Is 1+1=2 true by definition ?

Question

Is 1+1=2 true by definition ? Or, is there a way to prove it?

I'm trying to understand how do we know it's true, and how to reply if someone is skeptical or denies that 1+1=2.

Answer 1

Your sceptic must understand what the symbols 1+1 means otherwise he is not justified in claiming that 1+1 is two. For example there are number systems in which there isn't a 1, or certain operations are undefined, or 1+1=0. But one could also imagine that the symbol '1' means a drop of water, and '+' means physical addition, so that 1+1 means add one drop of water to another drop of water, so we get a another (larger) drop of water, so in this case 1+1=1.

Assuming it has the traditional sense, then if you start from the Peano axioms, which are roughly that there is a zero and that you can always add one to a number, then you can prove that 1+1=2.

But this is not the whole truth. Had these axioms shown us that 1+1 is not in fact 2, then Peano would simply have thrown his axioms away.

What he was attempting to do was find a set of axioms that accurately captures our intuition about how the integers act; and obviously 1+1=2 is an act of the integers that is true by intuition/observation which he has to incorporate for his axioms to meaningfully model the integers.

Now given Peanos formalisation and the mathematical logic introduced by Boole & Frege, Bertrand Russell attempted to derive Peanos axioms from logic. This is why it took him several hundred pages to reach the point of saying that 1+1=2.

Perhaps the most practical set of axioms in the sense that it mirrors our intuition is that it is a well-ordered ring. This means that it is a set with two operations called addition & multiplication and they are commutative, associative and have an identity; that multiplication distributes over addition; that there is an order relation on the set such that every non-empty set has a minimal element.

These three sets of axioms are connected by a simple dependency of deduction: Logic -> Peano Axioms -> Ring Axioms; but one should retain in mind that the other direction holds too as a process of historical reflection and label it as thus: Logic <- Formal (Peano) <- Intuition (Ring).

The mythical Amazonian tribe that can't do or understand arithmetic will also not understand what you mean by proof; but this is simply because they have no perception that arithmetic in the right context can be important; and this conceals the important point that the long development of arithmetic & measurement in Ancient Mesopotamia is to understand its use & importance; it is this acquaintenance that was bequeathed to Greece and by which Euclid first outlined a complete axiomatic system. It's hardly creditable that he was the first to conceive of an axiomatic one but he was the first to achieve something like a complete system. In India, and roughly contemporareus Panini developed a complete formalisation of Sanskrit Grammar.

Formalisation as a concept in mathematics only occurred in the early 20th Century after the revitalisation of mathematical logic. This differs from axiomatic systems in that the idea of truth is absent - self-consistency is the only requirement, whereas an axiom should be self-evidently true. That is in a formal system, you may 'prove' something, but because the 'axioms' are contigent rather than self-evident, one could argue that in fact nothing has been proved. In this case, proof has been reduced to syntax.

And this in fact, is the difference between the two earliest attempts at formalisation. After all, there is only one system of integers; whereas there are many languages other than Sanskrit.

Answer 2

A reasonable proof in ZFC would be to prove 1 + 1 = 2 for the corresponding ordinal numbers. The first few ordinal numbers in ZFC are 0:={}, 1:={0} and 2:={0, 1} with the order 0 < 1 on {0, 1}. The sum of two ordinal numbers is the disjunct union of the two well-ordered sets, with the concatenation of the well-orders as the well-order for the sum. For example, we would have {a, b} + {c, d} = {(a,0), (b,0), (c,1), (d,1)} with the order (a,0) < (b,0) < (c,1) < (d,1), if WLOG a < b on {a, b} and c < d on {c, d}. Note that the Kuratowski definition (x,y)={{x},{x,y}} is used here.

So 1 + 1 = {(0,0), (0,1)} with the order (0,0) < (0,1). How can this be equal to 2 = {0, 1} with the order 0 < 1? Well, two ordinal numbers are equal if there exists an order isomorphism between them. It's easy to check that {((0,0),0), ((0,1),1)} is the required order isomorphism. This concludes my informal proof that 1+1=2 for ordinal numbers in ZFC.

