Author Topic: 1+1=2  (Read 11473 times)

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Offline lighthorseman

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Re: 1+1=2
« Reply #15 on: January 21, 2012, 12:28:47 pm »
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.
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 Murdin

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Re: 1+1=2
« Reply #16 on: January 21, 2012, 01:10:19 pm »
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.
« Last Edit: January 21, 2012, 01:24:30 pm by Murdin »

Offline Da Rat Bastid

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Re: 1+1=2
« Reply #17 on: January 21, 2012, 01:37:11 pm »
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
« Last Edit: January 21, 2012, 01:39:13 pm by Rat of Steel »

Offline Sigmaleph

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Re: 1+1=2
« Reply #18 on: January 21, 2012, 01:50:25 pm »
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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Offline GLaDOS

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Re: 1+1=2
« Reply #19 on: January 21, 2012, 03:34:19 pm »
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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Offline Cataclysm

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Re: 1+1=2
« Reply #20 on: January 21, 2012, 03:37:48 pm »
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!
I'd be more sympathetic if people here didn't act like they knew what they were saying when they were saying something very much wrong.

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Offline Old Viking

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Re: 1+1=2
« Reply #21 on: January 21, 2012, 05:40:21 pm »
One plus one equals two, but I hate to be dogmatic about it.
I am an old man, and I've seen many problems, most of which never happened.

Offline lighthorseman

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Re: 1+1=2
« Reply #22 on: January 21, 2012, 07:29:44 pm »
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.
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 Captain Jack Harkness

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Re: 1+1=2
« Reply #23 on: January 22, 2012, 02:07:58 am »
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.
« Last Edit: January 22, 2012, 02:12:26 am by B-Man »
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Offline The Right Honourable Mlle Antéchrist

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Re: 1+1=2
« Reply #24 on: January 22, 2012, 02:49:58 am »
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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Re: 1+1=2
« Reply #25 on: January 22, 2012, 04:03:15 am »
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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Offline Sigmaleph

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Re: 1+1=2
« Reply #26 on: January 22, 2012, 04:40:30 pm »
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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Offline NonProphet

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Re: 1+1=2
« Reply #27 on: January 30, 2012, 09:14:07 pm »
I'm desperately trying not to argue the whole "it's possible the earth is flat" thing... >.<

Self-control, NP... self-control.

Offline Lithp

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Re: 1+1=2
« Reply #28 on: January 30, 2012, 10:16:43 pm »
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.

Offline cagnazzo

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Re: 1+1=2
« Reply #29 on: February 02, 2012, 12:39:55 pm »
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.