Lemma, Theorem, Axi0m, Statements

A statement is a sentence which has objective and logical meaning.

A proposition is a statement which is offered up for investigation as to its truth or falsehood.

The term axiom is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch.

Different fields of mathematics usually have different sets of statements which are considered as being axiomatic.

The term theorem is used throughout the whole of mathematics to mean a statement which has been proved to be true from whichever axioms relevant to that particular branch.

So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others.

A definition lays down the meaning of a concept.
It is a statement which tells the reader what something is.

A lemma is a statement which is proven during the course of reaching the proof of a theorem.


Logically there is no qualitative difference between a lemma and a theorem. They are both statements whose value is either true or false. However, a lemma is seen more as a stepping-stone than a theorem in itself (and frequently takes a lot more work to prove than the theorem to which it leads).
Some lemmas are famous enough to be named after the mathematician who proved them (for example: Abel’s Lemma and Urysohn’s Lemma), but they are still categorised as second-class citizens in the aristocracy of mathematics.

A corollary is a proof which is a direct result, or a direct application, of another proof.
It can be considered as being a proof for free on the back of a proof which has been paid for with blood, sweat and tears.
The word is ultimately derived from the Latin corolla, meaning small garland, or the money paid for it. Hence has the sense something extra, lagniappe orfreebie

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Difference between axioms, theorems, postulates, corollaries, and hypotheses

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.

Axioms

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.

Postulates

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.


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:

  1. Given any two distinct points, there is a line that contains them.
  2. Any line segment can be extended to an infinite line.
  3. Given a point and a radius, there is a circle with center in that point and that radius.
  4. All right angles are equal to one another.
  5. 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 x2 is an even integer” I am not asserting that x2 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.


  1. 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.
  2. 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).
  3. Now that we have our axioms and undefined terms we can form some main definitions for what we want to work with.
  4. 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.
  5. 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.
  6. 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.
  7. The net collection of definitions, propositions, theorems, form a mathematical theory.

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.

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Lemma is generally used to describe a “helper” fact that is used in the proof of a more significant result.

Significant results are frequently called theorems.

Short, easy results of theorems are called corollaries.

But the words aren’t exactly that set in stone.


A lot of authors like to use lemma to mean “small theorem.” Often a group of lemmas are used to prove a larger result, a “theorem.”

A corollary is something that follows trivially from any one of a theorem, lemma, or other corollary.

However, when it boils down to it, all of these things are equivalent as they denote the truth of a statement.

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Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results.

Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma, Sperner’s lemma).

Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).


An axiom is any assumption/statement, which cannot be deduced or proven from existing material on that topic. The axiom of parallelism in Euclidean geometry is one such example.

Axioms are used as building blocks or foundations to prove a certain statement, which is called a “theorem”. Again taking an example from the Euclidean geometry, let us consider the Pythagoras’ theorem. While providing supportive statements to the arguments we make to prove this theorem, we make use of the “axioms”.

Corollary to a theorem is a slight modification of the statement of the theorem, which can easily be deduced from the theorem itself.

A hypothesis is a statement, the correctness of which one wishes to test. Generally, in statistics, a statement is made about the data initially and is tested at a significance level. e.g. The distribution under observation is a normal distribution with μ=μo

is a hypothesis. I still have a bit of confusion with the exact meaning conveyed by the word “postulate”.


As far as geometry is concerned, “Axiom” and “Postulate” are essentially interchangeable. Thus, a postulate refers to as a proposition which is obviously true and hence need not be proven.

However,in modern mathematics there is no longer an assumption that axioms are “obviously true”. They can be looked at as the rules for a game. You cannot go against them, while playing a game, otherwise you will be disqualified.

What is a theory then ? Is it a set of theorems/results derived based on the postulates?


You have all the points covered pretty well, but I’ll just just point out that the usage of hypothesis in mathematics differs slightly from that in the sciences. A mathematical hypothesis isn’t treated as a statement to be proved but as a starting assumption that is made at the beginning of a proof. As a rule of thumb, anything that follows the word ‘if’ or ‘given’ in a theorem constitutes the hypothesis.

However, there is some overlap with its meaning in science. When you prove something by contradiction you make a starting assumption i.e. posit a hypothesis that the negation of the statement to be proved is true and then disprove it.


A scientific theory is a well-substantiated explanation of some aspect of the natural world that can incorporate facts, laws, inferences, and tested hypotheses. A scientific theory is not as clear cut as a scientific fact or law, in that facts and laws must be repeatedly confirmed through observation and experimentation and also be widely accepted to be true by the scientific community. Often it has happened that a theory had to be discarded as it could not explain the observed phenomena. Further a new theory came up, which explained all new phenomena as well as the older ones.

It is worth mentioning here that few of the results have a specific term attached with them for historic reasons. e.g. Riemann Hypothesis, Collatz conjecture. These do not always agree with the the usual usage of the words.

 


Words like “fact,” “theory,” and “law,” get thrown around a lot. When it comes to science, however, they mean something very specific; and knowing the difference between them can help you better understand the world of science as a whole.

In this fantastic video from the It’s Okay To Be Smart YouTube channel, host Joe Hanson clears up some of the confusion surrounding four very important scientific terms: fact, hypothesis, theory, and law. Knowing the difference between these words is the key to understanding news, studies, and any other information that comes from the scientific community. Here are the main takeaways:

  • Fact: Observations about the world around us. Example: “It’s bright outside.”
  • Hypothesis: A proposed explanation for a phenomenon made as a starting point for further investigation. Example: “It’s bright outside because the sun is probably out.”
  • Theory: A well-substantiated explanation acquired through the scientific method and repeatedly tested and confirmed through observation and experimentation. Example: “When the sun is out, it tends to make it bright outside.”
  • Law: A statement based on repeated experimental observations that describes some phenomenon of nature. Proof that something happens and how it happens, but not why it happens. Example: Newton’s Law of Universal Gravitation.
  • Essentially, this is how all science works. You probably knew some of this, or remember bits and pieces of it from grade school, but this video does a great job of explaining the entire process. When you know how something actually works, it makes it a lot easier to understand and scrutinize.

 

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Ref:

https://www.quora.com/What-are-the-differences-between-theorems-definitions-axioms-lemmas-corollaries-propositions-and-statements

http://functionspace.com/topic/3465/Axioms–Postulates–Theorems–Corollaries–Hypotheses–Theories

http://math.stackexchange.com/questions/7717/difference-between-axioms-theorems-postulates-corollaries-and-hypotheses

http://math.stackexchange.com/questions/463362/difference-between-theorem-lemma-and-corollary

http://functionspace.com/topic/3465/Axioms–Postulates–Theorems–Corollaries–Hypotheses–Theories

http://lifehacker.com/the-difference-between-a-fact-hypothesis-theory-and-1732904200

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