Non-Linear Thinking and Structures

The Design and the Designer – Part 5

In the last post, I made a brief mention of Nasruddin, Heuristics and Non-Linear Thinking. I shall continue in this post on the same theme and establish the role of semantics in Design and Non-Linear Thinking, and explore some techniques of practicing non-linear thinking.

First, here is a very brief synthesis of the previous posts on this subject: In the previous posts, I applied law of 2 and tried to demonstrate the underlying unity of the theme in various disciplines – which is referred to as Plausible and Demonstrative, Creative and Logical, Synthesis and Analysis, Right Brain and Left Brain, Linear and Non-Linear, Heuristic and Rational and so forth. Largely, the conclusion that one can reach is that there are two distinctive types of thinking – which are interdependent on each other, but at the same time have very different functions and applications. We termed the functions of these two types as – recognition of problems and finding solutions to problems. We also presented a few laws of synthesis – from various fields – particularly societies, management, computing science and consciousness.

Before we get into Semantics, let’s examine linear thinking and non-linear thinking in some detail – with some examples and techniques of how to switch from one to another.

Logic – at least the way Socrates and his disciples developed it – is largely linear. This is carried over into mathematics as well. Logic mostly depends on reasoning from evidence. The early Artificial Intelligence (which is neither artificial nor intelligence) applications ran into some problems with the linear reasoning and there are now several attempts to develop non-linear reasoning systems. Such logics in AI are called “non-classical” reasoning systems. Here is a classic example of linear and non-linear reasoning – it involves a very well known logic puzzle.

Three monks and three cannibals were traveling together because of some strange circumstances – even though naturally there cannot be any friendship and trust among such a diametrically opposite groups. If at any time, the cannibals outnumber the monks – they will eat them. Now, they came to a river bank and they had to cross the river. There was only one boat. The boat can only carry two people at a time. Now, your problem is to devise a strategy to transport the monks and the cannibals safely across the river.

The usual linear reasoning system will start with River Bank-1: 3M, 3C, River Bank2: 0M, 0C as the initial state of the system, and will try to devise a series of moves that will establish the desired final state - River Bank1:- 0M, 0C, River Bank2: 3M, 3C. It is a simple puzzle – you start with something like first two cannibals will travel, one will come back, two monks will then go etc., and you can easily arrive at the final state.

Suppose you give the problem to Nasruddin, what do you think he will do? He will listen to your descriptions, and even before you have completed your narration, he will jump from his chair and shout with the excitement of a child “I know, I know - the monks will take the bridge, and the cannibals will take the boat”.

It is a perfectly valid solution to the problem – isn’t it? Then you tell him “look Nasruddin, I did not say that there was a bridge”. Nasruddin will reply, “Well – you did not say that there was no bridge either”. Now, you modify the problem description, and you will add another constraint to the problem – there are no bridges. “Well, in that case, the monks take the helicopter” will be Nasruddin’s answer. By this time, you get very frustrated and shout back at him – “I want you to solve the problem – not avoid it”. But, for Nasruddin, there was no problem there to be solved. Can you see how he eliminates the problem instead of trying to solve it?

No matter, how hard you try, you cannot contain Nasruddin. He will always think of exploiting some constraint that is not included in the problem definition and use that as a means to eliminate the problem.

The AI community in the last thirty years tried and discovered several techniques for “programming Nasruddin type behavior” into the computer programmers – with some reasonable success. I have no intention of boring you with the Non-Monotonic Reasoning systems here. But what is the technique of Nasruddin?

One, Nasruddin does not depend on “evidence” as the only way of establishing the existence of something. All formal logical systems depend on evidence as the only means of establishing “truth”. For example, if I tell you that all visitors from Canopus are fools, and you know that Mr. X is a visitor from Canopus, you can confidently establish the truth that this particular Mr. X is a fool. But, there is a catch. Suppose, if this particular Mr. X is not a visitor from Canopus, then what are you going to do? You cannot establish anything. He may or may not be a fool. You can conclude Mr. X’s foolishness, only if you know that he is a visitor from Canopus.

