The Competitiveness of Nations in a Global Knowledge-Based Economy
Michael
Polanyi
Problem Solving
British Journal for the Philosophy of Science, 8 (30)
August 1957, 89-103.
THERE is a purposive tension from which no fully awake
animal is free. It consists in a
readiness to perceive and to act, or, more generally speaking, to make sense of
its own situation, both intellectually and practically. From these routine efforts to retain control
of itself and of its surroundings, we can see emerging a process of problem
solving, when the effort tends to fall into two stages, a first stage of
perplexity, followed by a second stage of doing and perceiving which dispels
this perplexity. We may say that the
animal has seen a problem if its perplexity lasts for some time and we can
clearly recognise that it tries to find a solution to
the situation which puzzles it. In doing
so the animal is searching for a hidden aspect of the situation, the existence
of which it surmises and for the finding or achieving of which the manifest
features of the situation serve it as tentative clues or instruments.
To see a problem is a definite addition to knowledge,
as much as it is to see a tree or to see a mathematical proof or a joke. It is a surmise which can be true or false,
depending on whether the hidden possibilities of which it assumes the existence
do actually exist or not. To recognise a problem which can be solved and is worth
solving is in fact a discovery in its own right. Famous mathematical problems have descended
from generation to generation, leaving in their wake a long trail of
achievements stimulated by the attempt at solving them. Accordingly, at the level of animal
experiments, we see the psychologist demonstrating to the animal the presence
of a problem in order to start it off in search of a solution. A rat in a discrimination box is made to realise that there is food hidden in one of two
compartments, both of which are accessible by pushing open its door. Only if he has grasped this will he start
searching for a sign which discriminates the door with food behind it from that
of the empty compartment.
* Received 12. xii. 55
Similarly, animals will not start solving a maze
unless they are made aware of the fact that there exists a path through it,
with some reward at its outlet. In Köhler’s ‘insight’ experiments his chimpanzees grasped
their problem from the start and marked their appreciation of the task by
composing themselves quietly to concentrate on it.
Accident usually plays some part in discovery and its
part may be predominant. Learning experiments
can be so arranged that, in the absence of any definitely understood problem,
discovery can only be accidental. [1] Mechanistically minded
psychologists who devise such experiments would explain all learning as the
lucky outcome of random behaviour. This conception of learning underlies also the
cyberneticist model of a machine which ‘learns’ by selecting a ‘habit’ which
has proved successful in a series of random trials. I shall disregard this model of heuristics and
continue to explore the process of discovery resulting from intelligent effort
irrespective of the neural model that may be proposed for it.
Intelligent problem-solving is manifested among
animals most dramatically in Köhler’s experiments on
chimpanzees, whose behaviour already presents the
characteristic stages through which, according to Poincaré,
discovery is achieved in mathematics. I
have already mentioned the first: the appreciation of a problem. A chimpanzee in a cage within sight of a bunch
of bananas out of its reach, neither makes any futile effort to get hold of it
by sheer force, nor abandons its desire of acquiring the prize, but settles
down instead to an unusual calm, while its eyes survey the situation all round
the target; it has recognised the situation as
problematical and is searching for a solution. [2] We may acknowledge this
(using the terminology of Wallas based on Poincaré) as the stage of Preparation. [3]
[1] Guthrie and Horton placed a cat in a cage in which a small pole
placed in the midst of the floor acted as release mechanism. Cats who had touched the pole by accident and
found themselves freed in consequence, quickly realised
the connection and proceeded to repeat their releasing action in an exactly
stereotype manner. The situation in
which the cat was placed offered no intelligible problem to the cat and the
solution, found accidentally, showed no clear understanding of the release
mechanism; the role played by intelligence in the whole process was negligible.
(Cf. E. Hilgard, Theories of Learning, New York, 1948, p.
68.)
2. “The greatest impression on the visitor (writes Köhler)
was made when Sultan made a pause, scratching his head leisurely and not moving
anything but his eyes and very slightly his head, scrutinising
the situation around him in the minutest detail”, The Mentality of Apes, London,
1927, p. 200.
3. G. Wallas, The Art of Thought, London,
pp. 40 sqq.
