these intuitions are valuable and may perhaps help others a little to come to
clearer formulations of their own images of what makes minds run. I could
not hope for more than that my own mind's blurry images of minds and
images should catalyze the formation of sharper images of minds and
images in other minds.A Self-Modifying GameA first variation, then, concerns games in which on your turn, you may
modify the rules. Think of chess. Clearly the rules stay the same, just the
board position changes on each move. But let's invent a variation in which,
on your turn, you can either make a move or change the rules. But how? At
liberty? Can you turn it into checkers? Clearly such anarchy would be
pointless. There must be some constraints. For instance, one version might
allow you to redefine the knight's move. Instead of being l-and-then-2, it
could be m-and-then-n where m and n are arbitrary natural numbers; and
on your turn you could change either m or n by plus or minus I. So it
could go from 1-2 to 1-3 to 0-3 to 0-4 to 0-5 to 1-5 to 2-5 ... Then there
could be rules about redefining the bishop's moves, and the other pieces'
moves as well. There could be rules about adding new squares, or deleting
old squares ...
Now we have two layers of rules: those which tell how to move pieces,
and those which tell how to change the rules. So we have rules and
metarules. The next step is obvious: introduce metametarules by which we
can change the metarules. It is not so obvious how to do this. The reason it
is easy to formulate rules for moving pieces is that pieces move in a
formalized space: the checkerboard. If you can devise a simple formal
notation for expressing rules and metarules, then to manipulate them will
be like manipulating strings formally, or even like manipulating chess
pieces. To carry things to their logical extreme, you could even express
rules and metarules as positions on auxiliary chess boards. Then an arbi-
trary chess position could be read as a game, or as a set of rules, or as a set
of metarules, etc., depending on which interpretation you place on it. Of
course, both players would have to agree on conventions for interpreting
the notation.
Now we can have any number of adjacent chess boards: one for the
game, one for rules, one for metarules, one for metametarules, and so on,
as far as you care to carry it. On your turn, you may make a move on any
one of the chess boards except the top-level one, using the rules which
apply (they come from the next chess board up in the hierarchy).
Undoubtedly both players would get quite disoriented by the fact that
almost anything-though not everything!-can change. By definition, the
top-level chess board can't be changed, because you don't have rules telling
how to change it. It is inviolate. There is more that is inviolate: the conven-
tions by which the different boards are interpreted, the agreement to take
turns, the agreement that each person may change one chess board each
turn-and you will find more if you examine the idea carefully.
Strange Loops, Or Tangled Hierarchies 687