Axioms of Euclid

Question

The axioms of Euclid are :

1. Things which are equal to the same thing are also equal to one another.
2. If equals be added to equals, the wholes are equal.
3. If equals be subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal to one another.
5. The whole is greater than the part.

I know that axioms are meant to be memorized and are not subject to questioning or proofs. It is said that one cannot prove the axioms as they are the starting points of Mathematics. However can anyone prove that axioms (for e.g. the above mentioned axioms) cannot be proved.

Answer 1

You have to start somewhere! If you want to prove your axioms, you would have to do so on the basis of simpler statements. If you are not willing to take these other statements for granted, you would have to prove them in terms of other statements again, and so on.

You would never stop. . . in fact it's worse than that, you would never even start!

I am not an expert on the history of mathematics so am certainly subject to correction, but here is my understanding of (a small part of) the story. In about the 18th century, Euclid's axioms were accepted as "obvious facts" about the real world. As far as I am aware nobody questioned their "truth", though various people questioned their independence, asking if the parallel postulate could be proved on the basis of other postulates and axioms. Eventually it was shown that this could not be done: without going into details, it was shown that one can find a self-consistent system of geometry in which the parallel postulate is in fact false.

This had a consequence which I feel was very surprising to many people, or perhaps I should simply say it led to a question: if this "non-Euclidean" geometry is self-consistent, how do we know that it does not actually describe the world better than Euclidean geometry? Later on, Einstein did in fact describe the universe in terms of a very different geometry. An anecdote: in the early 19th century it seems that Gauss did a bit of surveying, measuring angles between various mountain peaks. It may just have been a job, but some have suggested that he was interested in determining whether, in the real world, the angles of a triangle really do add up to $180^\circ$. Now there's a good example of questioning accepted "facts"!

The next point of view to come along - and in (pure) mathematics, though not say in physics, this has very much lasted to the present day - was that if axioms which do not describe the real world are self-consistent, there is no need even to ask whether or not axioms describe the real world. They are just "the rules of the game", and you can change them if you prefer to "play a different game". Doing things this way, axioms do not have to be "true": the only requirement is that they be self-consistent, in other words that they do not lead to any contradictory theorems.

All this of course is speaking from a very "pure mathematical" viewpoint: if you want to do mathematics which does have application to the real world, then you need to match your axioms to the real world.

One final comment: it is true that the axioms of Euclidean geometry cannot be proved in the sense of deriving them from other statements of Euclidean geometry. However, they can certainly be proved on the basis of facts about real numbers. If you model the Euclidean plane by defining a point to be a pair of real numbers, and if you define a line to be a set of the form $$L_{a,b,c}=\{(x,y)\in{\Bbb R}^2\,|\,ax+by=c\}$$ with $a,b$ not both zero, then you can prove Euclid's postulate that there is exactly one line which passes through any two given unequal points. But then you would need axioms for the real numbers. You could either just accept them, or define the reals in terms of the rationals. . . and so on.

Hope this is of interest.

Answer 2

Any system or approach has to have a starting point. To prove that these axioms can/cannot be proved would require assuming some other set of axioms from which we could frame these.

Where you start or what axioms you assume depends on what you are trying to prove/ discuss.

In answer to your comment, that depends which axioms you begin with. However, one of the results of Godel states (for arithmetic theory), that in any consistent (that is, does not contain any contradictions), there will exist a statement that is true but not provable within that system.

The general approach of mathematics (as I understand it at this point), is you start with some axioms and see where they take you.

For example, one of Euclid's geometric postulates was his parallel postulate, basically http://en.wikipedia.org/wiki/Parallel_postulate

While this is true in standard Euclidean geometry, if we do not assume these, then we end up with other forms of Geometry such as hyperbolic or spherical.

Answer 3

I know that axioms... are not subject to questioning or proofs.

It depends what you want to do with them. The axioms of Euclid are intended to model not just any old geometry, but rather, the geometry of the real world. Ergo, they are subject to questioning, like any other model. Indeed, General Relativity suggests that the axioms of Euclid give rise to a poor model of real-world geometry near black holes and other very massive objects.

But even in pure mathematics, axioms are questioned all the time. The questions tend to be of the form: "Do the specific examples I'm interested in satisfy these axioms?" and "Can so-and-so theorem be proved using fewer/weaker axioms?"

Never, ever stop questioning.

It is said that one cannot prove the axioms as they are the starting points of Mathematics.

No. Every axiom of a formal system is a theorem of that system. In fact, an axiom can be defined as a theorem with a one-line proof (or a zero-line proof, depending on how you formalize things).

But perhaps you are looking for a slightly different concept. The axioms of a formal system are said to be independent if no subset of those axioms can prove the remainder of the axioms. You often can prove the axioms of a formal system independent, but only by using a second, stronger system.

Answer 4

Strictly speaking, those listed are not Euclid's Elements postulates, but the so-called common notions.

The difference is this :

• the five postulates are intended to be specific about geometry

• the common notions are intended to be more widely "applicable".

About :

assuming other axioms can you prove that Euclid's axioms can not be proved ?

the discovery of non-euclidean geometries has shown that the fifth (or parallel) postulates was in fact independent from the other four; thus, it is not provable in the context of Euclid's Elements.

Regarding e.g. common notion 5 :

The whole is greater than the part

it can be proved to be flase in the context of set-tehory with regards to infinite set.

It is a standard result regarding cardinality of countable sets that :

the sets $A = \{ 1, 2, 3, \ldots \}$, the set of positive integers, and $B = \{ 2, 4, 6, \ldots \}$, the set of even positive integers, have the same size [in the sense made precise by set-theory]. They are both countably infinite.

Clearly, $B$ is a proper part of $A$.

Answer 5

In modern mathematical language: (1) is just the transitive property of equality. (2) and (3) are just specific applications of the substitution rule for equality. This would be true for any binary functions, not just $+$ and $-$. In (4), the word "coincide" is ambiguous, but probably corresponds to the reflexive rule for equality. Missing, of course, is the symmetric property of equality.

These ideas have more or less stood the test of time as self-evident truths.

(5) however, may be problematic. It seems to stem from the fact that the ancient Greeks prior to Ptolemy (the philosopher) had no notion of a zero. They wouldn't have had anything analogous to our $x+y=x$, with $x$ and $y$ being the parts and $x+y$ being the whole.

Translating (1) through (4) into the formal language of modern mathematics:

(1) Things which are equal to the same thing are also equal to one another.

$$\forall x,y,z:[x=y \land y=z\implies x=z]$$

(2) If equals be added to equals, the wholes are equal.

$$\forall w,x,y,z:[w=x \land y=z \implies w+y = x+y]$$

(3) If equals be subtracted from equals, the remainders are equal.

$$\forall w,x,y,z:[w=x \land y=z \implies w-y = x-y]$$

(4) Things which coincide with one another are equal to one another.

$$\forall x: x=x$$