God gave us integers !

mathematics

Mathematics has become such an integral part of our lives that we hardly stop to think about it. We check our bill at the grocery store;  even without realizing that the basic math we are doing is actually a great power we have to understand nature.

On one hand, mathematical constructs like series, functions, calculus sound like something which is fit for textbooks and in the hands of engineers and scientists. But the history of mathematical discovery is very rich and diverse; the various branches of algebra, geometry, calculus, probability, statistics etc have had very different origins.

Pascal and Fermat, two Frenchmen basically developed the modern theory of probability in a series of letters to each other. Pascal was a mathematician and Fermat a lawyer, both concerned with the question of fairness in splitting up the prize money of an unfinished game of chance. They calculated the probabilities of outcomes and gave the definition of expected value.

Calculus was formulated by Isaac Newton because the prevalent rules of math weren’t enough for him to explain his work in acceleration and gravitation. Calculus allowed him to explain gravity in quite a beautiful fashion and his science became the bedrock of physics for many centuries to come.

On the other hand, there are many mathematical concepts developed just for sake of logic with no immediate practical relevance in mind; but these turn out to explain physical phenomena in nature which are discovered years later. E.g. the brilliant Indian mathematician Ramanujan published numerous equations and theorems in his brief lifetime in first 2 decades of the 20th century. His work has pervaded many areas of modern mathematics and physics. This begs the question, is math a human invention or is it a discovery of some underlying principles of nature…just like other sciences ?

A perspective that can be useful is to see how math is essential to understanding the complexities of nature. Just consider the simple action of throwing a ball in the air in front of you. Newton’s laws will give us pretty much the exact position where the ball will land. So one might consider math as some sort of tool created by us which gives us the info we need.

But nature is not always as simple as throwing a ball in the air. When we consider complicated systems like nature, we start realizing that our basic laws do not seem to useful in predicting outcomes. So does math have a boundary in which it can’t operate? Well, that’s really not the case; math can actually help us understand why nature behaves so erratically and cause what mathematicians call “Chaos”.

We don’t even need a system as complex as the weather to understand chaos. Let’s consider a simple expression f(x)=1-2x^2. To introduce complexity, we loop this expression by itself in the foll way :

f[x]= 1-2x^2 ; f2[x]=f[f[x]] ; f3[x] = f[f2[x]] … f25[x] = f[f24[x]]…

Let’s graph values for this functions for many iterations. We will use a starting value of x=1/π but once with 16 decimals and once with 15 decimals. The results are presented in the graph below.

graph

We see a very surprising trend after about 60 iterations. Even though the difference between two x values is very small, the function values start fluctuating wildly. Thus we can see how even a small difference in the initial conditions can affect the final result. This is known as the butterfly effect. It is said that a butterfly flapping its wings in one part of the world could possibly kickstart a series of events which can lead to a tornado on the other side of planet. Such is the fine margin that nature operates on. It’s important to highlight that this type of chaos generated is completely deterministic unlike truly random events.

Well one might say that we could avoid such chaos by simply using a perfect value for the inputs in our calculations. But as we will show, that is often not possible. Let’s take the example of the number π.

π is defined as the ratio of the circumference of a circle to its diameter. It is calculated by integrating smaller chunks of circular motion along the orbit. But we can only get results of this integration by approximating it to the best possible decimal. π is an irrational number, meaning the digits after the decimal place do not follow any pattern infinitely. The only way to get a value of pi is to approximate up to as many decimal places as our computing system allows.

Thus there is always going to be an error in measurement of inputs when it comes to any calculation involving π. In applications like space orbits when magnitudes are huge, this can lead to big differences based on what level of approximation that we use for π.

The German mathematician, Leopold Kronecker, once said that “God gave us the integers, all else is the work of man”. It seems to imply that most math is just pure human invention. It might be so on paper, but it is beautifully ingrained in nature and is vital for us to understand it’s mysteries. As we have seen, the best simulator of nature is nature itself and it seems like nature is unaware of its future. But math helps us grasp the depths of this uncertainty; so we can almost say that the closest we can get to nature is through mathematical ‘discoveries’.

Reference:

Mystery of Pi

Chaos & Self-similarity

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Is the environment really in danger ?

Maldives – The sinking paradise

A treat for the eyes.. but only for some time now..

Maldives – the place of unrivaled luxury and pristine beaches. A dream destination for every traveler to enjoy few days of complete relaxation. But unknown to the bedazzled tourist, this wonderland is undergoing a ‘sea’ of change. The global rise in the sea levels is slowly eating up the coasts of Maldives.

Maldives, one of the world’s most low lying areas, is one of the few places where we can see the effects of global warming so strikingly. The coastline of the Maldives is being taken up inch by inch every passing day by the rising waves. The pearl of the Indian Ocean might not remain on our maps anymore. To have a chance of preventing this, we can resort to the well documented ways of reducing energy usage and carbon footprints. But primarily, we need to reform our mindsets about climate change and nature. Here are some of the ideas which we think are very interesting in that aspect.

Jurassic Park – Let’s save ourselves !

When dinosaurs ruled the earth..

We all know about Jurassic Park. Michael Crichton’s epic novel and Spielberg’s breathtaking movie has captured the imagination of people all over the world. But Jurassic Park  is so much more than just scary dinosaurs. It  has some very crucial lessons on the relationship between man and nature.

