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It is a famous bit of trivia that it took Russell and Whitehead about 300 pages to prove that $1+1 = 2$. However, this seems more like a definition rather than theorem. As far as I know, $2$ is just the symbol we use as shorthand for $1+1$, where $1$ is the multiplicative identity.

Ovi
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  • Yeah, this depends heavily on what your exact definition of the intuitive quantity symbolized by the squiggle "$2$" is, not to mention "$1$" and "$+$". – Arthur Nov 16 '16 at 16:16
  • If you work in $\mathbb Z_2$ then $1+1=0$ :) – Momo Nov 16 '16 at 16:16
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    Also not to mention "$=$" – Daniel W. Farlow Nov 16 '16 at 16:18
  • we say that $1+1=2$ to avoid repetition. – hamam_Abdallah Nov 16 '16 at 16:20
  • Yes, it's more of a proof that the chosen formal framework and definitions for the symbols $1, 2, +$, and $=$ are good than it is a proof that $1+1=2$ in the conventional sense. There's a quote from the authors where they say at much, I wish I could track it down. – Jack M Nov 16 '16 at 16:22

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You have to define what system you are working in. In Peano Arithmetic, PA, $1$ and $2$ are not part of the language. They are abbreviations for $S(0)$ and $S(S(0))$, where $S$ is intended as the successor function, so you are being asked to prove $S(0)+S(0)=S(S(0))$. You can follow the Wikipedia proof, which you may need to update a bit depending on how your version of the axioms is written. Basically this should look a lot like your axioms that define addition. PA does not specify a multiplicative identity, you have to prove that $S(0)$ is one by induction.

Ross Millikan
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    -1: The OP clearly states that the system of interest is that of Russell and Whitehead. – Rob Arthan Nov 16 '16 at 23:44
  • @RobArthan: I didn't take that to be the system of interest. It just made OP think that such a simple fact could require an enormous proof. In fact, I don't think the proof is 300 pages long, it just came that far into the book. – Ross Millikan Dec 30 '23 at 16:46
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In the kind of type theory that Russell and Whitehead were working on, the natural definition of a cardinal number is a maximal class of equipollent sets: $1$ is the class of all sets with $1$ element, $2$ is the class of all sets with $2$ elements etc. For finite $n$, we can write down a formula $\phi_n(A)$ which holds iff $A$ has exactly $n$ elements, so that $n = \{ A \mid \phi_n(A)\}$. $1 + 1 = 2$ then becomes a meaningful "problem to prove".

Rob Arthan
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  • Interesting from a historical perspective (like Euclid), but starting with Peano's Axioms is a more direct route. – Dan Christensen Nov 17 '16 at 03:37
  • Is there any advantage to the approach in PM? – Dan Christensen Nov 17 '16 at 03:46
  • @DanChristensen: we don't start from Peano's axioms when we develop arithmetic in set theory. Both set theory and type theories like that of Russell and Whitehead are trying to do much more than set up a theory of natural number arithmetic. – Rob Arthan Nov 17 '16 at 22:51
  • I can't comment on type theories, but with set theory, if you are given the existence of an infinite set $I$ as in ZFC, you may be able to derive the equivalent of Peano's Axioms with $N\subset I$ . In any case, to develop arithmetic, you must first either derive these essential properties of the natural numbers from other axioms, or you must assume them as axioms themselves. – Dan Christensen Nov 18 '16 at 17:26
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    This is the only answer posted so far that addresses the question. – Christian Chapman Nov 22 '16 at 02:28
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    I can't vote, but this should be the top answer. As @enthdegree points out, all the others jump to Peano Arithmetic, where it takes less than a page. – JonathanZ Dec 16 '16 at 15:33
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Long comment

Without comment, the issue is not so trivial as it seems ...

R&W's system, developed into the Principia, was aimed at the foundations of mathematics.

The first volume is devoted to the development of mathematical logic and the basic part of a sort of "class theory" : at that time, axiomatic set theory was at its very beginning.

Then the work goes on with the definition, on the base of logic alone and "class theory", of the arithmetical concepts : number, zero, successor, sum, etc.

At that point, well into second volume, it was introduced the "canonical" definition [see Rob's answer above] of : $1$ as the successor of $0$ and of $2$ as the successor of $1$, and thus :

$1$ is the successor of the successor of $0$.

Then follows the proof of the fact that :

$2=1+1$,

i.e., in un-abbreviated form :

the successor of the succesor of $0$ is equal to the sum of the successor of $0$ with the successor of $0$.

With first order-arithmetic, base on Peano axioms, the basic concepts (successor, zero, sum, product) are primitive, i.e. "implicitly defined" by the axioms, but then the approach is similar : $2$ is defined as $S(S(0))$ and thus we can (and we have to) prove that :

$2=1+1$.

