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## Main Question or Discussion Point

Is there a proof that 1+1 = 2 ?

or we just accept it as it is ?

or we just accept it as it is ?

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Is there a proof that 1+1 = 2 ?

or we just accept it as it is ?

or we just accept it as it is ?

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It surely as hell is...Read HallsofIvy's post hereafton said:Is there a proof that 1+1 = 2 ?

or we just accept it as it is ?

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

Daniel.

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

Daniel.

Daniel.

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matt grime

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

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ahrkron

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Please refrain from negative comments. It is not in the spirit of the forums, and this is most definitely NOT a stupid question.FulhamFan3 said:

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Gokul43201

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Not a proof, but 2 is defined as the successor of 1 (under Peano), and hence satisfies the above requirement.afton said:Is there a proof that 1+1 = 2 ?

or we just accept it as it is ?

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.

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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

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It's all a matter of definition. In most mathematical examples, 2 isBut 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.

Incorrect.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.

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

eg 1 apple add 1 apple equals two apples ?

Roger

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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.roger said:

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

eg 1 apple add 1 apple equals two apples ?

Roger

Daniel.

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matt grime

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roger said:

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?

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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.roger said:

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

eg 1 apple add 1 apple equals two apples ?

Roger

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.

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matt grime

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Even if the two drops merged, the volume of water is two units ? no ?matt grime said:

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2 drops are in the cupmatt grime said:

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

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loseyourname

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Still a circular argument :-) "being one more...." contains "one" in theloseyourname said:

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

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matt grime

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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.afton said:If we somehow can arrive at a definition of what is 'one' (in which Godel's Incompletness Theorem said we can't)

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loseyourname

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It isn't an argument at all - it's a definition. That's the whole point! This would be the argument: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

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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I'm sorry,pal,this is after all,a science forum and any little/huge mistake must be corrected.loseyour name said: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.

One mole of any substance (obviously,water included) contains exactly [itex] N_{A} [/itex] atoms/molecules,where [itex] N_{A} [/itex] is called "Avogadro's number" and is aproximately equal to [itex] 6.023\cdot 10^{23} [/itex].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 [itex] N_{A} [/itex] molecules.

IIRC,the 'mole' is one of the 7 fundamental units from SI and is defined as the substance quantity corresponding to [itex] N_{A} [/itex] atoms/molecules.

Daniel.

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FulhamFan3 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

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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.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

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