Can somebody explain with one example the concepts: Lemma-Hypothesis-Theorem-Assumption-Proof-Axiom-Thesis-Determination-Definition-Proof

Question

It would be great if someone can give me for each concept a simple explanatory example ! What is the difference between:

• Lemma
• Hypothesis (Hypothese)
• Theorem (Satz)
• Assumption (Annahme)
• Proof (Beweis)
• Axiom
• Thesis (These)
• Determination (Festlegung)
• Definition (Definition)
• Proof (Beweis)

Answer 1

A hypothesis is an educated guess, often a statement you want to show or prove (or disprove). A good example is the Riemann Hypothesis, which Riemann hypothesized in the mid 1800s. It says that the real parts of the nontrivial zeroes of the function $$\zeta(s) = \displaystyle \sum_{n \geq 1} \frac{1}{n^s}$$ all have real part $1/2$. We do not currently know if it is true or not, which is usually indicative of being a hypothesis.

A proof is an argument that shows some statement is true (or untrue). This is a fundamental concept of mathematics. We do not accept just any hypothesis as true - we usually want proof. Proofs can be simple or very complex. A simple proof might be

Claim: The square of an $\color{#AA0000}{\text{even}}$ $\color{#AA00AA}{\text{integer}}$ is even.

Proof: An integer $n$ is even if it is $\color{#00AAAA}{\text{divisible}}$ by $2$, which means that it is of the form $n = 2k$ for some other integer $k$. In that case, $n^2 = (2k)^2 = 4k^2 = 2(2k^2)$ which is divisible by $2$, and therefore it is also even. $\diamondsuit$

You'll notice that there are three words that are different colors. This is because proofs build off of definitions. It's important to know the meaning of both the proof and the statement in an unambiguous way. Each of the three words have a mathematical definition. For example:

• An integer $n$ is $\color{#AA0000}{\text{even}}$ if it is divisible by $2$.

• An integer $n$ is $\color{#00AAAA}{\text{divisible}}$ by an integer $k$ if $n = kl$ for some other integer $l$.

Defining the $\color{#AA00AA}{\text{integers}}$ can be a bit more complicated. On the one hand, we all know what the integers are. They're the numbers $\ldots, -2, -1, 0, 1, 2, \ldots$ along with their standard multiplication, addition, and subtraction that we've been learning in schools since we were young.

But it is possible to really dive in. What is a number? What does it mean to add, subtract, multiply? What does it mean for two things to be "equal?" At every step, we can ask how more and more things are defined. Where does it end?

This is a very good and deep question. How can we prove anything? A Greek mathematician named Euclid had a good idea around the 3rd century BCE. He thought there were a few statements about geometry so obvious, and so clear, that everyone could accept their truth. Then every other geometrical statement would be true if they could be reduced down to those universally accepted truths.

Although he split his statements into "common notions" and "postulates", we would now call them all axioms. So one axiom might be that a line can be drawn through two given points. This is Euclid's first postulate. It's very intuitive... sort of.

In other words, an axiom is a starting point for reasoning. But it's slightly more convoluted than this. The "axioms" that are basic "truths" are one type of axiom. Another type are non-logical axioms (which I would link to, but which I can't because this is my first answer and I'm only allowed 2 links), which don't describe fundamental truths but instead are usually defining properties of some system or structure.

For instance, the $\color{#AA00AA}{\text{integers}}$ are often defined axiomatically. This might sounds something like:

The integers are a set $\mathbb{Z}$ with two binary operations $+, \cdot$ satisfying

1. $a + b = b + a$ and $a \cdot b = b \cdot a$ for all integers $a,b$
2. There is an integer denoted by $0$ such that $a + 0 = a$ for all integers $a$
3. ... and so on

One could say that an assumption is very similar (interchangeable, even) to the non-logical axiom. It is a statement about structure, not about truth.

As you can see, there's quite a bit here. Sometimes, one breaks up larger proofs into smaller, more manageable pieces. To do this, one breaks up a proof of a big statement into smaller statements that come together to prove the big one. To that end, there is a (very and extremely loose) hierarchy of statements. Roughly speaking,

1. A Theorem is a big, standalone, desirable statement that has been proved.
2. A Proposition is a slightly smaller one. Perhaps it could stand on its own, but isn't "good" enough to be a theorem.
3. A Lemma is smaller still. Often lemmas are used to prove larger propositions or theorems.
4. A Claim is smaller still, but is used in such a range of places that we shouldn't try to pin it down.
5. A Corollary is a result that follows directly from a theorem, proposition, lemma, (or claim, I suppose).

The names are all organizational concepts that indicate logical structure and importance. It is not too dissimilar to refactoring code, if you happen to know some programming. Since they have no real meaning, they are used in different ways by different people. Sometimes the same statement has different names. For instance, Burnside's Lemma is also called Burnside's Counting Theorem. So the hierarchy is very loose. But they are all statements that have been proved.

Altogether, you might think that someone makes a hypothesis about something. Then he or she tries to prove it by reducing it to known statements (and ultimately, axioms). They might call the result a theorem and then organize the proof into lemmas and propositions for easier mathematical communications.