Foundations of Philosophy

(3) Given that D = 5 what are the values of the other letters in this addition sum:

D O N A L D

G E R A L D

R O B E R T

(4) Give the next number in the following sequences:

(a) 5, 15, 25, …..

(b) 7, 3, -1, …..

(c) 1, -2, 4, -8, …..

(d) 1, 7, 25, …..

(5) Are you looking at this box from above or from below?

(6) Identify and describe an insight that you have had recently.

1. The Experience of Insight

In 1993 Nigel Short qualified to play Gary Kasparov, the reigning World Chess champion, for the World Title. As the games were in London and Nigel was a local boy, the BBC decided to televise the series live. As chess does not present much visible action, they assembled a team of commentators and analysts to fill in the gaps. However, during the sixth game Kasparov spent fifty-five minutes thinking about one move: almost an hour, motionless, transfixed, impassive, staring at the board, uttering an occasional sigh. Commentators soon ran out of chatter and the chess experts [59] exhausted their analysis. Still no move. BBC does not have commercial breaks, so you could feel the tension rising. Finally, one of the commentators cries out in exasperation, What on earth is he doing?

That is precisely our question also in this chapter: what is thinking? What are we doing when we are thinking? Why does it take so long? What brings it to an end? What kinds of activities are involved in understanding? It is clear that it takes time: fifty-five minutes to be precise in this case. It is clear that it is an activity: Kasparov was not asleep, he was not relaxing, nor daydreaming. There is a limit to the time allowed; he wanted desperately to win; he had to use his time wisely. It is not a visible activity that you can catch on a television camera but it is an activity which we can describe, analyze, define, break into its parts and put them together again as a whole. That is what we will be doing in this chapter.

What was Kasparov doing for those fifty-five minutes? Well, he was thinking; he was considering possibilities, analyzing three or four main lines to a depth of six or ten moves, involving hundreds of possible end positions. He was using his imagination to picture the board with pieces removed or advanced; he was using his memory to call up similar games and situations; above all he was using his intelligence to relate, combine, put together sequences of possible moves and corresponding replies; he was being systematic, eliminating this line, exploring alternatives, looking at the situation from his point of view and that of his opponent. He was evaluating positions, advantages, material loss, attacking possibilities, balancing tactics and strategy. Finally, the whole movement coalesces into the judgment: this is the best move. Much to the relief of the commentators he moved a piece. The game ended in a draw but he won the series.

Our interest in Kasparov, chess, and what he was doing is marginal. Our primary focus is our own modest efforts to think things through, to understand correctly, to find correct solutions to problems. The exercises given at the beginning of the chapter are designed to provoke a simple insight.² The main purpose of doing the problems is to reflect back afterwards and to identify the experience of an act of understanding and the factors which help or [60] hinder understanding to occur. We are not primarily interested in the content, whether it is chess or mathematics or practical problems; we are vitally interested in the activity or activities involved in the process. We are not interested in the what is being understood; we are interested in the activities by which something is being understood. We are starting the work of intellectual self-appropriation, taking it step by step, slowly assembling all the pieces, until finally we will be able to identify and discriminate with ease our own cognitional operations.

Direct insights are simply acts of understanding; we call them ‘direct’ to differentiate them from inverse and reflective insights, which we consider later. Direct insights are the normal, straightforward acts of understanding involved in working out a chess problem, a mathematics question or a puzzle. Thinking normally involves a series of insights, a series of connections, relations, and possibilities. Thinking is like a conversation that we conduct within ourselves, considering, rejecting, arguing, referring to examples, invoking images, working things out. Sometimes we reach a solution; sometimes we remain frustrated.

We are now discussing the intellectual pattern of experience and so we can expect to discern a pattern of related activities that form some kind of a unity. At first there is puzzlement even in grasping what is the problem. What are the elements to be solved? What is the question? What has to be taken into account? It is tantalizing; it is a challenge to one's intelligence. If it is a game, there can be considerable aggression, a will to win, pride, fear and other emotions involved. If it is work, research, a practical problem, then all sorts of other factors motivate us to solve it. But in that first stage of puzzlement, we cannot see the solution; we can get very annoyed at the author or the teacher or the opponent, who has confronted us with such a challenge.

The next moment everything falls into place; you ‘see’ the solution. It is like a light going on in your mind. It is a release of tension, a spurt of adrenaline, something clicks. You 'see' what is relevant and what is irrelevant; you breathe a sigh of relief; you feel confident. The pieces on the chessboard remain the same, but you 'see' how they relate in a different way. You 'see' something that [61] you did not see before, even though it was staring you in the face. We use a variety of metaphors to express this experience. To ‘see’ is the most common but we also use the expressions, ‘to grasp the point’, ‘to get it ‘, ‘it dawned on me’, an ‘aha’ experience, a ‘eureka’ experience. These are merely metaphors, that is, loose ways of using what is familiar to talk about what is difficult less familiar.

The person of intelligence is the one in whom these acts occur easily, frequently and with regard to a variety of subjects. We are not geniuses, but we are all blessed with a minimum of intelligence and are performing acts of understanding all the time. The not-so-bright person is one who has difficulty understanding; it takes a longer time, and happens less frequently. We all know people who are very quick to catch on, to get the point, to grasp the essential. Others are dull, susceptible to having their legs pulled; we describe them as slow, dense, thick. The intelligent person learns faster, concentrates better, absorbs material easily, and gets ahead in his subject.

We are performing acts of understanding all our waking life; but for the most part we take them for granted. When we are reading, we are performing a complicated series of acts of understanding the letters, the words, the sentences, and the meanings. When we are involved in practical work we using means to achieve ends, applying the law of the lever, exploiting the principles of mass and momentum etc. The difficulty is to be able to isolate the experience of a single act of understanding so that we can describe it, identify it, and pick out its salient characteristics. The purpose of this chapter is to make a start in identifying this experience of insight in our own minds. Later we will be able to expand on the implications of this for understanding, generalization, abstraction and other processes.

Just a cautionary note on what insight is not. It is not an intuition; it is not a direct, single, simple vision of intelligible objects. Understanding involves is a series of related activities, all of which are part of the process of having an insight. Think of what was going on in the mind of Kasparov for those fifty-five minutes; why did it take so long? What was he doing?

Insight is not in itself a religious experience. Religious experiences can be very dramatic, overwhelming, conversion experiences; such experiences may involve activities of [62] understanding, a change of world view or of concepts, but of themselves they primarily refer to God, prayer, behavior, choice, etc. There is nothing mystical or esoteric about acts of understanding in themselves; they are very common place; we are performing them all the time.

Finally, insight is not just remembering. We can wander around for hours looking for our spectacles - especially when we are old. Suddenly, we remember that we left them in the garden. It comes suddenly and unexpectedly but it is not an insight. This is simply an act of remembering at the level of sense.

