Posts Tagged ‘reality’

The Seeds of Mathematics

November 23, 2015

Mathematics is an introspective science, as opposed to experimental sciences(Physics, Chemistry etc). Unlike experimental sciences, progress in mathematics doesn’t require getting data from deep sky, probing into an atom or colliding elementary particles at high speeds. Numbers and other concepts are in our head and all open problems are in our head. This suggests a possibility, that even if we locked ourselves in a cave, cutting off all communications with the nature, we may still be able to continue our progress in mathematics. The subject of this blog post is to examine if  this is really possible.

The converse of this question has a clear answer: Halting mathematical progress would halt progress in all experimental sciences. Mathematics develops ways of thinking, that are employed in understanding the nature. It appears that mathematics is on its own; it doesn’t depend on any of the other sciences. It can continue progressing without any other sciences. This view is expressed in this xkcd comic. But I am going to contradict this view (and I am going to contradict other aspects of that cartoon in my next blog post :D).

Although it is not apparent, progress in mathematics requires us to experimentally probe into nature. The ideas involved in mathematical research are seeded by our experimental probes. Exploration of nature is a key not only to develop physical sciences, but also to develop mathematics, although it is a purely logical subject. A problem is deemed solved, only after a logically consistent solution is found. However, the question “what is an interesting problem to solve?” doesn’t quite have a logical answer.  G. H. Hardy has described  this question to be akin to asking where do poets, writers and other artists get ideas for their work. Conceiving new problems is a work of imagination. And imagination is always seeded by reality.

The classic example for an interesting problem is the one that led to Fermat’s last theorem: ‘are there integers a, b and c such that an+b= cn for some integer n?’. Surely, the complete answer to this problem was profoundly useful, only because large number of people worked on it and it developed a great deal of understanding of numbers. However, the problem was presumably seeded by the Pythagoras theorem. We have integers, like (3,4,5) such that 32+42=52, so  a curious question like ‘what if we change the exponent to a number other than 2?’, would have been the origin of Fermat’s last theorem.

Another question in mathematical research that doesn’t have a logical answer is “what set of axioms should we choose?”. Axioms, like problems, are chosen by taking a cue from previous mathematical theories. The resulting structural similarity between different mathematical theories has been capitalized in a theory called category theory.

Where do these ‘previous mathematical theories’ get their axioms and problems from? There must be a starting point, a seed for every mathematical idea. These seeds come from outside- from our interaction with nature. Cutting off interaction with nature will cut off the supply of new seeds. But that doesn’t entirely stop mathematical progress; instead, ideas for new mathematical theories will be entirely dependent on the old mathematical theories. Over a timescale of several hundred years, this is a significant setback to mathematical progress. Seeding of mathematics by interaction with nature is a slow process. In fact, we are still benefiting from the seeds of Pythagoras theorem.

The seeds of Pythagoras theorem

‘Geometry’ stands for measurement of the earth. ‘Earth’ here doesn’t mean the planet earth or the globe; it means land; real estate. Geometrical ideas were developed as a result of extensive measurement of land, during early human settlements. The most influential of these was the Pythagoras theorem. Let me go through its development, in its three chronological stages: the content, the statement and the proof, to identify its seeds.

Given two sides of a right triangle, knowing how to calculate the third side is the essential content of Pythagoras theorem. This may be done using a formula, or using tabulated data or using similarity of triangles. All these methods carry the basic wisdom– “the third side of a right triangle is not an independent variable”. Any civilization that built large planned settlements knew the content of Pythagoras theorem.

Explicit statements probably came several thousand years after the content. Early statements of Pythagoras theorem were in terms of areas. There are records of statements in Babylonian scriptures(2000 BC), in the Vedas(Shulva Sutras, 800 BC) and Chinese scriptures. An explicit statement  could’t have brought any change in the applicability. Perhaps, the room/house in which the statements were written was constructed using the content of Pythagoras theorem :D. However, it brought big changes in theory. Geometrical shapes were understood by cutting them into triangles. Triangles were now understood by cutting them in to right triangles. Right triangles took precedence over other triangles, leading to a new branch- Trigonometry.

A partial proof was recorded in 800 BC(Shulva Sutras) and a complete proof in 500 BC(Pythagoras). Presumably, there were unrecorded proofs prior to this. There are some theories that Pythagoreans might have been communicating with Chinese schools of mathematics. The proof was seeded by two pieces of intuition, which were developed when planning settlements. One is that land can be measured in areas, which can be added and subtracted by joining and cutting pieces of land. The second is scaling; a big piece of land can be scaled down and represented on paper or a flat stone. All proofs of the Pythagoras theorem are based on areas of triangles or scaling of small triangles to big ones(A book, The Pythagorean Proposition lists 370 proofs). Scaling is in fact a logical implication of the properties of areas. But it is likely that it was developed independently.

The proof, of course, had no practical implications. Proofs generally store methods of thinking. Even today we feel its impact on our thinking. The proof raised the status of the mere formula, ‘a2 + b2=c2′ to a theorem, resulting in the discovery of irrational numbers. Furthermore, it redefined the whole of geometry in terms of a single quantity- the distance between two points. The more advanced forms of geometry- Riemannian geometry and even Differential geometry contain the germs of Pythagorean distance.

The Pythagoras theorem and all of its intellectual impact on Mathematics are seeded by man’s physical exploration in to measuring land. A writer, within his lifetime travels extensively to gain experiences of reality that can seed his imagination. Mathematics is also seeded by explorations of real world, but this seeding has a longer timescale, much longer than a mathematician’s lifetime.



The atheist debate

January 4, 2015

Debating the existence of God and the relevance of religion is the doorstep to understanding the role of imagination in reality. Imagination is a tool of dynamics of reality- Imagination, shaped by the past of reality, shapes the future of reality. It evolves reality in time.

To explain the above statement with an example, consider a chess game. The board, the pieces and the players are real. The game setup and rules are imaginary. In the imagination, the board is a war-field, each piece is a certain type of warrior, and so on. The future of reality, i.e,  the next move to be made by the players is entirely guided by this imagination.

The sense of loss or win is also determined by the imagination. Losing a pawn is a much smaller loss than losing the queen- although in reality, they are both just pieces of plastic or wood.

God is an imaginary entity. So are the rules of religion and the associated wins and losses, rights and wrongs. In what way does it impact the reality? What is the magnitude of this impact? Is it possible for a civilization to exist without religion?

A civilization without a religion is likely to collapse internally or remain primitive. We could have seen why is this true, if we had a chance to watch the formation of a civilization, and observe how they came up with God and religion.  We can do so, but such an experiment will take several thousands of years, and so, it better be a thought experiment. 😀

A thought experiment

Let us choose an inhabitable, but uninhabited island, far off from the rest of the world as the site of our experiment. Let us then initiate a civilization, with young children. For a few generations, we have to silently protect them, making sure that they survive safe. Later on, we can cut off all contacts with that island. A a few generations later, the people in the island will forget about us, and it will grow just like any natural civilization; no civilization remembers a time when they didn’t have a language of communication and a system of documentation. They will eventually find us, after they invent ships and start sailing, but this will take a very long time.

We can observe how the civilization develops, from a remote sensing satellite.  Of course, this will take several generations of observation in reality, and that is the reason why this is a thought experiment.

This setting can be used to analyze many things. Our question here is of relevance of religion and God: Will the civilization in the island necessarily develop a religion and a God?  Let us refer to our history. We know of a large number of civilizations that existed sometime in the past, somewhere in the world. How many of these didn’t have a god or a religion? Turns out, most of the known civilizations have a religion and god(s), with extremely sparse exceptions. Pirahã people is one such example. They don’t believe in any deity, but they do believe in spirits. However, they are not an independently grown civilization; they are a subtribe of a bigger tribe. So, this doesn’t really tell us how to evolve the civilization in our island without a religion.

Does this mean that no civilization can exist without religion and God? There are two possibilities: One, religion is a part of the growth of a civilization, or two, all those civilizations that didn’t develop a religion collapsed too soon to leave any footprints of their existence, and so we don’t know about them. Perhaps, they collapsed because of not having a religion.

For one thing, the civilization in our island should say something about death; something nice like, dead people become stars in the sky, or they become spirits or, they go to heaven/hell. Otherwise, the civilization will collapse internally. People are glued in to a society by an emotional attachment(relation, friends,, etc). This attachment also has a bad facet – it causes anguish, particularly over death, which is certain. If it is not dissipated, it can potentially crush the civilization. So, a strong civilization needs a strong attachment and a robust way of dissipating destructive emotions. Evidently, rituals associated with death and afterlife are a big chapter in every religion.

Moving ahead, the most prominent feature of a religion is, it creates God, as a protector of all :D. Is it really necessary to have an imaginary protector? Will the civilization in our island develop such an imaginary protector?. Well, if it doesn’t, it will never explore outside the island, and so, it will make a very slow progress in science!. Let us see why:

A civilization will attach value to life of a person(and many more things), not only that a person values his own life, but also, others value his life. Any prospect of loss of life will therefore induce an emotion called fear. It prevents the civilization from exploring too far away from their safe home. An imagination of a protector, can create a counter emotion to fear and therefore make it possible to explore. Knowing that this protector is not real does not alter anything!; Imagination can create real emotions.  One example where this method of evading fear is employed is, explorations in the ocean. Sailors are known to be superstitious, in order to evade the fear due to risks in their sailing. (Sailor’s superstitions. Why aren’t there similar superstitions with today’s astronauts? This has a simple answer  😀 ). Therefore, the people in our island may never find us, if they don’t imagine a protector!

Exploration is the key for scientific progress. Scientific progress is not a process carried out by scientists alone. It is carried out by the entire society. Scientists need a strong support from all sections of the society. As an example, let us consider the big revolution brought by Newton’s laws of motion(they partly caused the industrial revolution). What does it take for the civilization in our island to make this breakthrough?. It takes three things, in order of decreasing importance:

  1. A thorough documented knowledge of the objects in the sky. This is accumulated by a thousand years of night sky observers
  2.  A thorough knowledge of the surface of the earth, and how the sky looks when viewed from different locations on the earth. This is gathered by exploratory sailors.
  3.  A genius like Isaac Newton

The people in our island will never get to this without being able to explore. As paradoxical as it is, science has gained a  little from some superstitions too!. :P. 

So, the civilization in our island should have a method of dissipating destructive emotions, in particular, it should have something nice to say about death. And it should also have a protector(or a means to evade fear). Do these two complete a religion? I don’t think so. I have considered only those aspects that affect the stability and growth of the civilization. Religion also has another kind of value that is shared by the arts- music, dance, stories etc. In societies where religion is strong, it appears to influence the way people think(something I don’t understand). That is a subject of another blog post. I will conclude now by saying, man created God, and then God created man!. 

Action and Expression

June 23, 2014

Why should I be rational?- Part III

<< Part-I, << Part -II

Imagine, there is table in front of you. You placed sugar at one end of the table, honey at the other end and an ant at the center. The ant can make its personal choice of whether to go for sugar or for honey.

Suppose, in addition, the ant has moods- three possible moods- happy, sad and neutral. It’s mood changes randomly between these three at irregular intervals. Like we have an urge to express our feelings, let us assume that this ant too has an urge to express its mood. It has no means of verbal communication to do so; but there is still a way: It is walking towards its choice, honey/sugar. When it feels happy, it stops where ever it is, and takes a few steps towards the honey. When it feels sad, it stops and takes a few steps towards the sugar. When it is neutral, it has nothing to express; it continues walking towards its choice. This is a way to using the walking as a channel of communication to express its mood- just because it has an urge to do so.

