If, on the other hand, somebody asks me to talk about the geological foundations of Madrid, I shall make an embarrassed effort, and recall that the town stands on an undulating plateau and, by some inarticulate analogy, perhaps arrive at the conclusion that the soil consists chiefly of sand and clay. But that was an inference arrived at by the usual method of problem-solving, and not of a spontaneous association. My matrix of geological knowledge is scant; its code consists of the vague rule 'all that is relevant to the structure of the earth's crust'. This indicates the direction in semantic space of the search for an answer; but since the data of knowledge are lacking, there is little firm ground on which to move about. 'Madrid' was not a member of my geological matrix; after this embarrassing experience, it has been recruited to membership, but its ties to the matrix remain weak and tentative.
We have seen that the concept Madrid can be activated by any of the matrices to which it belongs (for instance 'Civil War'); and vice versa, that any of these matrices can be activated by it. If we assign to each matrix, metaphorically speaking, a specific 'wave-length' then the concept may be represented as an aggregate of several oscillation-circuits, each of which will receive and emit on the specific 'wave-length' of its matrix. We may call this the 'multiple attunement' of the concept to the various matrices of which it is a member.*
Now the aggregate of circuits, which is the concept, may receive on one wave-length and emit on another. 'Madrid' was evoked by the phonetic matrix 'initial M', and in its turn activated a different matrix, 'Civil War', which functions on a different wave-length. If the matrix of the incoming signal is of a complex or abstract character, the aggregate may tend to switch over to a circuit functioning on the wave-length of an emotionally more appealing matrix. Thus a concept is a member of several clubs, but it likes some clubs more than others. Its 'multiple attunements' may be represented as a line-spectrum of frequencies with a relatively stable energy-distribution. The frequencies of maximum energy -- like the dominant partials in a sound-spectrum -- would represent the concept's 'most-preferred' associative contexts. As the years go by, new lines would be added to the spectrum, while others would fade away, and the energy-distribution of associative preferences would change -- getting mellower perhaps, like an old Stradivarius, or croaking, like an un-tuned piano. The effort to 'concentrate' on an abstract problem is probably proportionate to the energy required to inhibit preferential associative contexts of high energy-potential -- i.e. 'habit strength'.
The preceding paragraphs may have given an exaggerated impression of the subjectivity of concepts -- rather on the lines of the Red Queen's 'a word means what I intend it to mean'. The connotations of concepts referring to individuals or places are of course largely personal; but on the other hand, there is experimental evidence to show that the associative priorities and the connotative 'aura' of concepts of a general character are surprisingly stable and standardized in individuals of the same culture. 'Marbe's Law' demonstrated the existence of a logarithmic relation between an individual's reaction-time in giving a certain response to a stimulus-word in an association test, and the frequency of the same response occurring in a group of people.** Osgood has invented an ingenious method of measuring 'semantic differentials'. The subjects were asked to assess the ratings of a concept -- e.g. POLITE -- on ten different graded scales: e.g. 'angular-rounded', 'cold-hot', 'good-bad', 'wet-dry'. Two groups of twenty subjects were used, and the mean ratings of each group were then plotted and compared. Surprisingly enough, the two graphs were almost identical; even more surprising, the greatest amount of disagreement (c. fifteen per cent) was found in the ratings of politeness as 'good' or 'bad': one group thought that to be polite deserved a 'goodness' rating of six points, the other of six and a half points, on a seven-point scale. [1]
To recapitulate: a concept may be regarded as a relatively stable aggregate or 'cluster' of receiving-transmitting circuits, with a kind of nuclear core: the verbal label ('MADRID'). Additional circuits may be recruited, others may fade with disuse, and the relative energy potentials of the circuits may be altered by long-term processes or the person's momentary mood; but the auditory-vocal (and visual) trace of the word MADRID remains unaltered, and thus preserves the identity of the concept through all these changes in time. We may further assume that any incoming signal, regardless through which circuit it is received, will activate the nuclear circuit -- the auditory-vocal trace. If all goes well the response output will be emitted on the same wave-length on which the input was received, and I shall happily go on playing the 'towns with M' game. But since the cluster 'hangs together' it is likely that some amount of excitation has nevertheless spread to other circuits not concerned with the game -- and thus caused the fleeting, fringe-conscious stirrings of my memories of the Civil War. This is a minor, permissible kind of distraction; and perhaps even a necessary one -- without such ripples the stream of thought would be linear, colourless, all-too single-minded. But if one of the circuits with a high energy-potential gets excited, the control centres on higher levels of the hierarchy must prevent it from taking over if chaos is not to result; i.e., the code of the 'distracting' matrix must be blocked. During strenuous efforts to concentrate, one seems literally to 'feel' inside one's head the expenditure of energy needed to suppress diversional thoughts which keep popping up like jacks-in-the-box. 'A great part of our fatigue', Maxwell once remarked, 'often arises, not from those mental efforts by which we obtain the mastery of the subject, but from those which are spent in recalling our wandering thoughts' -- particularly, one may add, during a long and boring lecture; while the reverse phenomenon is produced by the disinhibiting action of alcohol and drugs. All this seems to indicate that our preferential matrices of ideation are most of the time blocked by centres on higher levels -- which agrees well with the predominantly restraining function of the hierarchic controls in perceptual and motor organization.
