It's hard to prove it, as any proof is going to involve techniques that already ASSUME a lot more than that.
For example, we could use alittle algebra and rearrange the terms so we have : 1 - 1 =
As GwynforWeb said, some sentences propsitions are called axioms. An axiom is a proposition that is not susceptible of proof or disproof; its truth is assumed to be self-evident.
in other words, axioms cannot be proven nor disproven, they are facts . Every science has axioms, even reigions have something like axioms - eg. that God exists (for christians).
Another important point is that axioms are like groundwork of a building - if they were disproven, whole structure of a science built on this axiom would fall.
There were attempts to prove that 1=2, but most of them have one big catch - dividing by zero...
It's hard to prove it, as any proof is going to involve techniques that already ASSUME a lot more than that.
For example, we could use alittle algebra and rearrange the terms so we have : 1 - 1 = 0, but that already assumes we believe in the correctness of algebratic operations, and the orrectness of arithmetic, and that anything minus an equal amount equals "zero".
One possible approach is to say: If we are going to be able to prove anything at all, we have to make a few assumptions, such as the location of a symbol on a page does not change its value, and a symbol is always numerically equivalent to itself. If we can't make those assumptions, then nothing else we write is going to manke any sense, so we might as well just ASSUME that A = A, for all A, assuming A is a constant. There's probably a logical proof of this in Principia Mathematica.
> dkloeck: u can also elaborate an algebra that sais
Well, you can, but what sense has algebra that states that constant isn't equal to itself? It'll mean that anything is equal to anything!
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i'll give you one proof that you may want to look on. this is the story of euclid and his 5th geometric axiom. euclid, when made the book called "the elements" proposed 5 axioms and several theorems. the 5th axiom (which is something supposed to be so obviously true that it doesn't need proof) says something like "give a point and a line, only one line passing through the given point can be parallel to the given line." now, considering the first four axioms (i.e. "all right angles are equal to themselves") the 5th axiom is pretty long so a lot of mathematicians suspected that this axiom is actually a theorem (a mathematical truth proven by axioms for a certain domain). so they tried proving the 5th axiom using the first four. nobody succeeded in doing so and some of those math fanatics just gave up the fight and accepted that it may just have been a weird axiom, but it really is an axiom. however, some people are just too persistent. this is their "battle plan":
1. try to prove the opposite of the 5th axiom, that is:
a. assume that NO parallel line can be drawn through the point that is parallel to the line, and
b. assume that at least TWO parallel lines can be drawn through the point and parallel to it.
2. disprove assumptions a & b using the first four axioms
3. using a "double negative" attack, if all other possibilities were exhausted and proven to be false, the remaining (which is the original 5th axiom) must be true.
sounds like a master plan, right? the problem was, after doing #1, they were unable to do #2, which is understandibly the hardest part. as a matter of fact, these assumptions gave rise to two very important geometries, which are spherical geometry and hyperbolic geometry. they were unable to succeed in their mission, but it produced good results just the same.the end.
now, if you're thinking on how to prove an entire axiom, i am telling you two things:
a. you may want to try to assume that 1=any number greater than 1 and/or 1=any number less than 1 WITHOUT reflexivity then disprove both assumptions. you got yourself a proof. that is, if you can find something else to use withot reflexivity.
b. good luck on your search. this may sound sarcastic, but those mathematicians faced severe criticism by the math community before being able to come up with their discovery. come to think of it, who knows? you may be able to find that proof and topple down the foundation of mathematics just like einstein did with physics!
Thewizrd:
>I think maimran wants the logic behind the axiom
What logic do U mean? there's no logic behind axioms...
Just a thought:
two guys had both one dollar note in their pockets. Both of them have spent whole dollar. now they have both no money in their pockets.
So, they had both certain number of one-dollar notes (one), now they have none.
Conclusion: they had equal number of one-dollar notes.
for the kind comment of jares:
how do we know if the dollar one guy had has the same value with the very dollar the other guy had in his pocket? because a dollar is a dollar? isn't the assumption the conclusion that we want? you know what i mean.
but what he said is true. the axiom IS the logic and supposedly, it can no longer be broken down to smaller principles.
Well, a couple of years ago, it confused me too...
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