Mathematically speaking, what is an ‘Axiom’?
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An axiom is a logical statement that we accept as true, and can then deduce things from.
For example, we could create a set of axioms that describe how the real numbers work (saying things such as “There is an operation called the addition that has some properties”, “There is an element that we note 0 such that for every real number x, x + 0 = x” “There is an operation called the multiplication that has other properties”, etc….). Once we are satisfied with our description of the numbers, we can actually start proving things.
The key idea is that we don’t try to prove the axioms. We decide them to be true, and then math is deducing theorems from these axioms that we chose as ground-truth. With a different set of axioms, we would deduce different things.
The reason why we spend way more time studying the truths from one set of axioms over a different one, is that that set of axioms seems to match our understanding of reality well and we are able to create useful models using it. You could study what happens if you choose as ground truth 1 + 1 = 3, but you won’t get very useful results from it.
It’s a fundamental proof , that you reason up from.
It’s the start of the foundations of logic, great for critical thinking: what are the fundamental truths (axioms) in an arena? Do you have the right proven axioms? (not assumptions) Are they relevant to the question at hand? Are you making the right conclusions based on those axioms? Without violating any of the fundamental laws? eg. Are you conserving momentum energy? If you’re not then you’re probably not gonna be successful. Then reiterating to come closer to the fundamental unknown objective truth via. counterintuitive concepts
To my knowledge an axiom is a rule that cannot be further broken down. Something you just have to accept as given, upon which you can build the rest of your theory.
For example: picking any number and adding 1 to it results in the next highest number. So basically 2 + 1 equals 3 because 3 is the next number after 2.
An axiom is a basic assumption that underpins all your other reasoning, and that you aren’t likely to give up in different situations.
So in a way, it is ‘just’ an assumption, but it is one that you think is not likely to be very context dependent, and that you’ll stick with much of the time.
Most people want to live, and so for most people, a desire to live might be an axiom. You don’t make a logical argument that you want to live, you *just do* want to live (hopefully). (Maybe we can make some argument about evolution or something resulting in a self-preservation instance, but that begins outside of you, rather than internal to your logical reasoning.)
Many people might have some axioms in their ethics/morality. Maybe some people care about equality. Maybe some people care about suffering (or preventing suffering). Maybe some people care about the word of God. If they believe them as the foundation of their ethics, then we might be able to phrase those ethical principles as axioms. If someone thinks equality is important, it can be hard for them to give a *reason* they think that – it just is what they think matters.
For mathematics, axioms are the foundational assumptions of a branch of mathematics. I’ll give an example (now, it turns out that most mathematical axioms are more estoeric and weird than this example, but I think it is a decent example just to get the vibe here). Consider the assumption that “The number 1 exists.” It is hard to give *reasons* that the number 1 exists. We just assume that it exists, because well, chances are most people doing maths just need that to be the case, no questions asked. I think actual modern mathematics actually go to some deeper or more abstract level, so eventually the existence of the number 1 I think becomes something you need to prove once you get pedantic enough, but as a general idea, I think this example is ok.
Axioms are what mathematicians call “axioms”. Seriously. There are no strict mathematical definitions. Axioms is used as part of assumptions a proof can make, but assumptions can be classified differently depends on its purpose. The view that axioms are things that are self-evidently true is a very classical view traced back to Euclid, but it is not how axioms are treated in modern mathematics.
Mathematically, every proofs have assumptions, things that are assumed to be true for a conclusion of the proof to hold. Assumptions fall into a few kinds:
– Hypothesis. Something assumed to be true about a particular unknown object, because the proof is only meant to be applicable to that object. For example “If x is an odd number”.
– Axioms. Assumptions that you can use without mentioning it explicitly. People are sure that they should be true. There are at least 2 kinds of axioms, as mention below.
– Well-founded assumptions. Claim that can be conceivably false, but so far it was considered reasonable enough chance to be true to produce mathematics out of. Example, “assume integer factorization is hard”. Proof that use well-founded assumptions are considered “conditional” proof.
