Tuesday, April 2, 2013

Reflections

After reading William Byers' book, How Mathematicians Think, I was inspired by the author's position on the problem: Is mathematics discovered or created? Or, in other words, are mathematical entities and theorems objectively true or subjectively true?

 I think I have encountered this question in other forms before, albeit I had no idea how to tackle it. Byers' book has shown me one possible way to approach this deep, epistemological conundrum. I have studied a little bit of mathematics and physics but I used to take the relationship between the two subjects for granted. However, after being exposed to the differential equations of damped oscillations and other applications of classical physics, I became increasingly suspicious. Mathematics, in the form of newton's laws, seems to have an uncanny ability to predict the behaviour of physical systems accurately. It does not need to be this way. Certainly, laws of physics must model nature accurately to deserve the name but the very existence of such mathematical laws is quite inexplicable. Byers gives one possible explanation in his book. It is a combination of Nietzschean and Platonist thought.

The central question as manifested in the fields of mathematics and physics must be discussed separately as their respective objects of study are different.

Byers claims that Mathematical truths are both subjective and objective. For example, a theorem within an axiomatic system is objectively true because it can be proven with standards of logic that are universal. Also, mathematical entities such as 'infinity' or 'zero' are clearly conceptualised in our minds and we have a strong intuition about them. Therefore, these entities are objectively "real" in a Platonic world that is wholly artificial, a construct of our intellect. They are not really there in the physical world. Consider the concept of the counting numbers, such as 'one', 'two', 'three' and so forth. How do we know that there is 'one' of something when we observe one apple, for example? It is our brain which interprets the splashes of colours in the form of sensory signals from the eyes. Our eyes obviously have limited sensitivity. That aside, the way our brain interprets these signals has been honed for the purpose of survival-- identifying predators and food for example--via natural selection. Hence our sensory perceptions are not representative of reality because they arise from our sensory organs and brain, which are machines adapted to survival. Since the counting numbers (1,2,3) arise from these inaccurate sensory perceptions, these numbers do not exist in physical reality! But the do exist in the Platonic world of mathematical forms, which mathematicians are concerned with. We have such strong intuitions about the counting numbers or integers because of the way our brain is structured. Hence mathematical entities are objective in the sense that every mathematician can envision them existing in his or her mind as a concept with certain intuitive properties. Mathematical entities are not objective in the usual sense of existing physically.

A mathematical system is composed of a set of axioms that are assumed to be valid and various theorems that are logically proven to follow from the axioms. A very well-known example is Euclidean geometry. Godel's incompleteness theorem number one claims that no mathematical system that is sophisticated enough to express number theory, the properties of integers, will be complete. This means that such a system cannot prove certain true theorems that lie within its domain of study. To prove these theorems which lie beyond its reach, the system must expand itself via modifying its axioms. However, after such a modification, the system remains incomplete as there will inevitably be a new theorem that is true but not provable in the new system. Godel's result shows that every mathematical entity will have different subjective interpretations in the form of mathematical systems and each system, being incomplete, cannot encapsulate every true theorem about the entity. For example, 'infinity' is part of both calculus and Cantor's theory of cardinal numbers but neither can express its full richness.

Mathematics is both discovered and created. I will aim to explain the unreasonable effectiveness of mathematics in physics in a later post.

Wednesday, February 20, 2013

Wonder

"The most exciting we can experience is the mysterious. It is the fundamental emotion which stands at the cradle of true art and true science. He who does not know it and can no longer wonder, no longer feel amazement, is as good as dead, a snuffed-out candle." -Albert Einstein

How do we experience "the mysterious"? I think the excitement that we feel when experiencing the unknown comes from a deep desire to understand the world around us. For scientists, this usually occurs when they are confronted by a surprising experimental result that has no plausible explanation under known paradigms. It is the sense of wonder resulting from such an experience that fuels the mind of theorists as they struggle to subsume novel results under a more robust framework, consuming a lot of coffee in the process. Recent examples of unexplained phenomena include dark matter and dark energy. Some scientists believe that they are on the verge of discovering the constituents of dark matter: http://www.space.com/19845-dark-matter-found-nasa-experiment.html

Art and music have long been inspired by the mysterious. It provides the artist or composer with a theme but yet, due to its vague nature, gives their imagination sufficient space to explore a myriad of possibilities. Consider Dante's Inferno. The workings of hell are not elucidated in much detail in the Bible. This makes it possible for Dante to divide the subterranean space into nine circles, each with its unique sin. By dividing hell, Dante is able to dissect the vague notion of sin into its components. The Inferno can be seen as an exploration of evil from the perspective of a Christian mind.

