# 6.0 The Structure-Mapping Theory of Analogy

Induction, as we have analyzed it, requires a store of patterns on which to operate. We have
not said how these patterns are to be obtained. Any general global optimization algorithm could
be applied to the problem of recognizing patterns in an environment. But pattern recognition is a
difficult problem, and a mind needs rapid, reasonably accurate solutions. Not just any algorithm
will do.

One might propose to determine, by induction, an effective pattern recognition algorithm. But
although this might be possible, it would be very slow. Such a process probably occurred during
the evolution of intelligent species, but within the lifespan of one organism there is simply not
time.

In Chapter 9, I will propose that intelligent entities solve pattern recognition problems with a
"perceptual hierarchy" that applies the multilevel philosophy of global optimization sketched in
Chapter 2. Among other things, this perceptual hierarchy makes continual use of two processes:
analogy and deduction. And deduction, it will be argued in Chapter 8, is also essentially
dependent on analogy. Hence analogical reasoning is an essential part of the picture of
intelligence drawn at the end of the previous chapter.

WHAT IS ANALOGY
What I mean by analogy is, roughly, reasoning of the form "A is similar to B in respect X,
therefore A is also similar to B in respect Y." As with induction, it is difficult to say exactly why
analogy works as well as it does. But there is no disputing its effectiveness. Bronowski (1956)
has driven the point home so forcefully that an extended quote seems appropriate:
Man has only one means to discovery, and that is to find likeness between things. To
him, two trees are like two shouts and like two parents, and on this likeness he has built all
mathematics. A lizard is like a bat and like a man, and on such likenesses he has built the theory
of evolution and all biology. A gas behaves like a jostle of billiard balls,and on this and kindred
likenesses rests much of our atomic picture of matter.

In looking for intelligibility in the world, we look for unity; and we find this (in the arts
as well as in science) in its unexpected likenesses. This indeed is man’s creative gift, to find or
make a likeness where none was seen before — a likeness between mass and energy, a link
between time and space, an echo of all our fears in the passion of Othello.

So, when we say that we can explain a process, we mean that we have mapped it in the likeness of another process which we know to work. We say that a metal crystal stretchesbecause its layers slide over one another like cards in a pack, and then that some polyester yarns stretch and harden like a metal crystal. That is, we take from the world round us a few models of structure and process (the particle, the wave, and so on), and when we research into nature, we try to fit her with these models.

Even more intriguingly, Bronowski goes on to relate analogy with structure:
Yet one powerful procedure in research, we know, is to break down complex events into
simpler parts. Are we not looking for the understanding of nature in these? When we probe
below the surface of things, are we not trying, step by step, to reach her ultimate and
fundamental constituents?

We do indeed find it helpful to work piecemeal. We take a sequence of events or an
assembly to pieces: we look for the steps in a chemical reaction, we carve up the study of an
animal into organs and cells and smaller units within a cell. This is our atomic approach, which
tries always to see in the variety of nature different assemblies from a few basic units. Our search
is for simplicity, in that the distinct units shall be few, and all units of one kind identical.

And what distinguishes one assembly of these units from another? the elephant from the
giraffe, or the right-handed molecule of sugar from the left-handed? The difference is in the
organization of the units into the whole; the difference is in the structure. And the likenesses for
which we look are also likenesses of structure.

This is the true purpose of the analytic method in science: to shift our gaze from the thing
or event to its structure. We understand a process, we explain it, when we lay bare in it a
structure which is like one we have met elsewhere.

What Bronowski observed in the history and psychology of science, Gentner and Gentner
(1983) have phrased in a more precise and general way. They speak of the "Generative Analogy
Hypothesis" — the hypothesis that analogies are used in generating inferences. And in order to
test this hypothesis, they setforth a specific theoretical framework for analogical processing,
called "structure-mapping." According to this framework, analogical reasoning is concerned with
deriving statements about a target domain T from statements about a base domain B. Each
domain is understood to consist of a number of "nodes" and a collection of relations between
these nodes. Essentially, a node may be any sort of entity — an object, a color, etc. A structure-
mapping begins with a relation which takes certain base nodes into certain target nodes: if the
source nodes are (b1,…,bn) and the target nodes are (t1,…,tn), it is a map M(bi)=tj, where i ranges
over some subset of (1,…,n). Analogy occurs when it is assumed that a relation which holds
between bi and bk also holds between M(bi) and M(bk).