There are 2 kinds of axioms, depends on how mathematicians view them.
– Definitional axioms: axioms that define the limitation of what a field study. Essentially they are just definitions of terms. For example, group axioms define what a “group” is, and that is the object to be studied by “group theory”. There are no fundamental different between definitional axioms and a definition.
– Foundational axiom: axioms that just run in the background, and considered to be things that should be true. Most mathematics will use the same shared collection of foundational axioms. For example, “if something is true about number 1, and if it’s true for any natural number then it’s true for the next number, then it’s true for all natural numbers”.
Even that above is not a strict dividing line. What one mathematician considered to be foundational axiom could be definitional axiom by another mathematician.
I’ve given an answer before that might help:
To put it simply, math is about creating abstract “systems”. The rules that govern these systems are the axioms. You can create (again, to put it simply) __any__ abstract system with arbitrary rules, and as long as those rules are consistent and not contradictory, it is a valid mathematical system.
To give an example, suppose i create abstract system to “count”. I give this system a set of axioms, for example whether i have one amount and add another, or have the other amount and add the first, i should get the same result. These axioms (the fundamental axioms of algebra) let us create a system that I can, simultaneously, use to count apples, as i can to count distances, something completely different! How crazy is that!
When the real world gives us examples where this system breaks, (I went 3 miles, then another 4 miles, but the distance is only 5 miles from the beginning not 7!), is when we create a new abstract system with its own axioms – in the above example, we deal with vectors instead of numbers directly.
Another way to interpret an axiom is not as a statement assumed to be true, but rather as a conditional: “*If* this thing is true, *then* these other things (theorems) must also be true.” So, it’s not that anyone is claiming water is wet, it’s that *if* water is wet, *then* fish go blub blub. If water is *not* wet, we’re not making any claim at all and so we’re not wrong.
I like this interpretation because whenever I tried to explain axioms as assumptions, some wiseass would always say “but what if water actually isn’t wet?”
Mathematicians like elegance, so it’s generally preferred to keep axioms as simple as possible – anything more complicated than the equivalent of “water is wet” should probably be proven as a theorem.
One example of an axiom: 2 parallel lines will never meet.
And you have a bunch of axioms like these: given two points, there is exactly one line going through those 2 points. They all sounded like “duh of course it’s true” but it is useful to formalize them even if you can picture it’s truthfulness in your head. Once you formalize them, mathematicians can start building theorems using those axioms, and build more theorems on top of those first layer theorems, and so on.
But the second reason why it’s important to formalize them is because sometimes it’s not true. Take the first one I mentioned for example: 2 parallel lines will never meet. That may be true in Euclidean Geometry. But when you’re talking about Projective Geometry, the axioms include one additional point called “point at infinity,” and the axiom is modified as: 2 parallel lines meet in exactly 1 point, that is point at infinity. The list of theorems that can be derived from there are slightly different from Euclidean Geometry, although they smell the same.
Do you remember the game Lil Alchemy? It’s that game where you start with the “basic elements” (earth, wind, fire, water) and you can combine those elements, and the new elements you create, to create all kinds of things.
The starting four elements are like “axioms,” it is what you have to have to start with to make everything else.
These are your basic “undefined elements.” This is just like axioms are your basic undefined rules. You can put the axioms together to make all the other rules.
You can even make weird different types of math, “non-Euclidean geometry” if you decide different rules/axioms must be true (ex: if you decide Circle arcs are lines you get a different kind of geometry; the Poincaré half plane)
Readin and KurtWagner did a good job explaining, but I’ll add that it’s not strictly mathematic. An axiom is often taken in philosophy as well. Any statement which is assumed to be true is “axiomatic,” most notably Descartes’ “I think, therefore I am.”
His entire *Meditations* can be said to be an attempt to find the axiom of what it means to exist and/or personhood.
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