We need facts to recognize truth, but Nasruddin does not rely on such petty things like facts to recognize truth. According to Nasruddin, some thing does not have to be a fact for it to be true. Kurt Gödel proved this mathematically – it is known as Gödel’s incompleteness theorem. The incompleteness theorem states that in any formal system, there always exists a statement which is “known” to be true, but cannot be proved.

Rabindranath Tagore – with the brevity that a poet alone can command – says it beautifully.

“If I say that the earth is flat – you inform me that my near sight is false.
If say that the stars are the fireflies attracted to the moon, the candle of the sky – you inform me that my far sight is false. My dear logician – in your presence, I prefer to be blind”.

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The second principle that Nasruddin mastered is the System’s Principle – understand the problem by studying its relationship to the larger environment in which it is only but a part. I wrote about this principle in part 2 of this series of articles.

In the specific case of monks and cannibals – all Nasruddin does is to bring his knowledge of the larger environment to solve the problem. The minute you tell him about the river and the boat, he can think of a bridge. He can bring his knowledge of traveling in the air, or swim in the water. Or, he will say that the monks – using their magical techniques – will vanish and reappear on the other side. Or, he will say that the monks will use their secret herbs and put the cannibals to sleep. His possibilities to come up a solution are infinite.

The study of system level relationships requires knowledge of semantics – or the “meaning” of things. There are basically two types of relationships – structural and semantic. It is important to distinguish between these two types of relationships. Though, in logic and other sciences, we are mostly taught how to study the structural relationships, in fact, our mind works naturally very well with semantic relationships.

A few examples from various fields may be in order here.

In language, grammar defines the structural relationships. In English, a valid sentence must have a subject, a verb, an orbject/predicate. And, there are various rules that define the relationships between various parts of speech. These relationships are basically structural – they govern the structure of the sentence. We can say that water is triangular – it is a grammatically valid sentence, but completely meaningless.

In information systems there are basically four types of structural relationships. By information systems I do not mean just software applications. Any system that organizes knowledge and depends on that organization is an information system. By this definition, societies, cultures, various disciplines of study are all information systems.

The first – and simplest structural relationship is hierarchy. This relationship in information systems is called “IS-A” relationship. A “IS-A” B. This could be interpreted in several different ways:

• B is the parent of A
• B is the boss of A
• B is part of A
• B is subsumed by A
• B is the root of A

Entire systems, organizations, societies are modeled based on this one particular relationship. Most hierarchical structures use this relationship as the primary relationship. The different sections in this blog-page have a parent-child relationship with the master page. There are basically five sub-sections – header, footer, posts, author information, archives – each of them is a “child” of the main blog page. Similarly, many corporations have a hierarchical structure – with the owner of the corporation at the top. The “semantics” of this relationship connotes ownership and protection.

The second structural relationship is network relationship. In such a structure, elements, or objects at the same level can be connected. This is called “sibling” relationship. A “IS-A brother of” B. The relationship basically means that two things are at the same level. It connotes competition and/or collobaration.

There are two other types of structural relationships in information systems hyper-text and groups. Both these relationships are somewhat non-linear relationships, but nonetheless, they are structural relationships.

The Set, and Hyper-Text are more complex structural relationships than a simple hierarchy and a Network. A set uses a function to create a structure. For example, you can say that all integers belong to a set. You can then define a function that determines an integer and therefore, the membership to the set. The members in the set may not have any particular relationship – they have something in common – that’s all. This is how most social groups are formed. The function gives an identity to the set and therefore to all its members. All people “born” in US are “Americans”. “Being born” in the US determines the identity of the set called Americans.

A Hyper-Text relationship is much more interesting relationships. It does not relate two nodes, but instead it relates content from one node to another node. All web pages use this structure extensively. For example, I can provide a link to an article in wikipedia in one of my articles in this blog. This does not mean that this blog as a whole and wikipedia share any relationship; it also does not mean that this article and the wikipedia share any relationship, and it also does not mean that this article as a whole has any relationship to that article in wikipedia. This relationship connotes friendship. There is no function, there is no responsibility, and there is no ownership in this relationship.

