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”.