*infinite*.

For some reason, it really bugs me when people take a perfectly good word, a word like "infinite", and then apply it incorrectly.

You see, a desktop computer is finite.

Suppose your desktop computer is a Commodore VIC-20, with a whopping 3.5 kilobytes of memory. Then there are only 256

^{3,583}different states that your desktop computer can be in.

Sure, that's a lot of states, but it's certainly not infinite. You could, at least in

*principle*, enumerate them all. And you'd find that there are exactly 256

^{3,583}of them. That's the very definition of finite.

### Finite software runs on finite computers

Lets take a closer look at those finite states on the VIC-20. We know that the computer is finite, but maybe there is some magical technique in which we could write a computer program to have an infinite amount of state?

Unfortunately, no, we cannot. The pigeon hole principle forbids it.

### Fast forward to the Modern Era

Oh? Your computer has more memory than a VIC-20? 4 Gigabytes perhaps?

Well that's still just 256

^{4,294,967,296}states. It's still not infinite.

Oh, you have a 3TB hard drive as well?

Okay, so now you have access to an additional 256

^{3,298,534,883,328}different states.

That's a lot of storage. These numbers are large, but they're all still

*finite*.

The problem is that infinity is just so mindbogglingly larger than any number you could possibly store on your hard drive

*.*

You'd need a technology shift to be able to store infinite state.

### Bandwidth, over time, is Infinite

So hopefully I've managed to convince you that your computer, and by extension, the software running on it, is finite. Regardless of what that hardware is.But now consider, the curious case of your internet connection.

If you're like me, you have a bandwidth cap of 4GB per month. Then it is true, that for any particular month, your bandwidth is finite.

But consider your 4GB bandwidth extending over time.

I can send 8GB in 2 months. Or 40GB in 10 months. Or 400GB in 100 months.

Here's the curious thing, if we assume that time is infinite (a big assumption, granted), then for any amount of state, we can calculate how many months it would take to send that state on your internet connection by dividing by 256

^{4,294,967,296}.

Let me repeat that, given

**any**amount of state, we could send that state in a finite amount of time, over your internet connection.

And

**that**is what is meant by "Bandwidth, over time, is infinite".

This is actually a special case of the halting problem.

ReplyDeleteGiven any arbitrarily large, but *finite*, amount of state, and a fixed rate of transmission (4GB/month, whatever), you can send that much state in a *finite* amount of time.

ReplyDeleteYou can't send an infinite amount of state in a finite amount of time.

Bandwidth over time is infinite because time is infinite.

-seb