There are numerous examples in painting where the artist is trying to explore a mysterious concept. The genre of still life paintings invites the observer to contemplate the beauty of nature and of scenes in everyday life. Of course, simultaneously, the painting might have other themes such as death and the temporal nature of life (represented by a snuffed-out candle on a table, for example), which are also mysterious concepts.

In music, the performer also experiences the mysterious and is inspired by it. On one level, the composer compels the performer to create a coherent message out of meaningless pages of notation whose intentions are unknown and mysterious. The performer then transmits his own message or interpretation to the audience. On a sonic level, the performer has to deal with the mysterious  and unpredictable tone colour that is produced by the instrument he or she is performing on. The esteemed conductor Sir Colin Davis once said that a pianist has to tame and control the complex  soulless machine of levers and hammers that is the piano to produce a soulful tone colour. The process of acquiring a delicate touch on the piano or a sweet tone on the violin requires one to explore the mysterious concept of a good tone colour on these two instruments. It is an exploration of the concept of beauty in a sonic dimension. It is like seeing the refined shape of a sculpture within a rough piece of marble.

Einstein's quote expresses his keen observation that the sense of wonder one gets from experiencing the mysterious, or a desire to understand the unknown, is an important driver behind advancements in science and the making of great works of art. I hope the quote has given you some insight as it did for me.

Monday, February 18, 2013

Also Sprach Zarathustra

My favourite piece of music recently is Richard Strauss' Also Sprach Zarathustra. Strauss is an underperformed composer, especially in Singapore. I think I have never heard a live performance of any Strauss tone-poem. Sadly (because i'm supposedly an amateur musician), like most people, it is only after watching Stanley Kubrik's 2001 A Space Odyssey that I decided to find out where this snippet of music came from.

The music is beautifully orchestrated with contrasting sections of different style and form. After basking in the bright timbres of the introduction which brims with positive energy, one is plunged into momentary darkness before the intimate and sweet melody from the solo strings emerges.

I also appreciate the colourful and turbulent inner sections, where everything seems to be in flux. The grave fugue in 'Von der Wissenschaft' aptly presents logical thought as a slow, deliberate process. The music ends somewhat mysteriously in the 'Nachtwanderlied'. The alternation between pizzicatos on the double bass and the extremely high registers in the flutes and upper strings might be purposeful. Perhaps it serves to contrast the ultimate destiny of humankind against our humble beginnings; or it might depict the final few heavy steps in human evolution.

Anyway, I am listening to the piece at least once a day (along with many mental replays of random sections whenever I have nothing to think about) and I just cannot get enough of it... I think I would someday read an english translation of Nietzche's book, from which Strauss and Kubrik drew inspiration for their wonderful works of art. Wait, I might also want to research on Zarathustra himself and Zoroastrianism.

Sunday, February 17, 2013

Koch snowflake

I was just introduced to fractal curves, which are usually the boundaries of beautiful shapes made from iterated geometrical constructs. One fascinating example of such curves is the Koch snowflake. If a person were to view one portion of the snowflake at ever smaller scales, such as by putting the shape under increasing magnification, the curve would be symmetrical. In other words, the curve would superimpose on itself after magnifying it by three times, as each side of the smaller triangles in a new iteration is one third the length of one side of the original triangle. Because the number of iterations is infinite, each segment can be resolved into more and more triangles. Therefore, each point technically lies on a corner of a triangle, which is a kink. The curve is thus not differentiable at any point.
Even though the snowflake is made from an infinite number of iterations, it occupies a finite area, which depends on the area of the original triangle. It is a great exercise to find a formula of this area in terms of the length of one side of the original triangle. The process involves quantifying the relationship between iterations.



Saturday, February 16, 2013

First Post

I have just deleted all the older posts on this blog, which were unbearable to look at . From now on, this page would be a platform for me to practise writing. I would also like to share some ideas that I find interesting.

I am trying to view everything that happens around me in a more casual way. If I were to treat life somewhat like a game or experiment, then every 'bad' thing that happens might just be the effect of some 'variables' in my life that are not adjusted properly. If some of these 'variables' at fault are not under my control, the bad consequences should attributed to bad luck. By adopting this new mindset, I hope I could laugh at myself whenever I made some stupid mistake or was subject to misfortune. I used the word 'bad' because seemingly undesirable things might turn out to be fortuitous in hindsight.

I am still trying to transform my long and convoluted sentences into concise and effective ones.