# how axioms/postulates really work

#### PG1995

Joined Apr 15, 2011
816
Hi

When a certain subject is being studied, including a natural phenomena, certain fixed rules, characteristics are noticed as defining whatever is under discussion. These rules or characteristics are called axioms/postulates which are the foundation upon which the system rests. For example, one may observe that every dog hates cats (I'm just making it up). So, a postulate is established which very much defines one of the traits of a dog. If we see an animal which looks like a dog but likes to play with cats, then according to the postulate it couldn't be a dog. But if biological studies prove that it's really a dog, then the established postulate falls apart. Another postulate related to dog can also be established that every dog loves a bone. So, a set of postulates or axioms defines what a particular thing should be or how it should behave. These are reasonable assumptions which are made after careful observation of the thing under discussion.

I have read somewhere that in the past there used to be a distinction between the word 'postulate' and 'axiom' but now they stand for the same.

In Euclidean geometry there are certain postulates. Euclidean geometry can be described as highly idealistic version of some of the things in real word (I have been told when it comes to real world hyperbolic geometry maps the reality better). Then, circle is an object of Euclidean geometry. It can be said that the circle in itself has a set of axioms such as the sum of angle subtended by the diameter. The same can be said of a triangle, which is also an object in Euclidean geometry, that if its meets certain axioms such as internal angles sum is 180 degree etc. then it's a triangle. But none of the axioms related to a circle or triangle can contradict the axioms of the system (i.e. Euclidean geometry) of which they are members.

Sometimes postulates seem self-evident and very reasonable assumptions which holds true as far as one can confirm. But sometimes those postulates aren't that much clear. For example, one of the postulates/axioms of theory of the relativity is that lights travels at the same speed no matter what. One might wonder that why Einstein chose this postulate. Was he able to measure the speed of light under different conditions? I don't think so. I believe some of the equations etc. he was working with made him realize that the speed of the light had to constant otherwise nothing works. So, in this case the postulate isn't that much self-evident from observable point of view but math equations tells us so (and now even the experiments).

Further it should be noted that axioms/postulates themselves are just a set of rules. They do not describe how the system works, how it could relate to the natural phenomena, why we study the system etc. You can have a set of rule of the game of chess but those rules does' really tell what chess is. They only tell you how are expected to play. Likewise, you can have a set of axioms for vector spaces (in layman terms, vectors are members of the system called vector spaces). Those axioms of vector spaces say nothing why we study vectors, how those vectors relate to the physical world. They are just rules which have been formulated after close observation of the system. So, as a result they don't need any proofs.

Do I make some sense? Please let me know. Thanks

http://math.youngzones.org/Non-Egeometry/axioms.html
http://www.sfu.ca/~swartz/euclid.htm

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#### studiot

Joined Nov 9, 2007
4,998
Before you can have either postulates or axioms you need definitions.

Euclid 'Elements' has many more definitions than axioms.

But what exactly was your question?

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#### PG1995

Joined Apr 15, 2011
816
Thanks for the response, studiot.

There was no particular question. I wanted to know if I make some sense or not. Anyone like me who reaches this thread might find it useful.

By the way, could you please tell me little about the definitions which you say precede even the axioms/postulates? You can use Euclid geometry as the example system. Thanks for the help.

Regards
PG

#### studiot

Joined Nov 9, 2007
4,998
OK

Euclid's geometry is actually founded on 23 definitions, 5 axioms or 'postulates' and 5 'common notions'.

I will illustrate with reference to one version of the first axiom.

Between any two points only one straight line may be drawn
Clearly you have to have definitions of a 'point', a 'line' and 'straight' to be able to work with this axiom.

Incidentally the ancient Greeks distinguished between axioms and postulates in the following way.

They realised that you have to start somewhere, that is that there are some things you can't or don't derive from more fundamental things.

They further recognised as postulate, as statements that could be 'demonstrated', but not proved. For example you can balance a circular disk on a diameter (This was a postulate by Aristotle). Although they knew about levers they did not have a formal concept of moments.

They further recognised as axioms, such as the one above, as statements that could not be demonstrated but were claimed to be 'self evident'.

In modern times we do not make this distinction since we reckon the fundamental consideration to be that we 'take on trust' and do not derive in any manner.

We talk about a collection of axioms or postulates being the founding statements of a deductive system of logic that are not themselves deduced and from which subsequent statements are deduced.
We still devide the classification of subsequent statements into propositions, theorems, lemmas, etc.

go well