It takes over 25 thousand steps to prove that 2+2=4

a = b
a^2 = ab

(a^2) - (b^2) = ab - (b^2)

(a+b)(a-b) = b(a-b)

a+b = b

a+a = a

2a = a

2 = 1

Spot the problem, lol

I had to read some excerpts from Principia Mathematica when I was taking a Logic course. Hated that part of the semester...

I tried to read it. Normally I get confused by something within 25 seconds at least.

But, I got confused in a mere second by reading this. I think I just broke one of my records. I can't read that. or else my brain will explode.

Whoever wrote this has absolutely no life.

Originally Posted by Double A

a = b
a^2 = ab

(a^2) - (b^2) = ab - (b^2)

(a+b)(a-b) = b(a-b)

a+b = b

a+a = a

2a = a

2 = 1

Spot the problem, lol

That's your problem. If a=b, (a+b)(a-b) cannot equal b(a-b), because that would be implying (a+b) = b, which isn't true.

Originally Posted by Double A

a = b
a^2 = ab

(a^2) - (b^2) = ab - (b^2)

(a+b)(a-b) = b(a-b)

a+b = b

a+a = a

2a = a

2 = 1

Spot the problem, lol

The problem with this "proof" is that a - b is zero, so you can't divide both sides by (a - b) in this step here:

(a + b)(a - b) = b(a - b)
a + b = b

Edit: Damn, I got "ninja'd".

Well, actually, no I haven't been, because (a + b)(a - b) does equal b(a - b), because both sides are zero. The incorrect step is to assume that this implies that a + b = b.

Originally Posted by John

Aren't the first few hundred pages of the Principia Mathematica dedicated to proving that 1+1=2?

Proving it from basics is...not easy.


Because how do you know that one finger plus one more finger is two fingers? Why isn't it three? Or twelve? Or none?


Of course, my (exceptionally limited) understanding of this is that, despite such proofs, other proofs have popped up showing that it is impossible to prove something as basic as 1+1=2, but I could be mistaken on this point.

Originally Posted by John

And you know that that's four, how? How do you know it isn't five? Or 0.22?

How do you even know what addition is? 2+2 might be 0! or 99! How can you tell?

I always thought that one finger plus one more finger is two fingers and not three or twelve or none because that's what humans decided it would be.

all it means is when you have an object and put another object with it, you have two objects. That's really all the proof of 1 + 1 = 2 that you need.

we decided that one would be called one and two would be called two. It just so happens that as an inherent fact of nature, putting two ones together results in two.

Originally Posted by The Goron Moron

Whoever wrote this has absolutely no life.

Thats a really stupid thing to say.

Originally Posted by erinys

I always thought that one finger plus one more finger is two fingers and not three or twelve or none because that's what humans decided it would be.

all it means is when you have an object and put another object with it, you have two objects. That's really all the proof of 1 + 1 = 2 that you need.

we decided that one would be called one and two would be called two. It just so happens that as an inherent fact of nature, putting two ones together results in two.

Nope, that's not a proof.

Math doesn't work if every part of it is just arbitrary, so the point of such exercises is to prove things that have just been assumed without proof.

It is entirely possible that 1+1 could not equal 2 all the time, say, or that it wouldn't equal 2 at all, in which case we'd have a bit of a problem.

Math doesn't work if every part of it is just arbitrary, so the point of such exercises is to prove things that have just been assumed without proof.

Well note that in a foundational approach, like the OP's proof, the non-arbitrary part of math is a small set of logical axioms, not a set of claims about natural analogs to abstract ideas. Nothing at all in that 25 thousand step proof claims to describe a physical phenomenon; it simply builds entirely abstract definitions of "2", '4", and "+" and notes a consequence of those definitions. Human empiricism is then required to see an analog between a these abstractions and a process involving physical objects, and scientific determinism is required to "promise" that abstraction and physical occurrence will continue to be analogous... but both are well outside the domain of mathematics.

In a more "traditional" approach to the development of math, where the relationships between the physical and the abstract are used as axioms, erinys is essentially constructing arithmetic properly; observing a deterministic natural process and labeling its parts for the sake of convenience.

Originally Posted by John

Nope, that's not a proof.

Math doesn't work if every part of it is just arbitrary, so the point of such exercises is to prove things that have just been assumed without proof.

It is entirely possible that 1+1 could not equal 2 all the time, say, or that it wouldn't equal 2 at all, in which case we'd have a bit of a problem.

one plus one always equals two, all the time, because that's the way we have defined "one," "plus," "equals," and "two." It's not possible at all for one plus one to equal anything other than two, not without changing the definition of one or more of the parts.

to make proofs as complex as the one in the opening post seems superfluous and unneccesary.

If it took them that many steps to prove 2+2=4, I wonder how many it would take to prove that all of those steps really add up to 25,000....

Originally Posted by erinys

one plus one always equals two, all the time, because that's the way we have defined "one," "plus," "equals," and "two." It's not possible at all for one plus one to equal anything other than two, not without changing the definition of one or more of the parts.

to make proofs as complex as the one in the opening post seems superfluous and unneccesary.

Oh, sure, we could just say "One plus one equals two", but it wouldn't mean anything.

Proofs are required to, well, prove and define such things. Otherwise 2+2=4 would mean as much as "quiznor flurb quiznor fleeg canyig".