Mathematically speaking, what is an ‘Axiom’?
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It’s a statement that can’t be proven (or disproven), but is assumed to be true so that we have a starting point from which to build up the rules of mathematics.
Think of the game of tag. There are some basic rules that you need to accept in order to play tag:
1. Someone is “it”
2. others are not “it”
3. when “it” touches someone who’s not “it”, that someone is now “it”
If you do not accept those rules, you do not get to play tag.
Likewise, in order to do math you must accept certain things a a given. Sorry I don’t have a good example, but someone else might be able to.
A law that you have to assume to be true without proving it. A base law that you derive other laws from.
For example “if it isn’t true it must be false”. You can’t really prove that, it’s kinda a definition of what true and false mean. But from that you can construct more complex logic rules.
An axiom is a logical statement that you decide you’re going to just assume is true. For most people, these will be obvious and well-established things, like
a=a
and
two parallel lines do not intersect
You need these kinds of assumptions for logic to have something to build on. It’s possible to logically prove these statements, but only by taking *other* statements as axiomatic – and so on, forever.
Your everyday life is built on axioms like “there is a reality external to my mind” and “my senses are able to perceive information from that outside reality” and “my mental model of reality, is reasonably accurate”. You have to assume *something* to get anywhere.
Notably, axioms do not have to be true. The geometry you learned in school is Euclidean geometry. Euclidean geometry takes it as axiomatic that planes are flat, and lines are straight. You may have heard that space is curved, and Earth is a sphere. In real life, [parallel lines frequently *do* intersect](https://notesychs.weebly.com/uploads/1/8/5/4/18542518/8163074_orig.png), and the interior angles of a triangle [don’t have to add up to 180°](https://d2r55xnwy6nx47.cloudfront.net/uploads/2020/03/ShapeU_Sphere08.jpg).
Have you ever seen a child repeatedly ask a parent “why?”?
“Why do I have to wear a raincoat?” So you don’t get wet. “Why would I get wet?”
Because it’s raining. “Why is it raining?” BECAUSE IT IS!
That last one is an axiom. It’s raining, and there is no reason for it.
In math we can make a statement like “The square root of a prime number greater than 1 is always irrational.” Then you ask “why?”. Some Mathematician gives you a proof and for each step of the proof you ask “why?”, so he gives you proofs for each step and again you as “why?” At some point the mathematician runs out of reasons and says “because that’s the way math is.” That thing that doesn’t have a reason is an axiom.
There are a limited number of axioms. They are the building blocks for math. All math is made of combinations of those axioms.
1 + 1 = 2
But why?
Just accept it
In math, axioms are a set of statements which you simply assume to be true without proof. You then try to deduce what else is true, assuming that the axioms are. When you succeed in proving some interesting consequence of the axioms, you call it a theorem.
For example, there is an axiom in geometry called the triangle postulate. It states that the sum of the angles of any triangle is equal to 180 degrees. One interesting consequence of this axiom is the Pythagorean theorem – in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the two other sides. This theorem only holds true if you assume the triangle postulate first. If you don’t, then what you have is a non-Euclidean geometry, and the Pythagorean theorem does not work.
A logic system is roughly like a Lego set. You have bricks, and you have ways to connect the bricks into bigger assemblies. In logic you have rules of inference that tell you how to connect smaller things into bigger things, and axioms are the ground truths, the bricks you get out of the box.
In order to prove something in mathematics you need to have a chain with no weak links. Let’s say you prove that C is true, but in that proof you assume that B is true. Your proof that C is true doesn’t hold unless you also prove that B is true. Your proof that B is true assumes that A is true, but that A is true is so fundamental that it’s universally agreed upon, so you don’t have to prove that in order for your proof that B (and thus, C) is true to hold.
A in this case, is an axiom.
When we say that a mathematical fact is “true” or “false”, that’s not really a statement about the real world. Circles and triangles and numbers and equations don’t really “exist” outside of textbooks and the minds of mathematicians. Mathematics is *useful* for getting things done in the real world, like building bridges, but there aren’t any actual mathematical objects in a bridge, just bits of metal and stuff. We’ve figured out ways to do mathematics and then relate it to real life in such a way that it works out well.
Instead, mathematical “facts” only exist in the context of systems that are built on axioms. For example, everyday geometry is built on axioms like “you can draw a straight line between any two points” and “all right angles are the same”. Using these axioms, you can build up lots of theorems like “the angles at the base of an isosceles triangle are equal”. And that’s very useful for real life, because real life happens to correspond quite well to these axioms. But those theorems are only “true” in the sense that they can be derived from that particular set of axioms. If you use a different set of axioms, then different theorems are “true”.
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