90
In the most striking cases of ‘insight’ observed by Köhler, this preparatory stage is suddenly followed by
intelligent action. Sharply breaking its
calm, the animal proceeds to carry out a stratagem by which it secures its aim,
or at least shows that it has grasped a principle by which this can be done. Its unhesitating manner suggests that it is
guided by a clear conception of its proposed operation. This conception is its discovery, or at least
- since it may not always prove practicable - its
tentative discovery. We may recognise in its coming the stage of illumination.
The practical realisation of
the principle discovered by insight often presents difficulties, which may even
prove unsurmountable. The manipulations by which the animal puts his
insight to the test of practical realisation may be
regarded as the stage of Verification.
Actually, Poincaré observed
four stages of discovery: Preparation, Incubation, Illumination, Verification. But the
second of these, Incubation, can be observed only in a rudimentary form in
chimpanzees. Yet the observation
described in some detail by Köhler, in which one of
his animals sustained its effort by solving a problem even while otherwise
occupied for a while, [1] anticipates to
a remarkable extent the process of Incubation: that curious persistence of
heuristic tension through long periods of time during which the problem is not
consciously entertained.
An extensive preoccupation with a problem imposes an
emotional strain, and a discovery which releases us from it is a great joy. The story of Archimedes rushing out from his
bath into the streets of Syracuse, shouting ‘Heureka!’
is a witness to this; and the account I have quoted from Köhler
of the way his chimpanzees behaved before and after solving a problem suggests
that they also experience such emotions. I shall show this more definitely later. I mention it now
1. An ape which for a while had been searching for a tool to rake in a
bunch of bananas lying outside its cage, and had made various fruitless
attempts in this direction - such as trying to break off a board from the lid
of a wooden case or hitting out with a stalk of straw in the direction of the
prize - had apparently abandoned the task altogether. It went on playing with one of its fellows for
about 10 minutes without turning again to the bananas outside the cage. Then suddenly, its attention having been
diverted from its game by a shout nearby, its eyes happened to fall on a stick
attached to the roof of the cage and at once it went for the stick and by
jumping up a number of times finally secured it and hauled in the bananas by
its aid. We may take this to show that
even while otherwise occupied the animal kept its problem alive ‘at the back of
its mind’, keeping it ready to pounce on the instruments of a solution when
they happened to meet its eye. Köhler, op. cit., p.
184.
only to make clear that nothing is a problem
or discovery in itself; it can be a problem only if it puzzles and worries
somebody and a discovery only if it relieves somebody from the burden of a
problem. A chess problem means nothing
to a chimpanzee or to an imbecile and hence does not puzzle them; a great chess
master on the other hand may fail to be puzzled by it because he finds its
solution without effort; only a player whose ability is about equal to the
problem will find intense preoccupation in it. Only such a player will appreciate its
solution as a discovery.
It appears possible to appraise the comparative
hardness of a problem and to test the intelligence of subjects by their
capacity for solving problems of a certain degree of hardness. The intelligence of chimpanzees and the
hardness of certain problems were both successfully assessed by Köhler when he devised a series of problems which some of
his apes could solve with some effort while others among them usually failed
altogether to do so. The success of Yerkes in setting problems to earthworms (which these could
solve after about a hundred trials), shows that he could assess even such
extremely low powers of intelligence as were required here from the earthworm. [1] Editors of a crossword
column undertake a similar feat in supplying their readers with a steady stream
of equally difficult problems. We may
conclude that an observer can recognise a problem as
such in respect to identifiable persons - even though we admit that there are
no problems outside a relation to some kind of person.
If an animal who has solved a
problem is placed once more in the original situation, it proceeds
unhesitatingly to apply the solution which it had originally discovered at the
cost of much effort and perhaps many unsuccessful trials. This shows that by solving the problem the
animal has acquired a new intellectual power which prevents it from being ever
again puzzled by the problem. Instead,
it can now deal with the situation in a routine manner involving no heuristic
tension and achieving no discovery. The
problem has ceased to exist for it.
This irreversible character of heuristic acts is
important. It suggests that no solution
of a problem can be accredited as a discovery if it is achieved by a procedure
following definite rules. For such a
procedure would be reversible in the sense that it could be traced back
stepwise to
1. R. M. Yerkes, ‘The Intelligence of
Earthworms ‘,Jour. Anim. Behav.,
1912, 2, 332-352 ; cf. N. R. F. Maier and T. Schneirla,
Principles of Animal Psychology, New York and London, 1935, pp. 98-101.
