Proving 1 + 1 = 2 How can we prove that 1+1=2? We could say that 1 pencil + 1 pencil = 2 pencils, which is correct, but then we only know that that is true for pencils, but not for all of the other things in the world. For example, a person is trying to prove that y=x for any y and x. He tries y=2 and x=2, y=3 and x=3, and many numbers, being convinced that y=x because in all of those examples, y=x. However, we know that this is not the case since an easy example like x=3 and y=2 results in y not being equal to x. So we might think that 1 pencil + 1 pencil = 2 pencils, 1 computer + 1 computer = 2 computers, 1 fireplace + 1 fireplace = 2 fireplaces, etc., thus 1+1=2. But we might be as narrow-minded as the y=x guy; he is not thinking about any other cases, but only thought about the cases in which y is equal to x. So, is 1+1 really equal to 2? Always?

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As stated, x and y are variables with unknown values that can add to a sum. However, 1 is a constant, and added to itself it also produces a sum; because we know it's a constant, and we know the value of the constant, we know it always resolves to 2.

You are representing Whole Numbers when you assign one pencil, and another pencil totalling two pencils, and the properties of Whole Numbers are such that they cannot be fractions, negative, or irrational; they are simply constants of known value.


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The problem with this question is that 1 and 2 are numbers while x and y are respresenations of numbers.  The number 1 is always 1 while x and y could be anything.  Y does not always equal x because y and x might represent different numbers at different times.  A number will never change its value.  1+1 will always equal 2 because 1 and 2 will never change their value.  X and y might be equal if the numbers they represent are equal.  Since x and y can change their values, there might be times when they are not equal as well.  Comparing 1+1 to x=y is similar to comparing a fact to an opinion.  Facts cannot change just as number values cannot change.

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