Using these four relationships, we can explain almost all “order” in our world. We defined the basic family relationships (parent-child, sibling), groups and friendships.

So, where is the place for Semantic Relationships?

We conveniently left out the most important, most complex, wonderful, problematic and the most beautiful relationship of all– that of the lover and the beloved. None of the above relationships explain that relationship – do they? What is the relationship of a husband and wife? What is the relationship of a teacher and a student? Friendship comes closest, but these relationships transcend friendship.

This is the realm of semantics – which is the topic of next post.

Non-Linear Thinking, Nasruddin and Polya

The Design and the Designer – Part 4

In these series of articles on Creativity and Design, my intention is to explore if there is a “formal” model of creativity. I am not interested in the psychological aspects of creativity, how it works and so on. The Psychology of discovery and invention is wonderfully described by Mihaly Csikszentmihalyi in his classic book called Creativity.

My interest is to discover a method of consciously practicing it. In the last few posts, I tried to provide a basic framework that I am working on. Basically I am trying to bring together all my experience under one unified theme. Design is the name I gave it.

I studied mathematics and more importantly I studied how to ‘do’ mathematics. I studied computer science and information systems – I designed some very large and complex software systems. I studied design theory – product design, appliance design, communication design, aesthetics, user interfaces, ergonomics etc. I studied philosophy – Western Philosophy, Indian Philosophy and Sufi Philosophy. I studied systems thinking. Basically I am a problem-solver. Given any problem, I can come up with some kind of a solution. I met many people who are fantastic problem solvers. Many of these people can almost instantaneously identify a line of attack and can come up with a solution almost immediately.

In my experience, problem-solving can be taught and can be learned – even though we generally think some people are gifted with this ability. But, teaching problem-solving has been one of the toughest problems for the teaching community. There does not seem to be a discipline for teaching problem-solving. Even the most beautiful subject – mathematics – has not addressed this problem.

How are we taught mathematics? The mathematics teacher presents the proof of a theorem in a step by step manner. First, he defines the theorem, and he has the proof in front of him, and all the teacher does is to explain the ‘logical flow’ of the proof. In the course of several years, various problems and solutions are demonstrated as examples, and somehow the student is expected to ‘understand’ problem-solving from these examples. There is no conscious attempt at teaching problem-solving.

The ‘doing’ of mathematics is not taught. This is what makes Mathematics difficult to learn for many people. The best reference works ever written about this are Polya’s books on mathematical method: How to Solve It and Mathematics and Plausible Reasoning. For some strange reason – these books are not part of the mathematics curriculum.

Polya makes a very useful observation. According to him – rightly so – the process of problem solving is a heuristic process. The difficult part of problem solving is to ‘recognize the problem’ correctly. This recognition is not a “rational”, “logical” process. It is very irrational – if you are good at it – you can immediately recognize what the problem really is in an instant, and you decide on the line of attack. How do we normally recognize the problem? We generally use a set of heuristics. For example, you may recognize that the problem is “similar” to another problem you solved before, or you may recognize that the problem belongs to a class of problems that are already solved and so on. These are all heuristics.

Here is an illustration of how heuristics are applied. Archimedes first discovered the formula for calculating the area of the circle. The area of the circle is PI*r2. In Mathematics texts, the proof is presented using some complex coordinate geometric equations. Archimedes proof is rather very simple and can be explained in one line. He drew several lines from the center of the circle to its circumference. Now, the area enclosed by two such lines and the arc of the circle looks like a triangle. The area of the triangle is (base*height)/2. Now, the height in this case is r. And, the base of all the triangles put together is the circumference of the circle – which is 2*PI*r. Therefore, the area of the circle is PI*r2. Beautiful – isn’t it?

A mathematician will not agree to a proof like this – because, the way Archimedes divided the circle into different regions are not exact triangles – they only look like a triangle. Archimedes used a heuristic.