92
its beginning and repeated once more any
number of times, like any arithmetical computation. Accordingly, any strictly formalised
procedure would also be excluded as a means of achieving discovery.
It would follow that true discovery is not a strictly
logical performance. Accordingly, we may
describe the obstacle to be overcome in solving a problem as a ‘logical gap’,
and speak of the width of the logical gap as the measure of the ingenuity
required for solving the problem. ‘Illumination’ is then the leap by which the
logical gap is crossed. It is the plunge
by which we gain a foothold at another shore of reality. On such plunges the scientist has to stake bit
by bit his entire professional life.
The width of the logical gap crossed by an inventor is
subject to legal assessment. Courts of
law are called upon to decide whether the ingenuity displayed in a suggested
technical improvement is high enough to warrant its legal recognition as an
invention or is merely a routine improvement, achieved by the application of
known rules of the art. The invention
must be acknowledged to be unpredictable, a quality which is assessed by the
intensity of the surprise it might reasonably have aroused. This unexpectedness corresponds precisely to
the presence of a logical gap between the antecedent knowledge from which the
inventor started and the consequent discovery at which he arrived.
Established rules of inference offer public paths for
drawing intelligent conclusions from existing knowledge. The pioneer mind which reaches its own
distinctive conclusions by a leap across a logical gap deviates from the
commonly accepted process of reasoning, to achieve surprising results. Such an act is original in the sense of making
a new start, and the capacity for initiating it is the gift of originality; a
gift possessed by a small minority.
Since the Romantic movement
originality has become increasingly recognised as a
native endowment which alone enables a person to initiate an essential
innovation. Universities and industrial
research laboratories are founded today on the employment of persons with
original minds. Permanent appointments
are given to young scientists who are credited with signs of originality, in
the expectation that they will continue to produce surprising ideas for the
rest of their lives.
Admittedly, there are minor heuristic acts within the
power of ordinary intelligence and indeed continuous with the adaptive capacities
of life down to its lowest levels. The
interpretative framework of the educated mind is ever ready to meet somewhat
novel experiences
and to deal with them in a somewhat novel
manner. In this sense all life is
endowed with originality and originality of a higher order is but a magnified
form of a universal biological adaptivity. But genius makes contact with reality on an
exceptionally wide range: by seeing problems and reaching out to hidden
possibilities for solving them, far beyond the anticipatory powers of current
conceptions. Moreover, by deploying such
powers in an exceptional measure - far surpassing our own as onlookers - the
work of genius offers us a massive demonstration of a creativity which can
neither be explained in other terms nor taken unquestioningly for granted. In confrontation with genius we are forced to
acknowledge the originative power of life, which we might and commonly do
overlook in its ubiquitous lesser manifestations; for by paying respect to
another person’s judgment as superior to our own, we emphatically acknowledge
originality in the sense of a performance the procedure of which we cannot
specify.
In choosing a problem the investigator takes a
decision fraught with risks. The task
may be insoluble or just too difficult. In
that case his effort will be wasted and with it the effort of his
collaborators, as well as the money spent on the whole project. But to play safe may be equally wasteful. Mediocre results are no adequate return for
the employment of high gifts, and may not even repay the money spent on
achieving them. So the choice of a
problem must not only anticipate something that is hidden and yet not
inaccessible but also assess the investigator’s own ability (and those of his
collaborators) against the anticipated hardness of the task, and make a
reasonable guess as to whether the hoped-for solution will be worth its price
in terms of talent, labour, and money. To form such estimates of the approximate
feasibility of yet unknown prospective procedures leading to unknown
prospective results is the day-to-day responsibility of anyone undertaking
independent scientific or technical research. On such grounds as these he must even compare
a number of different suggestions and select from them for attack the most
promising problem. Yet experience shows
that such a performance is possible and can even be relied upon with a
considerable degree of probability.
There are three major fields of knowledge in which
discoveries are possible: natural science, technology, and mathematics. I have referred to examples from each of these
fields to illustrate the anticipatory powers which guide discovery. These are clearly quite similar
94
in all three cases. Yet the efforts of philosophers have been
almost wholly concentrated on the process of empirical discovery which
underlies the natural sciences. Ever
since the rise of empiricism at the turn of the sixteenth century philosophers
of science have been preoccupied with an attempt to define and justify the
process of induction, while by contrast, nobody seems to have tried to define
and justify the process by which technical innovations are made, as for example
when a new machine is invented. The
process of discovery in mathematics has received some attention, and has
recently been attacked both from the logical and psychological point of view,
but neither approach has raised any epistemological questions parallel to those
so sedulously pursued for centuries in connection with empirical induction. It seems to me that any serious attempt to analyse the process of discovery should be sufficiently
general to apply to all three fields of systematic knowledge, and so I should
like here to identify and acknowledge the powers on which we rely in solving
mathematical problems. For reasons of
space I shall exclude the history of major discoveries which often involve
modifications in the foundations of mathematics and shall attend only to the
type of problems that are set to students in teaching them mathematics. Since the solution of these problems is not
known to the student the process of finding it bears the marks of a discovery,
even though it involves no fundamental change of outlook.
The fact that the teaching of mathematics relies
heavily on practice, shows that mathematical knowledge can be acquired only by
developing an art; the art of solving mathematical problems. The same is true not only of mathematics and
formal logic, but equally also of all mathematical sciences, like mechanics,
electrodynamics, thermodynamics, and the mathematical branches of engineering;
you cannot master any of these subjects without working out concrete problems
in them. The art you strive for in such
practical courses is that of converting a language, so far only receptively
assimilated, into an effective tool for interpreting a new subject matter,
which in this case consists in solving problems.
Thus the process by which mathematics is taught shows
once more that the solving of mathematical problems is a heuristic act which
leaps across a logical gap. While we
cannot expect to find any strict rules for performing such an act, we may
expect to discover certain rules of art, the interpretation of which is itself
a part of the very art for the pursuit of which they offer us guidance. This is confirmed by the fact that the maxims
of problem solving can themselves be learnt only
by practice. It is indeed above all the art of heuristic
reasoning that the practical teaching of mathematics seeks to impart. This seems to me clearly proven by the
comprehensive studies of G. Polya on the subject of
mathematical heuristics, on which I shall lean heavily for this study. [1]
The simplest heuristic effort is to search for an
object you have mislaid. When I am
looking for my fountain pen I know what I expect to find; I can name it and
describe it. Though I know much more
about my fountain pen than I can ever recall, and do not know exactly where I
left it, the pen is clearly known to me and I know also that it is somewhere
within a certain region, though I do not know where. My knowledge of the thing I am looking for is
much less ample when I am looking for a word to fit into a crossword puzzle. This time I know only that the missing word
has a certain number of letters and designates, for example, something that is
badly needed in the Sahara or flows out of a central chimney. These properties are merely clues to a word
that I definitely do not know; clues from which I must try to gain an
intimation of what the unknown word may be. Again, a name which I know
well but cannot recall at the moment lies somewhere halfway between these two
cases. It is more closely present
to my mind than the unknown solution of a crossword puzzle, but less closely
perhaps than the mislaid fountain pen and its unknown location. Mathematical problems are in the class of
crossword puzzles, for to solve such a problem we must find (or construct)
something that we have never seen before, with the given data serving us as
clues to it.
A problem may admit of a systematic solution. By ransacking my flat inch by inch I may make
sure of eventually finding my fountain pen, which I know to be somewhere in it.
I might solve a chess problem by trying
out mechanically all combination of possible moves and countermoves. Systematic methods apply also to many mathematical problems though usually they are far
too laborious to be carried out in practice. [2] It is clear that any such
systematic operation
1. G. Polya, How to Solve it, Princeton,
1945, and Mathematics and Plausible Reasoning, Vols. 1 and 2, London,
1954. Penetrating observations on
problem solving have also been contributed by psychologists, mainly Duncker and Wertheimer.
2. A. M. Turing (Science News, 1954, 31) has computed the number
of arrangements that would have to be surveyed in the process of solving
systematically a very common form of puzzle consisting of sliding squares to be
rearranged in a particular way. The
number is 20,922,789,888,000. Working
continuously day and night and inspecting one position per minute the process
would take 4 million years.
96
would reach a solution without crossing a
logical gap and would not constitute a heuristic act.
