Another connection between the previous discussion and the MU-
puzzle is the two modes of operation which led to insight about the nature
of the MU-puzzle: the Mechanical mode, and the Intelligent mode. In the
former, you are embedded within some fixed framework; in the latter, you
can always step back and gain an overview of things. Having an overview is
tantamount to choosing a representation within which to work; and work-
ing within the rules of the system is tantamount to trying the technique of
problem reduction within that selected framework. Hardy's comment on
Ramanujan's style-particularly his willingness to modify his own
hypotheses-illustrates this interplay between the M-mode and the I-mode
in creative thought.
The Sphex wasp operates excellently in the M-mode, but it has abso-
lutely no ability to choose its framework or even to alter its M-mode in the
slightest. It has no ability to notice when the same thing occurs over and
over and over again in its system, for to notice such a thing would be to
jump out of the system, even if only ever so slightly. It simply does not
notice the sameness of the repetitions. This idea (of not noticing the
identity of certain repetitive event~) is interesting when we apply it to
ourselves. Are there highly repetitious situations which occur in our lives
time and time again, and which we handle in the identical stupid way each
time, because we don't have enough of an overview to perceive their
sameness? This leads back to that recurrent issue, "What is sameness?" It
will soon come up as an AI theme, when we discuss pattern recognition.
Applying AI to Mathematics
Mathematics is in some ways an extremely interesting domain to study
from the AI point of view. Every mathematician has the sense that there is a
kind of metric between ideas in mathematics-that all of mathematics is a
network of results between which there are enormously many connections.
In that network, some ideas are very closely linked; others require more
elaborate pathways to be joined. Sometimes two theorems in mathematics
are close because one can be proven easily, given the other. Other times two
ideas are close because they are analogous, or even isomorphic. These are
two different senses of the word "dose" in the domain of mathematics.
There are probably a number of others. Whether there is an objectivity or a
universality to our sense of mathematical closeness, or whether it is largely
an accident of historical development is hard to say. Some theorems of
different branches of mathematics appear to us hard to link, and we. might
say that they are unrelated-but something might turn up later which
forces us to change our minds. If we could instill our highly developed
sense of mathematical closeness-a "mathematician's mental metric", so to
speak-into a program, we could perhaps produce a primitive "artificial
mathematician". But that depends on being able to convey a sense of
sirAlplicity or "naturalness" as well, which is another major stumbling block.
These issues have been confronted in a number of AI projects. Ther.e(^614) Artificial Intelligence: Retrospects