Is there a way to refute '1+1 = 2'?
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Is there a way to refute '1+1 = 2'?
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Re: Is there a way to refute '1+1 = 2'?
"The mind is a wonderful servant but a terrible master."
I believe spiritual freedom (a.k.a. self-discipline) manifests as bravery, confidence, grace, honesty, love, and inner peace.
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Re: Is there a way to refute '1+1 = 2'?
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Re: Is there a way to refute '1+1 = 2'?
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
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Re: Is there a way to refute '1+1 = 2'?
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Re: Is there a way to refute '1+1 = 2'?
However, it might also have to make truthful predictions outside the system. For instance, if you were designing a system that depended on mathematical calculations - and many do, even the one you're reading this on - then the outcome can't be specifically internal to the system you have devised. It also has to yield accurate results. And they may not be at all arbitrary or fanciful. If you designed a bridge to meet at a point, then whatever system you use to calculate that figure has to be accurate with reference to the bridge, not simply with reference to itself, otherwise the calculation will be faulty, and the project will fail.A_Seagull wrote:Philosophically speaking, '1+1=2' is a string of 5 symbols that is declared to be 'true' within the system of 'mathematics'.
If you were to create your own system call it 'Raminatics' if you like, then you could set up symbols and axioms and means of generating theorems from that system which would then be 'true' within that system.
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Re: Is there a way to refute '1+1 = 2'?
In order to apply mathematics (or any other abstract system for that matter) to the real world you need to create a mapping between the two.
-- Updated September 13th, 2014, 6:07 pm to add the following --
Yes, I would say that 'either P is true or ~P is true' is only 'true' within the system of formal logic.Ramin22 wrote:Thank you everyone for the answers. I have one more question. Is 'either P is true or ~P is true' just true inside classical logic? I know it can be considered true just in that sense. But is there any other sense, or are there others who argue classical logic is true on other grounds?
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Re: Is there a way to refute '1+1 = 2'?
Contrary to other comments in this thread - there is no mathematical basis for this statement.
As has been noted, a mathematical statement has a specific meaning with respect to a set of axioms; change the axioms and the "truth" of a statement may change.
However, axioms themselves are statements. So axioms only have a specific meaning with respect to a specified set of axioms... which in turn only have a specific meaning with respect to a set of axioms...
In other words; it is impossible to define a set of axioms without having first defined a set of axioms.
Without axioms there is no 'proof' and no 'true or false'; there is no deductive logic.
Mathematics is justified purely by the extent to which it works. 1+1=2 is part of a group of relationships that help us design bridges that don't fall down in the first breeze. To this extent, 1+1=2 is justified... but we know that there are other useful groups of relationships within which 1+1 != 2.
So... no - 1+1=2 cannot be refuted... because refutation implies logic - and there is no logic (there is reasoning - but that isn't the same thing as mathematics logic). But neither can 1+1=2 be proven. It is just a statement that we sometimes find useful.
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Re: Is there a way to refute '1+1 = 2'?
The short proof that 1+1=2 (which, inexplicably, none of the respondents has seen fit to supply) is as follows:
1+1 = 1+s(0) = s(1+0) = s(1) = 2
It will be immediately apparent that this proof has little or nothing to do with scripts or movies, and I am not sure in which school of mathematics A_Seagull would have learned these terms.
This proof, like all mathematical proofs, is conditional upon an agreed set of axioms (and god, if only mathematicians could be made to remember this). It can easily be falsified by proposing a suitable modification to the axiom set.
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Re: Is there a way to refute '1+1 = 2'?
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Re: Is there a way to refute '1+1 = 2'?
Alan Masterman wrote:A most amusing and interesting thread. Very few of the posts have any connection with the question, but that is quite normal and predictable, and (in this case) forgivable because the question is a question in mathematical philosophy, a subject of which very few mathematicians have any understanding. .
But clearly, you are not one of them!
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Re: Is there a way to refute '1+1 = 2'?
