The Magic Of Mathematics

Mathematics! What a demonic word is this? Mathematics means different things for different people.

To some people, it is something to be avoided at all cost. It means pain, suffering, confusion and pure torment. Maybe they think that hell is a place where its inhabitants are forced to solve mathematics problem every day for eternity where heaven is the place where all traces of mathematics is banned forever.

Many others treat is as a necessity of life. They may need it in their jobs or to pass their examinations. They treated in with “professional” respect, learn what they can with the intention to use it to advance their career to the next level.

A small minority like myself see it as pure magic. It begins from nothing. You draw some lines, curves and numbers. Put them into equations, move the components around and a whole new universe is created. This is what I called “magic”. In this article, I shall elaborate more on this phenomena call “mathematics”.

Mathematics begin when people learned how to count. I don’t know much about tribes but based on what I have learned, some tribes do not have words for numbers above a certain limit. For example, they may have words for number 1, 2 and 3 only. Number 4 onwards are classified as “many”. Based on this limited vocabulary, one can deduce that such tribe could not have planned far. As far as they have the word for the largest number.

They could not have done much engineering and will be disadvantaged at war. Imagine this, when you organise an attack on them, what you have to do is to send in say 50 soldiers to set up a camp. The next day, hide 20 of them and order the remaining 30 men to leave. The tribesmen would have interpreted it as many people came and many left. They could not tell the difference between 50 and 30. Neither could they work out that there are 20 men still hiding.

No civilisation could have existed without the knowledge of basic mathematics which involves counting and arithmetic. If any of them really existed, they would have been replaced by the process of natural selection in mathematics.

In short, any civilisation that managed to exist knows the value of mathematics. Many of them developed it independently and yet it is consistent regardless of the culture it comes from. This means that regardless of which part of the world you are from, the rules of addition, subtraction, multiplication and division are all the same.

Mathematics is therefore absolute and independent of one’s culture, geography and history. This is the opposite of religion. All civilisations also created their religions which tell them how their world was created, what can or cannot be eaten, how to live their lives, fight their wars, what type of structures they have to build for their gods etc. However, all religions differ from one another and are dependent on the culture, geography and history that created them. Therefore, unlike mathematics, all religions are relative and non-absolutes.

What this means is mathematics is the anchor of stability in the midst of a changing universe. Through mathematics, we can be assured that there are absolutes in this universe that will not be changed and independent from changes in discoveries, technology breakthroughs and trends. In mathematics, we have the rights and wrongs clearly proven and defined. What was proven right will be right forever. It is the island of stability in the midst of chaos.

One may ask, “How about other branches of science?” The difference between mathematics and other sciences like physics, chemistry and biology is mathematics consists only of deduction and the manipulations of numbers. It does not depend on experiments or observations at natural processes. This means new discoveries have no impact on the established mathematical laws.

The above phenomenon is also a drawback for mathematics itself. Since it is independent from the natural processes in the world around us, it is useless by itself. All the equations and laws created from it are useless unless we can put meanings behind those variables. The only way to do it is to conduct experiments, observations and followed by precise measurements to produce raw data. Feed it in the equations and we will have working mathematics.

Then again, how accurate are those measurements from experiments and observations? This is where the problem lies. Whatever discovery we made based on the observations today may be proved false tomorrow because tomorrow we may have better instruments or new events. Science is in constant flux. What is true today may not be so tomorrow due to new discoveries.

However, I need to stress here is that the changing portion is in the raw data that comes from observations and not from the mathematical process. The mathematical laws, once proven true are true forever.

So much for the consistency.

What I find so fascinating about mathematics is, it is purely the product of the brain. It was conceived in our brains. The sketches and equations we see on papers and boards are merely the representations of it. When I study this subject, I don’t see numbers flying around. Instead I see the thought processes that transform one state to another in a series of logical steps. You don’t even need expensive equipment like particle colliders for that. Just a pencil and a few pieces of papers , the rest is up to your brain to create your own miracle.

