1. Oct 30, 2004

### drcrabs

I reckon addition is a basic skill. But can it be proved?
How?

2. Oct 30, 2004

### robert Ihnot

Good heaven! Have you ever gone through the check out at the store? I have 6 of this and three of that...etc. Notice how addition is Abelian. Could it be possible it is not? Like three pops + two soups is not equal to two soups + three pops? What would the world be like then?

3. Oct 30, 2004

### drcrabs

That kinda didn.t answer my question

4. Oct 31, 2004

### Gokul43201

Staff Emeritus
What is it you want to prove ?

5. Oct 31, 2004

### drcrabs

6. Oct 31, 2004

### StatusX

prove the color red

7. Oct 31, 2004

### drcrabs

DrCrabs

The visible red light has a wavelength of approximately 650 nm.
There are three types of color-sensitive cones in the retina of the human eye, corresponding roughly to red, green, and blue sensitive detectors.
Experiments have produced response curves for three different kind of cones in the retina of the human eye. The "green" and "red" cones are mostly packed into the fovea centralis. By population, about 64% of the cones are red-sensitive, about 32% green sensitive, and about 2% are blue sensitive.
So if a body emitts a wavelength of approximately 650 nm, the cones that correspond to that wavelength absorb that are hance seeing the color red.

8. Oct 31, 2004

### drcrabs

Maybe i should specify what im wanting.

Can someone prove that if

1 + 1 = 2

then

1 + 2 = 3

9. Oct 31, 2004

### StatusX

first of all, thats not a proof, your stating physical observations. second, wow, what a waste of time. and third, you missed my point. addition is such a basic, intuitive concept that it defies proof. we define formal systems that reflect our intuitve feeling of how addition works, and proofs can be set up within those formal systems. 2 is defined as the number equal to 1+1, and 3 is defined as the number equal to 2+1.

if youre asking me to prove the physical law that if i put two apples in a bag, and then put another three in, and then count the total number of apples in the bag, i will always get the number five, i cant do that. just like i cant prove the law of gravity is always true, even though overwhelming evidence shows it is. any rules we formulate about the physical world are made with the assumption that the world works in a regular, consistent way. this assumption cannot be proven.

10. Oct 31, 2004

### drcrabs

All i want to know is how to prove addtion.
Is that so hard to do?

It wasn't a waste of time. I know dat stuff like the back of my hand

11. Oct 31, 2004

### Gecko

omg, he's saying you CANT. its not something you can prove, it just is what it is. in other words, its not something that was built on foundations that show its true.

12. Oct 31, 2004

### StatusX

i thought i answered it. your question is extremely vague, and if i didnt understand exactly what you were thinking, then youll have to explain it better.

in math, addition is DEFINED to go along with how we feel the world works. look up peanos axioms for a formal way of defining addition. if you want to prove the physical property that our idea of addition is based on, (ie, the apples in a bag i described before), well, you cant.

basically, any proof is based on initial assumptions. you can never say something is absolutely true, you can only say it is true if some other statement is true, the assumption. the initial assuptions are usually based on human intuition or physically observed phenomana, but neither of these can be "proven." you can prove the moon will orbit the earth in an ellipse if gravity is an inverse square law, but you cant prove that about gravity. its based on observation, and theres no reason we couldnt wake up tomorrow and find theres a new law of gravity.

and if you dont think your question is vague, think about the fact that the statements 1+1=2 and 1+2=3 are meaningless until 1,2,3,+, and = are defined. and once they are, what do you have left to prove? (a better question would be to prove that addition is commutative, which is nontrivial and can be done from peanos axioms)

Last edited: Oct 31, 2004
13. Oct 31, 2004

### Math Is Hard

Staff Emeritus
Hi drcrabs,
I thought about your question, and there's not a simple answer, even though I'm sure many folks would dismiss this question as trivial. It occurred to me that you might be interested not simply in the "proof of addition", but in the general history of how human beings developed number systems and counting.
I have a wonderful book called The Universal History of Numbers by Georges Ifrah that you might enjoy. It covers human concepts of counting and its applications, from the counting of the number of fingers on our hands, to the invention abacus, to the underlying foundations of the modern computer.
It's really a fascinating read if you like that sort of historical data.

14. Oct 31, 2004

### Gecko

1+1 = 2
subtract one from both sides
1 = 1

lol.

15. Oct 31, 2004

### CrankFan

It depends on where you start from.

From something like the Peano Axioms you can prove the existence of all of the arithmetic functions commonly used with the naturals and ensure that they have the properties that we intuitively expect them to have.

After that you have to develop Z, Q, R, etc.. and do the same as described above at each step of the development. It's not something that is properly described in a single post but to give you a very basic idea of how this works:

In PA, the existence of 0 and a successor operation ' is postulated so that we can think of the natural numbers as the sequence 0, 0', 0'', 0''' and so on...

To prove addition exists we want to prove the existence of a unique function f: NxN->N, with the properties

(1) for all x in N, f(0,x) = x
(2) for all x and y in N, f(y',x) = (f(y,x))'

After this is proven, it's then shown that this function f, or addition has all of the properties we intuitively expect it to have: it's commutative, associative, etc.

This kind of approach (first oder PA) is older in style, you tend to see it more in older books like "Set Theory and Logic" Robert Stoll and "Foundations of Analysis" Edmund Landau. The modern approach is more set theoretical, and usually based on ZF rather than PA. You start our defining numbers as sets, and then define basic arithmetic operations as operations on sets.

I was cleaning out my links the other day and came across this, which describes the set theoretcal approach.

http://www.uwec.edu/andersrn/sets.htm [Broken]

Last edited by a moderator: May 1, 2017
16. Oct 31, 2004

### drcrabs

Yes thanx guys

17. Oct 31, 2004

### matt grime

The proof you want, that if 1+1=2, then 1+2=3 can be written as:

1+2
= 1+(1+1)
= 1+1+1
=3 {definition}

assuming that the integers are a ring.

if you want to prove that from first principles then you're welcome to, but don't expect many mathematicians to care these days.

18. Oct 31, 2004

### recon

Maybe this should be moved to the philosophy forum.

19. Oct 31, 2004

### jcsd

You haven't even said in which set, but if we were to talk abourt the natural numbers and Peano's axioms here is proof that 1 + 1 = 2:

Peano's axioms:

1) 0 is a member of N.

2) for each element n that is a member of N there exists an element n* (which is a meber of N also) called the succesor of n.

3) 0 is not the succesor of any elemnt in N

4) For each pair n, m in N where n is not equal to m, then n* is not equal to m*

5) if A is a subset of N, 0 is an elemnt of A and p is a member of A implies that p* is a member of A then A = N.

Let 0* = 1 and 1* = 2

n + 1 = n* (note this doesn't fully defoine additon, but it is sufficent for our purposes, to fully define additon we would also say n + p* = (n + p)*)

therefore:

1 + 1 = 1* = 2

Two thing to notice here, firstlythe proof is very, very trivial and in all honesty it proabably wasn't worth doing (more fool me)! The second thing to notice is that except the very last line, it is just a list of definitions, this is as additon is not something we prove it is something we define.

20. Oct 31, 2004

### Hurkyl

Staff Emeritus
Sorry, I feel the need to nitpick!

Addition is defined from axioms, but those axioms were selected because we feel they reflect how the world works.