The meaning is :

in an axiomatized theory, having assumed the axioms as well as the definition, all other "known facts" must be proved.

Mauro ALLEGRANZA
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Here is a proof in Fitch, starting with the relevant axioms from PA:

enter image description here

Bram28
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  • what is Fitch ? – Gabriel Romon Nov 16 '16 at 20:12
  • @LeGrandDODOM Fitch is a program that allows the user to create formal proofs using a 'Fitch-style' natural deduction system (it uses subproofs, and a pair of Introduction and Elimination rules for each logical operator). The software will check if the rules are applied correctly (hence the checkmarks!) – Bram28 Nov 16 '16 at 20:36
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    -1:L what's this to do with Russell and Whitehead? – Rob Arthan Nov 16 '16 at 23:44
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    @RobArthan I took the question to be 'why is there a need to prove that 1+1=2, when 2 seems to be defined as 1 added to 1'. I really don't think the OP wanted to know the specifics of RW's proof, but simply wanted to know why there is something to prove here at all. So, I showed that there really is something meaningful to prove, and that it is not a tautology. I note that most of the other answers took the same approach. – Bram28 Nov 17 '16 at 02:10
  • How does your answer show that there is "something meaningful to prove"? – Rob Arthan Nov 17 '16 at 02:57
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    @RobArthan In that it is not a tautology or axiomatic truth. In that 2 is not defined as 1+1, but that we have to prove that 1+1=2. – Bram28 Nov 17 '16 at 03:30
  • But the question is about Russell and Whitehead's development of arithmetic. Your answer and comments provide no useful information about that. – Rob Arthan Nov 17 '16 at 22:44
  • @RobArthan Well, you and I interpret the question differently: you see it as a question about Russell and Whitehead's work relevant to proving that 1+1=2, and I see it as a question as to why the statement 1+1=2 requires any proof at all, and why it is not just by definition true. We are not going to settle this without asking the OP what was really meant, though I do note that the OP accepted an answer that did not go into Russell and Whitehead's work. – Bram28 Nov 17 '16 at 23:27
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Axioms for the natural numbers usually define only a single number: either $0$ or $1$. The existence of any other natural number must be inferred from these axioms using the axioms and rules of logic or set theory (if applicable). If addition on $N$ is given in your axioms, it would be a trivial exercise to prove that 1+1=2.

Suppose, for example, you are given the axioms:

  1. $1\in N$
  2. $\forall x \in N: \exists y\in N: y=x+1$

Applying both of these axioms, we can obtain:

$\exists y\in N: y=1+1$

Applying the rule of existential specification (for $y = 2$), we could infer that $2 \in N$ and $2=1+1$. This inference can be thought of as the "definition" of $2$.

If addition on $N$ is not given in your axioms, you would have to construct (i.e. prove the existence of such a function) using the axioms of your set theory before you could "define" or infer that $2=1+1$ as above.

Dan Christensen
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    -1: You have obviously not read any part of Russell and Whitehead's Principia Mathematica or, if you have, you have completely misunderstood their approach to cardinal arithmetic. – Rob Arthan Nov 16 '16 at 23:30
  • Once you have some form of Peano's Axioms and set thoery, you don't need to talk about cardinalities to prove or infer that 1+1=2. – Dan Christensen Nov 17 '16 at 00:33
  • Have you read Russell and Whitehead? If not, see my answer for a brief outline of their approach. – Rob Arthan Nov 17 '16 at 00:47
  • No, but here is my formal construction of the addition function on N starting with a modern version of Peano's 5 axioms for the natural numbers: http://www.dcproof.com/ConstructAddN.htm At 727 lines, it is long but nowhere near the 300 pages of PM. And no notion of set cardinality was required. – Dan Christensen Nov 17 '16 at 01:29
  • I don't actually prove that 1+1=2, but at line 694, I have established the required property of the addition function. Another 11 lines would be required to prove add(1,1) = 2 where 2 = s(1). – Dan Christensen Nov 17 '16 at 01:51
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    So you haven't read Russell and Whitehead, but you are happy to lecture us about their work! – Rob Arthan Nov 17 '16 at 02:49
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    I did not take the OP's question to be about PM, but rather about what does it mean, in general, to "prove" that 1+1=2, something that seems to the OP more like a definition than a theorem. I outlined two different ways that it could be derived or inferred with no mention of PM. I have also written a detailed formal proof starting from Peano's axioms (see link in comments) that I think is more straightforward than PM and certainly much shorter. So, I think I have some insights to share on "proving" 1+1=2. – Dan Christensen Nov 17 '16 at 03:25