2. Examples from History

Fortunately, history provides us with many examples of acts of understanding. Let us take some dramatic, clear cases to see if they can help us to analyze and identify the activities involved in understanding. Although they are experiences of historical figures our questions is always, does this resonate in your own experience?

1. Archimedes, a Greek mathematician, was given a problem by Hiero, the king. The king had received a chalice of gold, which was beautifully decorated, from a visiting dignitary. The king wanted to know whether the chalice was made of pure gold or whether it was mixed with lead. He asked Archimedes to find out, without melting down the chalice. Archimedes knew that lead is heavier than gold. He could easily find out the weight of the chalice but he still had not enough information to work out how much lead and how much gold. We are told that he tired of the problem and decided to relax and go to the public baths. Floating in the water, perhaps wondering why he was floating, he suddenly realized that if he weighed the chalice in water he would get the further information he needed to solve the problem. He ran naked through the streets of Syracuse, shouting, 'Eureka, Eureka, I have found it, I have found it.' Later on, in his calmer moments, I suppose, he worked out the principles of specific weight and displacement, and realized that they could be applied to all and every material body to determine whether they would float.

2. A more modern example is that cited by the mathematician H Poincare. This is an insight into mathematics, but we do not have to understand the mathematics; what we are interested in is the experience of insight, which he calls illumination and describes quite clearly.

Just at this time, I left Caen where I was living, to go on a geological excursion under the auspices of the School of Mines. The incidents of the travel made me forget my mathematical work. Having reached Coutances, we entered an omnibus to go some place or other. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscience sake, I verified the result at my leisure.

Then I turned my attention to the study of some arithmetical questions apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness and immediate certainty, that the arithmetic transformations of indefinite ternary quadratic forms were identical with those of non-Euclidean geometry.³

Here again, we have a mathematician faced with a difficult problem and unable to solve it after much time and effort. So he takes a break, relaxes, thinks of something else, and goes for a walk. And quite suddenly, in a flash, the answer comes to him; he understands the solution. Later on, he can return to his work and iron out the details. This extract recounts two such experiences where insights occurred not in the midst of work but at a time of relaxation.

3. Let us look at the well-known story of Helen Keller (1880-1968). She was afflicted with a disease at the age of eighteen months, which deprived her of sight and hearing. She grew up a very unruly child, cut off from the world, helpless and dependent. When she was six years old her parents hired Annie Sullivan to be her tutor. Annie herself had suffered from neglect and cruelty in her youth and also experienced partial blindness. She was trained in the methods of the Perkins school to communicate with the blind and deaf by tracing the alphabet on the hand and associating these movements with common objects. She started using this method with Helen but was not meeting with much success even though Helen did learn to [64] imitate the movements on the palm of the hand. On March 3rd, 1887, all this changed, as Helen describes in her autobiography:

We walked down the path to the well-house, attracted by the fragrance of the honeysuckle with which it was covered. Someone was drawing water and my teacher placed my hand under the spout. As the cool stream gushed over one hand she spelled into the other the word w-a-t-e-r, first slowly, then rapidly. I stood still, my whole attention fixed upon the motions of her fingers. Suddenly I felt a thought; and somehow the mystery of language was revealed to me. I knew then that w-a-t-e-r meant the wonderful cool something that was flowing over my hand. That living word awakened my soul, gave it light, hope, joy, set it free! There were barriers still, it is true, but barriers that could in time be swept away.

I left the well-house eager to learn. Everything had a name, and each name gave birth to a new thought. As we returned to the house every object which I touched seemed to quiver with life. That was because I saw everything with the strange, new sight that had come to me........ It would have been difficult to find a happier child than I was as I lay in my crib at the close of that eventful day and lived over the joys it had brought me, and for the first time longed for a new day to come.⁴

Helen was seven years old when this incident took place. She had learned almost nothing in those seven years, her mind blocked by her blindness and deafness. But this is a liberation, a breakthrough. She discovered naming: things have names, can be referred to, fall into classes, can be identified and recognized. So we have the paradox of a blind girl who can now see - 'that strange new sight' - which we are calling insight or the act of understanding.

4. A more contemporary example is taken from Scientific American:

In 1951 David A. Huffman and his classmates in an electrical engineering graduate course on information theory were given the choice of a term paper or a final exam. For the term paper, Huffman's professor, Robert M. Fano, had assigned what at first appeared to be a simple problem. Students were asked to find the most efficient method of representing numbers, letters or other symbols using a binary code. Besides being a nimble intellectual exercise, finding such a code would enable information to be compressed for transmission over a computer network or for storage in a computer's memory.

Huffman worked on the problem for months, developing a number of approaches, but none that he could prove to be the most efficient. Finally, he despaired of ever reaching a solution and decided to start studying for the final. Just as he was throwing his notes in the garbage, the solution [65] came to him. "It was the most singular moment of my life," Huffman says, "There was the absolute lightening of sudden realization."⁵

He had been working from the wrong end of the coding tree; reverse the process and it works. Huffman's code is now one of the basic ideas of computer science and data communication.

These examples are somewhat dramatic but they do give good detailed descriptions of the experience of insight. It is an extremely important moment of transition from not understanding to understanding, from not seeing to seeing. A psychologist, E. D. Hutchinson, who has studied many examples of insights, divides the occurrence of an insight into four stages.⁶ There is a first stage of preparation when the question is posed but despite great effort nothing seems to happen. A second stage of incubation or retreat follows when we temporarily give up on finding a solution, relax, turn our mind to something else to forget our frustration. There is a third stage of insight when it just happens; we suddenly see, a light comes on, we get the point, we have a sudden illumination. Finally, there is a stage where we quietly verify, clarify and apply the insight. These stages can be perceived in all of the above examples.

However, what is important is not to study the insights of others but to recognize the same experience in your own intellectual activity. You may not be able to remember an experience as vivid as that of Helen Keller. For most of us the experience of insight is routine, taken for granted, not adverted to. But we are understanding whenever we are reading, speaking, listening to a lecture, taking notes, driving a car. The exercises at the beginning of the chapter are designed to make us stop and think; to jolt us out of a routine; to provoke the experience of puzzlement, of trying different solutions, being frustrated, and finally finding the correct answer. Our interest is not in learning mathematics, but in identifying the experience of understanding. Do some puzzle and, having found the solution, reflect back on the mental processes involved in solving it. What images did you use? What figures did you write down? What led you astray? What blocked you from understanding? What was the clue that helped you to solve it? Where did the insight come from? Each one of us has privileged access to the working of our own minds. Shift the focus from the objects on the paper to the activities [66] of the mind. There we can examine the intimate details of the experience of insight. The examples we have given are intended as a kind of mirror in which we can see the workings of our own understanding. The protracted struggle is only present when we are confronted with difficult problems. But you will usually find that our insights conform to the pattern of preparation, incubation, illumination and verification suggested above.

