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Community => Science and Technology => Topic started by: Sigmaleph on January 20, 2012, 11:42:46 pm
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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.
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In a related news... I educated myself about Graham's number. Rather insane.
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*slow clap*
Your point?
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*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.
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LHM requested a thread to reinvestigate the 1+1=2 thing.
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Fantastic.
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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.
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As Bertrand Russell said
The above proposition is occasionally useful.
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*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?
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As Bertrand Russell said
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.
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*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.
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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.
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I am considering using base two exclusively, just so I can tell people 1 + 1 = 10.
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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.
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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.
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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.
Any "proof" at some point rests on axioms, does it not?
Anyway, yes, ok, I'll admit that at one point I said something like "you cannot prove 1+1=2", when really what I should have said was "proving 1+1=2 is actually more difficult than simple 'common sense' tells us".
The context in which I said what I said was that things everyone "just knows" aren't always that simple, if one cares to look deeper. But rather than taking that meaning, most people's reaction was "look at this idiot, he thinks 1+1=/=2 !" which was absolutely NOT my point.
In hindsight, I wish I'd never made the comment, or at least expressed it very differently.
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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.
... Sigmaleph's entire point was to show you why there is absolutely no way for 1+1=2 to be false. It doesn't come from any external observations that are possibly falsifiable. It's because this result comes directly from the definitions of 1, 2, + and =.
And you are basically ignoring it wholesale.
I'm not saying that mathematics are an absolutely certain science, mind you. It's always possible for a mathematical system such as Peano's integers to be self-contradicting, and therefore nonsensical. But the possible "falseness" of 1+1=2 has nothing to do with that of the theory of evolution or the theory of rotundity. 1+1=2 is not unambiguously true if and only if its entire abstract "universe" is shown to be invalid.
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I am considering using base two exclusively, just so I can tell people 1 + 1 = 10.
(In before GLaDOS shows up and posts "Two plus two is ffffffffff.....ten, IN BASE FOUR. I'M FINE.") ;D
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Anyway, yes, ok, I'll admit that at one point I said something like "you cannot prove 1+1=2", when really what I should have said was "proving 1+1=2 is actually more difficult than simple 'common sense' tells us".
Then we are in agreement. My only problem was with the statement that one cannot prove 1+1=2. It is, indeed, somewhat more difficult than people think.
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.
... Sigmaleph's entire point was to show you why there is absolutely no way for 1+1=2 to be false. It doesn't come from any external observations that are possibly falsifiable. It's because this result comes directly from the definitions of 1, 2, + and =.
And you are basically ignoring it wholesale.
I'm not saying that mathematics are an absolutely certain science, mind you. It's always possible for a mathematical system such as Peano's integers to be self-contradicting, and therefore nonsensical. But the possible "falseness" of 1+1=2 has nothing to do with that of the theory of evolution or the theory of rotundity. 1+1=2 is false if and only if its entire abstract "universe" is invalid.
No, that was not my point. My point was that you can prove that 1+1=2. It remains possible, in an absolute technical sense, that 1+1=/=2, and not necessarily because Peano Arithmetic is inconsistent, thanks to Cartesian Demon arguments.
That is: On occasion, people think they have proven something and they have not, having made a mistake in some step. Given this, I cannot assign a probability of literally, exactly 0 to the hypothesis that there is a demon out there who confuses me every time I look at Peano's axioms and derive S(0) + S(0) = S(S(0)), while the correct result is S(S(S(0))).
The probability is not zero, but close enough for most purposes. We routinely make statements of fact that do not have probability 0 of being wrong, because the probability is so small it is not worth the effort of bringing up (that includes every statement I've made in this post).
Or, in other words: Is 1+1=3? No. I know this, you know this, and Lighthorseman knows this. Is it possible, in the most literal interpretation of the word, that 1+1=3? Yes. But only in the most literal interpretation of the word "possible", which is generally useless in any discussion. I don't know why Lighthorseman felt the need to bring up epsilon probabilities of some facts being wrong. I don't know if it was justified (it rarely is). But the only point I was trying to make is that you can prove that 1+1=2. Lighthorseman having acknowledged that, my work here is done.
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I am considering using base two exclusively, just so I can tell people 1 + 1 = 10.
(In before GLaDOS shows up and posts "Two plus two is ffffffffff.....ten, IN BASE FOUR. I'M FINE.") ;D
Hey! how did you know what I was going to say?
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The moon is made out of cheese, the sun is pulled by a chariot, and we're all brains in jars connected to a matrix.
Hey, it's not impossible!
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One plus one equals two, but I hate to be dogmatic about it.
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Anyway, yes, ok, I'll admit that at one point I said something like "you cannot prove 1+1=2", when really what I should have said was "proving 1+1=2 is actually more difficult than simple 'common sense' tells us".
Then we are in agreement. My only problem was with the statement that one cannot prove 1+1=2. It is, indeed, somewhat more difficult than people think.
Cool, happy we've managed to understand each other.
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Can't we just argue that 1+1 = 2 because we (humanity) created a language and assigned names to each number in base 10, and then proceeded to name various ways to handle these numbers. Really, I think this is a fight that is beat and most easily resolved in language. We defined a bunch of abstract shit in our language The end.
Why are any concepts and ideas called what they are? Because that's the way the language evolved! I know that's a tad circular, and I bet some of you think I'm missing the point, but I seriously think this is the most retarded fight ever, and I like math too.
I just don't see when you're getting anything done by providing a long winded explanation to prove something that's defined through the language itself.
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The Proboards forum. Just search for "1+1=2" and you should find it.
Maybe if you inhabit an alternate universe where the Proboards search function isn't a heap of crap.
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Well, I was gonna be very clever and use google's search with key terms but it's so pants on head simple-minded that it works even worse than the proboards' search.
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Can't we just argue that 1+1 = 2 because we (humanity) created a language and assigned names to each number in base 10, and then proceeded to name various ways to handle these numbers. Really, I think this is a fight that is beat and most easily resolved in language. We defined a bunch of abstract shit in our language The end.
Why are any concepts and ideas called what they are? Because that's the way the language evolved! I know that's a tad circular, and I bet some of you think I'm missing the point, but I seriously think this is the most retarded fight ever, and I like math too.
I just don't see when you're getting anything done by providing a long winded explanation to prove something that's defined through the language itself.
The argument was about whether you can prove that 1+1=2. You can. I gave the proof. Lighthorseman accepted the proof. The end. I got done exactly what I wanted to get done, which was showing that the statement "you cannot prove that 1+1=2" is false (and the related "it takes hundreds of pages to prove 1+1=2" is also false).
Also: You can define 1+1=2, or you can define one, two and addition separately. I prefer the latter approach, it makes more sense.
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I'm desperately trying not to argue the whole "it's possible the earth is flat" thing... >.<
Self-control, NP... self-control.
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Ewww, we had a redo of this? Hell, you had me at "natural number." Numbers exist. What we call them is irrelevent, but for the purposes of convention, 1+1=2. Case fucking closed, as far as I was ever concerned.
Which makes this post sort of redundant.
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Also: You can define 1+1=2, or you can define one, two and addition separately. I prefer the latter approach, it makes more sense.
It's also more useful, and allows us to abstract more things. You can get 1+1=2 from 1, +, 2 and =. You can't get the primitive parts from the structured whole.
You've got to start somewhere, and if you've got means of abstraction and means of combination, you'd best start at atomic primitives.