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

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Londoner
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### Re: Is there a way to refute '1+1 = 2'?

You said 'A' didn't stand for anything. So, it's just a logical variable. How do you falsify a variable? Add a tilda~
You have a different understanding of the meaning of 'falsify' to me. I think it involves argument or evidence. Just contradicting something is not to demonstrate it to be false.

But as I say, if that is what you understand by 'falsify' then there is nothing that cannot be falsified - therefore there is also nothing that can be verified.

And as I said, re the OP's question about '1 + 1 = 2', it would be 'falsified' by putting 'not' in front of the equals sign, but could be 're-verified' by adding a second 'not' to make a double negative, then 're-falsified' again with a further 'not'... Again, I do not think that process is what is normally understood by 'verification' and 'falsification'.
Especially this latter, an encounter with Being sans the particulars, sans the body of contingency that provides contextual sense in order for individual identity to announce itself. Roquentin experiences the world free of logical structure and the purpose of this bloated epiphany is to make a powerful point: the world is not logic; it has no categories, no Kantian synthetic concepts that divide and formalize. It is cognitively alien, amorphous, and entirely unspeakable.
I do not see how this relates to the question in the OP. Roquentin is not Pythagoras, he does not see the world as number, but he can still add one and one. We are discussing what we mean by terms like 'one'.
Sound arguments, which are supposed to be true about the world, are really about language and its rules for concatenation. I could say something "out there' causes me to say the book is on the shelf, but the equation of proposition rests with the rules and their principles of subsumption. And when I say "something causes" I am, as Wittgenstein will tell you, speaking nonsense.
It isn't about what might cause you to say 'the book is on the shelf'. Nor is 'the book is on the shelf' an argument, it is only a proposition that might be part of an argument.

If I wanted to know what you meant by 'the book is on the shelf' I might ask you what would falsify it. Would it be the location of the book? If the book was really on the floor, might the proposition 'the book is on the shelf' still be correct? I gather you think so. That the location of the book doesn't matter when it comes to the truth of 'the book is on the shelf'.

Rather you say it is a matter of whether 'the equation of proposition rests with the rules and their principles of subsumption'. (What is a 'principle of subsumption'? As far as I can see it that phrase refers to ideas in educational theory about how individuals learn from pictures or text, which I cannot see is relevant). So what then are 'the rules'? Is it more than the notion that 'the book is on the shelf' can be falsified with the addition of 'not'?

I don't think I should be diverted into discussing Wittgenstein, so I will just say that I do not think he was saying what you are saying.
I don't really follow this. Logic expounds?
A valid logical argument is one where the conclusions are consequent, or entailed, by its premise. That is to say the conclusion is already completely contained in the initial assumptions; it does not require anything external, for example empirical evidence.
My point has been, with some side steps, that 1+1=2 cannot be refuted because you can never get behind logic, and mathematics is a system of logic; that if you could refute the principle of addition (notwithstanding how you want to display it) you would be refuting a tautology (all mathematics is tautological. I think this is likely true.)
Pure maths is. 'Bachelors are unmarried men' is true because we have decided one term means the other. But there is no logic in it. 'X is X' is not logic. It is an axiom necessary in order for us to do logic, but it is not in itself something known through logic.

But if something cannot be refuted, then it can't be verified either. Thus we are back to the point I have been making all along, that '1 + 1 = 2' cannot be refuted or verified because it has no content - like a tautology it is circular.
...and if you are fine with this, then you must be equally fine refuting any and all occasions of logicality, which would apply to the logic you employ to make your refutation and the bleeding affair is self-refuting.
That doesn't follow at all. Once again, I am not saying 'logic doesn't work'.

I'm saying that a logical or mathematical formula in which the propositions do not relate to anything will produce conclusions that don't relate to anything either.

It is simply the ability to appreciate that pure maths is different to applied maths and that a logical argument being 'valid' does not make it 'sound' .

I think this exchange has run its course.

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

"I think this exchange has run its course."

Look, you can walk away, but you really don't get it. That book on the shelf is only a book because it is a logical construct. There are no books out there that applied logic is about. Applied logic is applied to itself. The whatever out there is pure ineffability. How it comes to "cause" an occasion for application is purely ineffable. What, you think rules are "applied" as you live and breathe? You ARE logic. All logic is self referential. You need to read some more serious philosophy. Too bad you didn't find the comments about Sartre interesting. They were.

Jam2289
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### Re: Is there a way to refute '1+1 = 2'?

Numbers are highly abstracted. All specifics other than quantity have been eliminated. It is a closed symbolic system, so one plus one always equals two unless you add another element, this can be in the equation or in the context. You have taken it into a biological context, this adds multitudes of other qualitatively different specifics. This is especially shown by the temporal relation. Over time one man and one woman would produce a child, then ten, then one hundred, then one million, it is just a matter of time. You can also add this into the equation, you just get a different equation to represent the new quality of the context that you have applied it to. There are computer programs written with the equations that produce simulations of population growth.

Zayl
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### Re: Is there a way to refute '1+1 = 2'?

I think 1+1 = 2 may be verifiable because you can simultaneously experience both facts. Also if you add a unit to another unit, it would be a contradiction to say that you added 2 units such that it is 3. However if you want to add 8+4=12 you will have to count each unit mentally to verify the number. Then maybe a demon is tricking you while counting so that you're counting wrong? Even so I think saying that the answer is anything else than 12 becomes a contradiction, since if you cannot end up with a different number than what you add per definition.

The interesting question is, could the universe possibly change in ways that are fundamental beyond our imagination such that 1+1=3? If so it does not mean that it is a 100% correct generalization to conclude that 1+1=2 always or when applied to describe nature.

Hereandnow
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### Re: Is there a way to refute '1+1 = 2'?

Zale:
Numbers are highly abstracted. All specifics other than quantity have been eliminated. It is a closed symbolic system, so one plus one always equals two unless you add another element, this can be in the equation or in the context. You have taken it into a biological context, this adds multitudes of other qualitatively different specifics. This is especially shown by the temporal relation. Over time one man and one woman would produce a child, then ten, then one hundred, then one million, it is just a matter of time. You can also add this into the equation, you just get a different equation to represent the new quality of the context that you have applied it to. There are computer programs written with the equations that produce simulations of population growth.
All sounds very good,that is, if you're talking about numbers apart from reality. But what if you ask a more basic question: What is a number? Or better, How can numbers and their quantifications (not to forget the distinct rules of their logical possibilities) be understood apart from the Real acts of arithmetic? Keep in mind that the thought of the number one even in the moment of its conception occurs in time and the number is a dynamic of the thought itself. I.e., something happens when the number is uttered, thought, conceived, executed in an equation, and so forth; and these are not abstractions at all. By this, of course, nothing can be an abstraction, for the moment it is present in a thought to be considered an abstraction, it materializes as a Real. There are no abstractions. But does this make sense? i would say it does.

-- Updated March 8th, 2016, 8:02 am to add the following --

that was Jam2289
Zale:
The interesting question is, could the universe possibly change in ways that are fundamental beyond our imagination such that 1+1=3? If so it does not mean that it is a 100% correct generalization to conclude that 1+1=2 always or when applied to describe nature.
A better question would be why is it not possible for 1+1 to be anything else? Why is the world bound to this? What makes it so? If we make our intuitions an object of inquiry, what does it mean that we can actually do this? Are numbers and the rules of their concatenation actually part of the world? Or do they have their grounding elsewhere?

Ormond
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### Re: Is there a way to refute '1+1 = 2'?

Philosophy Explorer wrote:Try this. One lump of clay combined with one lump of clay is just one lump of clay.
Oooh, oooh, I like this one. Bravo.

-- Updated March 8th, 2016, 10:34 am to add the following --

It seems that, in the real world, 1 + 1 = 2 means one thing plus another thing equals two things.

You might be able to refute the equation if you can argue that, in the real world, there are no separate objects which we can label "things". I would argue that the phenomena of "things" is, like math, a conceptual product of the inherently divisive nature of human thought, and not a property of reality itself. We seeing "things" does not automatically equal their existence.

In the real world, it seems more accurate to say that all of reality is functionally really just one big thing. If true, then the only valid equation would be 1=1.
If the things we want to hear could take us where we want to go, we'd already be there.

Excaljnur
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### Re: Is there a way to refute '1+1 = 2'?

Zayl wrote:
The interesting question is, could the universe possibly change in ways that are fundamental beyond our imagination such that 1+1=3? If so it does not mean that it is a 100% correct generalization to conclude that 1+1=2 always or when applied to describe nature.
What do you mean by this? If you are referring to the meaning of '2' as symbol that refers to any two things, then a universe where 1+1=3 exists is our world because we are merely changing the numerical symbol that refers to two things. For instance, the meaning of '2' is the same as the meaning of 'II' because they both refer to two things. In this way, the meaning of '2' and 'II' (mereologically speaking) is any two parts combined is a whole made of two parts. Given this, 1+1+1=3 would be any three parts combined is a whole made of three parts. So the symbolic representation, whatever it is, would refer to a specific number of combined parts. But if you are suggesting that "the universe could change in ways that are fundamental beyond our imagination such that" two parts combined is a whole made of three parts, then I would think something (maybe the combination) creates the third part out of nothing. Would this mean that a whole greater than the sum of its parts?

Jackson1982
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### Re: Is there a way to refute '1+1 = 2'?

It's apparently true because of definitions but it could be wrong if the definitions or laws of logic are incorrect.

The first definitions are the number system and counting where the quantities are in this order

1,2, 3, 4, 5, 6, etc...

First it must be assumed true that the law of identify is correct. Or X=X.

1 = o
2 = oo
3 = ooo
4 = oooo
etc...

By law of identity the reverse is true also

o = 1
oo = 2
ooo = 3
oooo = 4

If we put one "o" next to another "o" it looks like "oo" which is the same as 2.

But if the law of identity were not true then it's possible 1+1=3, 1+1=4 or whatever. The whole concept of sameness, differentness, and quantity would be meaningless and an illusion. Reality breaks down and nothing really exists as we know it since existence and non-existence would be the same thing.

Simplyhuman
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### Re: Is there a way to refute '1+1 = 2'?

Traditional math only adds up if you use it in the designed application.

As an earlier response pointed out: 1 lump of clay combined with another is 1 lump of clay. The same goes for a drop of water combined with another, and so on.

If I have a banana, and add another, on the surface, it would seem that I now have 2 bananas. But only if the banana is seen as an equal unit despite any other math that could be included, but has been previously agreed upon to ignore. Such as wight, surface space, volume, etc.. No two bananas are exactly the same (more than likely). One may consist of 7 billion atoms, the other may consist of 13 billion atoms. I have only reached the final sum of 2 if the restraints are predetermined and agreed upon.

Traditional/simple math is a vital tool, but is not absolute. I can build a wall with Lego blocks and it can work well. But if I use a mixture of Lego blocks and masonry bricks, the wall will fail. The intent and rules must be established ahead of time or else unforeseen variations will cause breaks in the logic.

Steve3007
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### Re: Is there a way to refute '1+1 = 2'?

Ormond:
I would argue that the phenomena of "things" is, like math, a conceptual product of the inherently divisive nature of human thought, and not a property of reality itself.
I agree with this. As a general rule, I would say that the divisions of nature are created for us for our purposes. Different purposes: different divisions. Just look at the arguments people have about whether Pluto is a planet or a dwarf-planet.
"When the seagulls follow the trawler, it is because they think sardines will be thrown into the sea." - Eric Cantona.

Jklint
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### Re: Is there a way to refute '1+1 = 2'?

If there were a way it would have to be done outside of math. I have no idea what that methodology would be like. It cannot be done in reference to math itself since that would annul the validity of every equation which are really complex and extremely complex versions of 1+1. To prove something within the same process which disproves it is a thorough non sequitur completely beyond the range of what we would otherwise classify as paradox. That's how I see it anyways. I haven't read all the posts on the subject or whether this as one scenario was already mentioned.

gimal
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### Re: Is there a way to refute '1+1 = 2'?

here is a disproof of 1+1=2
from the poet.
gamahucherpress.yellowgum.com/wp-conten ... ssible.pdf
It be said that 1+1=2 be a certain truth
Blah
1 number + 1 number = 1 number
1 number (2) +1 number (2) =1 number (4)

So 1 +1=2
And
1 + 1 = 1

Sealight
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Joined: July 19th, 2016, 11:25 am

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

Ramin22 wrote:Hi. Some claim that '1+1=2' is true.
Some claim it is true, some claim it is not. Looks like every truth is local and '1+1=2' is true in Arithmetic and Algebra. One can think that '1+1=3' is also true, and it is fine. But there are some problems with the last statement. The symbol '+' there cannot be an algebraic operation since that statement is wrong in Algebra. Then one has to redifine the symbol '+' and maybe symbols '1' and '3'. Then he gets a true statement with a clear meaning and which is not clear now, before all symbols are defined properly.
Everything is simple in Math.

-- Updated October 21st, 2016, 8:26 am to add the following --

And the previous example from gimal illustrates my post. There is a small flaw there. The obtained result, being false in Algebra, is not a contradiction in mathematics, but rather an appending, once all symbols in the equality(?) '1+1=1' are properly defined.

The Beast
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### Re: Is there a way to refute '1+1 = 2'?

Well. To the proposed 1+1 = 2 we add another proposition such as 1-1 =0; therefore, 1 is 1+0. Or 1 is 1 plus its container and 1+0+1+0= 2 in two containers. As such we defined an apple container as the container that can only contain an apple. This container is the specifications and history of apple. I am a consciousness in a container. The Universe is a container with the rules and regulations that are known or possible. If the container is gone then 1 = 1. We should define 1 as all that exist. If we exist then 1 exist and “nothing as container” is impossible the same is said for infinity as 1+1+n without containers since it is known that the Universe is a container.

Skakos
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### Re: Is there a way to refute '1+1 = 2'?

It's all a matter of axioms used. Choose a different set of axioms from the very beginning and you will end up with a different theory. However note that does not change the way you experience reality/ your environment. You are just altering the way you describe this environment via mathematics and science.
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