Author Topic: 1+1=2  (Read 11471 times)

0 Members and 1 Guest are viewing this topic.

Offline Sigmaleph

  • Ungodlike
  • Administrator
  • The Beast
  • *****
  • Posts: 3615
    • sigmaleph on tumblr
1+1=2
« on: January 20, 2012, 11:42:46 pm »
Today, I'm going to prove that 1+1=2. Why, you ask? Because I wasn't around last time the subject came up, and then it came up again, and now I am around.

We start with Peano's axioms. These are not arbitrary rules I'm pulling out of thin air, we need some axioms to make sure 1, +, =, and 2 mean something other than just being a set of lines on paper (or pixels on a screen), and in particular that what it means is the same we mean when we say "one", "plus", "equals" and "two". Behold:

  • 0 is a natural number.
  • For every natural number x, x = x.
  • For all natural numbers x and y, if x = y, then y = x.
  • For all natural numbers x, y and z, if x = y and y = z, then x = z.
  • For all a and b, if a is a natural number and a = b, then b is also a natural number.
  • For every natural number n, S(n) is a natural number. (S is called the "successor" function. It will be important.)
  • For every natural number n, S(n) = 0 is False
  • For all natural numbers m and n, if S(m) = S(n), then m = n
We will be ignoring the axioms relating to induction, because I am lazy.

Now, you'll notice so far we only have explicitly named one number, 0, but axiom 6 allows us to build more. We know that if n is a natural number, then so is S(n). Then, from 1 and 6 we know we have a natural number called S(0). And since S(0) is a natural number, so is S(S(0)). We could go on for as long as we wanted, but these will suffice.

But wait, how do I know I really do have different natural numbers, and not just 0 called by different names? Axiom 6 only specified that S(n) is natural, not that it's different from n. Fortunately, we have axioms 7 and 8 for that. 7 says S(n) cannot be equal to 0, so in particular S(0) and S(S(0)) are different from 0. How do we know S(0) is different from S(S(0))? With the help of axiom 8, which tells us that if S(S(0)) is equal to S(0), then S(0) has to be equal to 0. We already know that is not the case, so we have proven that 0, S(0), and S(S(0)) are all different numbers. We could give them different names, if we wanted to. The usual names for them are 1 and 2.

Now, we already have "1" [S(0)] and "2" [S(S(0))]. "=" has already appeared before and axioms 2 through 5 deal with how it works. So the only missing element is "+", the addition function. The addition function as we're gonna use it takes two natural numbers as arguments and outputs another natural number (in less words, that would be + : N × N → N), and we can define it (recursively) as:
  • a + 0 = a
  • a + S(b) = S(a+b)

Now that we know what every part of 1+1=2 means, we can say that:
1+1 would be written as S(0) + S(0)
By the second part of the definition of addition, S(0) + S(0) = S( S(0) + 0 )
By the first part of the definition of addition, S(0) + 0 = S(0)
Therefore: S(0) + S(0) = S( S(0) )

1+1=2

There you have it. To the extent that 1, +, =, and 2 are defined to be the same things we mean when we use the words, it can be proven that 1+1=2.

Now, I am not a mathematician. I know there are plenty of discussions about definitions and axioms and first- and second-order logic and ZFC set theory and who knows what else that go way above my head. But 1+1=2, and that can be proven.
« Last Edit: January 20, 2012, 11:46:34 pm by Sigmaleph »
Σא

Offline Shano

  • Bishop
  • ***
  • Posts: 136
  • Gender: Male
  • Trust me, I am a doctor.
Re: 1+1=2
« Reply #1 on: January 21, 2012, 01:28:00 am »
In a related news... I educated myself about Graham's number. Rather insane.
The unbreakable wall

Offline Captain Jack Harkness

  • Petter, Brony, and All-Around Cartoon Addict
  • The Beast
  • *****
  • Posts: 2868
  • Gender: Male
  • Or as a friend calls him, Captain Jack Hotness!
Re: 1+1=2
« Reply #2 on: January 21, 2012, 01:51:40 am »
*slow clap*

Your point?
My friend's blog.  Check it out!

I blame/credit The Doctor with inspiring my name change.

Art Vandelay

  • Guest
Re: 1+1=2
« Reply #3 on: January 21, 2012, 02:01:38 am »
*slow clap*

Your point?
It's related to some drama we had a few months back. On that note, I feel compelled to ask why this is being dug up again.

Offline Jack Mann

  • Gold Bugger Jihad Pony
  • God
  • *****
  • Posts: 603
Re: 1+1=2
« Reply #4 on: January 21, 2012, 03:14:45 am »
LHM requested a thread to reinvestigate the 1+1=2 thing.
اللغة العربية صعبة ، ولكنها جميلة جدا

Art Vandelay

  • Guest
Re: 1+1=2
« Reply #5 on: January 21, 2012, 03:26:02 am »
Fantastic.

Offline lighthorseman

  • Argumentative Contrarian
  • On Probation
  • Bishop
  • *
  • Posts: 155
  • Proud Y chromosome owner
    • Company of the Wolf
Re: 1+1=2
« Reply #6 on: January 21, 2012, 03:49:54 am »
I never said l don't think 1 & 1 equals 2, l merely said it requires a surprising degree of effort to prove. It's something that serious and well respected mathematicians far smarter than you or l have devoted vast amounts of time to trying to prove.

Most people can't prove it, but merely accept as axiomatic.
Let me make it real easy... if anyone is interested in my actual opinion, please ask, I'd love to talk to you. If you are interested in trying to catch me out in some sort of "gotcha, before you said 'many', but now you're saying 'lots', you totally shifted goal posts", then, I'm not playing.

