1x1=2

[Armoured]

You can't prove anything without a logical system, and a logical system needs axioms.

That's an axiom.

 

[FrumiousBandersnatch]

No they aren't. They logically follow from axioms.

It was Russell and Whitehead who sorted out the details in Principia Mathematica, using symbolic logic to explicate the logical foundations of mathematics, building up to 1+1 =2, and so-on. It only took a few hundred pages...

 

[The Cadet]

It was Russell and Whitehead who sorted out the details in Principia Mathematica, using symbolic logic to explicate the logical foundations of mathematics, building up to 1+1 =2, and so-on. It only took a few hundred pages...

They were then proven completely wrong.

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

Also this thread physically pains me.

 

[FrumiousBandersnatch]

They were then proven completely wrong.

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

I wouldn't say completely wrong. Their logical foundations were OK, but their objective of demonstrating by that logic that all mathematical truths could be proved was frustrated; i.e. the logic did build the foundations of maths, but couldn't prove what they were after (that it was complete & consistent).

Also this thread physically pains me.

Have an early night.

 

[quatona]

After high school, he attended Pratt Institute in Brooklyn, studying chemical engineering, until he got into an argument with a professor about what one times one equals. "How can it equal one?" he said. "If one times one equals one that means that two is of no value because one times itself has no effect. One times one equals two because the square root of four is two, so what's the square root of two? Should be one, but we're told it's two, and that cannot be."

I am just wondering what he thinks one times two is. Or one times three.

 

[Kylie]

I wouldn't be surprised if the very foundation of mathematics is based on falsehood.

What does it mean to 'times' something? Take an amount and duplicate it by another amount. So if you take 1 and duplicate it 1 time, you have to 2. If you take 2 and duplicate it 2 times, you have 4. Makes sense to me.

Edit: now that I look at that again, if you have 2 and duplicate it 2 times you have 6. Something doesn't seem right there...

It means that you have a certain number of groups, each contain a certain number of items.

A x B means that you have A groups, and each group has B objects in it.

So 6 x 3 means you have six groups, each group containing three objects. And if you collect six groups of three objects, you'll end up with 18 objects altogether.

And so, 1 x 1 means you have one group, and that group has one object in it.

I learnt this so early on in my education that I can't believe anyone who has completed basic primary school would be unaware of it.

 

[[serious]]

But that is not the problem with this skit.

So you think that the "redundancy" is what makes it wrong?

Let's do it again, then!

a = a (Axiom or not, it is correct)
a^2 = a^2 (Multiplying both sides of an equation with the same factor doesn't change the equality, so correct)
a^2 - a^2 = 0 (Subtracting the same value on both sides... still correct)
(a-a)(a+a) = 0 (valid transformation according the the binomic formulas... correct)
(a-a)(a+a)/(a-a) = 0/(a-a) (Dividing both sides with the same value... correct)
1(a+a) = 0 (reducing a fraction by eliminating equal factors in counter and denominator on the left side, using the fact that 0 multiplied by any fraction stays 0 on the right side of the equation... both correct.)
(a+a) = 0 (simplifying by just dropping the factor of one... correct)
1 + 1 = 0 (inserting a value... correct)
2 = 0 (computing the values... correct)
1 = 0 (not sure what was done here... division by 2? Still, correctly done.)
1 + 1 = 1 (using the last equation within a new calculation... if we were correct up to now, we can do this. So... correct.)

So, no redundancy anymore. Just one variable used.

Where is the problem?

Yeah, that's a fun one.

I like to resolve it this way:
(a-a)(a+a)/(a-a) = 0/(a-a)
0/(a-a)=1, making the next step
1(a+a) = 1
Now drop the 1 from each side and you get
(a+a) =
Divide both sides by parentheses and you get
a+a = )(
Now do the addition:
14 = )(
Wait a sec, let's back up a step,
a+a = )(
Those parentheses don't even make sense, let's move them back
4a+a1 =
Wait, what? ok, ok, I can fix this:
4a = -a1
Hmm, a positive can't equal a negative, let's square everything to get rid of it:
16a^2 == a^2 FALSE
...
dec(16a^2 == a^2 FALSE)
22 10^2 == 10^3 15LS14
Let's flip that around
ls1514 10^3 == 22 10^2
1514 10/^3 / =/= 22 10/^2 TRUE
wait, no, this doesn't even, is it treating those / as a variable? Um, divide them out of both sides?
1514 10/^3 /// =/= 22 10/^2 TRUE //
Why is that blue? let's try printing it again
1514 10^3
1514000

And there's your answer.