Mathematics had brought me much joy when I was in my pre-university days. I was fascinated by it. When I saw my teacher transforming equations on the black boards, I was mesmerised. After the final answer was out, I wanted to stand up, clap and shout in joy but I never did that because I was afraid others might think I am crazy or something. When I repeat the same process on paper and successful in completing the steps, I felt like a wizard. I have created my own magic spell. Doing mathematical exercises is like practising my magic. I felt like I was god.

Unfortunately, my priority then was to secure a place in the university, not to appreciate any subject. When the time came, I got into engineering. In case you are wondering on why not mathematics, the answer is, it is purely out of practicality. How much can an mathematics and physics graduate earn compared to an engineer?

I was not born in a rich family. So money is a high priority to me. As I think back, I wonder what would have happened if I opted for physics and mathematics instead? I don’t know.

Another quality of mathematics is it is universal and consistent even among species.

Suppose we encounter an alien civilisation. How do we deal with the first contact? We cannot assume their norm as the same as ours. We think of oxygen and water as the necessities of life but it might be toxic to their biology. We can see lights within a certain band of wavelengths but they might see it in a separate band. Then we can see what they cannot and vice versa.

The only common ground we can establish is through mathematics. The fact that these aliens can create a civilisation shows that they know some mathematics. They should have the same laws on addition, subtraction, multiplication and division. They should have the same value for pi and natural number e. They should know the meaning of prime numbers. All these can be used as the common ground to establish the first contact.

Enough of aliens. As we move forward, I believe we can progress at a faster rate if we can have a mathematician in every research team. The reason is because no other non mathematical faculties or departments in institutions of higher learning can teach all the mathematical topics to their students. They are bound to leave out certain topics that they think as irrelevant. Those irrelevant could become relevant in specific cases. It takes fully qualified mathematicians to identify that.

In having an in-house mathematicians we can have a higher chance of constructing mathematical models to analyse and confirm new discoveries. In fact, software companies and hedge funds have already begun to appreciate the importance of mathematicians in their teams. It is my hope that the rest of the economic sectors can do the same.

If you are a mathematics graduate or undergraduate reading this, I want to tell you that I am envious for being in the studies that I truly desire. However, I am not so envious of your economic prospects compared to your engineering counterparts. My wish is for the society at large to have better appreciation on our talents.

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The Process Of Science

In the previous post, I have given a brief introduction of science. I have checked the meaning itself from the dictionary and find the definition of science to be totally appalling. Somehow, they have the idea that science is just organised knowledge. The truth is science is more than that. Just because someone can organise some kind of knowledge, it does not make it “science”. For example, it is possible to organise the knowledge of astrology but it is still not science.

The question here is what makes science, science? The answer lies in the process. In this article, I am going to explain this scientific process that separates science from superstition.

Here we go. The process of science can be divided into the following steps:

  1. Identify a field/ phenomena to study and understand.
  2. Deduce or guess possible hypothesis to it.
  3. Conduct experiments and tests to verify the above hypothesis.
  4. The hypothesis that is able to pass the test will be taken as the true explanation until proven otherwise from the above process.

The above is very brief and therefore I will explain further.

 

Identify a field/ phenomena to study and understand.

There is no limit on what science can study. It is up to personal preference or the necessity of the time. Here are some examples:

 

  • Ancient civilisations devote resources to study hydrology and botany because they want to increase food production. As food productivity increased, they can support larger population, which in turn created more soldiers, artists, scientists and engineers which in turn make the civilisation stronger.

 

  • During the early stage of the Industrial Revolution, a lot of accidents happened due to explosion of boilers. Efforts were made to study these phenomena which gave rise to the creation of thermodynamics.

 

  • The study of medicine has always been important in all civilisations because no one likes to suffer from illness.

 

Apart from the above necessities, science was pursued out of pure interest. The most obvious example is astronomy. People have been staring at the night sky for generations wondering that is up there. It was only after some time that they conduct scientific study on it.

