0. Introduction (from imitation to understanding)

0 - 4: So far, we’ve learnt what subject and object are and how brain can create a concept based on abstraction. Through the last post the process of creating concept of red in a kid’s mind was explained. It was mentioned that we can’t transfer a concept; like agriculture, we cultivate it; and then we hope that the kid would understand it. If not, we need to change our method, changing the examples probably, until the kid could conceive it.

However, when we teach a concept to somebody, especially kids, before they understand it, they imitate how to apply it, and they might be trained to use the concept just by imitating.

Let’s study a case: you have a bunch of keys and locks. You teach a kid that key A can unlock lock 1, key B for lock 2, etc. Then the kid learns how to use the key to unlock the locks. Who can say if the kid was imitating because he memorized the correspondence between keys and locks or he understood the concept of specific (sometimes unique) key for specific lock? Nobody could answer the question, sometimes understanding a concept is not apart from its application. However, we can test the kid by giving them a new bunch of keys and locks and ask the kid to unlock them. If the kid tries driving every key through every keyhole one by one, we can say that they couldn’t understand the concept which is the key must fit into the keyhole, so their shape and size must look alike. If the kid just tries the key and keyhole with the same size, they are solving the problem intelligently, it means that they’ve understood the concept of key and lock. So we can conclude that unfamiliar problems can show how deeply people have understood a concept.

I don’t believe in categorizing people. I believe in a continuum such as a spectrum to show the diversity of thoughts and actions. To show people’s capability to solve a problem, I use the gradient of blue.

For the end points of this bar, I use operator and engineer. I redefine these terms.

Here, operator means somebody who solves problems exactly by imitating. Operator follows the instruction given and doesn't think about it personally. They don’t have the slightest idea what they are doing. On contrary, engineer means someone who can solve problems using their own intelligence solely. They can design or fix a system because they have understood how it works. They are fully aware of what they are doing step by step.

In real life, nobody is an absolute operator or an engineer; however this gradient can help us to determine how far we are from each endpoint, to see whether we’re closer to either engineers or operators. Normal people use combination of others ideas and their own intelligence to solve problems. Relying on your intelligence exclusively is as foolish as implementing what other people have prescribed without a single thought.

Jean Piaget, the Swiss psychologist and philosopher, stated that intelligence develops to adapt the humankind to the world. Two complementary agents of this process are: assimilation and accommodation.

When our mind processes a new object, it uses the information stored in memory to analyze it: at the first place to find the similarities and differences. Assimilation tends to fit the object into the subjects that our brain has created, so it tries to find the similarities. Imagine that you had seen a horse, then you saw a donkey for the first time, based on the concept of horse created in your mind, you might compare the horse and the donkey, and might find similarities, here assimilation tries to fit the donkey into the image of the horse, so you might conclude that this animal is a horse. The reason that kids confuse naming the things around can be assimilating it with something in their mind.

On the other side, accommodation tends to change the block of information that our brain has created to fit them into what we perceive. If we examined the donkey more closely, we could find the differences between donkey and horse, so donkey doesn’t completely fit into the concept of horse, therefore our mind creates the concept of donkey. That’s how it fits the subjects into objects.

Here assimilation looks clumsy. Obviously donkey is different from horse. Now let’s study another case.

If there were no shoes, accommodation would tell us to walk on the smooth paths without sharp or spiky stones, we would adapt ourselves to the ground we would walk on. If we hadn’t had assimilation, nobody could have invented shoes. By inventing shoes, humankind changed the world around to fit it into their image (previously the concept of a cover to protect the feet must have been created in the mind, then humankind tried to make it come true).

