How Far to Trust a Long Run
Last timeWhen Small Differences Grow
A walk forward is not the rule, and the rule is not the world. Two separate audits are needed, one on the arithmetic and one on the modelling, each with a test you can run first.
This course has been making claims about the infinite future from finite work. The
last lesson is about the two places such claims break, and the tests that catch them.
The first is the arithmetic. Every run in this course advanced a rule in steps,
assuming the rate held constant across each step, and it never does.
The second is the rule. Every rule in this course was a simplification of something,
and some of the conclusions drawn from them do not survive putting back what was
left out.
Behaviour that is in the arithmetic
Here is the failure in its purest form. Take a rule that should go round in a perfect
circle forever, like the spring, and advance it with the most obvious possible step.
The quantity that ought to stay fixed does not. It grows, and it grows by a factor
that can be written down exactly.
The growth is proportional to the square of the step, so halving the step quarters
the error per step. That sounds like a solution, and partly it is. But the run also
gets twice as long in steps, so the total drift over a fixed amount of simulated
time only falls by half, and the answer to a long enough run is still wrong.
The lesson stops here
4 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents