Is there a way to refute '1+1 = 2'?

[Gulnara]

Hi. Some claim that '1+1=2' is true. It seems to me that '1+1 = 2' is like a short movie in our head. 2 dots get toghether or something like that. Now if you say this is wrong in general and point out that sometimes '1 + 1 = 3', by pointing out that if you put a man and a woman together, after 9 months there is 3 of them. Then supporters of '1 + 1 = 2', will claim that it was a wrong application of the theory. So they have a winning strategy. There is no way they can lose. They have a movie script in mind, if you make a movie that doesn't end the way their script does, they say it was not based on that script. Which is true. But then saying that their script is true doesn't make sense. Or does it?

Ramin

1 + 1 means 1 and 1 and equals 2. What is 2? The mark called 2 was chosen and named so to be a shorthand when depicting, drawing or writing 1 and 1. Instead of writing 200 times 1 and 1 and 1..., we can shortly write 200. 1 is then shorthand for depicting actual object counted. Instead of drawing entire person, or building, or fruit, I can simply write the mark 1 which people agreed to go with something separate, individual, the thing of its own. It became anonymous mark that goes with anything, can be attached to anything. Same goes for every other number.

[Misty]

[quote="Ramin22"]Hi. Some claim that '1+1=2' is true. It seems to me that '1+1 = 2' is like a short movie in our head. 2 dots get toghether or something like that. Now if you say this is wrong in general and point out that sometimes '1 + 1 = 3', by pointing out that if you put a man and a woman together, after 9 months there is 3 of them. Then supporters of '1 + 1 = 2', will claim that it was a wrong application of the theory. So they have a winning strategy. There is no way they can lose. They have a movie script in mind, if you make a movie that doesn't end the way their script does, they say it was not based on that script. Which is true. But then saying that their script is true doesn't make sense. Or does it?

How about a person born with two DNA"s? 1 DNA+1 DNA = 4 DNA's

[Q7382]

1 + 1 means 1 and 1 and equals 2. What is 2? The mark called 2 was chosen and named so to be a shorthand when depicting, drawing or writing 1 and 1. Instead of writing 200 times 1 and 1 and 1..., we can shortly write 200. 1 is then shorthand for depicting actual object counted. Instead of drawing entire person, or building, or fruit, I can simply write the mark 1 which people agreed to go with something separate, individual, the thing of its own. It became anonymous mark that goes with anything, can be attached to anything. Same goes for every other number.

I think this answer is simplest and most direct. In essence, saying "1+1" is just another way of saying "2".

The idea of refuting "1 + 1 = 2" is then akin to refuting a definition. A definition may be useful or useless in given communicative context, or it can comport or not comport with common usage, but it can't properly be called correct or incorrect; hence it doesn't make sense to speak of a definition being proven or refuted.

However, to get a final satisfying answer on the overall issue requires a firm definition of each symbol. If, as someone mentioned already, we take the symbols as instructions on how to visualize various numbers of dots, we will regard the result as obvious: "Yes, if I imagine a dot on the left side of my visual field and I imagine another dot on the right side of my visual field at the same time, I am now clearly seeing two dots on my visual field."

In other words, I suggest that any uncertainty around "1 + 1 = 2" arises not from any empirical question, but from the fact that various people and various schools of mathematical thought will interpret those symbols differently. One person sees dots, while another envisions something much more complicated and abstract involving set theoretic concepts believed to be required for the foundations of mathematics. It is not so much that there is no clear answer to the OP's question as that there is no one agreed-upon way to specify the question, at least in a rigorous context.

[Ben Saint-Clair]

In group theory in mathematics if you are working with a cyclic group C3 under addition 2+2=1.

Boom.

all mathematical truth depends upon axioms and the implications and laws of inference used upon those axioms.

[Nemisisx]

Why would yo want to refute it? It's a pragmatic reality, you can give it the "man in the street test" based (very loosely) on Wittgenstein who suggested that a skeptical objection to an established fact must have reasonable grounds to be valid. That is the skeptical objection, weighed up against the weight of reasonable proof should be greater for it to be valid. Go into a store count out 1 dollar then 1 dollar ( indicating that you have handed over 7 dollars) to a shop assistant, for an article that costs 7 dollars. see what happens. Then run your movie script theory passed them. See what happens, that's a basic man in the street test.

[Philophile]

1+1 = 1+s(0) = s(1+0) = s(1) = 2

Please explain. s(0) is what? Do you mean s 'times' 0 or the function s at 0. Please define s.

[Ben Saint-Clair]

1+1 = 1+s(0) = s(1+0) = s(1) = 2

Please explain. s(0) is what? Do you mean s 'times' 0 or the function s at 0. Please define s.

Alan is using something called the successor function, which essentially just means that it represents the number that comes immediately after the number in the brackets. It follows from the Peano axioms on the natural numbers.

[Philophile]

(Nested quote removed.)


Please explain. s(0) is what? Do you mean s 'times' 0 or the function s at 0. Please define s.

Alan is using something called the successor function, which essentially just means that it represents the number that comes immediately after the number in the brackets. It follows from the Peano axioms on the natural numbers.

