# Tag Archives: infinity

## Story Probability

Yesterday I was re-watching Arachnids in the UK, the fourth episode of the latest season of Doctor Who, and a somewhat goofy idea popped into my head about how to respond to the charge that sometimes stories are just ‘too improbable’ to enjoy — or to have happened at all.

That certainly is an accusation that seems to apply in many cases. In order for some story to have happened at all, certain events had to happen just so and in the right order. It’s easy to shake your head and think, “Yeah, right. As if that could actually ever happen.”

For many years I’ve had a generic response to that accusation, but yesterday I realized it can be justified mathematically!

## Sideband #54: Cantor’s Diagonal

Be warned: these next Sideband posts are about Mathematics! Worse, they’re about the Theory of Mathematics!! But consider sticking around, at least for this one. It fulfills a promise I made in the Infinity is Funny post about how Georg Cantor proved there are (at least) two kinds of infinity: countable and uncountable. It also connects with the  Smooth or Bumpy post, which considered differences between the discrete and the continuous.

This first one is pretty easy. The actual math involved is trivial, and I think it’s fascinating how the Yin/Yang of separate units versus a smooth continuum seems a fundamental aspect of reality. We can look around to see many places characterized by “bumpy” or “smooth” (including Star Trek). (The division lies at the heart of the conflict between Einstein’s Relativity and quantum physics.)

So, let’s consider Cantor.

## Replicator Redux

Earl Grey. Hot!

I’ve written about the Yin-Yang of analog versus digital, a fundamental metaphor for how reality can be smooth or bumpy. I’ve applied the idea to numbers, where we see two types of infinity — countable (discrete, digital, bumpy) and uncountable (continuous, analog, smooth). There is also how chaos mathematics says that — the moment we round off those smooth numbers into bumpy ones — our ability to use them to calculate certain things is forever lost.

I’ve also written about Star Trek replicators and transporters, as well as the monkey wrench of the hated holodeck. According to canon, all three use the same technology (which raises some contradictions for the holodeck).

Today, for Science Fiction Saturday, I want to tie it all together in another look at transporters and replicators!

## Smooth or Bumpy

Last time I wrote about analog recording and how it represents a physical chain of proportionate forces directly connecting the listener to the source of the sounds. In contrast, a digital recording is just numbers that encode the sounds in an abstract form. While it’s true that digital recordings can be more accurate, the numeric abstraction effectively disconnects listeners from the original sounds.

In the first month of this blog I wrote about analog and digital and mentioned they were mutually exclusive Yin and Yang pairs (a topic I wrote about even earlier — it was my seventh post).

Today I want to dig a little deeper into the idea of analog vs. digital!

## Infinity is Funny

You probably have some idea of what infinity means. Something that is infinite goes on forever. But it might surprise you to know that there are different kinds of infinity, and some are bigger than others!

As an example, a circle has an infinite number of points. (Yet the circumference of the circle has finite length.) Compare that to a straight line with infinite length. Both have infinitely many points but does the finite length circle have fewer? [The answer is no.]

To understand all this, we have to first talk a bit about numbers.

## Big Bang? Let there be Light?

In an earlier post, I wrote that:

The problem for any honest theist is,
“What if it isn’t true?”
The problem for any honest atheist is,
“What if it is true?”

Ultimately both represent ways of looking at the universe. There is no factual conclusion, no proof, about either one; both are matters of faith and belief.

Science can argue all it wants that the Logic and Scientific Method is superior to believing in an ineffable reality, but given all we do know and all we don’t know, in the end it is still just a worldview.