Disorder/Order: Discovery or Invention? - Understanding Understanding: Essays on Cybernetics and Cognition - page 279

arbitrary arrangement, A , and its representation on the tape of a Turing
Machine by using a certain alphabet (language). What Turing showed is that
there exists another tape expression, called the “description” of A , which,
when used as the initial tape expression will allow the machine to compute
from it the arrangement A . Let me now draw your attention to three mea-
sures (numbers). One is the length L ( A ) (that is, the number of squares) of
the tape that is taken up by the arrangement A ; the second is the length
L ( D ) of A 's description (the initial tape expression); and the third figure is
N , the number of cycles the machine has to go through to compute the
arrangement A from its description D .
Now we can collect some fruits from our intellectual investment into the
notions of machines, demons, etc. I will describe just four:
(i) Order
If the initial tape expression, the description, is short, and what is to be com-
puted, the arrangement, is very long ( L ( D ) < L ( A )), then it is quite obvious
that the arrangement possesses lots of order: a few rules will generate A .
Take A to be 0,1,2,3,4,5,6,7,...,999,999, 1,000,000. A suitable descrip-
tion of this arrangement may be: Each follower equals its precursor + 1 .
(ii) Disorder
If the length of the description approximates the length of the arrangement,
it is clear that we do not understand this arrangement, for the description
just parrots the arrangement. Take A to be:
8, 5, 4, 9, 1, 7, 6, 3, 2, 0.
I challenge the mathematicians present, or any puzzle wizard, to come up
with a rule other than: write 8 , 5 , 4 ,...that generates this arrangement.
(iii) Complexity
I propose to use N , the number of cycles for computing an arrangement, as
a measure for the complexity of this arrangement. In other words, I suggest
that we associate with the complexity of an arrangement the time it takes
the machine to compute it. For instance, during this meeting a juxtaposi-
tion molecule/man was made with the suggestion—so I understood—to
learn about the properties of human beings from the known properties of
molecules. In computational jargon such computations are usually referred
to as computations ab ovo or, as in our case ab molecula . From this point
of view it may be not too difficult to see that N , the number of computa-
tional steps, will be so large (e.g., the age of the universe being too short to
accommodate N ) that N becomes “trans-computational.” That means, we
can just forget about the whole thing, for we shall never see the end of it!
(iv) Language
The choicest of the four fruits I have left to be the last for you to taste, for
it is the most crucial one in my narrative. It is the observation that all the
three quantities mentioned before: the length of an arrangement, the length
of its description, and the length of computing this arrangement, are dras-