How difficult is it to convert such an informal proof into a formal proof? For me, the first difficulty would already be that I'm not sure in which form I should specify the order. I guess the correct way is to use a set of pairs, similar to how I specified the order isomorphism above. The formal proof for 1+1=2 from metamath uses cardinal numbers instead of ordinal numbers (as DBK indicated in a comment, that's also what Principia Mathematica did), but that seems to make the proof even more difficult. Note however that already formalizing and proving a simple formula like (a,b)=(c,d) -> (a=c ∧ b=d) in ZFC is quite some work. So maybe the informal proof given above is not so bad after all.

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A simpler interpretation of 1+1=2 would use Peano arithmetic. Then 1+1=2 turns into the statement S(0)+S(0) = S(S(0)). Then we can use the axiom ∀x1,x2∈N. x1 + S(x2) = S(x1 + x2) to get S(0)+S(0) = S(S(0)+0) and then the axiom ∀x1∈N. x1 + 0 = x1 to get S(S(0)+0)=S(S(0)). We see here that 1+1=2 is true in this interpretation, as a consequence of two axioms and the two definitions 1=S(0) and 2=S(S(0)). Because there were two axioms involved (not even mentioning the first order logic deduction system implicitly used), it's pretty clear that the statement "1+1=2 is true by definition" is at least questionable.

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But if one really wants, one can exclude 0 from the natural numbers, and use 1+1 as the definition of 2. This was done for the 2+2=4 proof, which is explained under the 2+2=4 Trivia paragraph on the starting page for the Metamath Proof Explorer subproject. Then 1+1=2 is really true by definition, but so what?

Answer 3

It is true by definition, in fact i would write it like this 2=1+1 because you are defining number 2.

By the way, proves or demonstrations are just ways to simplify expressions to reach definitions, so we can be sure that premises were correct.

Answer 4

What about an explanation by Newton Laws (A mass can not be at the same place and same time of another mass -> mass + mass = 2 x mass)

Answer 5

When I was growing up I learned that in some situations the word "one" was to be used. Say the number of dots between the parentheses (.). After a while I encountered situations like (. .) and would say "one" "one" because that would tell someone what I had seen. This kept up with (...), (....), etc with just repeating the word appropriately for the situation. It became tedious and time consuming however and new utterances were substituted for the different situations. The utterance for the (..) situation was "two", and so on.

OK I lied. This wasn't exactly how I grew up but I bet it's an approximation as to how 1+1=2 came about.

This is a fundamental aspect of human cognition. We see/experience things and cast them into categories. The fact that we can dissect our sensory input into any parts at all is the origin of numbers. We had to differentiate between aspects of experience (sensory input) to avoid tigers and find edible fruit. We came up with utterances to match these discrimination. As soon as we had a way to identify single instances we could identify multiple instances. We then shortened saying "ugh" 16 times for a given herd of buffalo to whatever the Indian word for 16 was (depending on tribe). Well that's how we got 1+1 = 2. The rest of math was developed from that by inventing new notation (multiplying instead of repeated addings and such) and demanding that we didn't end up with contradictions because of all the inventing we were doing. It's how a relatively limited system can deal with too much input. Our senses provide far more actual data then we can effectively process so we use lossy compression to classify similar gobs of data. If an input stream approximates others fairly well we say close enough and say that we have "one" of those. We then get into the above scenario above leading to two etc.

Answer 6

In the binary system, we have 2 digits only: 0 and 1, where 1+1=10. In the decimal system, the one that we mostly use, we have 10 digits (0-9), and 1+1 is always 2.

We all made an agreement about a set of rules. Based on these rules, 1+1=2 for the decimal system. A sceptical person may have in mind to change these rules. There is nothing wrong about this. That person is just creating a new arithmetic system, and believe me, it's not as easy task.

To conclude, based on the known decimal system and its rules, 1+1 is always 2. The logic behind the result is the set of rules followed by the specific arithmetic system.