The above historical examples are all acts of discovering something for the first time; hence it is very dramatic, joyful, memorable. An educational system cannot afford the time to allow students to discover everything for themselves. The accumulated discoveries of generations of scholars are put together systematically and the student has to understand, remember, digest it all in one semester and repeat it in the exam. In an educational system the solution is often presented before the problem; learning loses the joy of insight and discovery and becomes dull and tedious. It is no so easy to isolate individual acts of understanding in that process of learning. Perhaps, an educational system which allowed more time for personal research and discovery would motivate and encourage students in the search for understanding.⁷

3. Five Characteristics of Insight

To further our objectification of the act of understanding we will discuss five characteristics of insight.⁸ Remember that we are not describing something that happens only in the minds of geniuses, but rather, what is going on in your own mind all the time when you are understanding.

3.1 Insight comes as a release of the tension of inquiry

Knowing begins in inquiry, in the asking of questions. The first sign of the emergence of intelligence is the tension of inquiry, the questioning of experience. We are all familiar with the incessant questioning of the child. But even as adults we understand in proportion to the strength of our desire to know. There is a great difference between the student who has a personal interest in a subject, and one who is there only because he or she has no choice. It [67] is that interest which motivates, directs attention, sharpens concentration, and dismisses distractions. An idle curiosity might move us to attempt a problem, but it may not be strong enough to keep us at it until it is solved.

Aristotle starts his Metaphysics with the cryptic statement, 'All men by nature desire understanding.' On that he built his metaphysics; on that we build our philosophy. We presuppose a deep, strong Eros of the mind, an inquiring spirit, and a thirst for knowledge. The stronger, deeper and more intense the desire, the better chance there is that we will persevere in the effort to understand. The questioning dominates the activities of our mind, our drawing of diagrams, our exploring of analogies, our search for relevant data, our testing of hypotheses. The many disparate activities are united by the one desire seeking the one solution. Our searching is purposive, determined.

Archimedes had a desire to know as a mathematician but he had a strong reinforcement of that desire with the request of the king, the possibility of a reward and the fear of failure. It became a deep drive in his psyche, such that, even when he was physically relaxing, somehow unconsciously he was still working on the problem. Tension is created by the elements that do not fit together, the clues that we cannot reconcile, the lines and dots that do not make sense. We cannot tolerate disorder, we do not like a mess, we want to put things in their place. Puzzles annoy and challenge us. This is the period of the struggle, the thrashing around, the frustration, as we explore possible solutions.

When we discover the correct solution we experience it as a physical, mental and emotional release of tension. The joy of insight is symbolized by the naked Archimedes rejoicing in the streets of Syracuse. Or perhaps we can think of Helen Keller, happy for the first time in her life, eager to begin a new day of learning and insight. It is the joy of a desire that is satisfied, that has finally reached its goal. It has its emotional side as frustration, anger, despair give way to joy, contentment, peace in the finding of the solution. It has an intellectual aspect as the mind too is at peace, the contradictions have been reconciled, the clues have been put together, we have hit upon the solution; we cannot comprehend why the others cannot see [68] it. There is even a physical component to such release; the tension of inquiry can disturb our sleep, affect our appetite, and interfere with our digestion.

The stream of images, examples, analogies, similar cases, possibilities that come into our mind are crucial for the generation of insights. Once we have the right image, clue or example, the insight will follow very quickly. The stream of images and possibilities come from our imagination and memory but under the influence of questioning. It is not a random stream. We dismiss immediately images and examples that have no relevance. The relevant image, data, example, come readily to mind for the intelligent person. Concentration means bringing this stream of images under control; distraction means that the images have taken over and we are being led far from the problem at hand. Even before insight occurs, intelligent inquiry is active, sorting out the relevant from the irrelevant, picking out clues, hints, suggestions, disposing the image so that it can become intelligible. Classical Scholastic philosophy recognized this activity as disposing or throwing light on the phantasm preparatory to the act of understanding.

Questioning represents the active aspect of understanding; we are doing the asking, constructing images, searching for relevant examples. We can control our imagination and memory and harness them for the purpose of understanding. Although the particular operations of constructing images and drawing on examples from memory are activities at the level of sense, they have been already been harnessed for the work of understanding. Higher animals have imagination and memory as well as an ability to coordinate responses; but we see no evidence of an ability to control or construct images. What is missing is intelligent inquiry; what is distinctively human starts with this inquiry.

You usually profit from a book or course to the extent that it responds to your personal questions. You will profit from this text to the extent that you are ceaselessly asking pertinent, critical, personal questions. What is he at? Is that true? What is the value of that? Surely that is wrong? How does that fit in here? Philosophy is about asking questions. Sometimes we think of science and philosophy as providing answers and so they do, but there are always further [69] questions, different questions, broader and deeper questions; the questions will always go beyond available answers.

3.2 Insight comes Suddenly and Unexpectedly.

It comes; you cannot force it; you have to wait and hope and keep trying. But when that moment comes, it is something like a gift, an intervention from outside, something that happens to us; we receive it, we are to some extent recipients; we are illuminated. When faced with a problem, we are active in the sense that we are looking for different images, putting different bits of information together, searching in the memory for similar examples, etc. But we are also passive in the sense of waiting, hoping and praying that the insight will come. Aristotle was on to something when he distinguished between an Active and Passive Intellect. We need to discriminate between these two aspects to the act of understanding, the active and the passive.

The act of understanding does not follow automatically after questioning. In this, we can contrast sensation and understanding. Sensation is automatic; if you open your eyes you see. If there is light and it is reflected off surfaces and your eyes are functioning normally, then automatically an image is cast on the retina and the reactions of the nerve endings send signals to the brain. If you are shortsighted or long-sighted, wear spectacles and automatically the image is corrected. If we are not seeing, then something is wrong with the functioning of the eyes, the nerves or the brain. But understanding is different. Presented with a puzzle, it does not follow automatically that you will solve it. Understanding is not like immediate automatic vision of objects. You cannot impose a timetable; you cannot force an insight to come; you can only create the conditions in which it is most likely to occur. Understanding is a highly complicated series of interrelated operations, in some of which we are active and in others passive.

There is a difference between working out a calculation and making a discovery. In working out a calculation, you have a problem but you have clear rules to be followed; if you follow them you will automatically get the correct result. The process of getting the square root of a number is a typical example of a calculation. No [70] matter how complicated the number, simply apply the rules and work out the steps and automatically you will get the result. But that is because somebody in the beginning figured out what were the correct rules and procedures to be followed to get the correct result. We learn the rules, practice how to apply them, and if we follow them we will get the correct result. Acts of understanding are involved; we understand the rules and how to apply them but we probably do not understand why these rules get this result; for that we trust the teacher. But the preliminary exercises we have suggested at the beginning of the chapter are different because you do not know what are the rules to apply; they are not calculations but discoveries. You have to work out which rules to apply; you are on new ground; it is original; there are no automatic results. Discovery is a new beginning. If there were rules for discovery the Greeks would have developed the computer. There is intelligence involved in following rules; but more creative intelligence in discovering what the rules are in the first place.