In short, there is no area of knowledge that we cannot apply scientific knowledge on it. This includes areas like astrology or God. How to do it? Let me show you some possible way.

In order to verify whether astrology can affect one’s luck, all you have to do is to gather the birthdates of all the lottery winners, tabulate them according to constellation at those dates and see if there is any “lucky” constellation.

If a believer is to argue that the luck component is month specific, then all we need to do is to group the winnings into various months. If the result of the study can show that there is a huge uneven proportion of winners in a certain month that were born under a certain constellation, then astrology is true.

The same can be done to the study on the affect of God. Gather a group of people with terminal illnesses and divide them into various subgroups. Each subgroup is to pray to a specific God. One of the subgroup is not to pray to anyone in order to serve as a control. If the result shows that a single subgroup has an excessive high rate of healing, then it is proof that the God it prays to is the true God or at least the most productive.

So, there is no area that science cannot be conducted on. The challenge is to determine the methods.

 

Deduce or guess possible hypothesis to it.

There are 2 ways to this:

  1. Deduce the most logical and simple hypothesis.
  2. Guesswork.

It is obvious that the first option is the preferred one. Everything else being equal, the hypothesis that is the most logical and simple will be preferred as the answer. It is only when the above does not work, will we result to guesswork and test it.

This is why the God Hypothesis will never be considered as an acceptable answer. In science, we use the known to explain the unknown. We do not use the greater unknown to explain the relatively lesser unknowns. The God Hypothesis is the greatest unknown ever existed in human history.  This means it is unfit to be use as an explanation.

This is something that religious fanatics cannot understand. In their minds “God did it” is the only answer. If all of us think the same, then there is no science because “God did it” can be literally used to explain everything.

We do acknowledge that there are many things that science cannot answer. Some of them might never be answered at all. However, that does not mean we must adopt the God Hypothesis for all the questions that we cannot answer. That is foolish, irresponsible, lazy and disrespectful to God. This is like telling God, “God I am putting you as the solution because I don’t know the real answer. After I find out what the real answer is, I will throw you out.” The God Hypothesis is pure blasphemy to both God and science.

Let me move on to guesswork. There is nothing wrong with guessing in science. In fact, many scientific breakthrough happen because there were scientists who guessed the answers which later found to be true. This is where the creative part comes it. Being creative and the ability to think out of the box are a part of the qualities to be scientists but this does not exclude vigorous testing.

 

Conduct experiments and tests to verify the above hypothesis.

This is the hard work part of science. Just because one can cook up an explanation, it must not be taken as true until tested. There are ways to test hypothesis. The best is through experiments where we create setups for the phenomena and measure the results. If experiments are not possible, then we will need to gather data and do statistical study.

The above studies will tell us one of the following:

  1. The hypothesis is completely true.
  2. The hypothesis is completely false.
  3. The hypothesis is true if conducted only under specific circumstances.
  4. Only a portion of hypothesis is true.

The first 2 possibilities are clear cut. It is either true or false. The next 2 are the ones that makes science interesting. When to do experiments and study the statistics, we will not only get the true or false result, we gather new information which can help us to enhance our understanding and thus add to our initial hypothesis. Not only that we can create new field of knowledge out of it. The Theory Of Relativity and Quantum Mechanics were discovered this way. These 2 fields are what I call “common sense unfriendly”. They were discovered from experiments that did not yield the expected results.

Experiments and statistical studies can add to our knowledge.

The main weakness to this step is, we may not have access to the complete data. As I explained earlier, we humans were never fully equipped to be scientists. Our senses are pitifully limited compared to what is available in the universe around us. As our technology improves we will gather more data that we could not get before. Some of these new data may even prove that the answers we got in the past as false. When it happens, we will have to revise those theories and form new ones.

