1+1=2 always true?

• Dec 21st 2010, 07:32 AM
tass
1+1=2 always true?
Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.
• Dec 21st 2010, 08:04 AM
emakarov
All mathematical facts are unmovable and absolute. The only way an appearance of change may come is from redefining concepts. For example, one may redefine plus so that x + y is always 1, regardless of x and y. Properly speaking, this is an abuse of notation because we did not bother to come up with a new symbol and instead reuse the same +, which has a different meaning in a different context. This is why it may seem that 1 + 1 = 2 is not true anymore.

To make more mathematical sense, we can consider the set {0, 1} instead of all natural numbers and define x + y = 0 if x = y = 0 or x = y = 1; otherwise, x + y = 1, by definition. This newly defined operation has all familiar properties of +, including 0 + x = x, x + y = y + x, (x + y) + z = x + (y + z), etc.
• Dec 21st 2010, 08:19 AM
Soroban

. . . . . $\boxed{\begin{array}{c} \\ 1 + 1 \:=\:3 \\ \\[-3mm] ^{\text{for large values of 1}} \\ \\ \end{array}}$

• Dec 21st 2010, 09:33 AM
wonderboy1953
In my universe
Quote:

Originally Posted by tass
Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.

If you combine one lump of clay with another bigger lump of clay, guess what? You still wind up with one lump of clay.
• Dec 21st 2010, 09:41 AM
chisigma
In $GF(2)$ is $1+1=0$...

http://digilander.libero.it/luposaba...ato&#91;1].jpg

Merry Christmas from Italy

$\chi$ $\sigma$
• Dec 21st 2010, 09:45 AM
DrSteve
1 polar bear + 1 seal = 1 polar bear
• Dec 21st 2010, 09:50 AM
wonderboy1953
Quote:

Originally Posted by DrSteve
1 polar bear + 1 seal = 1 polar bear

Good one.
• Dec 21st 2010, 10:20 AM
dwsmith
$\left\lfloor 1.9999999999\right\rfloor + \left\lfloor 1.999999999\right\rfloor = 2 \ \mbox{too}$
• Dec 21st 2010, 10:38 AM
wonderboy1953
0/0 is equal to one (and just about anything else you can think of).
• Dec 21st 2010, 12:09 PM
FernandoRevilla
Notations: $S(0):=1,\;S(S(0)):=2,\quad (S:\;\textrm{succesor}\;\textrm{function})$

Definition: $x+S(y)=S(x+y),\;\forall x,y\in \mathbb{N}$

Axiom: $0+z=0,\;\forall z\in \mathbb{N}$

Theorem $1+1=2$

Proof

$1+1=S(0)+S(0)=S(0+S(0))=S(S(0))=2$

Alternative proof:

Avoiding unnecesary simbolism:

$S+S=SS$

That is, time and/or space, or natural ability to create bijections as a consequence of time and/or space. That is all.

Fernando Revilla
• Dec 21st 2010, 12:34 PM
emakarov
Well, if we go to that level...

Quote:

Originally Posted by FernandoRevilla
Notations: $S(0):=1,\;S(S(0)):=2,\quad (S:\;\textrm{succesor}\;\textrm{function})$

It should be, 1 := S(0). The left-hand side is being defined. Usually what is defined is a new symbol. If the thing that is defined is a composite expression, then it may be necessary to prove that the thing is well-defined.

Quote:

Definition: $x+S(y)=S(x+y),\;\forall x,y\in \mathbb{N}$

Axiom: $0+z=\mathbf{z},\;\forall z\in \mathbb{N}$
The definition does not specify the + operation completely. It is not clear why one clause is a definition and the other is an axiom; they both should be either one or the other (as in Peano axioms).

Quote:

Theorem $1+1=2$

Proof

$1+1=S(0)+S(0)=S(0+S(0))=S(S(0))=2$
According to the definition, S(0)+S(0) = S(S(0) + 0). It does not follow that this is S(S(0)).
• Dec 21st 2010, 01:05 PM
FernandoRevilla
Quote:

Originally Posted by emakarov
Well, if we go to that level...
It should be, 1 := S(0).

Right (a lapse)

Quote:

The definition does not specify the + operation completely. It is not clear why one clause is a definition and the other is an axiom; they both should be either one or the other (as in Peano axioms).
Irrelevant. The functions $f_1^1$ (succesor) and $f_1^2$ (sum) are correctly defined. If I decide to name them as axiom or definition are only pedagogical strategies (good or bad).

Quote:

According to the definition, S(0)+S(0) = S(S(0) + 0). It does not follow that this is S(S(0)).
Too easy: correct it.

Fernando Revilla
• Dec 21st 2010, 03:08 PM
jgv115
I still recall a lot of my friends telling me 1+1 = window. So 1+1 isn't always 2
• Dec 21st 2010, 03:20 PM
CaptainBlack
Quote:

Originally Posted by tass
Hi,

I am not sure wether I am in the right forum for this question and it may even turn out banal.
I think to remember that the often cited, unmovable truth 1+1=2 is not so absolute as it seems to be. Could someone explain me, if there are mathematical systems or preconditions under which 1+1=2 is false or not entirely true anymore and how this works in simple terms?

Thank you very much for your patience with a math idiot.
Cheers.

It took Russell and Whitehead 379 pages before they could prove this in PM (1st edition, sightly fewer in later versions).

CB
• Dec 21st 2010, 07:46 PM
tass
These were very useful and unexpectedly many contributions. Thank you very much for your time and creative explanations.

Merry Christmas.
T.