The difference between the two kinds of problem
solving, the systematic and the heuristic, reappears in the fact that while a
systematic operation is a wholly deliberate act, a heuristic process is a
combination of active and passive stages. A deliberate heuristic activity is performed
during the stage of Preparation. If this
is followed by a period of Incubation, nothing is done and nothing happens on
the level of consciousness for this time. The advent of a bright idea (whether following
immediately from Preparation or only after an interval of Incubation) is the
fruit of the investigator’s earlier efforts, but not itself an action on his
part; it just happens to him. And again,
the testing of the ‘bright idea’ by a formal process of Verification,
is another deliberate action of the investigator. However, the decisive act of discovery must
have occurred before this, at the moment when the happy thought emerged.
Though the solution of a problem is something we have
never met before, yet in the heuristic process it plays a part similar to the
mislaid fountain pen or the forgotten name which we know quite well. We are looking for it as if it were there,
pre-existent. Problems set to students
are of course known to have a solution; but the belief that there exists a
hidden solution which we may be able to find, is essential also in envisaging
and working at a yet unsolved problem. It
determines also the manner in which the ‘happy thought’ eventually presents
itself as something inherently satisfying. It is not one among a great many ideas to be
pondered upon at leisure, but one which carries conviction from the start. We shall see in a moment that this is a
necessary consequence of the way a heuristic striving evokes its own
consummation. To the closer analysis of
this process I shall now turn.
A problem is an intellectual desire (a ‘quasi-need’ in
K. Lewin’s terminology) and like every desire it
postulates the existence of something that can satisfy it; in the case of a
problem its satisfier is its solution. As all desire stimulates the imagination to
dwell on the means of satisfying it, and is stirred up in its turn by the play
of the imagination it has fostered, so also by taking interest in a problem we
start speculating about its possible solution and in doing so become further
engrossed in the problem.
Obsession with one’s problem is in fact the mainspring
of all inventive power. Asked by his
pupils in jest, what they should do to become ‘a Pavlov’, the master answered
in all seriousness : ‘Get up
in the morning with your problem before you. Breakfast with it. Go to the laboratory with it. Eat your lunch with it. Keep it before you after dinner. Go to bed with it in your mind. Dream about it.’ [1] It is the unremitting
preoccupation with a problem that lends to genius its proverbial capacity for
taking infinite pains. And the intensity
of our preoccupation with a problem generates also our power for re-organising our thoughts successfully, both during the hours
of search and afterwards, during a period of rest. [2]
But what is the object of this intensive
preoccupation? Can we concentrate our
attention on something we don’t know? Yet
this is precisely what we are told to do: ‘Look at the unknown!’ says Polya, ‘Look at the end. Remember your aim. Do not lose sight of what is required. Keep in mind what you are working for. Look at the unknown. Look at the
conclusion.’
[3]
No
advice could be more emphatic.
The seeming paradox is resolved by the fact that even
though we have never met the solution we have a conception of it in the same
sense as we have a conception of a forgotten name. By directing our attention on a focus in which
we are subsidiarily aware of all the particulars that
remind us of the forgotten name, we form a conception of it; and likewise, by
fixing our attention on a focus in which we are subsidiarily
aware of the data by which the solution of a problem is determined, we form a
conception of this solution. The
admonition to look at the unknown really means that we should look at the
known data, not, however, in themselves, but as clues to the unknown; as
pointers to it and parts of it. We
should make every effort to feel our way to an understanding of the manner in
which these known particulars hang together both mutually and with the unknown.
Thus we make sure that the unknown is
really there, essentially determined by what is known about it, and able to
satisfy all the demands made on it by the problem.
All our conceptions have heuristic powers; they are
ever ready to identify novel instances of experience by modifying themselves so
as to comprise them. The practice of
skills likewise is inventive; by concentrating our purpose on the achievement
of success we evoke
1. J. R. Baker, Science and the Planned State, London, 1945, p.
55
2. ‘Only such problems come back improved after a rest whose solution
we passionately desire and for which we have worked with great tension’ writes Polya (op. cit. p. 172).
3. ibid., p. 112.