Being a scientist demands the ability to reject earlier “proven” theories if the current data proves otherwise. This is the characteristic that separates science and religion. Science is not ashamed of changing minds and beliefs. In science, being a heretic and blaspheme against earlier proven theories are healthy practices as long as one can produce sufficient evidence to support this heresy. No religion can do this.

This tells us that science and religion are at the opposite end of the logic spectrum. Does it mean no one can do both? The answer is, many have done both and the way to do it is adopt the discipline to separate both.  When you are in your religious mode, you are free to believe in the things in it like making the world in 6 days, talking snake, dinosaurs in Noah’s Ark, walking on water and other miracles but when you are out of in and step into the real world where science is the way of progress.

 

The hypothesis that is able to pass the test will be taken as the true explanation until proven otherwise from the above process.

This is the part where the scientists publish the results to be verified by their peers. If the results are significant enough, the results may even in incorporated into textbooks.

As explained earlier, just because a certain theory is verified as true, it does not mean that it will always be true. Future data may prove otherwise. Science is a dynamic process. It exalts us from mere surviving into progress, teaches us to be open minded and most importantly makes us truly civilised.

The Power Of Science

If I am allowed to use a single word to separate humanity from the rest of living things on earth, I can only think of “science”.  Through our understanding in science, we have risen beyond survival centred living into progress centred living. In other words, we no longer live our lives in trying not to die but to achieve the next level of progress.

 

Definition Of Science

What exactly is science?

This is what the dictionary says:

sci·ence  [sahy-uhns] Show IPA

noun

  1. a branch of knowledge or study dealing with a body of facts or truths systematically arranged and showing the operation of general laws: the mathematical sciences.
  2. systematic knowledge of the physical or material world gained through observation and experimentation.
  3. any of the branches of natural or physical science.
  4. systematized knowledge in general.
  5. knowledge, as of facts or principles; knowledge gained by systematic study.

Source: http://dictionary.reference.com/browse/science?s=t

 

In my opinion, the above definition is totally meaningless. Let me give you a better definition of science:

Science involves the following steps:

  1. Identify an observed event.
  2. Guess possible explanations to this event.
  3. Conduct experiments and tests to eliminate those explanations.
  4. The one explanation that remains, no matter how improbable will be accepted as the true answer until further contradicting data comes along.

 

Through the above procedures, our species had not only gained knowledge on what have been going on around us, we progressed to make use of this knowledge to elevate ourselves to the next level. Not only that we managed to eliminate many false ideas which we call “superstitions”.

 

Science As An Antidote For Superstition

We no longer live in fear each time we encounter an eclipse of the moon or sun. Eclipse is nothing more than shadows. There are no “sky-dogs” up there in the heavens wanting to eat up our sun and moon. We do not have to worry of getting strike by lightning if we do not follow some religious routines because we not only know how lightning works, we know how to control its powers. So if lightning is a manifestation of the judgement of God, then we have overcome this judgement by installing some lightning rods and copper tapes on our buildings.

 

When earthquakes, typhoons, volcanic eruptions, tsunamis, floods or any other natural disaster happens, we know they are the result of the law of physics. They were not caused by the “Judgement Of God” as some con artists claimed to be in order to scare the public into giving them money.

 

In short, our knowledge gained through science has set us free from the bondage of stupidity and superstition.

 

Here is a disclaimer. I do realise that there are still con artists out there who had successfully made a living by proclaiming the “Judgement Of God” after every disaster. The only reason this can happen is their victims did not pay attention in their science class. So, the moral of this story is, a scientific illiterate and his money are soon parted.

 

Limitations Of Common Sense In Science

Many people have this misconception that science is common sense. The truth is, such belief was true in the beginning of scientific pursuit. Earlier disciplines like classical mechanics, wave theory and thermodynamics are very common sense friendly.

We can throw up an object, drop it from height and slide it on a smooth surface to see Newton’s Laws at work. Ancient armies had been using it to hurl stones to their enemies’ fortifications even before they could fully understand the law behind it.

