Mathematically speaking, what is an ‘Axiom’?
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– Math is like games. Axioms are the rules of a game.
– You can design a game with with whatever rules you want. In math, you can develop a theory with whatever axioms you want.
– However, not all rules make fun games. Likewise, not all axioms make interesting or useful mathematical theories.
People tend to use [ZFC set theory](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) as an example of an axiomatic theory, but I think it gives the misleading impression that axioms are always these low level rules that are “fundamental” and set in stone.
In practice, axioms are much more fluid than that and they exist in higher level theories: [groups](https://en.wikipedia.org/wiki/Group_(mathematics)), [rings](https://en.wikipedia.org/wiki/Ring_(mathematics)), and [vector spaces](https://en.wikipedia.org/wiki/Vector_space) have way more relevance in practical applications.
An axiom is a statement that you accept as true, so that you can move on with your life, and continue working on whatever it is that actually interests you. If you accept the axiom as true, then you can use that to prove a bunch of other stuff is true as well.
Axioms are fundamental assumptions that we propose, in order to make a logical system sound.
If I propose that 1+1=2, and if I propose that 2+2=4, then it follows that 1+1+1+1=4 as well. That last step is a deduction that follows other evidence, but where did that first evidence come from? Who says 1+1=2 in the first place?
In most sciences we can simply look at the world and answer that “we’ve measured it.” But in maths we can’t do that; it’s abstract. There’s nothing in the world that forces the assumption that 1+1=2. You could look at one apple and then another, but there’s no force in existence that defines those as _2_ and not _aX7;h_.
Because maths is abstract, there are a lot of logical trails you can follow to get to basic rules of it, and when you do, at some point you’ll simply have no more deductions to make — only fundamental assumptions to propose, without which your mathematics wouldn’t work.
And those are axioms; proposed fundamental assumptions, without which a logical system wouldn’t be sound.
Formal logic, and maths, has about a dozen of them, IIRC. Literally all else in maths can be derived from those few axioms. Everything. Every single computation you’ve ever made or will ever make in your life.
If you want to read more on this, look into Russell as a starting point. Very gifted logician, and a very clear communicator too.
The term axiom refers to a fundamental fact. This is the case in philosophy as well (and in ancient times many mathematicians were also philosophers).
One equals one. It’s both apparent and true, and doesn’t really benefit from deconstruction
They are the parts if math that are true but can not be proven. However, if they were not true then nothing ales in math based in them would be true either. a=a is an axiom.
Wasn’t the ship in Wall-E called the Axiom?
You can’t make any claim without making at least 1 assumption.
That initial assumption is an Axiom (or a postulate). You can’t explain why this is the case, but it’s just a starting point to build logic off of.
For a math example:
1 + 2 is the same as 2 + 1
Why are these the same? There’s no reason. “It is, because it is”. No matter what order you add two numbers together in, they equal the same thing.
Axioms in math often seem like “no duh” rules, because they’re some of the first things we learn in preschool.
A few Axioms given by Euclid’s Elements include:
1) It is possible to draw a straight line from any point to any other point.
2) It is possible to extend a line segment continuously in both directions.
3) It is possible to describe a circle with any center and any radius.
4) It is true that all right angles are equal to one another.
If anyone asks “why” about any of these, you just say “because we said so”.
An axiom is simply something you assume to be true. You can put together any number of axioms you like and proceed. If your axioms contradict in any way you have an uninteresting and useless system. If they do not you can start proposing and trying to prove things from the axioms. If you look into geometry you can see alternative systems can coexist and illuminate truths in the proper context.
Axioms are essentially things we choose to hold as truth because there is no way to prove that they are true, and they are the building blocks we use to “prove” other things.
There’s a field of philosophy called *epistemology* which is basically the philosophical study of “how we know things” and one of the most famous conclusions reached in this study (and one of the most famous philosophical ideas ever) is that virtually nothing can actually be proven, the only thing that *can* be proven is “I think, therefore I am.” Basically saying that the only thing that you can know with absolute certainty is that because you have a consciousness, you exist in some capacity, everything beyond that can not be proven.
This, as I said, is a very famous conclusion, but it is also impractical, we want to be able to make sense of our world so to get around this idea we create “axioms” which are statements that we hold to be true without any actual proof that they are. These axioms become the building blocks of math, science, and philosophy that we use to “prove” other things. But those proofs rely on the axioms being true, if you can falsify an axiom, you will either falsify or bring into question any things that were “proved” using the axiom.
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