98
ever new capacities in ourselves. A problem partakes of both these types of endeavour. It is a
conception of something we are striving for. It is an intellectual desire for crossing a
logical gap on the other side of which lies the unknown, fully marked out by
our conception of it, though as yet never seen in itself. The search for a solution consists in casting
about with this purpose in mind. This we
do by performing two operations which must always be tried jointly. We must (i) set out
the problem in suitable symbols and continuously reorganise
its representation with a view to eliciting some new suggestive aspects of it
and concurrently (ii) ransack our memory for any similar problem of which the
solution is known. The scope of these two
operations will usually be limited by the student’s technical facility for
transforming the given data in different ways and by the range of germane
theorems with which he is acquainted. But
his success will depend ultimately on his capacity for sensing the presence of
yet unrevealed logical relations between the conditions of the problem, the
theorems known to him, and the unknown solution he is looking for. Unless his casting about is guided by a
reliable sense of growing proximity to the solution, he will make no progress
towards it. Conjectures made at random,
even though following the best rules of heuristics, would be hopelessly inept
and totally fruitless.
The process of solving a mathematical problem
continues to depend therefore at every stage on the same ability, to anticipate
a hidden potentiality which enables the student to see a problem in the first
place and set out to solve it. Polya has compared a mathematical discovery consisting of a
whole chain of consecutive steps with an arch where every stone depends for its
stability on the presence of others, and pointed out the paradox that the
stones are in fact put in one at a time. Again, the paradox is resolved by the fact
that each successive step of the incomplete solution is upheld by the heuristic
anticipation which originally evoked its invention: by the feeling that its
emergence has narrowed further the logical gap of the problem.
The growing sense of approaching to the solution of a
problem can be commonly experienced when we grope for a forgotten name. We all know the exciting sense of increasing
proximity to the missing word which we may confidently express by saying; ‘I
shall remember it in a moment’ and perhaps later ‘It is on the tip of my
tongue’. The expectation expressed by
such words is often confirmed in the event. I believe that we should likewise acknowledge
our capacity both to sense the accessibility of a hidden inference from given premisses and to
invent transformations of the premisses which increase the accessibility of the hidden
inference. We should recognise
that this foreknowledge biases our guesses in the right direction, so that
their probability of hitting the mark, which would otherwise be zero, becomes
so high that we can definitely rely on it simply on the grounds of a student’s
intelligence: or for higher performances, on the grounds of the special gifts
possessed by the professional mathematician.
The feeling that the logical gap separating us from
the solution of a problem has been reduced means that less work should remain
to be done for solving it. It may also
mean that the rest of the solution will be comparatively easy or that it may
present itself without further effort on our part, after a period of rest. The fact that our intellectual strivings make
effective progress during a period of Incubation without any effort on our part
is in line with the latent character of all knowledge. As we continuously know a great many things
without always thinking of them, so we naturally also keep on desiring or
fearing all manner of things without always thinking of them. We know how a set purpose may result in action
automatically later, as when we go to bed resolved to wake up at a certain
hour. Post-hypnotic suggestions can set
going latent processes which compulsively result after a number of hours in the
performance requested of the subject. [1] Mrs
Zeigarnik has shown that unfinished tasks continue
likewise to preoccupy us unconsciously; their memory persists after finished
tasks are forgotten. [2] The
fact that the tension set up by the unfinished task continues to make progress
towards its fulfilment, is shown by the well-known
experience of sportsmen that a period of rest following on a spell of intensive
training produces an improvement of skill. The spontaneous success of the search for a
forgotten name or for the solution of a problem, after a period of quiescence,
falls in line with this experience.
These antecedents explain also the manner in which the
final success of problem solving will suddenly set in. For each step, whether spontaneous or
contrived, that brings us nearer to the solution increases our premonition of
its proximity and brings a more concentrated effort to bear on a reduced
logical gap. The last stage of the
solution may therefore be frequently achieved in a self-accelerating manner and
the final discovery may be upon us in a flash.
1. Cf. N. Ach, ‘Determining Tendencies; Awareness’, in D. Rappaport, Organisation
and Pathology of Thought, New York, 1951, pp. 16 sqq.
2. W.
D. Effis, A
Source Book of Gestalt Psychology, New York, 1938, pp. 300-314.
100
I have said that our heuristic cravings imply, like
our bodily appetites, the existence of something which has the properties
required to satisfy us, and that the intimations which guide our striving
express this belief. But the satisfier
of our craving has in this case no bodily existence; it is not a hidden object
but an idea never yet conceived.
