How to prove the axiom is wrong?

Question

The father of math (Euclid) wrote a book named Elements. The book is full of axioms and here are some of them I am interested in:

1. Things equal to the same thing are also equal to one another.
2. And if equal things are added to equal things then the wholes are equal.
3. And if equal things are subtracted from equal things then the remainders are equal.
4. And things coinciding with one another are equal to one another.
5. And the whole [is] greater than the part.

Few years ago I came to the idea that Euclid could be (is..) wrong about the No.5.

Why I think he is wrong ? He is wrong because making decision on bigger/smaller (as a consequence: more/less, faster/slower etc.) is the (main) reason of all negative we have happening. Try to think about any conflict on the Earth and You will find at least two sides fighting for the same thing there. For example: fighting for bigger territory, fighting for more money, fighting for less problems, etc. I think that the concept of thinking (thinking that some bigger or smaller even exists) is false. However, it seams I cannot logically prove it being false or true, because the whole logic itself is made on top of such axioms...

So, let`s say (or just assume..) that some of axioms are wrong. The question is - what base should be used to prove axiom being wrong for the rest of the world? My own assumption is that I have to use the "what is valuable for society" base or even the "how it feels for society" base.

And, one more thing to mention. If I am not able to prove axiom being false or true (because false/true is part of the logic which is made on top of the axiom itself) then I would like to call "true" as "real" and "false" as "illusion", but would this be correct?

Answer 1

The correct question is not :

How to decide if an axiom is right or wrong ...

but :

Of what "domain" (of discourse or of reality) is this axiom true ?

Modern mathematics, after Georg Cantor, has found "domains" (infinite colelctions or sets) where it is not true that

the whole [is] greater than the part

for a "suitable" interpretation of greater than.

As was already known to Galileo (see Galileo's paradox) we can associate to each natural number n its double : 2n.

Thus, if we use this "procedure" to count the objecst in a collection, we can roughly say that the collection of natural numbers has the "same number" of elements as the collection of even numbers.

This result show us that, for infinite collection, we can roughly say that

the whole is not always "greater than" a proper part of it.

Answer 2

In Euclid's time, axioms were taken as basic, unquestionable truths about the world. In more modern times, however, we have less faith in the existence of such things, and axioms are defined less rigidly as the foundational building blocks of a particular system of thought. Thus, Euclidian axioms define Euclidian geometry, but there are also non-Euclidian geometries with different axioms.

As it so happens, the axioms of formal logic are not dependent on the axioms of Euclidian geometry, so you could attempt to disprove an Euclidian axiom using logic without any fear of a paradox.

In general, if you want to prove something to someone, the proper approach is to start from axioms that your target endorses. If you do your job correctly, you will demonstrate that the target cannot consistently hold the position you are trying to disprove and still endorse the axioms you started with.

Answer 3

The best way to falsify an axiom is to show that the axiom is either self-contradictory in its own terms or logically implies a deduction of one theorem that leads to a self-contradiction.