Sunday, 9 March 2014

Infinity and Beyond (2001)

Watching the Horizon documentary "Infinity and Beyond" had renewed my interests in the concept of mathematical infinity. And how the concept  is fraught with paradoxes.

One example of such paradox can be illustrated as follows (this is not mentioned in this documentary):

Let's start with the digits 1 and 2. Let halve these two digits to get 1/2. And then we halve that, and so forth to ad nauseum, so we get the series,

1, 1/2, 1/4, 1/8, 1/16 ...1/∞

Mathematician will tell you that this series has infinite members or fractions, and yet we know that it contains within the finite boundaries of 1 and 2. Finite, and yet infinite. This is in fact the well known Zeno's Paradox that originated in the ancient Greece.

Infinity symbol

Other seemingly mind-boggling concept that does appear in this documentary is the sets of equations:

∞ + 1 = ∞
∞ + ∞ = ∞
∞ - 1 = ∞
∞ - ∞ = ???

We assume that the ∞ in the equation belongs to a positive digit number.

The first three equations seems to make sense 'intuitively', and at the same time seems contradictory because how can z + 1 = z ? This equations only 'makes sense' if z = ∞.

Similar 'logic' applies to the next 2 equations.

What about the last equation (∞ - ∞ = ???) ? According to the mathematician in the documentary, it could be 1, it could 3,  it could be anything ! Nobody knows. Well, this is no good.

Here's what I think is wrong with the picture.

when you have the equation 1 + z = ?

It doesn't matter how big z is as long as it isn't ∞, I can give you a precise answer. If z = ∞, then I can' give you an answer. Why? Because ∞ isn't a number. While we think of it as an infinitely large number, but it is not! It's a concept, an idea. So it's nonsensical to apply an arithmetic operation to a concept.

The operation 1 + ∞ makes as much sense as 1 + democracy, or 1 + chaos.

'∞' isn't a number, so you can't apply mathematical operation to it.

Too simple? Or maybe the documentary had made a poopoo with that sets of equations, or that I've missed something.