The theory of structure-mapping analogy states that reasoning of this form is both common
and useful. This hypothesis has been verified empirically — e.g. by studying the way people
reason about electricity by analogy to water and other familiar "base" domains. Furthermore, the
evidence indicates that, as Gentner and Gentner put it, relations "are more likely to be imported
into the target if they belong to a system of coherent, mutually constraining relationships, the others of which map into the target." If a relation is part of a larger pattern of relationships which have led to useful analogies, people estimate that it is likely to lead to a useful analogy.

The structure-mapping theory of analogy — sketched by Bronowski and many others and
formalized by Gentner and Gentner — clearly captures the essence of analogical reasoning. But it
is not sufficiently explanatory — it does not tell us, except in a very sketchy way, why certain
relations are better candidates for analogy than others. One may approach this difficult problem
by augmenting the structure-mapping concept with a more finely-grained pattern-theoretic
approach.

INDUCTION, DEDUCTION, ANALOGY
Peirce proclaimed the tendency to take habits to be the "one law of mind", and he divided this
law into three parts: deduction, induction, and abduction or analogy. The approach of computer
science and mathematical logic, on the other hand, is to take deductive reasoning as primary, and
then analyze induction and analogy as deviations from deduction. The subtext is that deduction,
being infallible, is the best of the three. The present approach is closer to Peirce than it is to the
standard contemporary point of view. When speaking in terms of pattern, induction and analogy
are more elementary than deduction. And I will argue that deduction cannot be understood
except in the context of a comprehensive understanding of induction and analogy.

STRUCTURAL SIMILARITY
As in Chapter 3, define the distance between two sequences f and g as d#(f,g)=%(P(f)-
P(g))U(P(g)-P(f)%#. And define the approximation to d#(f,g) with respect to a given set of
functions S as
dS(f,g)=%[(S%P(f))-(S%P(g))]U[(S%P(g))-(S%P(f))]%#. This definition is the key to our
analysis of analogy.
A metric is conventionally defined as a function d which satisfies the following axioms:
1) d(f,g) % d(g,h) + d(f,h)
2) d(f,g) = d(g,f)
3) d(f,g) % 0
4) d(f,g)=0 if and only if f=g.
Note that d# is not a metric, because it would be possible for P(f) to equal P(g) even if f and g
were nonidentical. And it would not be wise to consider equivalence classes such that f and g are
in the same class if and only if d#(f,g)=0, because even if d#(f,g)=0, there might exist some h
such that d#(Em(f,h),Em(g,h)) is nonzero. That is, just because d#(f,g)=0, f and g are not for all
practical purposes equivalent. And the same argument holds for dS — for dS(f,g)=0 does not in
general imply dS(Em(f,h),Em(g,h)), and hence there is little sense in identifying f and g. A
function d which satisfies the first three axioms given above might be called a pseudometric; that
is how d# and dS should be considered.
TRANSITIVE REASONING
To understand the significance of this pseudometric, let us pause to consider a "toy version" of
analogy that might be called transitive reasoning. Namely, if we reason that "f is similar to g, and
g is similar to h, so f is similar to h," then we are reasoning that "d(f,g) is small, and d(g,h) is
small, so d(f,h) is small." Obviously, the accuracy of this reasoning is circumstance-dependent.
Speaking intuitively, in the following situation it works very well:
gf
h
But, of course, one may also concoct an opposite case:
f      gh
Since our measure of distance, d#, satisfies the triangle inequality, it is always the case that
d#(f,h) % d#(g,h) + d#(f,g). This puts a theoretical bound on thepossible error of the associated
form of transitive reasoning. In actual circumstances where transitive reasoning is utilized, some
approximation to d# will usually be assumed, and thus the relevant observation is that dS(f,h) %
dS(g,h) + dS(f,g) for any S. The fact that dS(f,h) is small, however, may say as much about S as
about the relation between f and h. The triangle inequality is merely the final phase of transitive
reasoning; equally essential is to the process is the pattern recognition involved in approximating
d#.

Kaynak: A New Mathematical Model of Mind
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