Question on Euclid’s 5 Postulates

Question

Here are the five postulates:

1. Each pair of points can be joined by one and only one straight line segment.
2. Any straight line segment can be indefinitely extended in either direction.
3. There is exactly one circle of any given radius with any given center.
4. All right angles are congruent 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 extended indefinitely, meet on that side on which the angles are less than two right angles.

Questions:

1. These to me sounds more like something that shouldn't require proving... does it?

2. Why is it important to stress things that are obvious? For example, what other answers can you get when extending a line segment other than it can be extended indefinitely in either direction?

3. Similarly, what space can allow two circle of the same radius and center to be not the same?

4. Saying all right angles are congruent ... isn't that the same as saying all 64.506 degree angles are congruent? Isn't it ANY angle are congruent if they are the same degrees measures from the same reference point (say x-axis)?

5. Why do we need the 5th postulate?

Answer 1

Your questions seem to have as common theme the underlying question "What is the point of Euclid's postulates?". To answer this, remember that these postulates where introduced in the context of Euclid's book Elements. Therefore, it makes sense to first consider the question "What us the point of Euclid's Elements?".

The point of Euclid's Elements is to collect statements and constructions concerning lines, points and circles in the two-dimensional plane, all of which are known to be absolutely true. To show the reader that these statements are indeed true, Euclid uses a technique which is even today still the basis of all mathematics, namely mathematical proof. This works as follows: you start with a set of statements, all of which you know to be true beyond doubt, and you show that some other statement is a logical consequence of this statement. It then follows that this last statement must also be true beyond doubt. As an admittedly contrived example, if you already know that the statements $x = 3$ and $y =5 $ are both true, then you also know that the statement $x \cdot y = 15$ is true.

So the point of mathematical proof is to expand the collection of statements of which we are absolutely sure. But there is a flaw in this system, namely that we already need to know certain things before we can start proving other things. So before we can use mathematical proof to show which statements are true, we must already have a non-empty set of true statements.

The point of the postulates is exactly to provide this first set of true statements. The point is that we only need to agree on the fact that these five postulates are true, and then the proofs in Euclid's Elements guarantee the truth of all other statements in the book.

So if the postulates feel like they are completely self-evident and almost too obvious to be worth writing down explicitly, then the postulates do exactly what they are supposed to do.

Answer 2

The Euclidean 5 Postulates in general shore up the sketchy introductory Euclidean Definitions. There are definitions of line, and straight line which are responded to by 1st and 2nd Postulate regarding straight line and extending the straight line. Similarly definitions of angles, surface, and plane surface and circle are responded to by postulates. A definition of parallel lines is met by the parallel postulate. But alas definitions of surface and plane surface depart from this trend as there is no postulate of the plane which seems like an oversight since we are dealing with plane geometry. Isn't that odd?