Question about Axioms and Theorems

[Salt]

Okey, this might be a silly question.
I know that theorems are deduced logically from the axioms. But I was just wondering is it possible to deduce an axiom from the theorems? In another words work backward, assuming the required theorems are known.

 

[Chronos]

Basically, they are identities. Some theorems do lead to axioms, by deduction. Theorems take an axiom and make predictions. Once confirmed by observation, they lead to more axioms.

 

[Hurkyl]

Generally you can deduce the axioms from theorems. In particular you can very easily deduce the axioms from themselves!

I suspect, however, you are attaching some extra (and unwarranted) meaning to the word "axiom".

The axioms of a mathematical theory are merely the collection of statements from which deductions are made in that theory -- nothing more, nothing less.

 

[HallsofIvy]

Generally, "A implies B" does NOT mean that "B implies A". you cannot generally reverse the "proving" process.

 

[Salt]

The axioms of a mathematical theory are merely the collection of statements from which deductions are made in that theory -- nothing more, nothing less.

Well, that does seem to be what I thought they are, with the addition view that it's they are the most fundamental.

Generally, "A implies B" does NOT mean that "B implies A". you cannot generally reverse the "proving" process.

I was sort of thinking.

if I know :
{
Axioms -

A implies B
B implies C
C implies D

theorem -

therefore A implies D

}

later I forget : C implies D

but remembering the rest, will I be able to recontruct C implies D?

 

[DeadWolfe]

I really wonder what you mean by deducing axioms from theoroms...

...certainly in physics, one might be able to deduce laws from a collection of experiences or equations, but since math is not directly based on reality, how could one possibly derive axioms?

 

[jcsd]

It is in some ways.

Lets say we are given all the axioms of a field except the axiom of additive identity and instead we are given the theorum 1*0 = 0. From the this theorum and the other axioms it is possible to deduce the axiom of additive identity (1*0 + a = 0 + a, 1*(a - a + a) = 0 + a, a = 0 + a).

 

[T@P]

isnt it also possible to create an experimentally *correct* theorem, and then taking the assumtions you made and making them "axioms"? is that right? I am not sure if I understand this correctly.

 

[NateTG]

Okey, this might be a silly question.
I know that theorems are deduced logically from the axioms. But I was just wondering is it possible to deduce an axiom from the theorems? In another words work backward, assuming the required theorems are known.

Proving an axiom in a logical system is, as Hurky pointed out, rather trivial and circular.

What does happen is that mathematicians do research to find out what kind of axioms make for an interesting system. That is, people look at a list of results that they want to be true, and then look at possible axioms that make those results true.

A familiar example of an axiom that was probably introduced is from geometry -- given a line, and a point, there is only one line parralel to the original line that goes through the point. It turns out that geometry can work without this axiom which is why there is the study of spherical and hyperbolic geometry in addition to plane geometry.

 

[Salt]

It is in some ways.

Lets say we are given all the axioms of a field except the axiom of additive identity and instead we are given the theorum 1*0 = 0. From the this theorum and the other axioms it is possible to deduce the axiom of additive identity (1*0 + a = 0 + a, 1*(a - a + a) = 0 + a, a = 0 + a).

Ah, nice. An example. At least now I know it's possible.

Well, thank you everyone for their replys.

This question actually came to me while I was browsing through Prof. Feynman - Character of Physical Law. In it he says that he does not remember all that much, what ever he forgets , he just derives from what he knows. Later should he forget the thing he use too know, he derive it backwards from the new stuff he just derive awhile back. So he can jump from theorem to theorem.

What I wonder was can one jump back to the fundamental axioms/first principles? Actually he feels that there are no fundamental axioms/first principles from what I read, and it doesn't matter where you start.

So I wonder if there is a situation where one can't get back to some theorem or axiom, if the required set of information is given.

Well after all this, I have come to the conclusion it's safe to assume that Prof. Feynman is right or at least most of the time. That with enough information one can derive any part out.

PS : Not to doubt him or anything, although he probably won't be all that offend, as I get the impression that he feel doubting stuff is a healthy thing. It's just that it's been 30+ year, i wonder if anyone found a counterexample.

 

[hypnagogue]

I was sort of thinking.

if I know :
{
Axioms -

A implies B
B implies C
C implies D

theorem -

therefore A implies D

}

later I forget : C implies D

but remembering the rest, will I be able to recontruct C implies D?

You won't be able to deduce C implies D in the above scenario. Let's simplify the problem by saying that we're given "A implies C" and "A implies D," and we're asked whether or not the statement "C implies D" logically follows. To show that it does not logically follow, we just have to conceive of a scenario where "A implies C" and "A implies D" are true but "C implies D" is false. Here is one such scenario:

A: X is a square.
C: X has 4 sides.
D: X is a rectangle.

In the wider context of your question, this doesn't imply that Feynman's techniques are flawed. For instance, it could be that Feynman's techniques are valid, but the specific way you framed the scenario above is not an accurate depiction of what Feynman was actually doing.

 

[Salt]

Well I suppose then what Prof. Feynman does might be more along the lines of:

Crude example:

square implies:

• has 4 sides
• all sides are equal
• angle between 2 side is 90
• 2 connecting sides times one another gives you the area of the figure

has 4 sides implies :

• square or
• rectangle or
• rombus or
• parrallogram

all sides are equal implies:

• square or
• rombus

angle between 2 side is 90 implies:

• square or
• rectangle

later we forget that :

square implies:

• 2 connecting sides times one another gives you the area of the figure

Remembering only the first 3.

And given the theorem :

unknown X is

• has 4 sides
• all sides are equal
• angle between 2 side is 90
• 2 connecting sides times one another gives you the area of the figure

Which we know for sure is true.

Since the only thing we know that has the first 3 properties is a square.
Therefore X = square, since it's equal:

square implies:

• 2 connecting sides times one another gives you the area of the figure

Or something along these line. I might have screwed up somewhere.