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#1




1+1 = 2 ?
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). 
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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 selfevident. 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?? 
#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 subatomic 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 unprovable 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.

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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 redefined. 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. 
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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.) 


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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. 
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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. 
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This whole thread makes me think of a tshirt one of my friends has. It says, "2+2=5, for exceptionally large values of 2." 


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(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.

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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.

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There are, however, statements which cannot be proven. Gödel famously proved that in any selfconsistent system sophisticated enough to encompass arithmetic, there must exist statements which are true, and yet which cannot be proven within that system. For instance, it is possible to express in the notation of such a system a statement G equivalent to "This statement cannot be proven within this system". If statement G is true, then by its own truth, we can deduce that G cannot be proven. If, on the other hand, G is false, then it cannot be proven by virtue of being false, since a selfconsistent system cannot prove a falsehood. Therefore, either way G cannot be proven. But that's exactly what G is saying: G says that it can't be proven, and in fact, it can't be. So G is true. Ergo, there exists at least one statement in any given system which is true, but which cannot be proven in that system. Now, that one's a pretty pathological case, and aside from exposing a quirk of mathematics, it doesn't really do much for us. But there are more useful examples. For instance, in geometry, exactly one of the following statements is true: The sum of the angles of a triangle is exactly 180 degrees The sum of the angles of a triangle is less than 180 degrees The sum of the angles of a triangle is greater than 180 degrees One of those three must be true, and the other two must be false. Which one's the true one? There's no mathematical way to know. Unless you take one of those statements, or its equivalent, as an axiom, you can't prove it. You can, however, answer the question scientifically (or at least attempt to do so), by measuring the angles of real triangles.
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(Man, I gotta find a tshirt with that on it...) 
#27




My question was a simple one, and I think that most of the answers handled it in a brilliant way, but to some respect none of you handled it. Or maybe I am looking for something that is not there. It feels like that soar on the top of your mouth that would heal if you could stop tounging it, but you can't.
Math has spawned most of the technical achivements in history, but that does not prove anything. The reason why I don't really think maths is the right way to go to prove this, at first glanze silly question, is that it is the very basis of maths. The only one who came close to any kind of answer was Gilles and Bathas, since they understood my gutfeeling approach to the problem. I have done more maths than average people, but I feel some kind of fundamental problem with 1+1=2. Why? I think that no one can please me, but I am so glad so many tried. The few halfwits that tried to joke it away I understand, and I can't think of any other place were I would pose such a question, than here. I guess I wish there was a word or sentence so elegant that it would sum it all up. But I guess that it is not possible to be more elegant than: 1+1=2 
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It may be the first thing you learn, but don't confuse that with it being fundamental. As Russell and Whitehead showed, arithmetic is actually pretty highlevel stuff. 


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 I'm sorry that those of us who might have taken the more technical approach weren't able to address your intuitive discomfort, but I can't think of anything more to say if you can't explain your problem a bit more clearly.  Hi, Opal! (Yes, I know I'm cheating. I don't care.)  I agree with those who said that math is by its nature abstract, with reasoning and axioms being the only true test of what is 'proven' within the system. Math is useful in modelling the real world, but that in and of itself doesn't make an equation true or false. It just makes it useful 
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#32




I must have misread a nickname, so it's whoever is closer to that then
I can't formulate a proper question, because what I looked for was discourse, and in this there might be some kind of answer. But I think if you read the last sentence of my last post, it sort of struck me what the real issue was. Some things are better left unexplained, since they are inheriently beautiful and elengant. Or something like that. G'nite 
#33




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To ask for "a proof that 1+1=2" is meaningless until you've defined the semantic content of the symbols, "1+1=2". There are various correct ways of defining this (and by "correct" I mean they are all both logically consistent, for one thing, and that they coincide with our intuitive notion of what "1+1=2" means). Russell and Whitehead did it one way, I'm sure others have done it other ways. If you're thinking in terms of the history of math, I'll first give a caveat that I'm no expert in the history of math, but I also feel pretty confident in saying that the original definition "1+1=2" was an intuitive one, and that it's true by definition. The individual symbols, "1", "2", "+", and "=" were taken to express (intuitively) two different quantities, an operation of "accumulating" (for lack of a better description off the top of my head), and an idea of "sameness", respectively. I'm being very vague here, I know, but historically the foundations of math are intuitive and vague (by modern standards of rigor in math). Given these intuitive definitions, 1+1=2 is a self evident truth: "2" is defined to represent the quantity (i.e., is "the same as", i.e., "=") when a single object (whose quantity is represented by "1") "accumulates" ("+") with another single object ("1"). In a modern, rigorous context, we need formal definitions of these symbols. As I mentioned, there are various ways of doing this so that we may then formally prove that 1+1=2. 
#34




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1.0 + 1.0 = 2.0 with a possible deviation of .099 = 1.90 to 2.10 exclusive. 1.3 + 1.4 = 2.7 with a possible deviation of .099 = 2.60 to 2.80 exclusive. 0.6 + 0.9 = 1.5 with a possible deviation of .099 = 1.40 to 1.60 exclusive. 1 + 1 = 2 with a possible deviation of .99 = 1 to 3 exclusive. So, with discrete units, 1 + 1 = 2 by definition, but with imprecise quantities, 1 + 1 = 2 on average. 


#35




sverresverre: You are approaching the entire field of mathematics with the wrong assumptions, and you are wondering why you can't get the answers you want to your questions. Your question has been answered just fine multiple times, but your reactions to those answers indicate to me that you aren't getting something very fundamental about the distinction between mathematics and science.
Try this on for size: Mathematics is just a game and in a game, things mean what we want them to mean. We can prove things in mathematics by shuffling symbols around in predefined ways because we have defined the concept of proof and all of those symbolshuffling techniques beforehand, and we can agree on the fundamentals of the symbolshuffling game. There is no reason why mathematics has to match up with anything whatsoever in the real world. It is, after all, just a game. Except it does. We've been jiggering the rules of the game long enough we've come up with certain ways of playing the game that yield useful predictions about how the real world will behave in certain situations. This is so effective that great minds have pondered why it should be so unreasonably effective. This is a paper on the unreasonable effectiveness of mathematics in the natural sciences. I think your skepticism comes from the fact that we've made up this game and somehow get to apply it to the real world, and that it's somehow very deeply intertwined with science to the point where you cannot reasonably discuss fields like particle physics without throwing around some pretty serious equations. That unreasonable effectiveness is why: Mathematics has graduated from being a game to being a language, and it is a damned useful one in all of the hard sciences. (In fact, a 'hard' science is one where the predictions and results can be expressed numerically and as equations.)
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"Ridicule is the only weapon that can be used against unintelligible propositions. Ideas must be distinct before reason can act upon them." If you don't stop to analyze the snot spray, you are missing that which is best in life.  Miller I'm not sure why this is, but I actually find this idea grosser than cannibalism.  Excalibre, after reading one of my surefire millionseller business plans. 
#36




A question related to .99999...
I'm not even sure how to ask this question, but I'll try.
I accept the fact that .999...=1. Which means, the numbers in the series ".9, .99, .999, ..." are all <1, but gradually approach 1, and ultimately (at infinity) =1. But there's another series of numbers, "1.1, 1.01, 1.001, ..." which are all >1, but gradually approach 1, and ultimately (at infinity) =1. If the <1 series is symbolized by the number ".999...", how do we symbolize the >1 series? Somehow, "1.000..." doesn't seem right. My brain is trying to put a "1" at the end, but obviously that's not right either. Help. 
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Oops, that was supposed to be a new thread.
I'm not quite that insane. Yet. 
#38




