1+1 = 2 ?

Factual Questions
1

My question is simple:

Can you prove that 1+1 = 2?
The answer everyone has ever given me, has been a long gaze, but the very fundaments of mathmatics is what makes me sceptical to science as a basis for everything. We have a strong scentiment that we can explain everything rationally (and if you are religion you tend to like to explain everything irrationally).

But since I can not see how one could prove 1+1 = 2, then science, like any other dogma is based on faith, and therefore I am automatically sceptical.

I would truly apprechiate some answers to my query, on the one place in the world where I think this question might be handled seriously and with as little mockery as possible (please:D).

2

Rough proof via peano arithmetic:

A few simple definitions in peano terms. 1 is the sucessor of 0, or S0. 2 is the successor of 1, or SS0

Any equation is equal to itself, so we can start with S0 + S0 = S0 + S0 (1 + 1 = 1 + 1)

Any expression of the form A + SB is equivalent to SA + B, so S0 + S0 is equivalent to SS0 + 0. Thusly we can prove that (1 + 1 = 2 + 0)

Since the expressions X + 0 and X are of equal value (by the Peano definition of addition,) SS0 + 0 can be reduced to SS0. Thus, we have proved that 1 + 1 = 2 according to Peano’s axioms. (I think… hopefully I haven’t skipped over anything or taken an illegal shortcut. It’s been a while since I’ve worked with Peano.)

Algebraically, we can approach the problem somewhat simpler:

Take 1 + 1 = 2

subtract 1 from both sides, and you see that statement is equivalent to 1 + 1 - 1 = 2 - 1

Simplifying both sides, you can get 1 = 1, which is pretty self-evident.

Basically though… the statement is nearly so trivial as to be true by definiton. When we use the numeral 2, we are using it as a symbol to represent the number we get when we add 1 and 1 together… or else what the heck is it supposed to mean?

Does that help at all??

3

I hesitate to post this, because I suspect that we can come up with as good or better answers here on the Dope (like chrisk has), but here’s one relevant link.

4

:: gives sverresverre a long gaze ::

Scientific proof of 1 + 1 = 2, ok, try this simple experiment. Take one bottle cap, observe that it is in fact one bottle cap and place it on a blank piece of paper. Take another bottle cap, observe that it is in fact one single bottle cap and place it on the paper next to the first bottle cap. Double check your observations to ensure that you should have one bottle cap sitting next to another bottle cap. Double count the bottle caps on the paper to ensure that there are, in fact, two bottle caps on the paper. As a control, place a seperate blank sheet of paper to the side and do not place any bottle caps on it. Count how many bottle caps are on the second one and record this. Repeat this experiment several times, then compair the results.

Science is based on observable* conditions that can be repeated. If an object, fact, condition, etc. fails to meet those two conditions then it is not scientific fact. This is the dogma of science, it must be observable and must be repeatable, thus no faith is required.

*Note that many scientific facts, objects, conditions, etc. are not observable but their effects on their surroundings are. If the observations of the effects are repeatable and consistant then the fact, object, condition, etc. is theoretical. For example sub-atomic particles, black holes, and Brittany Spears’ musical talent are all unobservable thus theoretical.

5

This is going to move to GD fairly quickly. Basically you’re asking if there are un-provable statements within mathematics that make it inconsistent. If math is inconsistent then anything that uses mathematics to model empirical data must also be inconsistent.

The thing is there are “unprovable” statements in mathematics called axioms. It’s from these axioms that the resulting structures hang. You can have a variety of mathematical structures, of course, all based on differing starting axioms. The thing is, within those structures the mathematical reasoning is consistent, even though there maybe statement not provable within that framework.

But in the end, if I give you 1 apple and then I give you another apple you’ve now got 2 apples.

6

Math is a language. 1+1=2 by definition.

7

sverresverre, you are comparing apples and oranges.

1 + 1 = 2 because addition is *defined * such that 1 + 1 = 2. Mathematics is a human construct, that has no external reality whatsoever. It is a thing of symbols only - have you ever held a 1 or a 2 in your hand? No, because there is no such thing as a 1 or a 2. It’s not a matter of taking it on faith that 1 + 1 = 2. It must be so because we have defined addition in that way. What math does is to postulate a certain set of “operations” such as addition, substraction, etc, and a set of rules.

Now, the interesting thing about this totally artificial construct mathematics is how useful it is in describing reality. We have never seen a situation that could be represented by 1 + 1 that did not equal 2. We don’t expect we ever will. So mathematics becomes an excellent predictor of what will happen under many different circumstances. Probably the vast majority of technology you use was developed using mathematics. Why? Because this set of rules and operations that human beings came up with fits our experience so completely. But it is still an artificial construct, and we know this. So we know, based on the millenia of experience we have with it, that correctly applied mathematics is extremely likely to conform to external reality. We call this modelling, and although the term is recent, the concept is very old.

Religion is describing an external reality. There either is a god or there isn’t. There either was a Jesus or there wasn’t. And so on. It is not, nor does it purport to be, an abstract set of symbols that accurately reflect external reality. It is external reality, or at least that is what it claims to be.