To say that insight comes ‘suddenly and unexpectedly’ is not the same as saying that it comes ‘immediately and easily.’ Sometimes it can be very difficult and comes only after much research and effort. Whether you get it or not depends not only on intelligence but also on imagination; an intelligent person can be lead astray by expecting something more than is there. There is a certain amount of luck involved in finding the right image, construction, clue or hint. A certain amount of guesswork is involved in choosing what seem to be the most fruitful alternatives to follow up. But even if we stumble on the solution after hours of sleepless work, when it comes, it still comes suddenly and unexpectedly. Even if we do not notice it, there is always this aspect of receptivity, passivity, emergence of order from confusion.

It does not always come when we are studying but, as with Archimedes, it can come when we are relaxing, even when we are walking or playing or working at something else. If the desire to understand, the tension of inquiry, is strong enough, then the unconscious seems to continue to work on the problem and we can wake up in the middle of the night and shout, 'I've got it.' So it is recommended that when you reach a dead-end in some work, you [71] take a break, do something else, go for a walk, come back to it later and try again from a new angle.

3.3 Insights depend on Inner Conditions rather than outer Circumstances

Even though we cannot force insights to come, we can create the conditions in which they are more likely to occur. These are mostly inner conditions in the mind rather than in outer circumstances. Primarily, it means to be continually asking questions; to be manipulating the data in the direction we think the solution lies; to be looking at the problem from different angles, dragging up new images, testing examples, remembering similar situations, exploring possibilities, trying analogies, starting again when we reach a dead end.

Students hear the same lecture but receive it in many different ways. They are in the same classroom, have the same equipment, the same general educational background; they hear the same words but grasp different meanings. There is an old Latin tag, Quidquid recipitur, ad modum recipientis recipitur; whatever is received is received according to the mode of the recipient. It is the inner conditions of interest, attention, questions, images, and habits, expectations, ability, which determine how a lecture is to be received.

The inquiring subject sets the context into which insights will be received. A person who is familiar with mathematics will easily solve mathematical problems but may not be so good at crosswords or trick questions. Someone with a good memory will be good at general knowledge questions but may have difficulty with math or logic. So much depends on previous education, state of development of the culture, age, etc. Many people frequented the baths at Syracuse and experienced the sensation of floating in the water but only one of them had an insight into the laws of displacement and specific weight. [72]

3.4 An Insight Pivots between the Abstract and the Concrete

On the one hand, an insight is dealing with data and images which are concrete and particular: Archimedes had one chalice, one King, and one particular problem to solve. On the other hand, what the insight grasps is an idea, a relation, a universal, a law; and that is abstract. The laws that Archimedes eventually formulated were universal, referring not only to this chalice but also to any other material body immersed in any other liquid at any time or any place. The insight is constituted precisely by 'seeing' the idea in the image, the intelligible in the sensible, the universal in the particular, the abstract in the concrete. We pivot back and forth between images and ideas as we search for the correct insight. First let us now clarify the difference between images, ideas and concepts.

3.4.1. Images

An image is a sensible presentation; the most typical kind of image is the visual image, an imaginary picture. If we imagine a tree, then, it is a mental picture of a particular tree, with a definite size and shape and color. Even if we try to neutralize distinguishing features, still, the image will have a minimum sensible shape and size so long as it is an image. It is the function of imagination to receive images and to produce images of things that do not necessarily exist. An image is a product of the imagination. All the senses give images so there are also tactile images: we can imagine what it feels like to touch a snake. There are images of hearing: when we write music we can imagine what it will sound like. We can study a cookery book and imagine what the different dishes will taste like. We can imagine the smell of freshly baked bread. All these images are sensible, they are visible or tactile or audio, images of smell or taste, products of imagination.

We can construct images of things never seen or sensed. We can combine images such as in a golden mountain or a pink elephant. We can imagine things that do not exist. We can fantasize about the future. We can do thought experiments such as imagining what it would be like travelling at half the speed of light. It is imagination that is imagining, but it is under the influence or control of intelligence, questioning, searching. But imagination must first [73] receive the raw material from the senses; a person totally blind from birth can hardly imagine color.

Animals have imaginations and can store sensible images. Dogs can be taught to recognize and react to the smell of drugs or explosives or to follow a trail. Animals remember places and times. But their imaginations are limited; they do not seem to have free images, to be able to create new images out of the material received in the senses. The human imagination can do this; it can construct new combinations of images, extrapolate from images, visualize new possibilities, write endless fiction. The chess player can look at the board as it is at the moment but imagine it as it will be after three more moves. This flexibility is essential for facility in understanding. We can imagine not only the actual but also the possible, what does not or could not exist.

3.4.2. Ideas

As an image is the content of an act of imagining, so an idea is the content of an act of understanding. An image is concrete and particular; an idea is abstract and universal; they are as different as chalk from cheese. If you take the example of the sequences of numbers given in the exercises you can see that each number is particular; the next number required is particular. But the only way to get what generates that number is to grasp the intelligible relation between the numbers. That is grasped in the formula (e.g. 2x+4); that is an idea. It is abstract, universal. It can generate an infinity of particular numbers but by itself it is an abstract, empty formula. Once we have grasped the idea behind the sequence, we can go forward or backwards as far as we like, but we quickly find this boring and pointless because it is the same thing over and over again. An idea is called abstract or universal precisely because in itself it is not concrete or particular but can be applied to an infinity of concrete or particular instances. The definition of a circle can be realized in an infinity of circles of different sizes and colors. The definition of a human person is a universal but it can be applied to an infinity of particular cultures, ages, colors, sexes.

There are many different kinds of ideas. We have defined images as products of imagination and we will define ideas as products of [74] intelligence. Definitions, laws, relations, intelligibilities, unities, qualities, quantities, classifications, virtues, numbers, are all ideas. Helen Keller had an idea when she grasped the connection between the lines traced out on one hand and the experience of water on the other hand. She grasped a relation between a word and a reality; this relation is abstract. The lines are concrete, the water is particular, but she grasped a relation, by which that word represents that reality. Archimedes grasped a combination of laws operating on all concrete particular things, the laws of specific weight and flotation. Poincare grasped the similarity of one branch of mathematics to another. Huffman grasped the idea of going backwards instead of forwards. There is an almost infinite variety of different ideas. The crucial point for us at the moment is to grasp the difference between images, ideas and concepts.

3.4.3. Concepts

Concepts are ideas that are formulated explicitly and expressed in words or symbols or definitions. A concept is also a product of conceiving. A direct, simple insight into a particular problem usually comes first but then we think about it, put it clearly into words and formulate it as a definition or law. Archimedes had the correct idea of how to proceed when he was in the baths, but only later defined the concept of specific weight and the laws of displacement. Ideas come first but are quickly followed by concepts. Concepts are formulated ideas. Most people have an idea of what constitutes a circle and would be able to distinguish between a circle, an ellipse and an ovoid. But many would not be able to define a circle correctly; when you formulate a definition of a circle explicitly, then, you have the concept of the circle. Both idea and concept are products of intelligence, not of imagination. Both concepts and ideas are universal and abstract. It is intelligence that grasps intelligible relations, meanings, solutions, laws, and expresses them in a formula or definition. We will discuss this activity of formulating concepts and definitions in more detail in Chapter five on Developing Understanding and Formulation.