 

Drop a pebble into a pond of still water and you can see the wave theory come to life. Thermodynamics can be seen in the kitchen when the cooking process is in progress.

 

In short, the above are clear and easy to understand. Then, we discovered electricity and magnetism. At first, we thought of them as separate phenomena but we have learned that they come together. So, we call the combined entity as electromagnetism.

 

While electromagnetism may give us the “magical” feeling, it is entirely scientific because it is predictable and controllable. We can switch on and off our electricity supply, control its properties and even use it to perform logical functions, which had given rise to the invention of electronic computers. We know electromagnetism is true even though we cannot understand it as well as we understood the Newtonian Laws of Motion because experiments, tests and applications had confirmed on what we know.

 

This is the part where we begin to rely on the discipline of science to identify the truth even if  we could not fully understand it.

 

Then, as if the world is not strange enough, we stumbled on even stranger truths in relativity and quantum mechanics which totally defy our common sense.

 

Why is that so? The above disciplines had revealed knowledge of the world beyond the world we live in (which consists of billions of atoms and moving at speed that very small compared to the speed of light) to the extremes like the world of very small, very fast etc.

 

In the world of the extremes, things work very differently from what we think as “normal”.

 

Here is an example.

 

We live in the world where what goes up must come down and water makes us wet. We take for granted of the water “wetness” and focus more on gravity because it gives us more problems.  However, if we are as small as the ants, things become reverse.

 

If you are as small as an ant, gravity is not much of a problem. The main problem is the surface tension of the water. At this size, water droplets will appear like a huge boulder in the middle of the road. If you do not careful, you might walk into the “boulder” of water and get drowned. On the positive note, you don’t need cups or buckets either. You can literally hold a sphere of water in your hands.  Instead of wiping your wet floor, you simply pick up the balls of water on your floor.

 

However, if you are to shrink even smaller into the subatomic level, the world is even stranger. You are no longer limited to a single space at a given time.  You can be present at more than one location at the same time. No one, including you yourself can identify your own location. Instead, you can only define the region of space where there is a high probability of you being there.

 

If you know where you are, you don’t know how fast you walk. If you know how fast you walk, you don’t know where you are. If you stand still, a huge cloud will surround you where you can be anywhere inside the “cloud”.

 

This is the world of very small. What happens when you move very fast until you are close to the speed of light? You will experience time dilation. Time flows slower for you than the rest of the world.  If you look out, you may think that the rest of the people are moving in a fast forward manner while they may see you as moving in slow motion.

 

It is important to note that the above ”strangeness” are strange only because they are not common in the world we live in. If we have been living in the substomic scale or have the facilities to travel at close to speed of light, then they are normal to us.

 

Our brains were evolved to find ways to survive. Not to understand the universe. Our diversion into science is more of a side track from the “purpose” of our existence.

 

Here is another fact. We like to think our senses as powerful but from the cosmic scales they are extremely limited. For example, our eyes are definitely better than nothing but they can only detect a very narrow range of electromagnetic waves (which we call “visible light”) that exist in this universe. We cannot see electromagnetic waves with higher and lower frequencies of this range like infra-red, radiowaves, gamma rays, X-rays etc.

 

This tells us that we humans are not adequately “equipped” to be scientists. Our brains and sensors are too primitive for that. Yet we did not give up and made great progress in understanding the universe. What we lack in our sensors, we use artificial sensors to “see” what we cannot see. What we lack in our brains’ imaginary function, we use mathematics to make up for it.

 

In conclusion, the enterprise of science has elevated us beyond our natural limitation. We are no longer subjected to the limitations imposed by our genes. If the Creator has a purpose for our existence, we must have exceeded it.

 

Suppose there is a God up there. What will he think? It depends. If God is an insecure, obsessed with self-glory and low self-esteem person, he must be feeling threatened by our progress. It is only a matter of time when he will want to sabotage our Tower Of Babel.

 

However, if he is an honourable, moral and with healthy self-esteem, I suspect he will applaud our achievements.