If you insist on verifying the existence of boundaries by repeatedly banging your head on them, you will end up bruised. That's just hearsay, of course. It's most certainly unrelated to the fact that I haven't found time to write here -- or 'most anywhere that's not a report to be handed in for marks -- in the last two months or so. Most certainly unrelated.

Having said that, I'm sure it won't seem out of the way to ramble a little about how running out of time relates to number systems. I've been hooking a few ideas together and while none of this is rigourous or even necessarily true, I do think it's interesting. The thing about time is that it has to be continuous, kind of like the real numbers. If we allow it to be discrete -- like the integers, say -- we end up with Zeno's paradox:

Suppose Achilles is chasing a tortoise. In the first moment of his chase, we can say that he covers half the distance to the tortoise. In the next moment, he covers half the remaining distance. In the third moment, half of what is left then. Achilles always needs to cover half of the distance left before he can reach the tortoise, but he can continue like this indefinitely without actually catching up. So, says Zeno, it is impossible for Achilles to catch up with the tortoise.

The flaw in this argument is the assumption that time is discretised. It's modelled using integers: moment 1, moment 2, moment 3. We can get a better description of time by using real numbers: between time 3 and and time 4 is time 3.5. Between 3.5 and 4 is 3.75. Between 3.75 and 4 -- well, I could go on forever, which is how Achilles manages to catch the tortoise. (What he does with it next is still up for debate.) However, despite time's going on forever, I still can't manage to get everything I want done.

I can sort of explain that by looking at a mathematical kind of density. Suppose

*S* is a set of numbers. If I can pick any two real numbers and find a third number that's between them and in the set

*S*, then I can say that '

*S* is dense in the reals'. The rational numbers (numbers that can be written as fractions) are dense in the reals, for instance, but the integers are not. If I pick 1/2 and 1/4, I can't find an integer between them. I

*can* find a rational number between them: 1/3 is perhaps the most obvious. (If you think about it, being dense in the reals actually implies that

*S* has an infinite number of members between any two real numbers.) It seems to me that my perception, or perhaps my experience, is not dense in time. There may, in some sense, be an infinite amount of time between now and tomorrow morning, but I'm certainly not going to get an infinite amount of stuff done in it! In the same way, there are infinitely many (real) numbers between 0 and 100, but only 101 integers (or 99 on the open interval).

I guess I could link in things like response time here too or the fact that movies, which are obviously finite, give the impression of being continuous simply by being more densely packed than our perception. Or I could ditch the biology and go play

Mouse Guard with the rest of the family. Mmm.