A concept is only a concept, that is, a creation of the mind in its purposive search for understanding. We can have strings of concepts [75] which cluster together to form a theory or explanation. We could invent concepts at will in the sense of science fiction. But normally concepts are a stage in the process towards knowledge of the real when we need to be quite clear about what we are talking about. Concepts are usually a means towards an end, rather than an end in themselves. Concepts in themselves are simply suppositions. But they can also be the means by which we understand correctly the working of the material world. Any technical discipline requires concepts, but they are not the object of the science in question; they are the means the science uses in order to know the concrete. Idea and concept, being both products of understanding, belong very closely together and hence we sometimes use these terms interchangeably.

The terms 'perception' and 'percept' are used in various ways: meaning either pure sensation, or full understanding, or anything in-between. It can be very confusing when people are using the same word in totally different senses. We would tend to use the word perception to indicate not a pure sensation, but a sensation that is patterned by understanding. Psychologists are fond of certain diagrams which challenge our perceptions; a stairway, which can be perceived as from above or from below; a sketch that can be either an antelope or a bird - see question five of the preliminary exercises. For them, these furnish proof that our knowing is subjective; that the same sensible data can be 'understood' in two different ways depending on the subjective dispositions of the knower. For us, this is simply an example of understanding imposing upon or controlling, to some extent, how we see the diagrams; there is no correct way of seeing such diagrams.

3.4.4. Images and Ideas

The crucial distinction we are making is between images, on the one hand, and ideas and concepts, on the other. David Hume and many empiricists after him use the words image and idea in vague and confusing ways. He defines an idea as a ‘less lively’ image.⁹ Ideas seem to be vague replicas of vivid images; they have less force but are the same kind of thing. His intention was to reduce all the contents of the mind to the level of the sensible. He was trying to say [76] that even the most abstract ideas could be traced to some image and so could be explained in terms of the sensible.

We are using the terminology of image and idea in a strictly defined sense. Ideas are not just less vivid shadows of impressions but have quite different properties as noticed above. Our appeal for the justification of this distinction is our own experience of the activity of understanding. What happens in your mind when you grasp the formula that generates an infinite sequence? What happened to Archimedes when he grasped the concept of specific weight? What happened to Newton when he grasped that a line or motion was a continuum? It is more than imagination. It is intelligence at work. The imagination produces images. Intelligence produces ideas and concepts. Although image and idea are clearly distinct they are also closely interrelated in the process of knowing.

3.4.5. We cannot think without images.

Having distinguished clearly between images and ideas, let us not make the mistake of separating them. It is a matter of common experience that we cannot think with out using images. Spontaneously if we are trying to understand something we appeal to examples, we construct a diagram, we refer to a particular incident; if we are teaching we similarly use examples, tell stories, apply metaphors, draw illustrations. By image here we include the vast store of memories, imaginations and sense data given to us in experiencing. Insights emerge when we question certain aspects of that data. Ideas emerge from the images. They are quite distinct from one another. Yet the development of our understanding continues to depend on appropriate images. It is not as if once we got the idea we were set free from the senses and imagination. The process of pivoting between the concrete and the abstract, the image and the idea, continues.

Images may become more and more rarified but never completely disappear. The mathematician needs appropriate symbols to facilitate his procedures. The Romans used an extremely awkward system of mathematical symbols; it would be extremely difficult to perform operations of multiplication, division, roots, etc using such symbolism. They had to be replaced by symbols which [77] were more flexible, more suggestive, more functional. In the empirical sciences, as we shall see, it is necessary to construct images of atoms, diagrams of forces and vectors, tables and graphs of data. Even in theology it would seem very difficult to think of God without some vague image of light, size, power. In the end we are perhaps left with the image of the word as the peg on which to hang the idea.

The more appropriate the image the sooner the idea will come; we need the images to reach the ideas. We have to manipulate, adjust, add to the images in order to get the insight. Solving problems in Euclidean geometry usually involves a construction, bisecting an angle, drawing a parallel line, etc. This is manipulating the image; when we hit on the right construction, we can usually grasp the solution. There will be much drawing and pencil work in solving some of the problems at the beginning of this chapter, but the act of insight that comes at the end produces, not an image, but an idea. It is intelligence grasping intelligible relations; it is seeing the connection; it is reaching the definition. Newton's laws of motion are a statement about a series of interrelated concepts. They are products of insight. The insights would not have been possible without the experience of motion and the images associated with the experience, but the insight goes far beyond that experience to grasp intelligible relations that explains universal qualities of motion. The laws are universal and abstract. They apply to all motion, of all material bodies, wherever they may be, past, present and to come, no matter how large or small.

Plato seems to have thought that human intellectual knowing was a purely spiritual activity not dependent on the senses or images in any way. To understand was to be able to ‘see’ the intelligible Forms directly and immediately. For him there was no need for this complicated business of manipulating images, asking questions, struggling with the data, testing possibilities, following up leads, and finally getting the point. For Plato ideas did not emerge; they were already there. It is much simpler to think of intellectual knowing as a kind of intellectual seeing, simple, single, immediate contact, but such does not seem to conform to the common experience of the struggle to understand and the complicated [78] interrelationship of activities which seem to be necessary for understanding.

David Hume seems to have held the position that human knowing was simply sensible experience; that ideas are the same sort of thing as images; that intelligence was the same as imagination. All these ideas and concepts we have talked about are merely faint images and they are related together because of certain laws of association. This simplifies matters considerably; we can eliminate the spiritual, the intellect, metaphysics, theology, and anything else we don’t like. But the question is, does this square with your understanding and my understanding as actually experienced in solving a problem or formulating a universal law? It seems, on the contrary, that human understanding does involve images and ideas, imagination and intelligence, the concrete and the abstract, the particular and the universal. It is a complicated process involving interrelated activities; the attempt to oversimplify simply distorts rather than clarifies.

Aristotle held the middle position. There is an active intellect which operates on sensible data to throw light on phantasms to prepare for the act of understanding.¹⁰ Ideas emerge from images; the thinking part, then, thinks the forms in the images. There is a difference between images and ideas, phantasms and forms. But they continue to develop together; for this reason the soul never thinks without images. Aristotle’s account of human understanding is quite complicated, expressed in difficult metaphysical language, but seems to square with the actual experience of understanding better than any other theory.