The idea of the park’s creators was to re-create living creatures who have been extinct millions of years ago and try and fit them into a restricted park setting. The point that they missed was that dinosaurs weren’t some species that disappeared because of deforestation or hunting. They have had their shot at dominating this planet. Nature selected that they become extinct. Trying to bring back the past amounts to playing around with nature’s rules which never yields success. This was demonstrated in the end where the dinos broke out of their cages and wrecked havoc on the whole system. If one tries to control nature, then nature will hit back.

Surprisingly the novel also emphasizes on the notion that even in difficult circumstances, life finds a way to survive and thrive as the dinos did even if the humans there didn’t manage to do so That gives us another very important lesson. When we say that we are damaging the “environment”, we should understand what that means. Even if there are catastrophic events on the earth, somewhere, in some tiny place on the planet, life will remain. And given enough time, nature will once again regain its full splendor. It’s not nature which is being put into danger by our actions; it’s us. We do not wield any power over the Earth. The faster we grasp that fact, the better we can plan for saving ourselves from extinction.

Siberia’s Happy people !

Survive in Siberia

How do you think life would be in the hellish cold of Siberia ? Quite taxing, right ? 

Werner Herzhog, the famous German Film director, gives us a very different perspective on one such Siberian community in his documentary “Happy People”.

Bakhtia is a region in the Siberian Taiga known for its extremely cold weather where a group of villagers have been living for centuries in their traditional way. Every year in winter when the Yenisei River freezes, the men of the village go hunting in the woods where they live a life of solitude. Every hunter has his own cabin and has to be completely self reliant in order to survive. Most hunters carry dogs that are tremendously helpful in hunting. Other than using some modern technology such as guns to kill larger prey and motorized snow sledge to cover vast distances of the Taiga, these hunters rely mostly on traditional traps to catch small animals. They then carry their game back to the village during spring time to sell and/or eat. Even after these hardships, they are quite content with this lifestyle and are glad to have their life so intertwined with nature.

Now these people who are dealing with natural uncertainties for centuries using more or less traditional methods are better equipped in dealing with climate volatility than people living in urban areas who are mostly just exposed to a constant controlled environment through their lives.

For most part of human history majority of human population has been living in rural societies that were more in touch with nature than ours. These communities have developed basic age-old methods to solve many problems nature poses them whereas urban societies have grown disconnected from natural forces. With the growing disconnect comes more and more dependence on technology that works best in normal times but it has forced us to keep our natural survival instincts aside.

Lets survive!

We have to realize here that there are parts of climate change that we can affect and there are parts that we cannot control or predict to a certain degree, but humans have tackled climate change all the time and what has been a powerful tool for our survival is our ability to think about and plan for the future.

Human behavior has been shaped by variations in climate change and the question of our survivability on this planet still remains, but we have a chance to answer it. For our own good, we better take it.  

Reference for the documentary: 
http://www.imdb.com/title/tt1683876/

 

Science vs Faith

Image

A guy sitting in a big dome shaped room staring intently into the eyepiece of a huge telescope; another person praying to his God hoping that it will give him some strength and bring happiness in his life. History shows that each thinks the other is foolish or misguided. But is it really so?

What is Science ?

A mind of a scientist is perhaps one of the most misunderstood things around. What actually is the motivation for a scientist to do what he does? Is it really about solving differential equations, writing pages of code or making complicated instruments? A casual observer might think so but it is certainly not the case.

The universe is an entity far bigger than a human mind can ever fully comprehend. We introduce hierarchies like an atom, a tree, a star or even an abstract concept such as love for clarity of thought but each time we move higher in complexity there are things we have not taken into consideration. But yes, the universe also happens to be such an awe inspiring place that it is almost impossible for a human mind to not wonder. Every child has looked up and wondered what the tiny twinkling lights that adorn our skies are; he has wondered about the beautiful colors of nature, about the wind and the rain, about the sounds and the smells. This is the essence of science –curiosity!

Is Science applicable everywhere ?

Agnosticism arises when humans have to deal with a universe that they do not fully understand. Science by its nature cannot prove or disprove anything. It can collect evidence supporting or rejecting a thesis but that is all it can infer. Science is an excellent method of acquiring knowledge. It teaches us how the world works. But science is inadequate when it comes to making decisions of what humans should do with that understanding. Human choices are extremely complex and humans do not solely rely on scientific methods to make decisions. Social phenomena are very different from laboratories in nature when it comes to internal interactions. For most parts, we can safely observe the phenomenon in nature without worrying about affecting what we observe. But in a social situation, our every observation, every move makes a difference to the whole picture; which in turn should feed back into our observations. Like financial markets; where our bets affect the market and people tend to act on instincts. Therefore it turns out that when there is inadequate information people do act on certain degree of beliefs. Faith has been important in development of arts across different human cultures and the role of arts in human evolution is undisputed. Faith is an important part of human psychology.

Science and Faith – A peaceful co-existence ?

But isn’t faith monopolized by organized religion? What if we put away the rules and customs of organized religion and look at what faith truly represents? What happens when a person prays to his God? Be it a Krishna or Jesus or Allah or even to an object or place, the intention is the same. One prays for getting strength to deal with one’s problems. Maybe for abstract things like hope, love, pain, this is what helps more than what science can do.

This is not a pitch for believing in God or religion. Both of us have never quite subscribed to the idea of a God in the generic sense. But there is definitely space for faith to co-exist with Science. We understand that we are trivial in front of the huge uncertainty that we witness in nature. Science and faith are our small efforts to understand these uncertainties. As Thomas Huxley said “The known is finite, the unknown infinite; intellectually we stand on an islet in the midst of an illimitable ocean of inexplicability. Our business in every generation is to reclaim a little more land, to add something to the extent and the solidity of our possession.”