1 + 1 = 2 actually needs/HAS mathematical proof? Please tell me hubby's kidding?

Question

In "homework help" question dated a couple days ago, one poster mentioned Peano postulates (??don't try and explain what they are; I nearly flunked HS algebra and geometry) and put down a whole BUNCH of scary looking proofs. I assume he was having a giggle with the OP and this is on a par with Nessie or Bigfoot.
My technogeek husband mentioned that some intrepid souls actually sat down once and tried to "prove" as many mathematical basics as possible. They were trying to reduce the number of axioms/givens in use. Then Godel (spelling?) came up with some kind of weird theory....
He is teasing me, right? Did someone with way too much time on their hands actually sit down and establish, with way too many equations and formulas, what any nitwit can.....?
Math history buffs, help!

Clos: can BELIEVE 1+1=2 . I'm from Missouri, simple demonstrations do just FINE. It's these d*mned equations that are giving conniptions.
Staring in utter shock at equations &
formulas y'all presented (will ask Yahoo if y'all can get points for the work; ty.).
I guess it ISN'T some sort of utterly weird technical joke. People actually do have reasons to prove out the smallest things----I guess...ty, themeindzye, that kinda makes sense.
Guess better question to ask (any takers?)---REALISTICALLY, how often do people CARE about this high level of proof? Is this level of precision really necessary, when you come down to it? (It seems like the math equivalent of "how many angels can dance on the head of a pin?")
Going to talk to Yahoo now---take care. Thanks!

Answer 1

I found the proof online:

The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn't 0, then let c' = b, with c in N, and define
a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)'
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

- Doctor Rob, The Math Forum
http://mathforum.org/dr.math/

Answer 2

Yes, many people, not just Peano, have tried to create a set of axioms from which all of mathematics can be derived. Russel tried to do this early last century with his super logic, but failed to define number in logical terms. Godel actually proved that if any system that contains arithmetic contained a proof of its own consistency, it would also contain proof of its inconsistency, thus thwarting the formalists from establishing Mathematics as an Absolute system of eternal Truths.

Answer 3

Actually, 2" is defined as "1+1". (The "2" has a meaning because it represents the "1+1").

Similarly, we can define "3" as "1+1+1" (or using the definition for 2 :
3 = 2+1).

Well, you may accept this theory or try to prove that it does not hold.
If it is the second case define me the sympol "2" using simplier sympols or meanings than "2" (e.g. "1").


If there is not something better than this, I'd like you to give me, at least, the 10 points. (Hey, I'm sitting here trying to find out the anwser since you have submitted this question)!

Answer 4

Math is all about accuracy and irrefutable proof. Saying 1+1=2 doesn't make it true. You have to have reasoning behind it, and a way to prove that it is true. Notice the debate over the existence of God. There is no hard proof. This is a back-up to make sure that all math is correct math and that no math is wrong. That would defeat the purpose of math. So no, he is not kidding. Yes, there is proof that 1+1=2.

Answer 5

Yup. Kurt Gödel (probably the greatest logician who has ever lived) did, in fact, prove that mathematics could not be axiomatized. Don't worry about Peanos' postulates. They just define the integers and give rules for describing them and, in particular, counting with them. As far as 'proofs' that 1+1=2, they invariably involve a 'hidden' division by 0 (or some such other forbidden operation) at some point.


Doug

Answer 6

Here is the link to the mathematical proof that 1+1=2

http://mathforum.org/library/drmath/view/51551.html

Answer 7

I SO wanted to answer this question as i love math, but i couldn't make out wheather it was a question, a stetement, a wonder...What do you REALLY want to understand? that 1+1=2? please clarify your question so i'll give it a shot :)

Answer 8

To many big words. My head hurts.

Answer 9

I am so lost.