External reality lends itself to evidence, to experiential proof. You know the table in front of you is solid (at the macro level) because you can rest your hand on it, and so can any other person who tries (assuming they have a hand). In essence, that what the scientific method is. It postulates something: the table is solid. And then it sets up replicable tests: you rest your hand on the table, and so does the next guy and the next and the next. *Anyone * who attempts to rest his/her hand on the table can do so. And other people can see them doing so. There is evidence.

Mathematics isn’t the same kind of thing. It can’t be seen, felt, heard, or sensed in any way. It’s simply a humanly defined tool, a set of abstract concepts that, thus far, have accurately corresponded to our experience of external reality. There’s no faith about it, except the faith that because it has always done so in the past, it always will correspond to external reality. If, in fact, it fails to do so, then the chances are that the abstract definitions we created and called mathematics are simply wrong, and need to be re-defined.

Religion claims to be external reality. It therefore lends itself to evidence, proof. Mathematics is a symbolic construct. They are not comparable.

8

Here’s a sketch another proof that I think is a bit clearer:

You have a number 0, and you have an operator ’ such that whenever n is a number, n’ is too. Define + as n + 0 = n and n + m’ = (n + m)’.

Now, what are 1 and 2? Let’s call them 0’ and 0’’, respectively. 0’ + 0’ = (0’ + 0)’, and 0’ + 0 = 0’. So 0’ + 0’ = 0’’. In other words, 1 + 1 = 2.

I’m glossing over quite a bit here. For the full proof, I would have to define what 0, ', and = mean, but I don’t think that’s of any pedagogical value here.

9

If we are talking in mathematics, then we need to agree on what a few things mean, including “prove”, “1”, “+”, “=” and “2”.

chrisk has given a good outline of a proof in which 1 is defined as S0 and 2 is defined as S1. However, that’s not the only possible axiom system for the natural numbers, and in other axiom systems, the proof would be different. So Oy! has assumed a system in which “2” is defined as being “1+1”, and of course in that system the proof that 1+1 = 2 gets to be a lot shorter.

At another extreme, Alfred North Whitehead and Bertrand Russell’s book Principia Mathematica notoriously says on p. 360: “From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.” That is, PM takes 360 pages building up an axiomatic system for the natural numbers, at the end of which you can say that you have proven that 1+1 = 2. (I’ve only skimmed PM, but I can assure you that those 360 pages are heavy reading indeed.)

10

That’s nothing. Can you prove that .999… + .999… = 2?

11

The current dominant philosophy, either explicitly or implicitly, is “structuralism”. That is, the natural numbers are a structure (described by the Peano axioms mentioned elsewhere in the thread) which can be instantiated in various models (like the set theory models mentioned elsewhere in the thread).

Basically, 1 + 1 = 2 by definition. “1”, “2”, “+”, and even “=” have no meaning by themselves. They’re slots in the structure, and one of the properties of the structure is that “1 + 1 = 2”. This means that for any model of the structure, you can apply the model’s version of “+” to the model’s version of “1” and itself, and the result with be equal (in the model’s sense of “=”) to the model’s version of “2”.

This fact can be proven from the Peano axioms, as other commenters have done, so any model which satisfies the Peano axioms satisfies this property.

So sure, you can have a situation where 1 + 1 does not equal 2, but then you’re not talking about an instance of the natural numbers structure.

12

Now that’s being naughty, because in that equation “2” has to be a real number, not just a natural number. And you could ask, “Can you prove the 1/1 + 1/1 = 2” – and then 2 becomes another kind of number, a rational number.

Of course, there’s an obvious and natural isomorphism between a subset of the real numbers and the rational numbers, and another obvious and natural isomorphism between a subset of the rational numbers and the natural numbers. However, in the three different number systems, there are different axioms and definitions, and hence different proof methods.

13

Yep, that what I came in to reference. Don’t I remember correctly too the after writing this Whitehead was never again the same mentally? I believe the inference I’d received was that proving 1 + 1 = 2 had in effect used him all up and broke his brain.

14

Better yet is 3/3=1? 1/3=.33333… and 2/3=.66666… Now, assuming standard addition rules hold true, we know that 1/3+2/3=3/3, but .33333…+.66666… should equal .99999…, so does 3/3 really equal one, or does it just approach one?

This whole thread makes me think of a t-shirt one of my friends has. It says, “2+2=5, for exceptionally large values of 2.”

15

That’s a repeated subject of debate here: see 1 = 999… and .999~ – and those threads have references to earlier threads on the subject.

(And the correct answer is that the infinitely long decimal fraction .999… is equal to 1).

16

1+1=4 for very large values of 1.

17

More seriously,
1+1 = 10 in binary arithmetic, i.e., arithmetic base 2
1+1 = 0 in arithmetic modulo 2

18

1 + 1 = 1, for cases of love. :cool:

19

Yep. This is just a corollary of the theorems that .999… = 1 and 1 + 1 = 2. Q.E.D. :stuck_out_tongue:

20

Forget the birth control, and 1+1=3. :wink: