# Proof of 1+1=2

#### afton

Is there a proof that 1+1 = 2 ?
or we just accept it as it is ?

#### dextercioby

Homework Helper
afton said:
Is there a proof that 1+1 = 2 ?
or we just accept it as it is ?
It surely as hell is...Read HallsofIvy's post here

Or u can just google for Giuseppe Peano and natural numbers axiomatical construction.

Daniel.

#### irony of truth

I believe this type of proving falls under mathematical logic... I could recall that this has more than 20 statements

#### dextercioby

Homework Helper
What do you mean??Explain why you (or probably somebody else) think(s) Giuseppe Peano's construction falls when logically interpreted.

Daniel.

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#### FulhamFan3

I'm going to answer this question with a question. What are Numbers? What is "one"? What is "two"? When someone answers that I got a simple one line proof to your stupid question.

#### matt grime

Homework Helper
falls and fails aren't necessarily synomymous, dex, if I may be so familiar

#### ahrkron

Staff Emeritus
Gold Member
FulhamFan3 said:
I'm going to answer this question with a question. What are Numbers? What is "one"? What is "two"? When someone answers that I got a simple one line proof to your stupid question.
Please refrain from negative comments. It is not in the spirit of the forums, and this is most definitely NOT a stupid question.

#### Gokul43201

Staff Emeritus
Gold Member
afton said:
Is there a proof that 1+1 = 2 ?
or we just accept it as it is ?
Not a proof, but 2 is defined as the successor of 1 (under Peano), and hence satisfies the above requirement.

In any case, this is a question of definition, not proof. If 1+1 is single-valued, and different from 1, we can give it a name. This name is 2.

There was a giant thread on this somewhere...got moved to philosophy, I think.

#### afton

Why 1+1=2?

Why 1+1=2? Mankind built F-16, nuclear bomb, red Ferrari,
skycrapers and Apollo satellite based on the assumption
of the fact that 1+1=2.

But why is it so? I asked a mathematics professor who wrote
a book on Proof in Mathematics for university students pointed me
to read the Russell and Whitehead's Principia Mathematica.
Somewhere in the 500 pages of axioms and theorems
they're trying to prove by extending Peano postulates that
1+1 = 2.

But in 1931 Kurt Gödel with his Incompleteness Theorem
demonstrated that within any given branch of mathematics, there
would always be some propositions that couldn't be proven either true
or false using the rules and axioms.
[http://www.miskatonic.org/godel.html] [Broken].
So in effect disproved the whole Principia Mathematica.

The difficulties in proving that 1+1 = 2 or one plus one is two stems
from the fact that it's so difficult to define what is "one"?
Online Cambridge Dictionary [http://dictionary.cambridge.org] [Broken]
define "one" as "a single thing; not two or more",
but if we look further it also define "single" as "one only". So definitely
a circular argument (makes a conclusion based on material that has
already been assumed in the argument).

So why is it that one plus one become two? Simply because when
we're young we trust our primary school teacher that one plus one
become two. ("Johnyyy, oneee plus oneee is twooo.... you got to
believe me Johnyy.... if not you can't graduate from my class") ;-)
But why? We actually don't know the answer.
We just believe it that 1+1=2.
Quod Erat Demonstrandum :)

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#### Hurkyl

Staff Emeritus
Gold Member
But why is it so? I asked a mathematics professor who wrote
a book on Proof in Mathematics for university students pointed me
to read the Russell and Whitehead's Principia Mathematica.
Somewhere in the 500 pages of axioms and theorems
they're trying to prove by extending Peano postulates that
1+1 = 2.
It's all a matter of definition. In most mathematical examples, 2 is defined to be 1+1, so the proof is rather trivial.

But in 1931 Kurt Gödel with his Incompleteness Theorem
demonstrated that within any given branch of mathematics, there
would always be some propositions that couldn't be proven either true
or false using the rules and axioms.
Incorrect.

#### roger

Actually, I have a question..

Can i use physical objects to demonstrate the notion of addition ?

eg 1 apple add 1 apple equals two apples ?

Roger

#### dextercioby

Homework Helper
roger said:
Actually, I have a question..
Can i use physical objects to demonstrate the notion of addition ?
eg 1 apple add 1 apple equals two apples ?
Roger
It would be thesame thing.It would have to do with our perception of addition.For example if in school the kid were taught that 1+1=3 and 1+3=2 (that is to say the order 1,2,3 would be changed),then he would be convinced that one apple+one apple=3 apples.

Daniel.

#### matt grime

Homework Helper
roger said:
Actually, I have a question..

Can i use physical objects to demonstrate the notion of addition ?

eg 1 apple add 1 apple equals two apples ?

Roger

this reminds me of soemthing russell, I think, wrote, though I don't have access to any decent books of quotations.

to (mis)quote, it's something about using dogs to demonstrate addition, and it ends with

why, he may even find himself pondering if dogs exist,

or something like that anyway, anoyone got the proper version?

#### FulhamFan3

roger said:
Actually, I have a question..

Can i use physical objects to demonstrate the notion of addition ?

eg 1 apple add 1 apple equals two apples ?

Roger
I would think so. Each number represents a certain quantity of something. We defined the digits 0-9 to represent a fixed quantity. Hence if you put those quantities together the new quantity is your total. there's your proof.

