Proof of 1+1=2

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Hurkyl

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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.
 
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

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FulhamFan3 said:
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

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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

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

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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

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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

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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

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VietDao29 said:
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'?
 

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