Offline shadowpanther

  • Sod it, this'll do.
  • Bishop
  • ***
  • Posts: 103
  • Gender: Male
  • FOR SCIENCE!
Re: 1+1=2
« Reply #7 on: January 21, 2012, 05:35:21 am »
As Bertrand Russell said
Quote
The above proposition is occasionally useful.
Fight Spam! Click Here!
"Kick logic to the curb and do the impossible!"- Kamina, Gurren Lagann.

"Telling someone they know a lot is like telling a man with a few grains of sand clinging to him that he possess much sand."-Stephen Fry

They say an Englishman's home is his castle; which is handy as my home IS a castle and I LAUGH from the top of the turrets.

Offline The Bright Angel

  • Apprentice
  • **
  • Posts: 70
  • Just Your typical Bi possible Trans kid (not sure)
Re: 1+1=2
« Reply #8 on: January 21, 2012, 06:03:46 am »
*slow clap*

Your point?
It's related to some drama we had a few months back. On that note, I feel compelled to ask why this is being dug up again.

Really  ??? were?
"Can you imagine on judgement day, when God tells you your wife was secretly a lesbian who liked you but never really loved you. That she wasn't your soul-mate after all. Better if both of you were free to find your true love."-Evil Roy Slick of Yahoo News

Offline lighthorseman

  • Argumentative Contrarian
  • On Probation
  • Bishop
  • *
  • Posts: 155
  • Proud Y chromosome owner
    • Company of the Wolf
Re: 1+1=2
« Reply #9 on: January 21, 2012, 06:29:12 am »
As Bertrand Russell said
Quote
The above proposition is occasionally useful.
Bertrand Russel being a prime example of a greatly respected mathematician who put a vast amount of time trying to prove the proposition.
Let me make it real easy... if anyone is interested in my actual opinion, please ask, I'd love to talk to you. If you are interested in trying to catch me out in some sort of "gotcha, before you said 'many', but now you're saying 'lots', you totally shifted goal posts", then, I'm not playing.

Art Vandelay

  • Guest
Re: 1+1=2
« Reply #10 on: January 21, 2012, 07:07:03 am »
*slow clap*

Your point?
It's related to some drama we had a few months back. On that note, I feel compelled to ask why this is being dug up again.

Really  ??? were?
The Proboards forum. Just search for "1+1=2" and you should find it.

Offline Vene

  • AKA Vene-Eye the Science Guy
  • Pope
  • ****
  • Posts: 324
  • Patronizing Know-It-All Snotnose
Re: 1+1=2
« Reply #11 on: January 21, 2012, 10:19:10 am »
I never said l don't think 1 & 1 equals 2, l merely said it requires a surprising degree of effort to prove. It's something that serious and well respected mathematicians far smarter than you or l have devoted vast amounts of time to trying to prove.

Most people can't prove it, but merely accept as axiomatic.
No, you didn't.
It is still possible the Earth is flat and that 1+1=/=2.

Offline N. De Plume

  • Mysterious Writing Implement
  • The Beast
  • *****
  • Posts: 1936
  • Gender: Male
  • Nom, nom, nom…
Re: 1+1=2
« Reply #12 on: January 21, 2012, 10:32:37 am »
I am considering using base two exclusively, just so I can tell people 1 + 1 = 10.
-A Pen Name

Offline Sigmaleph

  • Ungodlike
  • Administrator
  • The Beast
  • *****
  • Posts: 3615
    • sigmaleph on tumblr
Re: 1+1=2
« Reply #13 on: January 21, 2012, 10:48:05 am »
I never said l don't think 1 & 1 equals 2, l merely said it requires a surprising degree of effort to prove. It's something that serious and well respected mathematicians far smarter than you or l have devoted vast amounts of time to trying to prove.

Most people can't prove it, but merely accept as axiomatic.
I distinctly recall reading "you cannot prove 1+1=2" on one of my occasional visits to the old forums. If anyone has a link to the original thread, I'd appreciate it.

In any case:
The difficulty of the proof depends on where you're starting from. From Peano's axioms, once you've already defined the relevant terms, the proof takes only a couple of lines. As shown above. Pretty much anyone who bothered to try could arrive at that proof.

The proof that 1+1=2 doesn't appear in Principia Mathematica until page 300-and-something, yes. That doesn't mean that it hadn't been proven before. It just hadn't been proven from the system Russel and Whitehead were using.
Σא

Offline lighthorseman

  • Argumentative Contrarian
  • On Probation
  • Bishop
  • *
  • Posts: 155
  • Proud Y chromosome owner
    • Company of the Wolf
Re: 1+1=2
« Reply #14 on: January 21, 2012, 11:47:51 am »
I never said l don't think 1 & 1 equals 2, l merely said it requires a surprising degree of effort to prove. It's something that serious and well respected mathematicians far smarter than you or l have devoted vast amounts of time to trying to prove.

Most people can't prove it, but merely accept as axiomatic.
No, you didn't.
It is still possible the Earth is flat and that 1+1=/=2.
It IS possible, but I don't think it's the case. Clear now? One can acknowlege something is possible without believing it to be true. Right? It is POSSIBLE, for example, that evolution is wrong. I don't think it is though.

For the record, I don't think the Earth is flat, either, even though I acknowlege the POSSIBILITY.
Let me make it real easy... if anyone is interested in my actual opinion, please ask, I'd love to talk to you. If you are interested in trying to catch me out in some sort of "gotcha, before you said 'many', but now you're saying 'lots', you totally shifted goal posts", then, I'm not playing.