The Issue with Math

OK, got this sorted and ready to go.  Enjoy!

Mathematicians can do some amazing things, but some have been accused of living too much in the world of theory rather than fact. Here's a neat video:







Basically, it says that:

1-1+1-1+1-1+1-1+1... = ½

Here's the "proof," in a nutshell:

1-1+1-1+1-1+1... = S

1-S = 1-(1-1+1-1+1-1+1-1+1...)

1-(1-1+1-1+1-1+1-1+1...) = 1-1+1-1+1-1+1... = S

Therefore,

1-S = S

1 = 2S

S = ½

Pretty straightforward, right? Plus we know the series oscillates between 0 and 1, so a number of 1/2 makes sense (as it would be the average, as it were).

Not so fast, Mathicus Maximus!

It's a new superhero I'm working on.

The issue here lies in treating S like a natural number; their premise in finding S assumes that S indeed exists and is not simply conceptual. This is a problematic assumption, which I'll demonstrate.

So, let's do us some math wrangling. Buckle up; it's math time!

Please, Hammer, don't infinite Riemann sum them.

It's assumed in the video that one can take S and perform basic arithmetic on it like an algebraic variable. Let's apply some of the same logic and play with the equation some more.

First, let's take 2S. In the video, this is treated like 2*S, thus we can solve:
2S = 1
S = ½

OK, all fair and good. Or is it? Let's do some fandangling with 2S and see what we can see.

S = 1-1+1-1+1-1...
2S = 2(1-1+1-1+1-1...)
2S = 2-2+2-2+2-2...

OK, here's the fun part:
2-2+2-2+2-2... = (1+1)-(1+1)+(1+1)-(1+1)...
(1+1)-(1+1)+(1+1)-(1+1)... = 1+1-1-1+1+1-1-1...
1+1-1-1+1+1-1-1... = (1+1-1-1)+(1+1-1-1)+(1+1-1-1)...
(1+1-1-1)+(1+1-1-1)+(1+1-1-1)... = (1-1+1-1)+(1-1+1-1)+(1-1+1-1)...
(1-1+1-1)+(1-1+1-1)+(1-1+1-1)... = 1-1+1-1+1-1+1-1...

And, by definition:
1-1+1-1+1-1+1-1... = S

Therefore,
2S = S

If this is true (which it is, based on the definition of S and the logic used in the video), then we have:
1-S = S = 2S
1-S = 2S
1 = 3S
S = ⅓

We could get any number of possibilities for S as logic dictates that nS = S (n being 1 or an even integer, at least; haven't tried it with odd integers).

The video treats the equation 1-S = S like we can algebraically manipulate it, but we can't do that and still get real results. This is because they assume that there exists a value S and then go about using that value, but they never actually prove that S exists. If we try the same thing with other abstract concepts/numbers, like 0, we get:

4(0) = 8(0)
4(0)/0 = 8
0/0 = 8/4 = 2
0 = 2(0)

Now to substitute 0=2(0) in our original equation. We still haven't committed any math ills, here.
4(2(0)) = 8(0)
2(0) = 8(0)/4

Now let's substitute again.
2(0) = 8(2(0))/4
0/0 = 16/8 = 2
1 = 2

In this way, we have "proven" that 1 = 2.

Since this is false, one or more of our premises or calculations must be false. We know that it's the division by 0; you can't take 0/0 = 1. The premise from the video commits a similar math sin. Namely, it assumes that S+S = 2S (that S is a number which can be added like that).

However, we have shown that S+S = S, and is in keeping with how the sequence is defined. We cannot simply treat S as a number or we run into the same issue we did above with 0. The defined definition of S will not allow us to simply manipulate it like an algebraic variable. We arrived at the conclusion that 1-S = S through non algebraic means; it was part of the concept of what S is. Similarly, we arrived at the equally valid conclusion that 2S = S. There is obviously more going on here than simple addition or multiplication when we perform arithmetic with it. Therefore, we must either reject the premise that S = ½, or we must accept that S = ⅓, ¼, ⅙, or any other value we can manipulate it to give.