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Prove 1+1 = 2

  1. May 30, 2013 #1
    I dare anyone to prove 1+1 = 2, in one less page than already is; not reducing the font size or changing font, either.

    Whoever does so will receive US$5 off their next psychic appointment/end-of-world date prediction email subscription. Only redeemable at Evo and I's business to be set up at some point in the future.
  2. jcsd
  3. May 31, 2013 #2
    How can you prove a stipulation? 1+1=2 is a statement that stipulates 1+1 may also be referred to as "2". There's nothing to prove.
  4. May 31, 2013 #3


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    Actually, I seem to recall that in an advanced math graduate class I took some 50 years ago, one of the exercises was to use Peano's Postulates to prove exactly that. 1+1=2

    I'm an engineer, not a mathematician and I remember thinking the whole thing seemed stupid from my practical point of view but I understood that as an exercise in mathematical formalism it could be useful to folks who care about that sort of thing.

    Stevie, is there some reason for your rant or did you just need to get that off your chest?
  5. May 31, 2013 #4
    Why not spend the energy on something applicable to the real world instead?
    Last edited: May 31, 2013
  6. May 31, 2013 #5


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    (Edit in bold)
  7. May 31, 2013 #6
    If I hold up one finger, then another one finger, I get two fingers.
  8. May 31, 2013 #7


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    The "full" proof can be found in Russell and Whiteheads "Principia Mathematica", and no it can not be done on one page (it takes them about 300 pages to get to the proof).

    But then Gödel came along and showed that the whole exericise was futile:uhh:
  9. May 31, 2013 #8
    Some people like reading Harry Potter. Some people like doing math exercises.
  10. May 31, 2013 #9

    D H

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    What, exactly, do you mean by "1", "+", "=", and "2"? The proof is incredibly trivial or incredibly complex depending on how deep you want to go.

    Some claim you cannot prove a definition. That's not quite true. Better said, the proof of a definition is trivial: Cite the definition. End of proof. Given that, here's the trivial proof:

    Proof that 1+1=2: The definition of 2 is that it is the natural number that satisfies 1+1=2. QED.​

    Defining "1", "+", "=" as per the Peano axioms and defining 2 as the successor of 1 requires a tiny bit more work, but not much. On the other hand, going the full nine yards as was done by Whitehead requires an immense amount of work and won't fit on a page.

    So what do you want? The proof is either trivial and requires but a line or two, or it's incredibly non-trivial and cannot be reproduced without writing a book.
  11. May 31, 2013 #10

    jim hardy

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    This one has been kicking around since at least 1966 when I saw it on a math dep't bulletin board at U of Miami.
    Today I found it many places by a search on "unknown but astute source"

    This is a pdf from U of Chicago:

    http://www.oekonometrie.uni-saarland.de/oekonometrie/oeko2011/FirstLesson.pdf [Broken]
    Last edited by a moderator: May 6, 2017
  12. May 31, 2013 #11


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    That's funny :smile:
    Last edited by a moderator: May 6, 2017
  13. May 31, 2013 #12


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    I was fine until we made zero a vector.
    Last edited by a moderator: May 6, 2017
  14. May 31, 2013 #13


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    If this is the overarching mentality then much of pure math is a waste of energy. Regardless, intellectual stimulation is intellectual stimulation regardless of how "applicable" it is to the real world.
  15. May 31, 2013 #14
    Which is why there are so few workers in the world doing it. Still, its certainly funner than watching a basketball game (IMO).
  16. May 31, 2013 #15
    Since no one noticed this, let me expand.

    1 finger + 1 finger = 2 fingers

    So "1+1=2" is true for concrete, physical objects like fingers, therefore it cannot be false for numbers. Numbers are just abstractions of real things like fingers.

    Furthermore, when you add 1 the result gives the next number, and "2" is simply the name of the number after 1, therefore "1+1=2" is a tautology.
  17. May 31, 2013 #16
    If you hold one finger then cut one of your other fingers of and hold it in another hand.Do you still have two fingers?
  18. May 31, 2013 #17

    D H

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    Nonsense. 1 drop of water + 1 drop of water = 1 bigger drop of water.
  19. May 31, 2013 #18
    You want to experimentally verify mathematics? I don't think so. It works the other way around. You experimentally verify science, not math.
  20. May 31, 2013 #19
    And 1 bigger drop = 2 drops. You are just giving two drops another name
  21. May 31, 2013 #20
    Thats the same thing you did when you appealed to fingers... You arbitrarily choose a naming scheme that would support your preconceived conclusion.
  22. May 31, 2013 #21
    What if it was made up of 4 smaller drops? Your system does not give unique labels to your objects. How do define your '1' drop?
  23. May 31, 2013 #22
    I thought about it, and the problem is actually that "drop" is a vague word, it describes something but doesn't define it. In "1 drop + 1 drop = 1 bigger drop" each time you use "drop" it actually has a different meaning.

    I don't understand. Was my meaning of "finger" arbitrary and chosen to support my conclusion?

    Then tell me where the abstract numbers "1", "2" etc. come from.

    I think you're getting at the same thing as me.

    And everyone has conveniently forgotten this:

  24. May 31, 2013 #23
    Yes, I think so.

    Why do they have to "come from" anywhere? They are postulated or defined axiomatically. When you start to attribute these abstract concepts onto "real world" objects like fingers and water drops you start doing science rather than mathematics.
  25. May 31, 2013 #24

    Ha ha ha. Yes. Indeed. Ho Ho Ho. Good one. Yes. Very good.
  26. May 31, 2013 #25
    Afaik, Boolean arithmetic is just as valid as standard arithmetic. In Boolean arithmetic 1+1=1
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