I've heard all these terms thrown about in proofs and in geometry, but what are the differences and relationships between them? Examples would be awesome! :)
In Geometry, "Axiom" and "Postulate" are essentially interchangeable. In antiquity, they referred to propositions that were "obviously true" and only had to be stated, and not proven. In modern mathematics there is no longer an assumption that axioms are "obviously true". Axioms are merely 'background' assumptions we make. The best analogy I know is that axioms are the "rules of the game". In Euclid's Geometry, the main axioms/postulates are:
- Given any two distinct points, there is a line that contains them.
- Any line segment can be extended to an infinite line.
- Given a point and a radius, there is a circle with center in that point and that radius.
- All right angles are equal to one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (The parallel postulate).
A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective. Sometimes what an author thinks is a 'corollary' is deemed more important than the corresponding theorem. (The same goes for "Lemma"s, which are theorems that are considered auxiliary to proving some other, more important in the view of the author, theorem).
A "hypothesis" is an assumption made. For example, "If $x$ is an even integer, then $x^2$ is an even integer" I am not asserting that $x^2$ is even or odd; I am asserting that if something happens (namely, if $x$ happens to be an even integer) then something else will also happen. Here, "$x$ is an even integer" is the hypothesis being made to prove it.
Based on logic, an axiom or postulate is a statement that is considered to be self-evident. Both axioms and postulates are assumed to be true without any proof or demonstration. Basically, something that is obvious or declared to be true and accepted but have no proof for that, is called an axiom or a postulate. Axioms and postulate serve as a basis for deducing other truths.
The ancient Greeks recognized the difference between these two concepts. Axioms are self-evident assumptions, which are common to all branches of science, while postulates are related to the particular science.
Aristotle by himself used the term “axiom”, which comes from the Greek “axioma”, which means “to deem worth”, but also “to require”. Aristotle had some other names for axioms. He used to call them as “the common things” or “common opinions”. In Mathematics, Axioms can be categorized as “Logical axioms” and “Non-logical axioms”. Logical axioms are propositions or statements, which are considered as universally true. Non-logical axioms sometimes called postulates, define properties for the domain of specific mathematical theory, or logical statements, which are used in deduction to build mathematical theories. “Things which are equal to the same thing, are equal to one another” is an example for a well-known axiom laid down by Euclid.
The term “postulate” is from the Latin “postular”, a verb which means “to demand”. The master demanded his pupils that they argue to certain statements upon which he could build. Unlike axioms, postulates aim to capture what is special about a particular structure. “It is possible to draw a straight line from any point to any other point”, “It is possible to produce a finite straight continuously in a straight line”, and “It is possible to describe a circle with any center and any radius” are few examples for postulates illustrated by Euclid.
What is the difference between Axioms and Postulates?
• An axiom generally is true for any field in science, while a postulate can be specific on a particular field.
• It is impossible to prove from other axioms, while postulates are provable to axioms.
Since it is not possible to define everything, as it leads to a never ending infinite loop of circular definitions, mathematicians get out of this problem by imposing "undefined terms". Words we never define. In most mathematics that two undefined terms are set and element of.
We would like to be able prove various things concerning sets. But how can we do so if we never defined what a set is? So what mathematicians do next is impose a list of axioms. An axiom is some property of your undefined object. So even though you never define your undefined terms you have rules about them. The rules that govern them are the axioms. One does not prove an axiom, in fact one can choose it to be anything he wishes (of course, if it is done mindlessly it will lead to something trivial).
Now that we have our axioms and undefined terms we can form some main definitions for what we want to work with.
After we defined some stuff we can write down some basic proofs. Usually known as propositions. Propositions are those mathematical facts that are generally straightforward to prove and generally follow easily form the definitions.
Deep propositions that are an overview of all your currently collected facts are usually called Theorems. A good litmus test, to know the difference between a Proposition and Theorem, as somebody once remarked here, is that if you are proud of a proof you call it a Theorem, otherwise you call it a Proposition. Think of a theorem as the end goals we would like to get, deep connections that are also very beautiful results.
Sometimes in proving a Proposition or a Theorem we need some technical facts. Those are called Lemmas. Lemmas are usually not useful by themselves. They are only used to prove a Proposition/Theorem, and then we forget about them.
The net collection of definitions, propositions, theorems, form a mathematical theory.
Axiom: Not proven and known to be unprovable using other axioms
Postulate: Not proven but not known if it can be proven from axioms (and theorems derived only from axioms)
Theorem: Proved using axioms and postulates
For example -- the parallel postulate of Euclid was used unproven but for many millennia a proof was thought to exist for it in terms of other axioms. Later is was definitively shown that it could not (by e.g. showing consistent other geometries). At that point it could be converted to axiom status for the Euclidean geometric system.
I think everything being marked as postulates is a bit of a disservice, but also reflect it would be almost impossible to track if any nontrivial theorem does not somewhere depend on a postulate rather than an axiom, also, standards for what constitutes 'proof' changes over time.
But I do think the triple structure is helpful for teaching beginning students. E.g. you can prove congruence of triangles via SSS with some axioms but it can be damnably hard and confusing/circular/nit-picky, so it makes sense to teach it as a postulate at first, use it, and then come back and show a proof.
Technically Axioms are self-evident or self-proving, while postulates are simply taken as given. However really only Euclid and really high end theorists and some poly-maths make such a distinction. See http://www.friesian.com/space.htm
Theorems are then derived from the "first principles" i.e. the axioms and postulates.