Human knowing combines elements of animal knowing with intellectual knowing in a new synthesis not just a mixture. It is animal knowing to the extent that it is tied to the sensible and particular by way of the external and internal senses. It is intellectual knowing in the sense that the human intellect can grasp the intelligible, the universal, the necessary, the abstract, the forms. But human knowing is not just a mixture of these two forms of knowing; it is a synthesis in which both are involved in a new unity, a new interrelationship, or interdependence. It is unique; it is complicated. Any attempt to reduce human knowing to the level of the sensible [79] alone will fail, because it cannot account for the grasp of the universal. Any attempt to maintain that human knowing is purely intellectual will fail when we notice the dependence on the sensible and images. Human knowing involves both images and ideas, both the concrete and the abstract. It is, perhaps, because it is so peculiar and so complicated that so many theorists have been tempted to oversimplify and thus failed to give a comprehensive account of human knowing.

3.5 Insights pass into the habitual texture of the human mind

If you have solved one of the puzzles at the beginning of the chapter, it is not easy to forget the solution. Whenever you discover something for yourself, it is not easily forgotten. There is a great difference between understanding and memorizing. Something we do not understand is usually very difficult to memorize; nonsense sequences are the hardest to memorize. But something we have understood becomes part of our mind in such a way that it is difficult to forget.

This constitutes the basis for the possibility of learning. We take it for granted that we can read and do not remember the years of learning involved in acquiring that skill. Learning the meaning of each letter was an effort, and when we finally learned it was an act of understanding the relation between a shape and a sound. Then, we were faced with the further task of associating letters together to form words. This presupposed the identification of the individual letters but called for a further step in associating letters together to form meaningful words. Again, each slow step forward was an insight, which passed into the habitual texture of the mind. Finally, there was the challenge of sentences, long sequences of words linked together to form a meaningful whole. This presupposed the words but went beyond the words. A new insight was required and when it was attained, then we could say that we could read. After years it has become so easy that we take it for granted. But that is only because each of the individual insights have coalesced and become part of the very texture of our minds.

Much the same thing happens in any area of competence when the learning process builds up a context of principles, laws and [80] relations, that are taken for granted as a background when approaching any concrete problem. A professionally trained mechanic sees and hears the same as the lay person, but because of his habitual store of principles and theory can attach significance to data which for the layman is insignificant. Similarly, a doctor, an astronomer, a physicist, an exegete, a historian, possess a store of theory, concepts, procedures which has become habitual.

4. Distinguish Experiencing and Understanding

4.1 Distinct

Our identification of the five characteristics of the activity of understanding has made us aware of the complexity and subtlety of this activity. We have started with direct acts of understanding of simple problems or puzzles. Much remains to be said about developing understanding and knowing but at least we have made a start on self-appropriation. Now we can distinguish clearly between two different sets of cognitional operations or activities, that of experiencing and that of understanding. We will use the image of levels hoping that it will not be taken too literally; experiencing we put on the first level and understanding on the second. By experiencing we mean any activity at the level of the external or internal senses; seeing, hearing, remembering, imagining, feeling, are all operations of sense. These activities of the senses we share with the higher animals. They can be explained and accounted for in terms of animal psychology. In contrast, understanding is a different kind of activity; it is grasping the intelligible in the sensible, the abstract in the concrete, the universal in the particular. It is both active and passive. It involves questioning, struggling, searching for relevant data. Finally it comes; we grasp the idea, the relation, the law. Experiencing and understanding are quite dissimilar; it is impossible to just lump them together; they are distinct activities.

The products of these activities are also distinct; images, sensations and memories are not the same as ideas and concepts. An image is a picture or sensible replica of touch or hearing or taste; it is concrete, sensible, and particular. Our image of a house involves certain dimensions, colors, and shapes. Without some of those [81] sensible qualities there is no image. An idea is abstract; the law of specific density states an abstract relation between weight and volume. To apply the law requires other further activities; the law itself is an idea, the content of an act of understanding. Images and ideas are distinct products of the distinct operations of experiencing and understanding.

Let us explore in what ways these activities are different. The senses operate immediately on sensible objects. The senses require direct physical contact between the object seen and the seeing eye. It is immediate in the sense that there is no intermediary, no other actions involved; it is direct physical contact. Similarly for the other senses of smelling, touching, tasting and hearing. Understanding, on the other hand, is mediated through operations of imagination, memory and questioning. Questions operate on data to make what is potentially intelligible to become actually intelligible. Kasparov spent fifty-five minutes on one move; Huffman spent months working on his problem before he got the simple insight to work backwards instead of forwards. Why does it take so long to understand? What are we doing before it comes? Understanding is mediated by questioning, selecting relevant data, organizing suggestive images, constructing helpful diagrams.

Operations of sense are simple, single activities; understanding is a complex series of interrelated activities finally issuing in insight. In experiencing one sense operates on one object, in one act, and constitutes one seeing or hearing. In contrast, understanding requires many activities in a sequence to produce one act of understanding. Human intelligence is discursive, it is a conversation with oneself; it is a jumble of activities; we rarely go directly to the correct solution; more often we take many by-ways, make many mistakes, get distracted, come back to it again. We tried to reconstruct what was going on in Kasparov’s mind for fifty-five minutes; review your work in writing a paper; the original vague idea, the research in the library, talking with friends, suggestions of the professor, preparing notes, writing the first draft, bring order out of chaos. Why does it take so long to get things clear, to understand? Because it is not one simple, single activity but a complex series of interrelated activities. [82]

Operations of the senses are passive whereas understanding is both active and passive. The senses are receptors. The eyes receive light waves; the ears are designed to receive vibrations; if there is no light there is no seeing; if there are no vibrations there is no hearing. They are purely passive; open your eyes and you automatically see what is there to be seen. Understanding is both active and passive. It is active in the sense of questioning, manipulating, organizing data, etc; but it is passive in the sense that when it comes, it comes; it is a reception. There is the aspect of work, concentration, research, and effort in understanding; but there is also the aspect of gift. It comes; it is received.

4.2 Not separate

Although these levels of activity are distinct, they are not separate; they do not normally operate independently of one another. Often if you claim two things are distinct, it is assumed that they can be physically separated from one another. We are claiming that experiencing is a distinct activity from understanding; but understanding cannot occur without the senses. Let us illustrate some of the many ways by which they are related.

It is possible for a subject to operate at the level of experience alone at least for a time; there is a sense in which experiencing can be separate from understanding. We start at the level of sensation, of experiencing. The child is mostly a bundle of sensations, desires, needs. At first there is no flicker of intelligence. But the child is seeing, hearing, tasting and smelling. Activities of experiencing can occur apart from the operation of understanding.

