Is "1+1" provable?

[Christer]

Sorry for starting a new thread, but SY suddenly closed
the thread where this was discussed and I was not allowed
to email millwood through the forum, thus leaving me no
other option.

Millwood said something about proving "1+1" and I commented
that this cannot be proved or disproved since it is not a
statement and suggested that maybe he meant proving
whether "1+1=2". To this he responded:

sure you can. both 1+1 and 1+2 come from a very old mathematical conjecture. and 1+2 has been proven (partially) but 1+1 remains.

I am puzzled what you mean by this. you can only prove or
disprove a logical proposition and neither "1+1" nor "1+2"
are logical propositions (claims, statements, whatever you
wish to call it. Could you please elaborate a bit on what you
mean, especially since you made the claim of one of these
being partially proven. Assuming the usual algebraic system
(which is not obvious to do) you could just as well say that
it is not proven whether "2" but it is partially proven that "3",
which makes clearer what the problem is, I think.

[Peter Daniel]

You also said:

Originally posted by Christer
Perhaps he meant
proving whether "1+1 = 2", which is or isn't true depending
on our choice of algebraic system.

This reminds me of cables discussions. Why would you like to choose the system for which it is not true?

[Peter Daniel]

I don't want to start another thread here, but millwood also said;

Originally posted by millwood

Again, awareness is an acquired taste and not everyone is gifted with it.

As far as I understand, awarness is a consciousness and knowledge, and this, unlike taste, is not biased. It is also not a gift, but something one gains if he follows the right path.

[millwood]

Christer:

1+1, 1+2 (and 1+3) are all different forms of a mathematical conjecture, Goldbach conjecture (actually strong Goldbach conjecture as presented by Euler). Essentially, it states that any even number can be expressed as the sum of two prime numbers. (thus 1+1).

in the 1970s, it was proven that sufficiently large even numbers can be expressed as the sum of a prime number and the product of no more than two prime numbers. (thus 1+2);

1+1 remains unproven at this point.

that should be sufficient as starters for this discussion.

[SY]

You're right, 1 + 1 = 2 can't be proved any more so than any other axiom. That's why it's an axiom and not a theorem. You can construct all kinds of arithmetical systems where this isn't true (one trivial example is binary, where 1 + 1 = 10).

BTW, I got a reference wrong in the other thread; treatment of these issues is in Courant and Robbins, "What Is Mathematics?".

[Sch3mat1c]

<Not one to delve into soon-to-be-hugenormous threads but whatever>

Seems easy enough to me.. define a number line with counting numbers 0, 1, 2, 3, . . . (on to infinity, and also going to negative infinity left of zero if you want to be complete) and then we define addition as moving a point on the line an according number of spaces to either side. (Add algebraically, i.e. 4 + (-2) = 2.) If it is a real object like a ruler and the numbers are evenly spaced, then it's simple to measure a length and put it either side of the point, mark the new point, and there you have it.

But then again I get the feeling I missed the point. But I don't get the point to begin with; 1+1 is obvious enough. If you have two equal quantities (1 = 1), then adding them together will double the quantity (algebraically, x + x = 2x (adding common terms), and let x=1, 1 + 1 = 2*1 = 2).

I think I still missed the point. But then I'm no philosopher, I tend not to argue the logic itself, because I know logic works.

Tim

[Won]

1+1=2 is basically a definition. Goldbach's conjecture does not "prove" it. In fact, it has very little to do with "proving" 1+1=2.

You can't just define the numbers like that (referring to the "ruler" comment). It is more involved than that. Your "axiom" usually starts with the definition of a set, and the empty set.

1+1=10 is the same arithmetic, you're just redefining what the symbols mean.

1+1=0 (arithmetic mod 2) IS a different arithmetic system. Note that in this system, 2 = 0, which is not a coincidence.

Whatever. I dealt with this a long time ago.
Won

[JOE DIRT®]

Heck??...what has all this thread have to do with DIYAUDIO>???....shall we get into boolean algebra and hex code...or is it that someone has a chip on their shoulder that needs to be slapped off....all I can say is stick with the theme guys,we have some smart minds here and I can tell the ones who are fresh out of school (or are in!)...but rifling off this verbatim in THIS FORUM is fruitless because your wasting the ink in your keyboard

DIRT®

[planet10]

Originally posted by Peter Daniel
This reminds me of cables discussions. Why would you like to choose the system for which it is not true?

There are lots of situations. For instance if you are a binary computer then 1+1=10

Millwood, 1+1 by itself cannot be proven, when you add the context that makes it the Goldbach conjecture it is a completely different beast.

A Christer says, proving 1+1 is topologically equivalent of proving "spot"

dave

[Prune]

Originally posted by JOE DIRT®
Heck??...what has all this thread have to do with DIYAUDIO>???

Well, this is the Everything Else forum, so implying that any post here is offtopic doesn't make sense.