Pondering a Proof

Re: Pondering a Proof

Unless 1 is just a variable, I don't really see how it can't be 1.

Re: Pondering a Proof

1 group of 1 unit = 1 unit...makes sense to me.

Re: Pondering a Proof

I did some googling and found the proof for 1 + 1 = 2

It looks quite complicated with fancy postulates that i don't understand:



i was expecting for something like that...but with 1*1 = 1

-edit-

lolz i found a video of it


oh wait, it's 1 + 1 = 0...and it's flawed. *sigh* googling attempts foiled yet again by misleading titles!


-edit-
For every simple understanding, there's a very complicated reasoning behind it.

It's like sex.

Re: Pondering a Proof

Wouldn't it just be:

1*1=1
1=1

Re: Pondering a Proof

well for 1 + 1 = 2...

the same logic is as you said

1 = 1

YET THERE HAS BEEN A PROOF DERIVED

Re: Pondering a Proof

There are postulates out there. I'm guessing if there isn't a proof for 1 X 1 it's just a postulate.

Re: Pondering a Proof

So, a lot of mathematics really is just built on a set of axioms that we deem is true in some system. In most cases in some structure, the existence of some "unit" 1 such that 1*g = g for any other element g in this structure is one of the given axioms. There really isn't much to say about this - it's just something that's assumed to be true and which allows us to study structures without having to really deal with what this unit "1" actually means.

In terms of a proof, it really goes down to breaking down mathematics in terms of whether or not certain axioms are correct or not, and whether or not the mathematical system we've developed is consistent in all cases. This gets VERY, VERY convoluted, which is why the proof to 1+1=2 is quite long, since all of mathematics essentially has to be dissected and put back together. If you're interested in stuff like this, definitely look out for a class in Mathematical Logic when you go to college, or some equivalent. There you'll formally break down what mathematical systems mean and prove a number of important results that will help you get a firmer grasp on what all these systems actually represent.

Re: Pondering a Proof

In the group of multiplication with real numbers, 1 is considered the identity element. Therefore for all n, n * 1 = n. Granted even that proof doesn't dig into why 1 is the identity element, but we're getting closer.

The word "axiomatic" is sometimes meant to mean, "taken for granted." Basically, we take axioms in math for granted. As far as 1 * 1 = 1 or anything like that, you don't need a proof. It's obvious. We can take it for granted.

Re: Pondering a Proof

Well I think it generalizes even further - in ANY group with any law of composition, there must exist some unique identity element, denoted "1", such that 1*g = g for all elements g in the group. (Note that * does not denote multiplication - it just denotes whatever law of composition exists on the group structure - this could be addition, multiplication, function composition, etc) Yes, we do take this for granted, and it's very convenient to do so. However, it is still a fair question as to WHY this is a convenient choice and HOW this affects the structure of mathematics and how we think about these things. There are plenty of ways I could define a set of axioms for a new structure that would be completely wrong and fail to be consistent with the rest of mathematics. So, the question becomes, why is the existence of such an identity element consistent? Is mathematics a consistent system anyways? Is it possible to "break" mathematics?

So yes, on the surface this seems like a question that requires a one line answer - "well, of COURSE 1 * 1 = 1, that's how we defined it!" But on a deeper level this is a very important issue that has plagued mathematicians, and more specifically logicians, since the establishment of mathematics as a rigorous system.

I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?

Re: Pondering a Proof

"I think it boils down to - is there really any reason to have complete faith in mathematics as an absolute system?"

Yes, if and until if becomes broken. It becomes broken when using it wrongly describes some aspect of the universe. If it doesn't describe it properly, it simply hasn't been 'invented' yet.

Re: Pondering a Proof

Let's try it with x*1.

x*1 = x
Why? Because it's one of x.

1x*1 = 1x

Anything that exists tangibly or theoretically exists in the form of 1, possibly manipulated by an action or equation.

So if x = x then 1x = 1x and 1 = 1.

Any one of something is that same something. It's one of it. This reasoning implies that any negative number has a positive counterpart though, as something must exist before it can unexist. Physically speaking though, it always existed, just in a different form.

I'm confusing myself. xD
Good luck.

Re: Pondering a Proof

1+1=2
1(1)=1

I know this is CT, but why are you debating over something so clear and obvious? I am not really what you would call 'mathematically oriented', but this is definitely some logic I have never pondered..

Re: Pondering a Proof

by definition

peace

Re: Pondering a Proof

1 times 1 times 1 times 1 times 1 = 1.

It's like an infinite loop!!11!!1!one!!11!