Now, let’s look at the process that Archimedes used to solve this problem. First, for Archimedes, the problem of calculating the area of a circle is a real problem; it is not a theoretical problem. He had to calculate the area of agricultural land for calculating taxes. Second, given a particular circle, he knew how to come up with answer. So, he has some data in hand. He has the areas of different circles in front of him and he was looking for a common “pattern” that explains all the answers. Third, he knew that the two fundamental properties of the circle are its radius and its circumference. The last step is the important one. He asked himself a question – can I formulate this problem in terms of problems that I already know how to solve? In other words, he knew how to calculate the area of a square, a rectangle and a triangle. Now, he is trying to “reduce” this problem to the problem of a square, a rectangle or a triangle. He also knew another important fact about areas – they are always expressed as a product of the two different lengths. He used four different heuristic techniques – generalization, analogy, reduction and induction.

Polya calls this type of reasoning “plausible” reasoning as opposed to “demonstrative reasoning”. Different disciplines may have different names for this. Psychologists call it right brain thinking, designers call it creativity at work, scientist call it intuition at work, systems people call it “systems thinking”. In popular literature, this is called “non-linear thinking”, “out of box thinking” and by many other similar names.

No matter what name you give it – the process of recognizing the problem, and the process of actually solving it involve two different kinds of thinking altogether.

Polya’s books are the best reference works on this subject. He tried to formulate a systematic method of “problem-recognition”. Unfortunately – thanks largely due to the American experiments with modern mathematics – the world today suffers from cultural isolation of mathematics. Polya’s achievement is remarkable. With the precision of a mathematician, he formulated a “dictionary of heuristic”. This means – if you have some patience – you can learn the heuristic thinking like learning the vocabulary from a dictionary.

Another dictionary of heuristic that is particularly enjoyable to read is the collection of Nasruddin Stories. These stories help us to break our usual linear and cause-effect thinking. Even just remembering the stories has the remarkable effect of constructing a different kind of associative memory. The stories ‘just’ pop up by themselves when we need them most.

More than 700 tales of Nasruddin are collected by Idries Shah and are published by Octagon Press. Here is one story of how Nasruddin describes the “two modes of thinking” that we have been talking about:

One day Nasruddin stormed into the tea house and announced with lot of excitement that he discovered a very important truth.

People asked him what it is.

Nasruddin said – “The moon is more useful than the Sun”

Everyone was taken aback, and asked him why he thinks so.

“Because we have more need for light in the night”

Nasruddin Stories make sense at many levels, and it is said that there are at least seven interpretations for each story. Here is one interpretation that is relevant in our context. Moon is the agent of synthesis. He synthesizes the sun light and reflects it back to earth. Sun is the source of light – when he is there – we have no need for light because it there everywhere. Your need is more when something is not available – right?

There are people who are experts at synthesis. Polya synthesized mathematical method, Will Durant synthesized history, Ackoff synthesized management, and Christopher Alexander synthesized Architecture. I am trying to synthesize Design. In order to synthesize a domain, you need to be a super-specialist in that domain and you need to be able to absorb the entire domain completely.

Both modes of thinking co-exist and compliment each other. I am not indicating that synthesis is more “important” or “superior” to analysis. Like the Sun and Moon, they are interdependent on each other. The linear and the non-linear, the creative and the logical, the plausible and the demonstrative co-exist together.

In order to synthesize Design – we need an understanding of Semantics. We have to understand the meaning of things, and how different things relate to each other.

This will be the topic of the next post in this series of articles.

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Nasruddin on cause and effect:

One day Nasruddin was waling along a street with his students. As Nasruddin was walking past a two story building, a man fell down from the first floor – and he fell on Nasruddin. Nasruddin’s neck was badly hurt.

His students asked him what lesson they can draw from this incident.

Nasruddin was in pain and he was now angry at such stupid questions. He shouted – “fools – can’t you see? He falls from the building, and it is my neck that is broken”.