onionmon

Alrite, we all know that 1+1=2 rite. BUT, how do u prove it?

My buddy's dad ask me that some time ago and said something about defining the cardinal numbers and defining the operations + and =. So if any of you know how to prove this, please show! I and probably many would like to see.

Hell prove 1+1=3 if u gots the time.

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BonesCPA

How about having 1=2?

LINK

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forseti-6

I know it says step 4 is the incorrect part, but how do you go from
b = a + b
to
b = 2b?

Isn't haveing b = a + b lead to
a = 0? thus proving 0 = 0....

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onionmon

Well see, here's the deal, u can't prove it with algebra. Its not something that u can prove with algebraic or ?calculus-ic? methods. Its more of a realization proof than anything else. For example, you would actually need to define the number 1. What is the definition of "1". Then you would need to define the operation "+". I remember that defining that would require explaining how the conjuction of two items creates the results. You also need to show what "1+" means and "+1" means ,etc. Thats the general outline, but i would like a PROOF. like just slap down on paper on a test.

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Akira

My teacher told me about 2 British brothers who tried to prove all the theorems in math. According to him it took them 2 books to prove the number 1. So I guess thats some heavy reading.

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PorscheBunny

Bertrand Russell and Alfred North Whitehead aren't brothers, and it was done in three volumes.

http://www-gap.dcs.st-and.ac.uk/~his...s/Russell.html

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filtherton

How hard is it to prove one?
Take an apple.
Look at apple.
Count apple.
There is one apple.

To prove 1+1=2:
Take one apple.
Add one apple.
Now you have two.

I don't understand how you would need to prove any more than that.

onionmon

What does "one" mean? For us humans, it means a single entity. a apple. BUT logically, if there was no picture, no image, no thing to show that "yes that is one", how would u prove that "something" is "one". on the page by porche bunny, the number one is classified as the set of all items that have the property of being single. --- i think... and yes it does have something to do with cardinal numbers. heres proof, chek it

http://www.cut-the-knot.org/selfreference/russell.shtml

lots of crazy stuff

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Simmons! I want you to poison Grif's next meal!
Yes sir!

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stingc

I doubt more than ten people have ever read those books. They are probably the most obtuse things in the english language (if you can call it that!). There's maybe one or two english words per page. The rest is written entirely with the symbols of formal logic.

The books were important, though, because they investigated the precise use of logic, and the minimal set of axioms necessary for mathematics (if such a thing existed). Its failure to accomplish those goals were a very strong statement about what mathematics (and logic) means.

Giltwist

What you guys are talking about is number theory, which Iam taking this semester, I will let you know when I find out.

Conclamo Ludus

I've read a lot about these guys lately and I checked Amazon.com for a copy.

Its only a merely $642.00 for the 3-vol set. I think I'll find a library instead.

Principia Mathematica At Amazon Link

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onionmon

yea see, i care about it, but i dont care that much to spend 600 dolalrs. btw, look under the hardcover edition and there is a paperback version -- 50 bux. not bad!

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Slims

It is obvious, but in order to show that your system of math is able to accurately represent this truth, you have to be able to prove it using that system.

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Corneo

It's funny how there is math and then there is math that makes sense for only a few. They latter is something I don't think anyone would need to know in real life. I'll stick to my caluclus thank you very much.

Tempboy

Yeah, I remember how the math they showed in Good Will Hunting made me really scared of entering University.

But then I got there and I didn't have to do it.. so I calmed down...
and ended up having to do it in my 2nd year anyway.
Grrr..

filtherton

I don't know. It seems like proving "one" would be similar to trying to prove "blue". It is defined by the definition we apply to it. If it is red, it is not blue. If it is two, it is not one. It is blue because it is blue. It is one because it is one. It is only one because it meets the conditions that we use to define one.

Incidentally, i once read a book called "The Mathematical Experience" and there was a brief chapter on "pure" mathematics, that is, mathematics with no possible application to anything. It seems there are elitest mathematicians who look down their noses at math that can be applied to science and the human condition. These mathematicians will study their little area for years, perhaps decades, carving out their little niche. Eventually they get to a level of understanding of their particular specialty that is equaled by only a handful of other people on this planet. Then they die and maybe a handful of people will ever be able to understand and appreciate what they did in their lifetime. Their life's work probably amounting to nothing more than a footnote in some math history book.

Corneo

Hurray for Applied Mathematics then! Math with an actual use!!!

Hunnar

From this point of view, we define things by what the are "not". The definitions are a limiting factor, and depend on the commonality of experience of the reader. From a different point of view an apple might not be an apple any more than a 1 might not be a 2.
Here is something to consider. Zero and Infinity. Obviously two different things. However, when dealing with math, the application of either one has a neutral effect on the equation. One is too small to be invasive the other too large. So if a person did not have a commonality of experience they might agree that Zero=Infinity.
And that can't be right, can it?

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KnifeMissile

A brief google search reveals that the natural numbers can be defined by the Peano axioms as such:

Let N be the set of natural numbers.

• There exists a natural number which we will call 1.
• For all x in N, there exists a successor, called x + 1.
• For all x in N, 1 != x + 1.
• For all x and y in N, x+1=y+1 => x=y.
• Mathematical Induction works.

Except for the last bullet point, Wikipedia has a more English explanation than the one given here. No $650 dollars needed, go buy yourself a new pair of pants!

If we were talking about fields, there exists one where 1 + 1 = 1.

However, we're talking about the natural numbers. So, for the natural numbers, 1 + 1 != 1. We can define a number, 2, to be the successor of 1. So, 1 + 1 = 2.
QED.

Back in school, we called this powerful technique proof by definition.

KnifeMissile

There are no elitist mathematicians, we're all elite.

The idea that there are mathematics that can't possibly have any application is such an exaggeration that I don't mind calling it a fallacy. Even for the most abstract branches of mathematics, we simply say that there is currently no known application. The idea that it is interesting and gives us confidence in our reasoning is useful enough, for the moment. However, too often in the past has some branch of abstract mathematics become applicable that we dare not say that something is useless. Two very important examples are complex numbers and calculus!

wipeout

numbers are something that man made to count and figure out the answer--but what is the point counting on man to answer something they made up

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filtherton

Read the book. "The Mathematical Experience" by Philip J. Davis and Reuben Hersh.

KnifeMissile

Read my post, written by me, under the pseudonym KnifeMissle.

filtherton

Okay, i read it again. I guess i mispoke. I wasn't saying that these elitest mathematicians aren't creating math that can't be applied ever. I was saying that the mathematicians in question wouldn't bother themselves with mathematics that are currently applicable, and indeed may even look down their noses at mathematicians who do. It is math for math's sake compared to math for the sake of physics or industry. Yes, eventually there may be some use for whatever thoerems they devote their lives to, but maybe not. They don't care either way.
It seems they are kind of like indy musicians who look doen their noses at commercial musicians for "selling out". Granted there is some indy rock that proves to be immensely relevant. Certainly there is also a lot of indy rock that sucks immense ass and isn't worth the medium it is recorded on.