Understanding presupposes the level of experience. We cannot ask questions, unless there is some content; there must be something about which we ask questions. We ask about what we have heard and seen, imagined and remembered. We do not just understand; we understand something. What is that something and where does it come from? All the preliminary exercises present sensible data; the data have to be seen before they are understood. First there are data that are seen and remembered; initially there is no meaning or sense or explanation for the phenomena. Later, there follows the moment of seeing the connection, the meaning, the solution. [83]

However, in a normal adult the sensitive side of us has been so penetrated and influenced by intelligence that it is quite difficult to identify a pure sensation, i.e. a sensation that is in no way influenced by understanding, naming, defining. Compare a cow and a man looking over a fence. Both have their eyes open, are conscious and seeing in the physical sense. Are they seeing the same things? The same images are being transferred to the brain along an optical nerve in much the same way. The cow sees in a physical sense within the context of its biological pattern of experience and, with its imagination and memory, can perceive and react to things related to its needs and instincts; the cow can recognize grass, smell an approaching fox, hear and fear rumbling thunder in the distance.

The man sees, but he adds identification and naming to his seeing. He sees five different species of trees; he sees the smoke of the village in the distance; he sees the lack of nitrogen in the pasture; he sees one of sheep limping and speculates on the cause. The experience of the adult is so patterned by organizing intelligence, so shot through with definitions, identities, relations, so transformed by the influence of insights that have become habitual, that it is difficult to separate out a sensation that is purely a biological sensation. It is this synthesis of animal experiencing and emerging intelligence that constitutes the uniqueness of human knowing.

5. Heuristic Structure

Plato was puzzled that a person, who genuinely asks a question not knowing the answer, can still recognize the correct answer when it comes to him. How can he recognize that the answer is correct, if, at first, he really did not know the answer? This seems to be a contradiction. Either he knew the answer all along, or he can never find the answer. In the Meno Plato presents the example of Socrates teaching a slave to solve a geometrical problem simply by asking questions.¹¹ The problem was to construct a square that would be exactly twice the area of a given square. The slave had no previous training in geometry; at the beginning, he genuinely did not know the answer. Socrates asks a series of rather pointed questions, and by answering them the laborer was able to discover the solution for [84] himself. He could now recognize that the answer was correct. What is this mysterious process from not knowing, to knowing?

One of the major questions in epistemology is, where do our ideas come from? Roughly speaking, there are three answers. One group says that we always possessed them, we have innate ideas, we were born with them, they were always there; these are usually idealists or rationalists. Another group claims that there are no such thing as ideas, and that what we call ideas are really less vivid sense images; these would be called empiricists. Then, there is the Aristotelian tradition, which claims that the human mind produces new ideas out of images. Let us consider each of these positions in turn.

Plato had to resort to the theory of Reminiscence and Innate Ideas in order to solve this dilemma. He had to suppose that we already know all the answers, already have possession of the Forms, from a previous life; but now that we are embodied, this knowledge has been hindered, buried, has become unconscious. The learning process is simply the unconscious becoming conscious on the occasion that the right question is put, or the appropriate teaching is presented. He had such a high notion of the immutability and infallibility of intellectual knowing that he could not make it dependent on sense objects which are always changing.

For the empiricists the difficulty is that if you admit of truly universal and abstract ideas it is only a short step to admitting a reality which is beyond sensation. But their position is built on the premise that there is only sensation and the sensible; therefore, ideas must be the same as images and these come from sensation.

Our answer, the third position, is that we can generate ideas out of images. There is a structure in the mind which enables us to move from images to ideas, from the sensible to the intelligible, from the unknown to the known. We call it a heuristic structure. 'Heuristic' comes from the same Greek verb which Archimedes used in his exclamation of joy at his insight, 'Eureka,' I have found it. A heuristic is a device that helps us to find something, to move from the unknown to the known; it gives direction to our search. It is an anticipation of the known while it is still unknown. What is this heuristic? [85]

At the most general level, we have already identified the activities of experiencing, questioning and understanding. We can do self-appropriation and recognize in ourselves the process from images to ideas, from not knowing to knowing. We can recognize a direction, a movement, a method in the unfolding of knowing. We have recognized many different mental activities going on in the mind, when we are understanding and particularly emphasized the three categories of activities, which we have called questioning, experiencing and understanding. These are not random activities. The first manifestation of intelligence is in the purposiveness of asking questions. But questions have to have a content, are directed at some data given in experience. The aim of the question is understanding; that is reached when we have the experience of insight. Is there a method to guide us from the question to the answer? In the most general sense we can answer yes; it is not a set of rules but general guidelines which orientate our search. We have identified many different mental activities going on in the mind when we are understanding. We can distinguish striving, remembering, writing, drawing, imagining, feeling, following clues, exploring, separating, rejecting, trying again, giving up, etc. But these are not random activities. There is a basic method in our madness, there is a direction in which we are moving.

The unknown is never completely unknown. We can ask questions about it so we must know something. A question represents a combination of the known and the unknown. The more clearly we can formulate our question the nearer we are to answering it ourselves. The first step would seem to be to give the unknown a name; naming seems to be the first and simplest kind of insight. We do not know what it is but at least we can refer to it and talk about it. This technique is particularly effective in algebra where the standard method is to let 'x' equal the unknown. But it also occurs in the empirical sciences; instead of 'x' we tend to use such a term as 'the nature of.' What is the nature of fire? What is the nature of a free fall? What is the nature of AIDS? We have a name but we do not yet know the intelligibility or law to which it refers.

The second step would seem to be to sort out and relate the knowns and the unknowns. What do we know about the unknown? [86] What do we want to know? What are the terms of the question? Write down all that you know. Is there any way that information can be manipulated in order to reach the unknown? In algebra the technique consists of forming equations; once you have a sufficient number of equations, combine them and find the value of 'x'. In the physical sciences it will be more complicated but the same process is at work. Much is known about AIDS but there are also many unknowns; call conferences to share what is known, isolate and pin down what needs to be known and how to find out. Be clear about everything that is known. State as clearly as possible the unknown that is being sought. Combine that with what is known. Where is the significant new data to be found? There will be many different techniques for finding further information; many complicated calculations involved; many sophisticated instruments to be used. But the guiding heuristic is the same. This is the period of struggle, of manipulating the data, of waiting for the insight to come, of creating the conditions in which it is most likely to occur.

The third step is solving the equations, doing the necessary observations, working out the solution. The example of the addition sum in the preliminary exercises is a good example where you must write down all that you know about each letter. Be perfectly clear about what you know and what you don’t know. You can infer certain conclusions from the laws of addition, i.e. an even number plus one gives an odd number; any number added to itself gives an even number.