Answer 7

Beware that there are different kinds of truths:

• Objective truths assert something about the objective world: "Paris is a French city" is an objective truth.
• Mathematical truths assert implications: "if pigs can fly then I am Pope" is a mathematical truth.
• Postulated truths are not necessarily mathematical truths or objective truths; they are simply assumed to be true. All of Euclid's postulates are postulated truths.

1 + 1 = 2 is an arithmetic rule, just like the rules of a game, which are propositions assumed to be true before proof. Like Euclid's postulates, it is a postulated truth. Mathematics used to have a great many postulates until someone speculated that mathematics can be reduced to a very small number of postulates.

The creation of a rule like this is an inductive process starting from counting, measurement and many other daily activities: people first figured out how to count and knew what one thing and two things mean. Then they discovered that one horse and another horse are two horses; one mile in addition to another mile are two miles, etc. Then people generalized that one and one equals to two with an aura of mystery. The conception of numbers detached from things is a great leap forward; it is very likely that a very small number of individuals, perhaps only one, made this breakthrough.

For a long time this rule is used to solve problems without precise definitions of 1, 2, + and =. Since people have no problem with understanding one thing or two things, for a long time, no one ever questioned what 1 or 2 means - anyone who raised a question like this would have been considered laughable.

This type of fuzzy thinking is not peculiar to arithmetic. Take "yellow and blue make green" for example, everyone understand this proposition, but few know the precise definition of yellow, blue and green. As a matter fact, no one ever saw yellow or blue or green independent of things; no one ever mixed yellow with blue; what they actually did was mixing yellow paints with blue paints. People are so familiar with blue things or yellow things, they unconsciously think they know what blue or yellow mean, but no one raise silly questions like these until some great minds think precise definitions are are necessary for the sake of clear thinking.

At first it was thought 1 + 1 = 2 has some objective truth in it until one day people realized that it was not always the case: if an emperor sends out 2 tax collectors and tells each to bring back a tael of silver, he has right to expect 2 taels of silver at the end of day, but if he tells them each to bring back a variety of exotic plant, there is no guarantee that he will have 2 varieties of plants after each of them brings one variety back. One had to admit that 1 + 1 = 2 was just a rule, which was sometimes applicable to the real world and some other times not.

Mathematics consists of many rules like 1 + 1 = 2. Some people discovered that some rules can be derived from other rules; some other people speculated that the entire mathematics can be deduced from a very small number of rules which were called the foundations. They speculated what these foundations (rules) were and tried to deduce ordinary mathematics from them - this process had the appearance of proving 1 + 1 = 2, but, as a matter of fact, ordinary mathematics has greater degree of self-evidence than their foundations. If 1 + 1 = 2 can be deduced from a speculated foundation, it only gives reasons for believing the validity of the foundation, rather than believing 1 + 1 = 2, which is already self-evident.

Similarly, if someone's theory prophesied a eclipse and an eclipse was observed as he predicted, his theory did not make the eclipse more true. Quite the contrary, it was the eclipse that gave reasons for believing his theory.

Another analogy is the creation of by-laws of an organization. At first, ad hoc rules were added to address specific scenario. Later on, people discovered that some rules were already implied by other rules, and the whole book of regulations was equivalent to just a small number of primitive rules.

Answer 8

1 + 1 = 2 is at its core a belief. This is based on our tendency to see separate things as one. Without this tendency our perceptions disintegrate. Such beliefs allow us to communicate/ use language and one might argue this leads to consciousness. There are no to ultimate truths. We always have to start off with some basic axioms which are again at core an agreed upon belief. Someone can disagree if they don't want to accept these starting axioms. It normally leads to a circular discussion where we realise that what are are talking about is whether there is internal consistency in our own system. The systems are circular and self defining, this can be seen by trying to define things and if done rigourously enough we may realise we define them in relation to one another.

So in short 1+1=2 is just what it is and how it is defined. It is internally consistent. With beliefs there is a leap of faith past the infinite regression.