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An axiom is a concept in logic. It is a statement which is assumed to be true without question, and which does not require proof. It is also known as a postulate (as in the parallel postulate). The axiom is to be used as the premise or starting point for further reasoning or arguments, usually in logic or in mathematics.
This means it cannot be proved within the discussion of a problem. So inside some discussion, it is thought to be true. There are many reasons why it has no proof. For example,
- The statement might be obvious. This means most people think it is clearly true. An example of an obvious axiom is the principle of contradiction. It says that a statement and its opposite cannot both be true at the same time and place.
- The statement is based on physical laws and can easily be observed. An example is Newton's laws of motion. They are easily observed in the physical world.
- The statement is a proposition. Here, an axiom is any mathematical statement that serves as a starting point from which other statements are logically derived. This means the emphasis is on what happens if the axiom is true. Whether the proposition is empirically true is not the goal of logic. This is a more modern definition of an axiom.
Logic can be used to find theorems from the axioms. Then those theorems can be used to make more theorems. This is often how math works. Axioms are important because logical arguments start with them.
Euclid of Alexandria was a Greek mathematician. Around the year 300 BC, he made the earliest list of axioms which we know of. The following are just a few of them:
- Two numbers that are both the same as a third number are the same number.
- If A and B are two numbers that are the same, and C and D are also the same, A+C is the same as B+D.
- If A and B are two numbers that are the same, and C and D are also the same, A-C is the same as B-D.
- Two shapes that fill exactly the same space are the same shape.
- If you divide a number by anything more than 1, the quotient (result) will be less than the original number.
- "The Definitive Glossary of Higher Mathematical Jargon" (in en-US). 2019-08-01. https://mathvault.ca/math-glossary/.
- "A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a. Oxford English Dictionary Online, accessed 2012-04-28. Cf. Aristotle, Posterior Analytics I.2.72a18-b4.
- In logic and philosophy, axioms are propositions rather than statements. See the page "proposition".
- "Definition of AXIOM" (in en). https://www.merriam-webster.com/dictionary/axiom.