The Act of Creation (93 page)

p. 644
. 'Wave-length' is of course used
metaphorically for much more complex processes, including both structural
and functional characteristics of nervous tissue. 'Excitation-clang'
or 'frequency-modulation signals' or Hebb's 'phase-sequences in cell
assemblies' would be closer approximations.

 

 

To
p. 644
. For a summary see Woodworth (1939)
pp. 360 seq., Osgood (1960) pp. 722 et seq. Osgood (p. 722), discussing
the relative frequencies and reaction times of verbal responses in
association tests, speaks of a 'hierarchical structure of associations;
but he uses the word 'hierarchy' to refer to a linear scale of gradations.

 

 

 

 

 

XVIII

 

 

HABIT AND ORIGINALITY

 

 

Problem-solving is bridging a gap between the initial situation and the
target. 'Target' must be understood in the widest sense -- it may be an
apple hanging high up on a tree, or a formula for squaring the circle,
or inventing a honey-spoon which does not drip, or fitting a fact into
a theory, or making the theory fit the facts.

 

 

Strictly speaking, of course, problems are created by ourselves;
when I am not hungry, the apple ceases to be a target and there is
no gap. Vice versa, the insatiable curiosity of Kepler made him see
a problem where nobody saw one before -- why the planets move as they
do. But the motivational aspect of problem-solving, and the exploratory
drive in general, have already been discussed.

 

 

There is also a different way of creating problems -- for others to
solve. Economy in art consists in implying its message in the gaps
between the words, as it were. Words, we saw, are mere stepping-stones
for thoughts; the meaning must be interpolated; by making the gaps just
wide enough, the artist compels his audience to exert its imagination,
and to re-create, to some extent, the experience behind the message. But
this aspect has also been discussed already, and no longer concerns us.

 

 

 

Bridging the Gap

 

 

The process of bridging the gap between the perceived problem and its
solution is described in an oft-quoted passage by Karl Mach:

 

The subject who wishes for a tree to be laid across a stream to enable
him to cross it, imagines in fact the problem as already solved. In
reflecting that the tree must have previously been transported to
the river, and previously to that it must have been felled, etc., he
proceeds from the target-situation to the given situation, along a road
which he will re-trace in the reverse direction, through a reversed
sequence of operations, when it comes to actually constructing the
bridge. [1]

 

This quotation has a long ancestry. It goes back -- as Polya (1938)
has shown in a remarkable paper -- to Pappus' classic distinction
between the analytical method, which treats the unknown solution of
a geometrical problem as if it were already known, then inquires from
what antecedent it has been derived, and so on backward from antecedent
to antecedent, until one arrives at a fact or principle already known;
and the synthetic method which, starting from the point reached last in
the analysis, reverses the process.

 

 

However, the traditional distinction between analytical and synthetical
method is full of pitfalls, and, though 'thinking backwards' from the
unknown to the given plays an important role in mathematical reasoning,
this is by no means always the case in problem-solving; moreover
'forward' and 'backward' are often quite arbitrarily used by taking
topological metaphors too literally. If I aim my rifle at the target
and then pull the trigger, it would be ridiculous to say that I was
'thinking backward' from target to trigger; I was merely demonstrating
the trivial fact that in all goal-directed activities one always has to
'keep one's eyes on the target' -- which can be taken either literally
or metaphorically. The chess player's aim is to capture the opponent's
king, either by directly attacking his defences, or by gaining such an
advantage in material that the king will be at his mercy. But the player
rarely reasons backward from an anticipated mate position -- this happens
only at dramatic combinative stages; as a rule he looks around the board
to see 'what's in the position', explores the possibilities, and then
considers what strategical or tactical advantages he can derive from it.

 

 

If I wanted to find out whether I am a descendant of Spinoza (as a
crackpot uncle of mine once asserted), I could follow one of two methods,
or a combination of both. I could trace Spinoza's descendants as they
branch out downwards, or I could trace my own ancestors branching out
and up; or start at both ends and see whether the branches meet. The
example is a paraphrase from the
Logique de Port Royal
, whose
authors seem to equate the upward process with analysis, the downward
one with synthesis. Spinoza, incidentally, had no descendants.

 

 

Returning to Mach's example, the following would perhaps be a more
realistic way of approaching the problem. To get to my target I must
cross this stream. This conclusion is arrived at by keeping my eyes
both on the target and on my own position -- by glancing in alternation
forward and back as it were. Since I must cross the stream, let's look
for a bridge. There is no bridge. Is there perhaps a boat somewhere? No,
there is not. Can I wade across? Yes -- no, it's too cold. I have found
three analogies with past methods of solving a problem, in my repertory of
simple routines. If I wish to be pedantic I can say that the rules of the
river-crossing skill allowed me three choices, three different stratagems,
each of which I tried out implicitly, as a hypothesis. But surveying the
lie of the land I find all three obstructed. What can be done? I must
search for some other routine which fits the situation. Mach's suggestion
to fell a tree is not a very practical one -- I saw only once a bridge
across a swollen gully built that way, by natives in Uzbekistan -- but
I have no axe and we are not in Uzbekistan. So, roll up your trousers,
and let's hope the water is not too icy.

 

 

This is the hum-drum routine of planning and problem-solving in every-day
life. It means, firstly, searching for a matrix, a skill which will
'bridge the gap'. The matrix is found by way of analogy (or 'association
by similarity'), that is to say, by recognizing that the situation is,
for my present intents and purposes, the same as some past situations. I
then try to apply the same skill which helped in those past situations,
to the present lie of the land. In the above example I have tried three
successive stratagems -- I have made three hypotheses -- and after two
have failed completely, I reverted to the third, which offered the only
solution, though a far from perfect one. In other words, I have settled
for an approximation. Most problems in practical life and in the history
of science admit of no better solution.

