b = c, then a = c. (Equality axiom)
In law, it refers to a fundamental legal principle or maxim that is widely accepted as a basis for justice or legal reasoning.
Examples
- Innocent until proven guilty.
- No one should be a judge in their own case. (= Nemo iudex in causa sua)
- Where there is a right, there is a remedy. (= Ubi jus, ibi remedium)
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FAQs
An axiom is a self-evident principle or statement accepted as true without proof, forming the basis for reasoning in fields like mathematics, philosophy, and logic.
Examples include:
- The shortest distance between two points is a straight line. (Geometry)
- Things equal to the same thing are equal to each other. (Logic)
Synonyms include:
- Principle
- Postulate
- Maxim
- Truth
- Fundamental rule
The belief in God is not typically considered an axiom in philosophy or theology, as it is not universally self-evident and often relies on faith or argumentation, unlike axioms which are universally accepted starting points.