There are many specific methods proper to each discipline; we are often taught the rules of how to proceed, for instance, to identify an unknown chemical substance. Some disciplines will favor deduction, some induction; some will favor analysis, some synthesis; some will favor observation, others experiment. The basic heuristic we are identifying underlies them all. A heuristic guides our questioning of data to a fruitful and correct understanding. We have seen that this process is not automatic and cannot be the subject of rules. But there is a direction, a guidance, an orientation that can promote the emergence of the idea from the image, the insight from the phantasm. [87]

Algebra is a particularly clear illustration of heuristic techniques. The basic technique in algebra is to give the unknown a name; let x equal the required answer. Now that it has a name it can be manipulated. Combine it with the data given in the question. Imagine you knew the answer, what could you conclude? Get equations containing x. Solve the equations. If Fred leaves the house at three o'clock traveling at fifty miles an hour; and Bob leaves the house at five o'clock traveling at eighty miles an hour in the same direction, at what time will he overtake Fred? Let x equal the number of hours after three o'clock that they will be traveling. Fred will be traveling at fifty miles an hour for x hours and therefore will cover a distance of 50x miles. Bob will be traveling at eighty miles an hour for (x-2) hours and therefore will cover a distance of 80(x-2). But they both travel the same distance, therefore 50x = 80(x-2). Solve the equation and get the value of "x".

The great mathematician, G. Polya, specialized in heuristic techniques for solving mathematical problems.¹² He taught mathematics not in terms of rules, but in terms of general strategies to be implemented to work towards a solution. He formulated his broad strategy in four maxims. First, understand the problem; analyze what you given and what you are looking for; be clear, write it down. Secondly, use experience from related problems to devise an attack; techniques which worked elsewhere might work here; identify similar problems; remember previous successes and failures. Thirdly, carry out the attack. Apply the techniques, do the calculations, work out the equations, find the coefficients, differentiate, integrate, etc. Fourthly, ask yourself whether you really believe your answer. Check backwards, verify, look for mistakes, and search for loopholes. This is the bare bones of a heuristic which need not be confined to mathematics.

Empirical scientists use the same technique even though they may not be aware of it. Galileo was searching for the nature of the free fall. What were the mathematical relations that pinned down the nature of a free fall? Well, set up an experiment to find out. Chart the series of results of the experiments. Get the insight that satisfies all the relevant data. Check the insight back against the data; repeat the experiments. [88]

There is a general heuristic involved once we start asking questions; the question suggests the direction in which to look for images, data, further information; the question sets the criteria that decide whether it has been answered correctly. There is often a scissors-like movement in the unfolding of understanding. On the one hand, we may be helped by more and better data, so we look up encyclopaedias, perform experiments, observe more closely. On the other hand, we need hypotheses, ideas, possibilities; we have to think as well as look for more data. At some moment the hypothesis fits the data and we have it.

Let us note, then, that there is a method implicit in these activities of questioning, experiencing, and understanding. There is the questioning, the initial puzzlement, the grasping of the problem, the challenge to understanding, the unrest of an unanswered question. This sets off a striving, a determination, and a stream of activities aimed at a solution. Our external senses are involved in looking at the book or blackboard, hearing the challenge, writing it down on a piece of paper, and beginning to draw lines, fill in spaces, doodle. Our memory is involved as we try to remember similar examples, similar problems, analogous puzzles, previous experience of dealing with this type of challenge. Our imagination is involved as we try to manipulate the data, construct better diagrams, write down all we know, dissect the question to sort out the knowns and the unknowns, imagine possibilities. Finally, hopefully, comes understanding, the breakthrough; things fall together, the excitement of seeing the point. But that is not the end, as there remains the task of formulating the insight and checking the solution against the data or against the criteria set by the question.

It seems then that our minds are dynamic, the source of new ideas, concepts and theories and that these are derived from the matter of sensation. The mind is a heuristic structure by which it can proceed from the unknown to the known, from images to ideas, from the vague to the precise. The dynamic aspect is represented by questioning which points us in a direction, sniffs out clues, suggests possibilities and finally hits on the solution. There is a general method implicit in the very structure of human understanding. We can work out solutions for ourselves; we do not have to wait to be [89] told; we do not need to appeal to the Authority of Aristotle or anybody else; we can recognize when we are right and we can learn from our mistakes.

We reject Plato’s innate ideas and reminiscence because it does not seem to conform to what actually happens. Why should we be obliged to call upon such unlikely and complicated presuppositions to account for the activity of understanding? Intellect does work in tandem with sense and ideas do emerge from images. We disagree with the empiricist position that ideas are the same as images. A little self-appropriation seems to indicate that they are quite different. Our position, then, is that human knowing can be progressive and cumulative. We can start with primitive insights into data; we continue to differentiate and develop as our education makes possible; there is no end to the cyclical development of questioning, sensing, observing, understanding, reflecting and, finally, knowing.

How then do we recognize an answer as correct if at first we did not know the answer? The question sets the criteria which must be satisfied if the answer is to be correct. You recognize that it is correct precisely because it satisfies these criteria. You check the correctness of the solution by working backwards to the question to see if it is satisfied. There is a closing off between the question and the answer. This checking procedure and other aspects will be explored in detail later when we deal with reflective insight.

In conclusion, then, this is our description of the experience of having an individual insight. We have started the work of intellectual self-appropriation. Already, I hope we have made important discoveries about the working of the human mind, of your mind. It is just a beginning and there is much more to be explored. It is enough for the moment that we recognize this activity as it occurs in our own consciousness; it is enough that we focus our awareness on this activity to see how it unfolds over time. It is an activity in which we are constantly engaged. It is the activity of grasping sense from nonsense; separating what is important from what is unimportant; distinguishing the essential from the non-essential; picking out the relevant and leaving behind the irrelevant; focusing on the significant and ignoring the insignificant. It is intelligence [90] that enables us to do these mental activities. The intelligent person does them easily, quickly, expeditiously. The unintelligent person does them with great difficulty, great effort and often with numerous errors.

Comments on Exercises.

(1) One straightaway assumes that the 'tail' of the fish will become the new 'head'. But it doesn't take long to see that that does not work. You have to force your imagination to consider other possibilities. Then it becomes easy.

(2) Looks deceptively simple but it doesn't seem to work. Understand the rules correctly: no going over lines once they are drawn; lines must be drawn continuously. It's impossible. We usually presume we must stay within the 'box' of points; but nobody said that the lines must remain within those parameters. Then it becomes reasonably easy.

(3) This necessitates a series of insights. Very clear diagrams are necessary to distinguish what you know for certain and what you are experimenting with. Write down everything you know clearly. The key insight is the peculiarity of the fifth vertical line: how can you add E to O and still get O? What number can you add to another number to leave it unchanged? (Don’t get confused between the letter O and the number zero.) At a certain point you may have to switch from deducing, to assuming something and trying it out

(4) (a) The answer is 35. The formula generating an infinity of numbers is (x + 10).

(b) The answer is -5. The formula is (x - 4).

(c) The answer is 16. The formula is (X x -2).

(d) This is more difficult as it combines both multiplication and addition. The answer is 79. Two formulas generate this sequence, (3x + 4) where x represents the value of the previous number in the sequence; or (3ⁿ - 2) where n is the number of the term in the sequence.

 

(5) You might be able to shift from one perspective to another.

(6) Even though we are understanding all the time, it is quite difficult to isolate one act of insight and identify its characteristics and genesis. But it is important to find your own examples in your own experience. The more detail you can provide the better.