That’s some mighty fine humanist-leftist word trickery you’ve got there.
You’re using words to argue. Words are slippery things. Philosophers have argued about reality for centuries and finally crawled up their own asses in pursuit of contradictions and paradoxes. After that, they looked at the issue again.
Today, mathematicians, philosophers, and logicians understand that some things cannot be proved within a system. Gödel’s incompleteness theorems apply to all mathematics:
This does not imply that those statement can never be proved. They can, at the expense of adding an additional axiom to form a new meta-system. But that meta-system has its own unprovable statements. And so on up forever.
Word paradoxes such as yours can be treated as if they were a mathematical function, and so are also doomed to incompleteness.
All that is in addition to the fact that neither you nor Thudlow Boink have given a rigorous definition of disputable and indisputable, so we don’t even know whether you are arguing about the same exact concept and therefore whether any possible solution exists.
In short, you can’t handle this stuff casually. Only the most rigorous treatments can approach settlements. Assertions on either side are meaningless and lead to back-and-forth gainsaying like a Monty Python sketch.
avolongood posted once or four times?
Did warinner reply once or twice?
Math is hard.
Take two piles of dust. Sweep them together. How many piles of dust do you have?
We can even create a perfectly fine mathematical abstraction of this. We just define a modulo-1 number system:
0+0 = 0
1+0 = 1
1+1 = 1
The counting numbers are a useful way to model certain features of the universe. But their utility depends upon us imposing arbitrary definitions of what constitutes an object on the world around us. In everyday life the “thingness” of objects seems intuitively obvious, but if we start looking closely at it, it breaks down. And if we assign thingness differently, the counting numbers may no longer be a useful tool.
I am still trying to figure out how two plus two factorial is supposed to equal 5…
“!=” is sometimes used for “not equal to”.
zombie or no
… there are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns – the ones we don’t know we don’t know. …
Yeah, that would be less ambiguously written as “2+2 != 5” to avoid factorial confusion. An exclamation to indicate logical negation (“not”) is pretty common in computer languages.
No you can’t because modulo 1 stands for one element group. Threre are no 1 i Z1.
Anyways in Z2 1+1=10 but this is the very same result as 1+1=2. You just have a different set of numbers to work with but te outcome is the same. It is like you would say that Clinton did not sleep with Lewinski, beacuse in fact he was actually спал с Моникой (which is the very same statement written in cyrylic russian).
What is that for a proof? Your objects of addition are being transformed which make it hard to see the result, but if you would weigh one pile and the other and then weigh the two combined you would get w(1)+w(1)=w(1+1)=w(2). Assuming they are the same weigh
Regards
Ziom
Yes, I meant modulo-2.
Uh. No. In a modulo-2 system 1+1 is not equal to 10 or 2. It’s equal to 1.
It’s not a proof. It’s an example, just like your billiard ball example. The fact that placing one billiard ball next to another billiard ball gives you 2 billiard balls is not proof that 1+1 = 2. Just like the fact that combining two dust piles into one is proof that 1+1 = 1.
Aha, your original example said nothing about weight! Yes, mass is conserved in the universe, so if we’re adding together masses then adding real numbers is a good way to model that. But there are plenty of other situations where different maths are better suited to the problem at hand.
Basically, we can construct axiomatic systems and prove things within them. Sometimes these axiomatic systems make accurate predictions about the behavior of the real world. The sorts of predictions we want to make determine which axiomatic system (if any) it’s best to use.
no…
In the group of integers modulo 2, 1+1=0. What you’re describing is the commutative monoid of order 2 that isn’t a group (I don’t know if it has a special name).
There is a lot of nonsense (and some sense) in earlier posts to this thread. Let me ignore it and try to explain what is the case.
Ultimately, what we prove is based on our axioms. Everything proved is a tautology. Mistaken proofs do get published and, if the mistake is discovered, retracted. Back arounf 1870 or so, a “proof” of the 4-color theorem was published. Ten years later, an error was found in the argument and the question remained open until it was finally resolved around 1975 with the help of computer. Some years ago, I heard of theorem that was found to be wrong after 30 or 40 years. And it was something that had been used regularly, but that is unusual. Everything the classical Greeks did is still correct after 2 1/2 millennia.
Could there be a contradiction in the usual axioms? Unfortunately Goedel showed that it is impossible to show consistency in any set of axioms strong enough to have ordinary arithmetic. So it is an article of faith to suppose that the usual axioms (generally called ZFC, Zermelo-Fraenkel with choice) are consistent. One thing: if there is any inconsistency, any at all, then 2 + 2 = 5 (or any other statement you like) becomes provable. Every practicing mathematician believes that ZF is consistent. And if it is, no inconsistency can be introduced by adding choice.
The predicate calculus can be proved consistent, but it is too weak to allow arithmetic.
The bottom line is that any time some claim is wrong, errors have been found in the proof. It is estimated that about half of all published papers in mathematics contain an error, mostly unimportant (and frequently obvious).
Well, this thread is 15 years old after all.
Math today just ain’t what it used to be back then.
(Cite: Ask any third-grader to explain it to you.)
Actually, in modulo 1, 1+1=1. But in this system, it’s also true that 1+1=0, and 0+0=1, and a bunch of other things. This is because, in a modulo-1 system, 0=1. There’s only one integer, and it can equally well be called by the name of any integer. It’s a very boring system.
Conversely, there is a method of proving that a number is “probably” prime, with a probability you can make arbitrarily small (like 1 chance in 10^100).
A former prof of mine was justifying using this algorithm by saying, “What is the chance that Fermat’s little theorem* is untrue because of a flaw that that no mathematician has noticed in the 400 years since it was stated? The probability of the algorithm being wrong because Fermat’s little theorem is actually false and no one has noticed the error in the proof is greater than the probability of it declaring a non-prime number to be prime.”
This is the first and only time when I have seen “probability that math is wrong” compared to a more conventional probability in a result.
*(*note, not the more famous last theorem where an error in the proof would not be surprising. Fermat’s *little theorem can be comfortably proved on a single sheet of paper with room to spare).
But the axia used in math is not of this physical world, but a ideal mathmatical universe. It works good enough for earth, but that is already intentionally introducing errors from real to idea model. So the math is not wrong, even near a black hole, You are just using one model which does not relate to the physical reality at that location.
No, math cannot be proven wrong, because it is not a science. It is a language. You might as well ask if English could be proven to be grammatically incorrect. It may change with time, but it is always correct. Math may change with its frame of reference or its axioms, but it is always correct.
Right. 1+1 = 2, with logical certainty. But that doesn’t mean that it’s necessarily a good model for anything in the real world. It works pretty well for whole apples: put one apple with one apple and you have two. But if you mix a gallon of water and a gallon of ethanol, you don’t end up with two gallons of mixture. That doesn’t mean that 1+ 1= 2 is wrong; it just means that numeric addition isn’t a good model for mixing volumes of alcohol and water.
Just this past week on PBS:
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[li]http://www.pbs.org/wgbh/nova/physics/great-math-mystery.html[/li][/ul]