panache45, probably the simplest way to express such a series would be simply as:
lim 1 + 10^{n}, where the limit is as n goes to infinity. We sometimes write "0.999..." to denote the limit of the .9, .99,... sequence, but there is no analogous way of writing this sequence. The only way to write the limit as a decimal is as 1 or 0.999...; these are the only two decimal representations of 1. There is no decimal representation "1.000...1" (with a 1 after infinitely many zeros), it wouldn't make sense. There is no "omegath" digit in a decimal expansion (for simplicity, "omegath" digit can be taken to mean a digit with infinitely many digits preceding it) 
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Mathematics doesn't say anything about the real world. Mathematics can go from the Peano axioms to "1+1=2". Mathematics then says that for any model satisfying the Peano axioms the model also satisfies "1+1=2". Mathematics doesn't say anything about realworld examples of such models. If you mean this as a real, physical question, ask a physicist. If you mean to ask the (deeper, unanswered) question of why mathematics is so "unreasonably effective" at describing the physical world, ask a philosopher. Mathematics has nothing to say. 
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This is a relatively new push that's based on the idea that we shouldn't talk about identity, but rather only about isomorphism. Equivalence is not Equality. Confusing the two leads to decategorification, and most of mathematics is decategorified. The natural numbers were "invented" to simplify (decategorify) the category of finite sets. To be exactly accurate here, I'm talking about cardinal numbers, since we identify sets with their cardinalities. Two sets with the same number of elements have the same cardinality. Given sets A and B with cardinalities a and b, the naturalnumbers statement "a=b" really means "there exists a bijection between A and B". Addition is disjoint union of sets, while multiplication is Cartesian product. So, within this context: prove that the disjoint union of two sets with one element each is in bijective correspondance with any set of two elements. Now it comes down to set theory again. 
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#43




So for you math philosophers  recently I heard a creationist talking, and he said that the laws of logic are evidence of a creator. This got me to thinking that the laws of logic are the same kind of thing as the laws of mathematics. These creationist guys say that our beautiful and "unreasonably effective" mathematics are the way they are because they were defined by God. Which leads to the observation that if God wanted to make them some other way, I suppose like a universe where 1+1=3, he could damn well do that.
Other thoughtful people, when asked the silly question about whether God could make a rock so heavy that he could not lift it, point out that an omnipotent God could do anything that wasn't logically contradictory, such as a square circle is a contradiction. So I think these people are saying that the laws of logic (and math) would predate even God and he himself would be bound by them. So is there any good observations about which would be right for the theist? I'd like a convenient way of refuting this argument for the existence of God, and this one seems pretty hollow to me, but I had not come across it before and was at a loss. What do you think? 
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I'm not a professional philosopher, but my take on it is that the laws of logic and of mathematics are true for any possible universe, and they are true regardless of whether there is a creator. (At least in part that's because a creator would have to follow the laws of logic). In any case, evidence of a creator is not the same as evidence of the Jewish/Christian God in the first few chapters of Genesis. 


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All this doesn't really get to the original question. Is math based on faith, just like religion. My answer is tentatively yes, but with a major difference. Math is tested all the time and passes all tests. Religious faith makes no testable predictions and so remains purely a matter of faith. When tests are attempted, such as the recent test of prayer, religious leaders insist that religion makes no such promises. I do think that some religions do and some don't. At any rate, may god strike me dead on the spot if he wishes to demonstrate his power. Incidentally, a colleague of mine claims that 2 is defined as the cardinality of a pair of platinumiridium balls held in a termparature controlled vault in Sevres, France. Ultimately, as many have said, give me your definition of 1, 2, +, and = and, i they are at all familiar ones, I can give you a proof. For a category theorist, 1 is a terminal object and 2 an object of truth values in a certain topos model of set theory and that fact that 1 + 1 = 2 is implicit in the model. Regardless, it makes the prediction that one ball taken with another ball gives two balls and that is verified regularly in everyday life, so it works. 
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