We hope that as we work at the problem this idea will
come to us, whether all at once or bit by bit; and only if we believe that this
solution exists can we passionately search for it and evoke from ourselves
heuristic steps towards its discovery. Therefore
as it emerges in response to our search for something we believe to be there, discovery,
or supposed discovery, will always come to us with the conviction of its being
true. It arrives accredited in advance
by the heuristic craving which evoked it.
The most daring feats of originality are still subject
to this law: they must be performed in the assumption that they originate
nothing, but merely reveal what is there. And their triumph confirms this assumption,
for what has been found bears the mark of reality in being pregnant with yet
unforeseeable implications. Mathematical
heuristics, aiming at conceptual reorganisation
without reference to new experience, once more exemplifies in its own terms
that an intellectual striving entails the conviction of anticipating reality. It illustrates also how this conviction finds
itself confirmed by the eventual solution, which ‘solves’ precisely because it
successfully claims to reveal an aspect of reality. And we can see again, finally, that this whole
process of discovery and confirmation ultimately relies on our own accrediting
of our own vision of reality.
To start working on a mathematical problem, we reach
for pencil and paper, and throughout the stage of Preparation we keep trying
out ideas on paper in terms of symbolic operations. If this does not lead straight to success, we
may have to think the whole matter over again, and may perhaps see the solution
revealed much later unexpectedly in a moment of Illumination. Actually, however, such a flash of triumph
usually offers no solution, but only envisages a solution, which has yet to be
tested. In the verification or working
out of the solution we must again rely therefore on explicit symbolic
operations. Thus both the first active
steps undertaken to solve a problem and the final garnering of the solution
rely effectively on computations and other symbolic operations, while the more
informal act by which the logical gap is crossed lies between these two formal
procedures. However, the intuitive
powers of the investigator are always dominant and
decisive. Good mathematicians are usually found capable
of carrying out computations quickly and reliably, for unless they command this
technique they may fail to make their ingenuity effective; but their ingenuity
itself lies in producing ideas. Hadamard says that he used to make more mistakes in calculations
than his own pupils but that he more quickly discovered them because the result
did not look right; it is almost as if by his computations he had been
merely drawing a portrait of his conceptually prefigured conclusions. [1] Gauss is widely quoted as
having said: ‘I have had my solutions for a long time but I do not yet know how
I am to arrive at them.’ Though the
quotation may be doubtful it remains well said. [2] A situation of this kind
certainly prevails every time we discover what we believe to be the solution to
a problem. At this moment we have the
vision of a solution which looks right and which we are therefore
confident will prove right. [3]
The manner in which the mathematician works his way
towards discovery by shifting his confidence from intuition to computation and
back again from computation to intuition, while never releasing his hold on
either of the two, represents in miniature the whole range of operations by
which articulation disciplines and expands the reasoning powers of man. This alternation is asymmetrical, for a formal
step can be valid only by virtue of our tacit confirmation of it. Moreover, a symbolic formalism is itself but
an embodiment of our antecedent unformalised powers;
it is an instrument skillfully contrived by our inarticulate selves for the
purpose of relying on it as our external guide. The interpretation of primitive terms and
axioms is predominantly inarticulate and so is the process of their expansion
and re-interpretation, which underlies the progress of mathematics. A formal proof proves nothing until it induces
the tacit conviction that it is binding. Thus the alternation between the intuitive and
the formal depends on tacit affirmations both at the beginning and at the end
of each chain of formal reasoning.
This brings us back to the purposive tension that we
share with
1. J. Hadamard, The Psychology of Invention in the Mathematical Field, Princeton, 1945,
p. 49
2. Agnes Arber, The
Mind and the Eye, Cambridge, 1954, p. 47
3. Archimedes describes in his Method a mechanical process of
geometrical demonstration which carries conviction with him, though he regards
its results as still requiring proof, which he proceeds to supply. B. L. Van der Waerden, Science Awakening, Groningen, 1954, p. 215.
102
animals, as the primordial root of
problem-solving at all levels. But the
rooting of our highest articulate powers in the inarticulate has far wider
implications. If everywhere it is the
inarticulate that has the last word, unspoken and yet decisive, then a
corresponding reform of the whole conception not only of problem-solving but of
truth in general, is itself inevitable. The ideal of an impersonally detached truth
must be abridged to allow for the inherently personal character of the act by
which truth is declared. It is the endeavour to effect such a revision of which the preceding
inquiry into problem-solving forms part.
The University
Manchester 13
103