Now, as an observer, you don’t know the ants choice- sugar or honey. Neither do you know of its changing moods. There no way to find out the ant’s choice or its mood just by watching it. If at some point, the ant is moving towards the honey, it could either be an action: i.e., it has chosen to get the honey or, it could be an expression of its mood.

So far we have assumed that the ant itself knows which of its movement is an action and which is expression. If it didn’t consciously know, and it tried to judge from its own motion, it would be impossible to do so. The ant doesn’t know what it wants!.
This is an analogy I used to define action and expression. We too use every available channel to express what we feel. And, like the ant, sometimes we ourselves are unable to identify an expression or an action.

This urge to express is one reason behind non trivialities in a language- figures of speech. Many metaphors are born because the mind uses multiple channels to express itself. Abusive terminologies(that claim the untrue 😛 ) are an expression of anger.

However, all these are instances where we consciously know that it is an expression. There are examples where we are unaware of this. A day before every exam, we “decide” to be better prepared next time. But this “decision” never gets acted. :D. This is an instance where the mind uses the process of deciding, as a channel to express that it is repenting. This is often a case where we don’t consciously know that it was an expression and not an action. An action too, is quite common. A decision to turn left while driving, for instance, is an action. It gets enacted without any hassle.

Many of our decisions are mere expressions. Being unable to identify actions and expressions is quite common. While both action and expression are essential, it is important to know which is an action and which is an expression. Before, this, we need to understand all the differences between them.

An action is a thought that is born in the mind and flows out through the body. It can’t be stopped in between, by a purely internal force. The thought is not complete until it is acted. It is a single piece- it can’t be broken in to thinking part, and enacting part. Action belongs to the deterministic part* of the future. Like all other deterministic parts of the future, it already exists in the present as a thought, but is invisible. It becomes visible in the future. Therefore, action is a part of reality. Therefore, questioning an action or an inaction is questioning the existence/non existence of a part of reality- it is an existential question.

An expression is always preceded by a strong emotion, which is to be expressed. It is not a part of reality. So, an obvious way to distinguish between action and expression: if a chain logical deductions leads to visibly absurdity, it is an expression. Expression is itself absurd, but its absurdity is usually invisible. So,, many times, we don’t identify expressions as expressions(like the confused ant). This is a state of illusion. The illusion is broken by a chain of logical deductions starting from it, that reaches an absurdity. We are a part of reality, and everything that we want to call reality must be deducible from it. Expression/ surreal objects can’t replace the reality.

* “the sun will rise up tomorrow” is a deterministic part of the future to our present knowledge

Why should I be rational?

June 23, 2014

Part I: Truth and logic

Trying to be rational is placing restrictions on oneself. If I don’t want to be rational, I can be sometimes rational and sometimes irrational :D. Like playing a game without observing its rules, not being rational is easier. So why should anyone try to be rational? One observation is: we are more peaceful when we are rational. This is just an observation, not an answer. Moreover, it could be that it is the other way round: we are rational when we are peaceful :P.

Being rational is a way of accepting the reality. Reality is anything that is either verified through senses, or deduced from another reality(that is verified through senses). A rational argument is a link connecting two realities. Reality is interconnected within, through logical deductions. In other words, the set of realities is closed under deductions. Therefore, no untrue statements can be deduced starting from a true statement. Also, starting from an untrue premises, some deductions will be untrue. Some of them might be visibly absurd, and this way, deduction can be used to identify untrue statements. Thus, logic is used to keep our self within reality.

Therefore, the question really is, why should I restrict myself to reality?. Is it possible to live totally in an imagination, by believing it is true?. If not, what is the role of imagination? Also, why do we feel more peaceful when rational?

To answer these questions, we need to understand the origin and nature of all surreal objects that we can think of. I have broken my thoughts on this in to two other posts, due to its length :D.

Part -II >> , Part-III >>

%d bloggers like this: