Is there a way to refute '1+1 = 2'?
 Ramin22
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Is there a way to refute '1+1 = 2'?
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?
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Re: Is there a way to refute '1+1 = 2'?
The mathematical concepts that leads to the inherent truth of the equation '2 + 2 = 4' or more broadly the deductive rules that make a statement like 'the bachelor is unmarried' or 'the cat is a mammal' is irrefutable, in context, and thus in that context those who would argue in favor of the truth of such statements have a winning strategy. I don't even think it is correct for you to refer to the equation or the fact that it is true as a theory. What you say about "their script" doesn't make sense, but that is your analogy so the fact that it doesn't make sense proves nothing.
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Re: Is there a way to refute '1+1 = 2'?
Try this. One lump of clay combined with one lump of clay is just one lump of clay.
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Re: Is there a way to refute '1+1 = 2'?
Philosophically speaking, '1+1=2' is a string of 5 symbols that is declared to be 'true' within the system of 'mathematics'.
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
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 Ramin22
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Re: Is there a way to refute '1+1 = 2'?
Thank you everyone for the answers. I have one more question. Is 'either P is true or ~P is true' just true inside classical logic? I know it can be considered true just in that sense. But is there any other sense, or are there others who argue classical logic is true on other grounds?
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Re: Is there a way to refute '1+1 = 2'?
However, it might also have to make truthful predictions outside the system. For instance, if you were designing a system that depended on mathematical calculations  and many do, even the one you're reading this on  then the outcome can't be specifically internal to the system you have devised. It also has to yield accurate results. And they may not be at all arbitrary or fanciful. If you designed a bridge to meet at a point, then whatever system you use to calculate that figure has to be accurate with reference to the bridge, not simply with reference to itself, otherwise the calculation will be faulty, and the project will fail.A_Seagull wrote:Philosophically speaking, '1+1=2' is a string of 5 symbols that is declared to be 'true' within the system of 'mathematics'.
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
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Re: Is there a way to refute '1+1 = 2'?
Quotidian, I was referring to (as I think the OP was too) pure mathematics, not applied.
In order to apply mathematics (or any other abstract system for that matter) to the real world you need to create a mapping between the two.
 Updated September 13th, 2014, 6:07 pm to add the following 
In order to apply mathematics (or any other abstract system for that matter) to the real world you need to create a mapping between the two.
 Updated September 13th, 2014, 6:07 pm to add the following 
Yes, I would say that 'either P is true or ~P is true' is only 'true' within the system of formal logic.Ramin22 wrote:Thank you everyone for the answers. I have one more question. Is 'either P is true or ~P is true' just true inside classical logic? I know it can be considered true just in that sense. But is there any other sense, or are there others who argue classical logic is true on other grounds?
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Re: Is there a way to refute '1+1 = 2'?
1+1=2 is merely a convention.
Contrary to other comments in this thread  there is no mathematical basis for this statement.
As has been noted, a mathematical statement has a specific meaning with respect to a set of axioms; change the axioms and the "truth" of a statement may change.
However, axioms themselves are statements. So axioms only have a specific meaning with respect to a specified set of axioms... which in turn only have a specific meaning with respect to a set of axioms...
In other words; it is impossible to define a set of axioms without having first defined a set of axioms.
Without axioms there is no 'proof' and no 'true or false'; there is no deductive logic.
Mathematics is justified purely by the extent to which it works. 1+1=2 is part of a group of relationships that help us design bridges that don't fall down in the first breeze. To this extent, 1+1=2 is justified... but we know that there are other useful groups of relationships within which 1+1 != 2.
So... no  1+1=2 cannot be refuted... because refutation implies logic  and there is no logic (there is reasoning  but that isn't the same thing as mathematics logic). But neither can 1+1=2 be proven. It is just a statement that we sometimes find useful.
Contrary to other comments in this thread  there is no mathematical basis for this statement.
As has been noted, a mathematical statement has a specific meaning with respect to a set of axioms; change the axioms and the "truth" of a statement may change.
However, axioms themselves are statements. So axioms only have a specific meaning with respect to a specified set of axioms... which in turn only have a specific meaning with respect to a set of axioms...
In other words; it is impossible to define a set of axioms without having first defined a set of axioms.
Without axioms there is no 'proof' and no 'true or false'; there is no deductive logic.
Mathematics is justified purely by the extent to which it works. 1+1=2 is part of a group of relationships that help us design bridges that don't fall down in the first breeze. To this extent, 1+1=2 is justified... but we know that there are other useful groups of relationships within which 1+1 != 2.
So... no  1+1=2 cannot be refuted... because refutation implies logic  and there is no logic (there is reasoning  but that isn't the same thing as mathematics logic). But neither can 1+1=2 be proven. It is just a statement that we sometimes find useful.

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Re: Is there a way to refute '1+1 = 2'?
A most amusing and interesting thread. Very few of the posts have any connection with the question, but that is quite normal and predictable, and (in this case) forgivable because the question is a question in mathematical philosophy, a subject of which very few mathematicians have any understanding.
The short proof that 1+1=2 (which, inexplicably, none of the respondents has seen fit to supply) is as follows:
1+1 = 1+s(0) = s(1+0) = s(1) = 2
It will be immediately apparent that this proof has little or nothing to do with scripts or movies, and I am not sure in which school of mathematics A_Seagull would have learned these terms.
This proof, like all mathematical proofs, is conditional upon an agreed set of axioms (and god, if only mathematicians could be made to remember this). It can easily be falsified by proposing a suitable modification to the axiom set.
The short proof that 1+1=2 (which, inexplicably, none of the respondents has seen fit to supply) is as follows:
1+1 = 1+s(0) = s(1+0) = s(1) = 2
It will be immediately apparent that this proof has little or nothing to do with scripts or movies, and I am not sure in which school of mathematics A_Seagull would have learned these terms.
This proof, like all mathematical proofs, is conditional upon an agreed set of axioms (and god, if only mathematicians could be made to remember this). It can easily be falsified by proposing a suitable modification to the axiom set.
 Ramin22
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Re: Is there a way to refute '1+1 = 2'?
Actually I still believe that it is based on a script. Not the proof of it, but the reason we believe in '1+1=2'. The proof as you point is some algebraic string. But the fact we accept '1+1=2' is not because there is a proof for it. Actually I believe it is the other way around. We use '1+1 = 2' as a test case to see whether our proof system is working correctly or not. The truth of '1+1 = 2' comes from our experience. The thing that made me think of a script is that if you tell someone that you doubt this fact, they will put one object on a desk, and say one. Then hold another object in their hand and say one, then put the new object next to the other one and then count the 2 objects and conclude 2. Now if in the middle of their demonstration, anything unexpected happens, like a kid coming and stilling one of the objects, they simply say 'that doesn't count'. It counts only when everything goes according to the picture they have in mind. If anything is different it doesn't count. That is how I see it as a script.
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Re: Is there a way to refute '1+1 = 2'?
Alan Masterman wrote:A most amusing and interesting thread. Very few of the posts have any connection with the question, but that is quite normal and predictable, and (in this case) forgivable because the question is a question in mathematical philosophy, a subject of which very few mathematicians have any understanding. .
But clearly, you are not one of them!
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Re: Is there a way to refute '1+1 = 2'?
A relevant and amusing post on an argument about number theory, from Ed Feser's blog:
You: Suppose you’re in the garden and you see two worms crawling around. Then two more worms crawl over. How many worms do you have now?
Peter: “Crawling” means moving around on your hands and knees. Worms don’t have hands and knees, so they don’t “crawl.” They have hairlike projections called setae which make contact with the soil, and their bodies are moved by two sets of muscles, an outer layer called the circular muscles and an inner layer known as the longitudinal muscles. Alternation between these muscles causes a series of expansions and contractions of the worm’s body.
You: That’s all very impressive, but you know what I meant, Peter, and the specific way worms move around is completely irrelevant in any case. The point is that you’d have four worms.
Peter: Science is irrelevant, huh? Well, do you drive a car? Use a cell phone? Go to the doctor? Science made all that possible.
You: Yes, fine, but what does that have to do with the subject at hand? What I mean is that how worms move is irrelevant to how many worms you’d have in the example. You’d have four worms. That’s true whatever science ends up telling us about worms.
Peter: You obviously don’t know anything about science. If you divide a planarian flatworm, it will grow into two new individual flatworms. So, if that’s the kind of worm we’re talking about, then if you have two worms and then add two more, you might end up with five worms, or even more than five. So much for this a priori “arithmetic” stuff.
You: That’s a ridiculous argument! If you’ve got only two worms and add another two worms, that gives you four worms, period. That one of those worms might later go on to be divided in two doesn’t change that!
Peter: Are you denying the empirical evidence about how flatworms divide?
You: Of course not. I’m saying that that empirical evidence simply doesn’t show what you think it does.
Peter: This is wellconfirmed science. What motivation could you possibly have for rejecting what we know about the planarian flatworm, apart from a desperate attempt to avoid falsification of your precious “arithmetic”?
You: Peter, I think you might need a hearing aid. I just got done saying that I don’t reject it. I’m saying that it has no bearing one way or the other on this particular question of whether two and two make four. Whether we’re counting planarian flatworms or Planters peanuts is completely irrelevant.
Peter: So arithmetic is unfalsifiable. Unlike scientific claims, for which you can give rational arguments.
You: That’s a false choice. The whole point is that argumentation of the sort that characterizes empirical science is not the only kind of rational argumentation. For example, if I can show by reductio ad absurdum that your denial of some claim of arithmetic is false, then I’ve given a rational justification of that claim.
Peter: No, because you haven’t offered any empirical evidence.
You: You’ve just blatantly begged the question! Whether all rational argumentation involves the mustering of empirical evidence is precisely what’s at issue.
Peter: So you say now. But earlier you gave the worm example as an argument for the claim that two and two make four. You appeal to empirical evidence when it suits you and then retreat into unfalsifiability when that evidence goes against you.
You: You completely misunderstand the nature of arithmetical claims. They’re not empirical claims in the same sense that claims about flatworm physiology are. But that doesn’t mean that they have no relevance to the empirical world. Given that it’s a necessary truth that two and two make four, naturally you are going to find that when you observe two worms crawl up beside two other worms, there will be four worms there. But that’s not “empirical evidence” in the sense that laboratory results are empirical evidence. It’s rather an illustration of something that is going to be the case whatever the specific empirical facts turn out to be.
Peter: See, every time I call attention to the scientific evidence that refutes your silly “arithmetic,” you claim that I “just don’t understand” it. Well, I understand it well enough. It’s all about trying to figure out flatworms and other things science tells us about, but by appealing to intuitions or word games about “necessary truth” or just making stuff up. It’s imaginary science. What we need is real, empirical science, like physics.
You: That makes no sense at all. Physics presupposes arithmetic! How the hell do you think physicists do their calculations?
Peter: Whatever. Because science. Because I @#$%&*! love science.
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Re: Is there a way to refute '1+1 = 2'?
1+1=3 in binary language
So refutation occurs through varying context, convention, and logic.
So refutation occurs through varying context, convention, and logic.
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Re: Is there a way to refute '1+1 = 2'?
The real trouble with 1+1 =2 is not that we are intuitively committed to it. We are and as much as one would like to play with terms and logical relationships among them, you never really touch this. The question that is begged goes to whether an intuition can be called into question. And it can: What is necessity? See Peirce on belief: We find the unwelcome position of unbelief which draws us toward belief. But what is this coersive force? Is it written in the mind of God, in which case, were this true then, who could argue with omniscience? But apodicticity is not so tied to this God premise and we are free to reject it; after all, if we can question modus ponens and the like, we can certainly question the God premise. So what IS the compelling force of belief (and belief is the issue here, for the logic detached from belief is like a baloney sandwich detached from taste, or something. Better: the principles of logic detached from belief is like detaching meteorology from the weather, the former being an abstract study of weather principles, the latter being the rain falling on my head.)Anyway, It is this force of belief that is at the very heart of logic, necessity and the like. It coerces, moves the rational agent toward stability, away from instability. So the next time you entertain a logical notion in your head, just remember the logic IS the doing of logic. Apart from this, you are living in a makebelieve land of reified abstraction.
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Re: Is there a way to refute '1+1 = 2'?
Yes, with Orwellian doublethink. 1 + 1 = 2, but it also equals 7 if and when it must.