Types of Association
I have used the word Association loosely, as it is nowadays mostly used. In Drever's Dictionary of Psychology, for instance, we find:
Association: used generally of the principle in accordance with which ideas, feelings, and movements are connected in such a way as to determine their succession in the mind or in the actions of an individual, or of the process of establishing such connections.
Or take Humphrey (1951):
The term "association", or mental association, is a general name often used in psychology to express the conditions under which mental events, whether of experience or behaviour, arise. [2]
Hebb (1958) speaks at length about the association areas of the cortex, but does not define association. Other authors, and textbooks, differ widely in their definitions of the word, or prefer, wisely, not to define it at all.
Hobbes was probably the first to draw a distinction between what came later to be called 'free association' and 'controlled' association:
The train of thoughts, or mental discourse, is of two sorts. The first is unguided, without design, and inconstant . . . in which case the thoughts are safe to wander, as in a dream. . . . The second is more constant, as being regulated by some desire and design. [3]
However, free association is never entirely free: there are motivations, conscious or sub-conscious, which give it direction. On the other hand, association controlled by some rule of the game, such as 'towns with M', is 'free' to the extent that the rule allows alternative choices between permissible moves. The degrees of freedom of a matrix vary from rigid automatism to the great adaptability of complex mental skills; and the flow of associative thought will accordingly vary in character: it may move along fixed canals, or follow, like a rivulet, the accidents of the terrain and make detours round obstacles with an air of earnest goal-directedness.
Since the attempts of the classic associationist school to reduce thinking to association by contiguity and similarity (plus the 'secondary laws' of facilitation) had to be abandoned, the principles 'supposedly underlying association have
been classified and re-classified over and again. Thus Wells [4] once made a catalogue of eighteen types of association adapted from Jung, such as: 'egocentric predicate' (example: lonesome -- never); 'evaluation' (rose -- beautiful); 'matter of fact predicate' (spinach -- green); 'subject-verb' (dog -- bite), and so forth, through 'object-verb', 'cause-effect', 'co-ordination', 'subordination', 'supraordination', 'contrast', 'coexistence', 'assonance', etc. Woodworth (1939) suggested four classes: definition including synonyms and supraordinates; completion and predication; co-ordinates and contrasts; valuations and personal associations. He also suggested an independent classification cutting across the one just mentioned, according to a scale from 'meaningfulness' to 'superficiality'. [5] Most of the experimental work refers to association tests where the stimulus is a single word and the response is restricted to one other word -- a condition not exactly typical of ordinary verbal discourse outside the laboratory.
The lesson which emerges from these elaborate and painstaking attempts at classification is that the principles underlying associative thinking are determined by the matrix in which the thinking takes place; and that there are as many types of association as there are codes which control verbal behaviour. In bilingual countries like Switzerland, the response to a German stimulus-word will often be its French' equivalent; some people are addicted to metaphor, others to punning; the chess phyer and the draughts player's associations follow the rules of their respective games.
To sum up: associative thinking is the exercise of a habit. It may be rigid or flexible, with a wide range of adaptability; yet it remains a habit in so far as it observes certain invariant rules of the game. Association, qua exercise of a skill, is thus distinguished from learning, which is the acquisition of a new skill, and from bisociation, which is the combination, re-shuffling and re-structuring of skills. But these categories overlap -- as discussed in the next chapter.
NOTES
To 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 fai
led 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.