Or in other words i have a bucket with one apple. I put another apple in it. I look in the bucket and it's 2 apples.

#### matt grime

Homework Helper
The usual "counter argument" goes: I have a cup with a drop of water in it, I add another drop of water to it, then I look inside. How many drops of water are there in the cup

#### roger

matt grime said:
The usual "counter argument" goes: I have a cup with a drop of water in it, I add another drop of water to it, then I look inside. How many drops of water are there in the cup
Even if the two drops merged, the volume of water is two units ? no ?

#### FulhamFan3

matt grime said:
The usual "counter argument" goes: I have a cup with a drop of water in it, I add another drop of water to it, then I look inside. How many drops of water are there in the cup
2 drops are in the cup

like I said. Numbers have fixed quantities. Drops are not a fixed quantity.

#### loseyourname

Staff Emeritus
Gold Member
Did no one notice Dexter's post about this? If we called a collection of one object and another object three objects, then 1+1=3. The only reason it equals 2 as it stands is because that is what 2 is defined as. In an integer series, each integer is defined as being 1 more than the one it follows (roughly put).

#### afton

loseyourname said:
Did no one notice Dexter's post about this? If we called a collection of one object and another object three objects, then 1+1=3. The only reason it equals 2 as it stands is because that is what 2 is defined as. In an integer series, each integer is defined as being 1 more than the one it follows (roughly put).
Still a circular argument :-) "being one more...." contains "one" in the
sentence. Imho the difficulties is from defining what is 'one'. If we somehow
can arrive at a definition of what is 'one' (in which Godel's Incompletness
Theorem said we can't) proving 1+1=2 would be much easier

#### FulhamFan3

there is no difficulty in defining 1. 1 is defined as an isolated quantity. 2 is defined the whole number quantity after that.

#### matt grime

Homework Helper
afton said:
If we somehow can arrive at a definition of what is 'one' (in which Godel's Incompletness Theorem said we can't)
This is the second time you've introduced Godel for no reason, and incorrectly. Godel's theorem actually requires that the system is strong enough to contain "the natural numbers" in any model of it.

#### loseyourname

Staff Emeritus
Gold Member
afton said:
Still a circular argument :-) "being one more...." contains "one" in the
sentence. Imho the difficulties is from defining what is 'one'. If we somehow
can arrive at a definition of what is 'one' (in which Godel's Incompletness
Theorem said we can't) proving 1+1=2 would be much easier
It isn't an argument at all - it's a definition. That's the whole point! This would be the argument:

2 is defined as 1+1
Therefore, 1+1=2

Here's the form:

x is defined as y
Therefore, y is x

Some instances:

A bachelor is defined as an unmarried man.
Therefore, an unmarried man is a bachelor.

A recycling bin is defined as any container that contains trash designated for recycling.
Therefore, a container that contain trash designated for recycling is a recycling bin.

Water is defined as at least one molecule containing two moles of hydrogen and one mole of oxygen.
Therefore, any molecule or collection of molecules containing two moles of hydrogen and one mole of oxygen is water.

Do you have an objection to any of these? If you're looking for a proof from first principles, I suppose we can expand the argument to this:

Any symbol that is defined is equal to its definition.
The symbol "2" is defined as "1+1."
Therefore, 1+1=2.

Is that good enough for you? Or do you see some logical paradox in defining the word "definition?" "1" is defined simply as the difference between any integer and an integer next to it on the number line.

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#### dextercioby

Homework Helper
Water is defined as at least one molecule containing two moles of hydrogen and one mole of oxygen.
Therefore, any molecule or collection of molecules containing two moles of hydrogen and one mole of oxygen is water.
I'm sorry,pal,this is after all,a science forum and any little/huge mistake must be corrected.

One mole of any substance (obviously,water included) contains exactly $N_{A}$ atoms/molecules,where $N_{A}$ is called "Avogadro's number" and is aproximately equal to $6.023\cdot 10^{23}$.In the case of water,the molecule has 2 atoms of Hydrogen and one atom of oxygen and one mole of water weighs approximately 18 grams and contains $N_{A}$ molecules.
IIRC,the 'mole' is one of the 7 fundamental units from SI and is defined as the substance quantity corresponding to $N_{A}$ atoms/molecules.

Daniel.

#### roger

FulhamFan3 said:
there is no difficulty in defining 1. 1 is defined as an isolated quantity. 2 is defined the whole number quantity after that.

But why should quantity even enter into the argument ? ( when trying to define 1)

After all quantity is physical, so shouldn't we be able to define 1 purely abstractly, without resorting to physical principles such as a quantity ?

Roger

#### FulhamFan3

roger said:
But why should quantity even enter into the argument ? ( when trying to define 1)

After all quantity is physical, so shouldn't we be able to define 1 purely abstractly, without resorting to physical principles such as a quantity ?

Roger
nope. Numbers can never have an entirely abstract basis. How would you teach someone to count just using numbers? You'd have to show somewhere the one is singular and two is twice that. They have to represent some sort of quantity whether its a unit of length, area, volume or apples. It's only abstract in the sense that it's a general unit basis.

"Proof of 1+1=2"

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