Mathematically speaking, what is an ‘Axiom’?
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Are there any axioms in Math that initially described the universe wrongly at the time they were adopted but have later proven to be an accurate description of the universe we observe now that we’ve studied more of it?
Humans discover physics. Physics exists out there already and we stumble upon it, test it, and try to understand it. The line between discovery and invention in mathematics is much more fuzzy. Negative numbers didn’t always exist. At some point, somebody thought “wouldn’t it be cool if there were numbers less than zero to signify owing money” (I don’t even know which came first, 0 or negatives, but just go with it.) And that person said “let there be negative numbers” and thus, negative numbers now exist. That’s an Axiom. Once someone decided negative numbers exist, we have a brand new playground of math to probe and explore.
Same thing happened with imaginary/complex numbers and infinity and a host of other things. What does it mean for 2 things to be equal. Defining certain operations that can be done like a union or replacement. Things like that. They are the way they are because we said so. Mathematicians throw a bunch of axioms in a blender together to discover what math drops out the other side… More or less, it’s obviously much more difficult than that.
Now, sometimes, we assume too many things at once or not specific enough. This gets us into trouble because we may end up being able to prove paradoxical inconsistencies like 1=2. At which point, we need to go back to the axioms and reword them, add prohibitions for certain activities (like dividing by 0), or throw them out. Doing so over and over for the hundreds of years modern algebra has been around has made math more and more eerily efficient at describing the natural world around us.
An axiom is a rule or a statement that is considered true without proof. For example, in planimetry you can’t draw two or more lines through two points, only one. Why? Well, because that’s how it is.
Another example is that if an object one is equal to object zero, then object zero is equal to object one.
All of mathematics rests upon such axioms.
Math is like chess. There are rules like “the board is like this,” “you get these pieces, and they move like this,” etc. There are only a few rules, which are like axioms. But there are lots and lots of different sets of moves you can make.
It’s an assumption you make. You can’t prove it, but math doesn’t work without it
Like, there’s no reason why 2 is bigger than 1 (i think). Sure, in reality 2 cherries are more than one cherry, but math isn’t based in reality. And there’s no mathematical reason for 2 to be bigger than 1. But we have to define the relationship between 1 and 2 *somehow*, otherwise nothing else works
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.
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.
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”.
Wasn’t the ship in Wall-E called the Axiom?
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.
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