Now it seems that assimilation is more attractive. It is the power that scientists, philosophers, inventors, and leaders use to change the world based on what they have had in their mind. However, assimilation without accommodation can be dangerous and destructive. Don Quijote is the most dramatic example of a person who just assimilated. He fit the reality into his imaginary world. That’s why he fought a windmill which he imagined to be a giant. In English there’s an adjective “procrustean” derived from Procrustes (Greek Mythology) who stretched the victims or cut their hands or legs to fit them into his bed, so this adjective is used for person, attitude or solution which uses other ideas but fit them into their own philosophy. The dictators are other examples of uncontrolled assimilation; and religious people who try to interpret everything, especially new discoveries, to fit them into their beliefs. So you see that the only difference between the insanity and innovation is the way we use accommodation. If Einstein’s theory hadn’t been verified by experimental findings (reality), everybody could have assumed him as an insane for the relativity theorem. 

Piaget mentioned that intelligence develops if assimilation and accommodation complement each other in a balanced way. If assimilation dominates, the brain tends to the game, if accommodation dominates, the brain tends to the imitation.

In my opinion, dominance of either assimilation or accommodation takes place while the other agent can’t develop accordingly. Brain evaluates itself statistically. If  kids are forced to do what they ought to, or their thoughts or opinions are belittled or devalued, in their mind assimilation doesn’t develop as much as accommodation, so they fall in imitation, consequently being a good follower. When we don’t tell our kids the reason of the expectations or duties, we don’t give them the material to think of, therefore both assimilation and accommodation can’t grow.

When we teach numbers to 5 year old kids, we can’t explain what the number is. We just teach them how to use them to count, then step by step other arithmetic operations come after. Like teaching words, we don’t tell them that orange is a noun and eat is a verb, we just teach them how to use the words appropriately. Kids ought to follow what teachers or parents ask them to do. This teaching method is supposed to transform to a conceptual method, and if it doesn’t, kids will fall in imitation. In mathematics if they’re just required to solve problems correctly, regardless of understanding it, kids don’t learn to think independently and critically. That’s why the problem arises in math when they go to higher classes where they are expected to use the concepts that they were supposed to create previously.

Based on my experience most of the kids who had problems with math told me that they loved it in elementary school or in the first or second year. Why? Because it was easy for them at that time like the key and lock problem, and their brain had developed enough to cope with addition or subtraction. But they just learnt how to imitate the approaches teacher applied to solve the problem without understanding it. Now the solutions can’t be applied to solve unfamiliar problems.

In conclusion, imitation is the first step of learning, but we are supposed to leap from imitation to understanding which can’t be achieved in one step. Concepts can grow, or they can become more profound when we move forward. When you walk through a city, you just see the houses and shops and streets. If you climb a mountain next to the city, you can see how the houses and shops and streets are related to each other. So in your mind a bigger integrated image is created (what maps can do for us). When you climb higher, you can see more of the city; likewise, the concepts can grow in our mind when we find the relation or the interaction between concepts. Therefore, we should equip ourselves for this journey.

0. Introduction (concept)

0 – 3: After abstracting the images, brain groups them based on at least one similarity. Each group is called a concept. In other words, concept enables us to classify things around us.

For example: Tree is a plant with an elongated stem, or trunk, supporting leaves or branches (Wikipedia), so tree is a concept for a group of plants having characteristics above-mentioned. 

So dog is a concept, doctor is a concept; eating and liberty are concepts too.

Concept is totally subjective (exists in mind). Can you portray tree? You can portray one example of trees not the tree as a concept.

Besides, one thing can be categorized based on different concepts. A man can be a father, a doctor, a citizen, a costumer and so on.

Sometimes, we create a concept for other concepts. For example concept of motion for running, walking, falling, flying and so on. So the concepts are developed or built upon other concepts. Another example for hierarchy of concepts is classification of animals.

We are supposed to know the basic concepts to conceive higher ones. That’s what we do frequently in mathematics, for example the concept of exponent  had been developed to negative or fractional exponents. Nobody can understand the concept of negative exponent unless they have understood the concept of natural exponents. 

The more we abstract, the higher concept we create. To create a concept for Oedipus the King and Macbeth, one can classify them as drama related to kings, however if we abstract it more, we’ll see that the similarity is a failure regardless of being a king or an ordinary man, so the way they fail, makes a new concept called Tragedy.

