1+1=2: Is There a Proof or Acceptance?

[afton]

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

 

[dextercioby]

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]

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

Daniel.

 

[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]

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

 

[ahrkron]

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]

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] .
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]
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 :)

 

[Hurkyl]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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]

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.

 

[dextercioby]

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]

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]

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.

 

[Hurkyl]

Numbers can never have an entirely abstract basis.

Wrong. Try checking out Peano's axioms for natural numbers, or the meaning of "complete ordered field".

How would you teach someone to count just using numbers?

Well, there are several things one could mean by "counting" -- if you simply mean naming the terms of the sequence 0, 1, 2, 3, ... in order, then it's fairly straightforward.

They have to represent some sort of quantity whether its a unit of length, area, volume or apples.

Nope. They don't have to represent anything, and even when they do, it doesn't have to have anything to do with quantity. For example, integers can represent proofs of formal logic.

 

[FulhamFan3]

Well, there are several things one could mean by "counting" -- if you simply mean naming the terms of the sequence 0, 1, 2, 3, ... in order, then it's fairly straightforward.

In that case your just telling them to memorize abstract symbols without telling them the meaning. You can do the same with the alphabet but it's useless unless they understand the letters represent sounds in speech.

 

[matt grime]

In that case your just telling them to memorize abstract symbols without telling them the meaning. You can do the same with the alphabet but it's useless unless they understand the letters represent sounds in speech.

"meaning" is dubious, after all i don't need to speak out loud, or even have sounds for an 'alphabet' in order for it to convey meaning.

instead of meaning, if we could even agree on what meaning means, forgive the semi-unintentional pun, how about its use. You are free to argue, obviously, that its usage is its meaning, and i'd agree, and people tell me Wittgenstein would have agreed, but i don't think that's the sense of meaning you mean. sorry again.
I don't need to know what the symbol sqrt(2) means, all i need to know is that it's positive, and when i square it i get 2.

 

[Hurkyl]

In that case your just telling them to memorize abstract symbols without telling them the meaning.

But the point still holds -- it can be done.

You can do the same with the alphabet but it's useless unless they understand the letters represent sounds in speech.

A common misconception. Abstraction is not a silly notion math geeks like to use to feel superior to everyone else; it has proven itself to be an extremely powerful tool. One of the more recent examples is the transformation of algebraic geometry -- the process of abstraction has turned it from the stuff you learned in Algebra II into one of the most powerful and pervasive subjects in mathematics.

But enough talk. There's an example sitting right in front of you: your computer.

 

[loseyourname]

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.

True. I should have said the molar ratio of hydrogen to oxygen is 2:1.

 

[FulhamFan3]

But the point still holds -- it can be done.

If you tell that person you taught to count to get 4 sticks of wood how would he actually know to get 4? They need a number to correspond to a quantity somewhere in the education. When I say teach someone to count I mean that they know the meaning of the numbers they are learning.

 

[matt grime]

Well, not really. 4 is the successor of 3, which is the successor of 2 which is the succesor of 1. I think even dictionaries define 4 as one more than 3. Perhaps you could call that the meaning of "4". Who's got the definition of definition?

Blackadder: I have two beans, and I add two more beans, what do I have?
Baldrick: A very small casserole.

 

[Hurkyl]

When I say teach someone to count I mean that they know the meaning of the numbers they are learning.

Or, are you merely teaching them the art of labelling objects with an initial segment of positive integers?

 

[VietDao29]

Hi,
I really don't think that it's worth arguing here.
We must accept that the natural numbers are: 0, 1, 2, 3, 4, 5, ...
And, if I had an honor to be the inventor of the natural number, I can make it whatever I like and my descendents just have to accept it. I can make it like:
1, 0, 5, 7, 10, 100, 20,... Or I can even create some more symbol to make different numbers.
Why 1 + 1 = 2 is you look at the array of natural number. Search where the 1 is and simply count from that number 1 more value, and you get 2.
And 2 + 3 = 5. Just do the same...
It's acceptable, and must be accepted, as you cannot do anything to change it.
It's basically correct... as I think.
Viet Dao,

 

[Alkatran]

Hi,
I really don't think that it's worth arguing here.
We must accept that the natural numbers are: 0, 1, 2, 3, 4, 5, ...
And, if I had an honor to be the inventor of the natural number, I can make it whatever I like and my descendents just have to accept it. I can make it like:
1, 0, 5, 7, 10, 100, 20,... Or I can even create some more symbol to make different numbers.
Why 1 + 1 = 2 is you look at the array of natural number. Search where the 1 is and simply count from that number 1 more value, and you get 2.
And 2 + 3 = 5. Just do the same...
It's acceptable, and must be accepted, as you cannot do anything to change it.
It's basically correct... as I think.
Viet Dao,

Are you arguing that "1" has no meaning, or that the meaning referenced by "1" cannot be defined? You seem to skip from "look I can change symbols!" to "you need '1' for '1'"

You always have to assume some rule before you can proceed in math. For example, "Math is logically consistent" is one of the main rules IMO ... Can you prove that math is logically consistent? well, it's one of the axioms, so if it isn't logically consistent it isn't math thus math is logically consistent.

Basically put: "1 + 1" can be represented the symbol we use to represent the quantity equal to "1 + 1", which is "2". Or am I being too 'superficial'?