Is there a proof that 1 + 1 = 2 or is it just true by definition?
You’re not the first to ask this: 1+1 = 2 ?
Maybe a little of both, depending on exactly what you mean by the question.
On one hand, it’s axiomatic (i.e., a defined property) that the natural numbers are well-ordered. This means there’s a smallest element, then a next smallest object, then a next…and so on. We define the symbol “0” to represent the smallest, “1” the next smallest, “2” the next…and so on.
The symbol “n+1” stands for the next smallest natural number after n. And “=” has its usual meaning. So “1+1=2” depends only on the definition/axiomatization of the natural numbers, along with the relevant meanings we give those particular symbols.
On the other hand, it’s not uncommon to see references to a proof that 1+1=2, such as the one given in Russell & Whitehead’s Principia Mathematica.
What this means is that Russell & Whitehead (for example) have developed a formal system in which the natural numbers can be represented. Of course, it can’t be taken for granted that the representation is a correct one–maybe it doesn’t match the axiomatization of the natural numbers. It needs to be shown, for example, that the representation is well-ordered (just like the natural numbers are supposed to be). Or that “1+1=2” is true, however those symbols may be defined in that particular formal system.
As stated in a previous thread: How many feet do you have? Can you count them? “2” is by definition 1+1. If you want to redefine “2,” then go ahead.
In response to the OP’s question, the answer is “both”. A mathematical proof proceeds from the relevant definitions, after all. Or, more generally, a proof proceeds from some premises (you can call them “axioms”, but don’t get too hung up on what justifies an axiom; you can investigate the consequences of any axiom system you want), and definitions are particular kinds of premises.
People like to trot out huge, multi-page proofs of “1+1 = 2”, like a famously touted example from Principia Mathematica, but these are really just examples of unwieldy indirect axiomatizations of arithmetic (via representation in some other system, generally, as Cabbage said). With different, more direct formalizations, the proofs are, well, more direct.
But many formalizations are possible, depending on the situation at hand. And keep in mind, there’s more than one system in which a statement like “1 + 1 = 2” could be interpreted: natural number arithmetic, integer arithmetic, real arithmetic, cardinal arithmetic, ordinal arithmetic, a statement about datatypes in programming languages, a Boolean expression, a schematic meant to hold in every ring.
Yes. Proving 1+1 =2 in Peano arithmetic is fairly trivial.
Here are some example “proofs” of 1+1 = 2. I put “proofs” in scare quotes only to remind you of the multitude of possible formalizations under which one may choose to examine the statement.
The sort of proof you’ll see in systems like ring theory: We take a natural number numeral n to be short-hand for the addition of n-many 1s. Then, by definition, the equation “1+1 = 2” is just short-hand for “1+1 = 1+1”, and falls out trivially proved. The equation holds in all rings.
The sort of proof you’ll see in systems with names like Robinson/Presburger/Peano Arithmetic: We could take the following axioms for natural number arithmetic: there’s a type of natural numbers, 0 is one of them, there’s a unary successor function on them (I’ll denote it by S, to save space), when we write a numeric constant n, it’s just a short-hand for S applies n times to 0, and there’s a binary + function on them making the following recurrence relations valid:
0 + x = x
S(x) + y = S(x+y)
Then we have that 1 + 1 is short-hand for S(0) + S(0), which translates by the second equation to S(0 + S(0)), which translates by the first to S(S(0)), which is the unabbreviated form of 2, proving the equality. It holds in all models of Robinson Arithmetic.
Most other “proofs” will essentially follow one of the above paradigmatic forms, depending on whether a numeral is interpreted as repeated additions of 1 (in which case the equality is just short-hand for “1 + 1 = 1 + 1”) or as repeated successors of 0 (in which case you have to do a tiny bit more work based on the interaction of addition and successor).
Incidentally, in the Frege-Russell logicist model, where arithmetic and everything else gets reduced to “logical” concepts, construed set-theoretically, you’ll see things like “The numeral n really stands for the cardinality of n-element sets, and addition is just the cardinal-level operation of disjoint union”, and the proof proceeding from there with much obfuscation and blather. But, depending on how “n-element set” is interpreted, this is really just one of the above two styles of proof at the core.
Someone’s parents didn’t expose them to Rosenshontz:
Broken Link.
To address the OP:
1+1=2 is inherently true by definition. I’d compare it to “why is the woman who gave birth to you your mother?” She is because that is the definition of mother.
As others have said, it depends on what you mean by “1”, “+”, “=” and “2”. What’s your definition?
D’oh.
The trick, of course, is coming up with the right definitions of 1, 2, + and =. That’s what the proofs are really for–to show that the very formal descriptions we have in the form of Peano arithmetic actually correspond with our intuition.
I can’t believe no one made the obvious joke yet!
1 + 1 = 2.5
Even past ring theory (though you do this sort of thing there too) is field theory, where you not only prove 1+1=2, but that A=A (which is NOT axiomatic, by the way!).
In what axiomatization of field theory is the reflexivity of equality not taken as axiomatic?
THAT is why I love this forum: a question with so many Latin roots, I’m at a loss for words.
Let us not forget, without specifying the number base, 1+1 might well equal 10.
Even if binary notation was our normal convention, no one would consider “1 + 1 = 2” to be false. They would just consider “2” to be an esoteric way to denote the number two, and “10” to be a more conventional way to denote the number two. (Just as no one would consider “14 + 17 = 0x1F” to be false even in our decimal world)
But I suppose it does tie into the point, if one wants a formal proof of “1 + 1 = 2”, it’s necessary to fix a formalization and say what exactly this is supposed to stand for in that formalization, whether it be “1 + 1 = 1 + 1” or “S0 + S0 = SS0” or “There exists a bijection from the disjoint union of {{{}}, {{}}} to {{}, {{}}}” or just plain “1 + 1 = 2” in a suitably direct representation or whatever else.
Indistinguishable, A=A is nonaxiomatic in the math I took in junior high. I’ve long since forgotten the proof, though it’s no more complicated than proving that 2^1/2 is irrational. It involves set theory, you know, 0=the set of all empty sets? I’d write that in proper math language if I could get my character map to work.