Through the next posts it will be discussed how we create concepts in mathematics and how we create higher concepts out of basic concepts.

The last point about concept is that the basic concepts can't be defined, however our mind can conceive them. For example, nobody could define time. You can set a unit for time, but can you define the time itself? (If you use words such duration, to define duration you will need concept of time, so there's a loophole which doesn't lead us to any form of defining time)

We don’t need a definition for time. We just use its concept in our daily life. When you teach the concept of red to a kid, you might show them a red apple, a red rose, a red curtain and you repeat the words red apple, red rose and red curtain, WE HOPE THAT the kid’s mind would grasp the similarity which is RED. Then the kid tries to apply the concept to name the next red object such as red tomato, and you show that it’s correct or you give them a reward, so your approval consolidates the concept of red in their mind.

As long as we understand a concept and we know how to apply it, we don’t need a definition. A swimmer might not be able to define swimming, but they've understood it physically and mentally.

In mathematics you will see that the basic concepts such as number, point and line, can’t be defined. But we can understand them easily, so we build the higher concepts upon them.

0. Introduction (Abstraction)

0 - 2. Through the previous post we discussed how a subject is created in our mind (the word "image" might be misleading since it refers to the visual aspects of things, so from now on we use the term "subject" instead.); and we saw the difference between a flower as object and subject. After identifying the existence of a single thing and creating its subject, we need to analyze it, so the first step to analyze something is comparison through which we conceive similarities and differences that help us to categorize subjects. Everything is unique in the world, even these two laptops which are identical; because each has occupied a unique space. Moreover, they have their own unique materials.

Our brain understands the similarities. For some objects such as these two laptops, it is easy to do so, but what can we say about these two horses?

We see that they are different and the first important difference is their color. So what our mind does to conceive the similarity is ignoring the difference in color and probably some negligible differences between other features, apart from that they look alike.

What are the similarities between a tree and a horse?

  

So you see that in order to find the similarities, our mind should disintegrate the components of things to find the similar components which might not be visible. This process is called abstraction. 

Picasso has shown us the finest example of abstraction. He might have aimed to see how much he could simplify the details of a bull such that people could find the similarity between the drawing and a bull.

You will see that in mathematics, we deal with abstract ideas. For example, if we’re asked to calculate the volume of a bottle, we ignore its color, material, mass, price and other dimensions and we just include the shape and size in our calculation. So to understand mathematics, we should grasp the abstract ideas which are apart from the reality. It will be discussed later thoroughly.

0. Introduction (how is a subject created?)

0 – 1. There are two different worlds: the world that we perceive and the world that we conceive, or the external world and the internal world. What are the differences between them?

First of all, the existence of an image is not depended on the corresponding object. Imagine that you saw a wonderful flower. After a year, you could remember it, but the real flower might be completely decayed. So it would exist in your mind, but not in the real world.

Secondly, our brain pays attention to details as much as needed. It doesn't deal with unimportant pieces of information because they are not needed to be processed. So if the flower had 10 petals, our brain would probably store a bunch of petals. The brain might memorize the number of petals for a short period, but it would forget it after a while, if it weren't that significant.

Thirdly, the image is processed on and off by our brain (frequency of process depends on its importance), and it's enriched by our feelings, beliefs, notions and occurrences. The image of the flower might remind you of a very memorable day when you had a lot of fun, so when you remember the flower, you'd feel happier. The imaginary flower would do so, not the real one.

And the last but not the least difference is the limitation which has characterized the real world unlike our imagination in which we can do whatever we want: our mind can disintegrate the elements of an image and combine with other images. Sphinx, unicorn, mermaid and many imaginary characters in mythologies or fairy-tale prove the ability of our brain to mix several things up in order to create a new creature which doesn't exist in the real world.

In conclusion, image, though its raw material is reality, is processed in our mind so the outcome is not the same as real object.

In philosophy, image of any object created in our mind is called subject.

In mathematics we move from objects to subjects frequently which we’ll discuss how it happens, through the next posts.