 

 

A point to be noted is that even in this trivial example, the solution
does not proceed in a single line from target to starting point, or vice
versa, but by a branching out of hypotheses -- of possible strategies --
from one end, or both ends, until one or several branches meet, as in
the Spinoza example. Furthermore, in a real geneaological search, the
expert would eliminate unlikely branches and concentrate on those which
for geographical or other reasons seem more promising. We have here, on a
miniature scale as it were, that groping in a vaguely sensed direction,
towards the 'good combination', the 'hooking of the proper
atoms' (Poincaré), which I have discussed in Book One,
VIII
. However, in these trivial examples the
groping and searching is done on the conscious or fringe-conscious
level, and what we are looking for to bridge the gap is merely some
routine trick in our repertory; a practical skill which will fit the
particular lie of the land. In other words, the fanning out of hypotheses,
the trial-and-check procedures in simple thinking routines, reflect the
flexibility of the skill, which can operate through several sub-skills or
equipotential lines of action, according to feedback from the environment.

 

 

These sub-skills of symbolic thought have been discussed in various
contexts. They range from the implicit codes of grammar, syntax, and
commonsense logic, through the operational rules of extrapolation,
interpolation, transposition, schematization (exaggeration and
simplification), and so forth, to the special rules of such special
games as vector-analysis or biochemistry. But even these very special
and complex skills can be practised by sheer routine; and vice versa
some of the most original discoveries arose out of relatively simple
problems. Complexity of thought is no measure of originality.

 

 

 

Searching for a Code

 

 

Polya defines a routine problem as one 'which can be solved either by
substituting special data into a formerly solved general problem, or by
following step by step, without any trace of originality, some well-worn,
conspicuous example'. [2] He contrasts these routines with the 'rules
of discovery': 'The first rule of discovery is to have brains and good
luck. The second rule of discovery is to sit tight and wait till you
get a bright idea.' [3] And he defines a 'bright idea' as 'a sudden
leap of the imagination, a flash of genius'. [4]

 

 

However, most practical and theoretical problems are solved at some
level between these two extremes. Polya's definition of routine is too
narrow and rigid; it does not take into account the great flexibility,
for instance, of sensory-motor skills such as rock-climbing, or
glass-blowing, or playing an instrument in an uninspired but technically
accomplished manner. In symbolic thinking we find equally flexible
skills which are nevertheless routine: solicitors dictating a document
or brief; interpreters at public congresses dictating ad hoc into the
multi-lingual earphone-circuits; politicians on whistle-stop tours reeling
off variations on well-worn themes. These are routine performances in
Polya's sense of 'substituting special data into general equations';
but the equation, the code, leaves in these cases many more degrees of
freedom than does a mathematical formula, and the act of 'substituting
special data' has varying degrees of trickiness. Even to a skilled
and expert translator, for instance, it often happens that he has no
ready-made idiom or turn of phrase in his repertory to substitute for
the original. He must improvise some approximation, perhaps a metaphor,
to cover the meaning -- which is a much higher skill, involving a degree
of originality.

 

 

Turning to more difficult problems, of the type which Duncker and
Maier put to their subjects, we find routine solutions combined with
intimations of originality, and we shall recognize an increasing number
of those factors, in embryonic shape as it were, which we saw at work
in the creative process.

 

 

The degree of originality which a subject will display depends, ceteris
paribus, on the nature of the challenge -- that is, the novelty and
unexpectedness of the situation. Familiar situations are dealt with by
habitual methods; they can be recognized, at a glance, as analogous in
some essential respect to past experiences which provide a ready-made
rule to cope with them. The more new features a task contains, the more
difficult it will be to find the relevant analogy, and thereby the
appropriate code to apply to it. We have seen (Book One,
VIII
,
XVII
) that one of
the basic mechanisms of the Eureka process is the discovery of a hidden
analogy; but 'hiddenness' is again a matter of degrees. How hidden is a
hidden analogy, and where is it hidden? And what does the word 'search'
mean in this context? In the terms of the present theory it means a
process of scanning, of bringing successive, perceptual or conceptual
analyser-codes to bear on the problem; to try out whether the problem will
match this type of filter or that, as the oculist tries out a series of
lenses in the frame before the client's eyes. Yet the word 'search', so
often used in the context of problem-solving, is apt to create confusion
because it implies that I know beforehand what I am searching for, whereas
in fact I do not. If I search for a lost collar-stud, I put a kind of
filter into my 'optical frame' which lets only collar-studs and similar
shapes pass, and rejects everything else -- and then go looking through
my drawers. But most tasks in problem-solving necessitate applying the
reverse procedure: the subject looks for a clue, the nature of which
he does not know, except that it should be a 'clue' (
Ansatzpunkt
,
point d'appui
), a link to a type of problem familiar to him.
Instead of looking through a given filter-frame for an object which
matches the filter, he must try out one frame after another to look at
the object before 'his nose, until he finds the frame into which it fits,
i.e. until the problem presents some familiar aspect -- which is then
perceived as an analogy with past experience and allows him to come to
grips with it.

 

 

This search for the appropriate matrix, or rule of the game to tackle
the process, is never quite random; the various types of guidance at the
fumbling, groping, trying stages have been discussed before. Among the
criteria which distinguish originality from routine are the
level
of consciousness
on which the search is conducted, the
type
of guidance
on which the subject relies, and the
nature of the
obstacle
which he has to overcome.

 

 

 

Degrees of Originality