So Alan is proving 1+1=2 by using the fact that 2 is the next number up from 1 on the natural number scale. I'm sure Peano had in mind the fact that 1+1=2 when setting up the axioms, thus using a successor function does not prove or refute 1+1=2.

One can set up any axiomatic system where 1+1= any number. The interesting question lies in relating 1+1=2 to the real observable world. Since in the real world it seems in almost all everyday cases that 1 thing + 1 thing = 2 things. This suggests a mapping between Peano's abstract axioms and the observable universe. What mapping that is i'm not sure..... Any suggestions?

[Conway]

Might I suggest...

It is possible to have 2 very small apples, it is possible to have 2 large apples. In both cases you have the physical number 2. But you do not have the same value. It is not that the small apples are .2 of an apple, they are full apples only containing smaller space. So then unless the space and value of all given numbers are all defined in the equation 1 + 1 = 2, then there will always be more than one sum. Albeit the sum is always the same symbol.

[HZY]

Hi. Some claim that '1+1=2' is true. It seems to me that '1+1 = 2' is like a short movie in our head. 2 dots get toghether or something like that. Now if you say this is wrong in general and point out that sometimes '1 + 1 = 3', by pointing out that if you put a man and a woman together, after 9 months there is 3 of them. Then supporters of '1 + 1 = 2', will claim that it was a wrong application of the theory. So they have a winning strategy. There is no way they can lose. They have a movie script in mind, if you make a movie that doesn't end the way their script does, they say it was not based on that script. Which is true. But then saying that their script is true doesn't make sense. Or does it?

Ramin

In quantum physics, 1+1 could theoretically equal to 3 so long as it is not affected by observation, because the act of observing 1+1 brings it back to 2.

[Londoner]

Might I suggest...

It is possible to have 2 very small apples, it is possible to have 2 large apples. In both cases you have the physical number 2. But you do not have the same value. It is not that the small apples are .2 of an apple, they are full apples only containing smaller space. So then unless the space and value of all given numbers are all defined in the equation 1 + 1 = 2, then there will always be more than one sum. Albeit the sum is always the same symbol.

We get into problems if we confuse objects with numbers. 'Two large apples' or 'a lump of clay' or the cast of Ramin's short movie are not numbers; there is no such thing as a 'physical number', numbers are abstractions.

We could use words that can also refer to physical objects as numbers, but only if we have those words refer to a class rather than objects. 'Apples' in the sense of 'members of that class of fruit' does not refer to any specific apples. We can never encounter 'apples' in the sense of 'the class of apples'; we can only encounter specific apples. Similarly, 'two apples' in the sense of 'two apples plus two apples equals four apples' does not describe any specific apples.

We may use phrases that confuse the two; that treat an apple both as an individual physical object ('very small' or 'large' or 'you see two worms') but also as an abstract class. But any confusion is in that description, not in the maths.

[Fooloso4]

There are several different issues being bandied about. The question cannot be addressed without reference to application. If you are counting then as long as you know how to count and no special rule is applied, as would be the case for example when counting a baker’s dozen, then 1 +1 will always equal two. One could attempt to ground this logically, either deductively or inductively, but all we need to do is count.

Now if I buy a breeding pair of rabbits I start with 1 male + 1 female which equals 2 rabbits. If I count again a few months later I will have more than 2 rabbits but this does not mean that 1 +1 equals more than 2. I cannot determine how many rabbits I have if it is no longer the case that 1+1 equals 2. While we are adding the rabbits are multiplying.

Now suppose I have 2 apples and 3 oranges. How many do I have? In order to answer the question we must identify the unit being counted. No matter how many oranges I add I still only have 2 apples. Once the unit that is being counted is identified then the answer can be determined unequivocally.

[Conway]

I am not debating the separation of abstract mathematics and application thereof. What I am saying is that the abstraction of a "number" does not define both the value and space, that the given thing, that it is intended to represent actually contains. The purpose of math is application. Therefore anything that is represented by a number does not have it's totality represented by that number. For this to occur all numbers then must contain both a value and a space. Both must be labeled. This is why the original equation can be refuted. Identifying the unit is not the totality of the truth. The unit and or "space" along with the value must both be identified.

[MHopcroft1963]

Let me see if this clarifies the situation.

Let's say you have one object on a table. It's doesn't matter what the object is. Then you find another object (again it doesn't matter what) and place it on the table. Now there are two objects on the table. You can repeat this process as often as you want, but not once will you be able to place one object and then another object and end up with three objects.

Every time you add one to one, you will always get two.

Why is it important that 1+1=2? Because mathematics is one of the principal tools by which we observe and quantify the physical universe. Math works.

[Conway]

Let's then consider the above analogy. If I place one thing on the table, and then place another like it but only smaller the sum is two. But then if I placed the same thing , and then placed another like it but only larger the sum is still two. However it is clear to all observers that in both cases the sum is not the same "value", albeit the same symbol. What the object "is" does matter. More specifically the space and value of the object. Math does not work for all scenarios, all doctorates in the subject that I have talked to have agreed on that at least.