You: Suppose you’re in the garden and you see two worms crawling around. Then two more worms crawl over. How many worms do you have now?
Peter: “Crawling” means moving around on your hands and knees. Worms don’t have hands and knees, so they don’t “crawl.” They have hair-like projections called setae which make contact with the soil, and their bodies are moved by two sets of muscles, an outer layer called the circular muscles and an inner layer known as the longitudinal muscles. Alternation between these muscles causes a series of expansions and contractions of the worm’s body.
You: That’s all very impressive, but you know what I meant, Peter, and the specific way worms move around is completely irrelevant in any case. The point is that you’d have four worms.
Peter: Science is irrelevant, huh? Well, do you drive a car? Use a cell phone? Go to the doctor? Science made all that possible.
You: Yes, fine, but what does that have to do with the subject at hand? What I mean is that how worms move is irrelevant to how many worms you’d have in the example. You’d have four worms. That’s true whatever science ends up telling us about worms.
Peter: You obviously don’t know anything about science. If you divide a planarian flatworm, it will grow into two new individual flatworms. So, if that’s the kind of worm we’re talking about, then if you have two worms and then add two more, you might end up with five worms, or even more than five. So much for this a priori “arithmetic” stuff.
You: That’s a ridiculous argument! If you’ve got only two worms and add another two worms, that gives you four worms, period. That one of those worms might later go on to be divided in two doesn’t change that!
Peter: Are you denying the empirical evidence about how flatworms divide?
You: Of course not. I’m saying that that empirical evidence simply doesn’t show what you think it does.
Peter: This is well-confirmed science. What motivation could you possibly have for rejecting what we know about the planarian flatworm, apart from a desperate attempt to avoid falsification of your precious “arithmetic”?
You: Peter, I think you might need a hearing aid. I just got done saying that I don’t reject it. I’m saying that it has no bearing one way or the other on this particular question of whether two and two make four. Whether we’re counting planarian flatworms or Planters peanuts is completely irrelevant.
Peter: So arithmetic is unfalsifiable. Unlike scientific claims, for which you can give rational arguments.
You: That’s a false choice. The whole point is that argumentation of the sort that characterizes empirical science is not the only kind of rational argumentation. For example, if I can show by reductio ad absurdum that your denial of some claim of arithmetic is false, then I’ve given a rational justification of that claim.
Peter: No, because you haven’t offered any empirical evidence.
You: You’ve just blatantly begged the question! Whether all rational argumentation involves the mustering of empirical evidence is precisely what’s at issue.
Peter: So you say now. But earlier you gave the worm example as an argument for the claim that two and two make four. You appeal to empirical evidence when it suits you and then retreat into unfalsifiability when that evidence goes against you.
You: You completely misunderstand the nature of arithmetical claims. They’re not empirical claims in the same sense that claims about flatworm physiology are. But that doesn’t mean that they have no relevance to the empirical world. Given that it’s a necessary truth that two and two make four, naturally you are going to find that when you observe two worms crawl up beside two other worms, there will be four worms there. But that’s not “empirical evidence” in the sense that laboratory results are empirical evidence. It’s rather an illustration of something that is going to be the case whatever the specific empirical facts turn out to be.
Peter: See, every time I call attention to the scientific evidence that refutes your silly “arithmetic,” you claim that I “just don’t understand” it. Well, I understand it well enough. It’s all about trying to figure out flatworms and other things science tells us about, but by appealing to intuitions or word games about “necessary truth” or just making stuff up. It’s imaginary science. What we need is real, empirical science, like physics.
You: That makes no sense at all. Physics presupposes arithmetic! How the hell do you think physicists do their calculations?
Peter: Whatever. Because science. Because I @#$%&*! love science.
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Re: Is there a way to refute '1+1 = 2'?
So refutation occurs through varying context, convention, and logic.
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Re: Is there a way to refute '1+1 = 2'?
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Re: Is there a way to refute '1+1 = 2'?
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