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How do I convince someone that $1+1=2$ may not necessarily be true?

I once read that some mathematicians provided a very length proof of $1+1=2$.

Can you think of some way to extend mathematical rigor to present a long proof of that equation? I'm not asking for a proof, but rather for some outline what one would consider to make the derivation as long as possible.

EDIT: It seems the proof I heard about is a standard reference given here multiple times :) I stated that the proof itself is less useful than an outline for me as I know only "physics level maths". Can someone provide a short outline what's going on in the proof? Some outline I can look up section by section in Wikipedia to at least get a feel of what could be needed to make such a proof?

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Gere
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    Usually, it is defined to be that $2 := 1 + 1$, $ 3 := 1 + 1 + 1$, etc. What definition of $2$ are you using? – Andrew Maurer Jan 15 '13 at 00:23
  • What is there to prove? Thats how we define the integer two, its doesn't require a proof because its a definition. – Ethan Splaver Jan 15 '13 at 00:24
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    Here's a page from Russel and Whitehead's Principia Mathematica proving that 1+1=2, on page 379.

    http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0001.001&frm=frameset&view=image&seq=401

     – user7530 Jan 15 '13 at 00:25
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    @andybenji Often, $2$ is defined as the successor of $1$, not as $1+1$. – Thomas Andrews Jan 15 '13 at 00:25
  • @ThomasAndrews By successor, do you mean $\min{n \in \mathbb{N}: n \gt 1 }$? You could think of $2 \in \mathbb{R}, \mathbb{C}, \mathbb{Q}$ and this definition would fail to hold (and even exist). – Andrew Maurer Jan 15 '13 at 00:27
  • You might be thinking about the proof by Russell and Whitehead from Principia Mathematica. IIRC, it took ~300 pages and you can view those online (if you can read them). Regards – Amzoti Jan 15 '13 at 00:29
  • @andybenji No, that's an equivalent, but that requires a lot more notation. The most standard axioms of the natural numbers starts with axioms about one operation: The successor of a natural number. Technically, it is the same as "+1," but we start without "+" defined. – Thomas Andrews Jan 15 '13 at 00:30
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    Have you read these two answers (http://math.stackexchange.com/questions/95069/how-would-one-be-able-to-prove-mathematically-that-11-2/95070#95070 and http://math.stackexchange.com/questions/243049/how-do-i-convince-someone-that-11-2-may-not-necessarily-be-true/243059#243059) of mine? – Asaf Karagila Jan 15 '13 at 00:31
  • @ThomasAndrews So you're saying that 2 is the successor of 1, where successor of $n$ is $n + 1$. (So $2 = 1 + 1$) But then there is nothing to prove when saying $2 = 1+1$. – Andrew Maurer Jan 15 '13 at 00:32
  • No, successor is the basic operation, and addition is defined in terms of the successor, not the other way around. @andybenji – Thomas Andrews Jan 15 '13 at 00:32
  • @ThomasAndrews Ah I see. – Andrew Maurer Jan 15 '13 at 00:35
  • $54.43$ on page $362$ has the final result. Go here in case the above link doesn't work for you. –  Jan 15 '13 at 00:40
  • @Asaf: I actually had seen that answer some time ago. It's clear. But the referenced proof seems to go a step beyond?! Your proof isn't that "long" :) – Gere Jan 15 '13 at 00:44
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    Note that the point of the work of Russell and Whitehead wasn't to make the derivation as long as possible. The point was to put the foundation of mathematics on firmly rigorous ground in a highly axiomatized system. The "long proof" is more accurately a book that builds up this system before eventually proving that $1+1=2$. – Code-Guru Jan 15 '13 at 00:45
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    On the Russell and Whitehead quotation in question, I don't believe that there was ever any doubt that $1+1+2$, nor do I imagine that either Russell or Whitehead expected that this would be a groundbreaking result. As a matter of fact, if I recall, they do not actually prove that $1+1=2$, but rather, simply state that "it will follow" from the definitions which they had laid out that $1+1=2$. The point was not to prove that $1+1=2$, but rather, to show that within the framework which they had laid out, addition worked in a familiar way and that well known true statements were provable. – crf Jan 15 '13 at 00:52
  • It's "Whitehead and Russell", not "Russell and Whitehead". – MJD Jan 15 '13 at 16:44

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You are thinking of the Principia Mathematica, written by Alfred North Whitehead and Bertrand Russell. Here is a relevant excerpt:

Principia Mathematica *54.43

As you can see, it ends with "From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2." The theorem above, $\ast54\cdot43$, is already a couple of hundred pages into the book (Wikipedia says 370 or so)

I wrote a blog article a few years ago that discusses this in some detail. You may want to skip the stuff at the beginning about the historical context of Principia Mathematica. But the main point of the article is to explain the theorem above.

The article explains the idiosyncratic and mostly obsolete notation that Principia Mathematica uses, and how the proof works. The theorem here is essentially that if $\alpha$ and $\beta$ are disjoint sets with exactly one element each, then their union has exactly two elements. This is established based on very slightly simpler theorems, for example that if $\alpha$ is the set that contains $x$ and nothing else, and $\beta$ is the set that contains $y$ and nothing else, then $\alpha \cup \beta$ contains two elements if and only if $x\ne y$.

The main reason that it takes so long to get to $1+1=2$ is that Principia Mathematica starts from almost nothing, and works its way up in very tiny, incremental steps. The work of G. Peano shows that it's not hard to produce a useful set of axioms that can prove 1+1=2 much more easily than Whitehead and Russell do.

The later theorem alluded to, that $1+1=2$, appears in section $\ast110$:

Principia Mathematica *110.643

MJD
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I suspect you are referring to the Principia Mathematica. I direct you to a quotation from Wikipedia about how the proof doesn't appear until page 379.

davidlowryduda
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A proof is a finite sequence of formulas (see here), where each formula is either an axiom or follows from the previous ones by some inference rule. So, if you wish to make your proof very long, just repeat an appropriate axiom a very large number of times.

Dejan Govc
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