On Mathematics

Mathematics is both a language and a form of artistic expression. Language because it has all its characteristics: it is symbolic, has a morphology and a syntax. It also represents something as an idiom does, in this case our natural world (or rather, our limited interpretation of it).

But Math is not tied to reality, unlike Physics for example.  There is no need for experiment to validade its work. As an important mathematician one wrote:

The essence of mathematics lies in its freedom.

One is free to conjure any kind of creation to suit his needs. That is the great power of this field. It can be something that was motivated by a practical problem: How to keep track of a herd of sheep? How to measure the area of a irregular terrain? Will there be enough corn to feed the population? For practical reasons, this comprises most of it.

And it can be simply a thought experiment: What would a 9-th dimension cube be like? The problems in this pure Mathematics have a higher degree of abstraction, and are considered a larger set of the problems that arise from the real world. It’s possible for one of these problems to be found later having a counterpart in the physical.

The more you come close to this century Mathematics, the stepper the abstraction curve goes. That is the reason why most people will only have the practical knowledge of the Classical Era. And that is a bad thing in my humble people, but I guess Math just isn’t everyone’s cup of tea.

People could benefit from the gain in coping skills from studying it, though. It’s trying at times, but the satisfaction of grasping a new math tool is beyond words. And the variety of problems you can solve increase rapidly with more studying. I consider my Math skills the ultimate measure of knowledge, as everything I ever hope to learn has it in its foundation.

And it’s still under construction, even when it comes to the basics. Much is done of course, but one can easily find knots to tie or a new problem to tackle. A example of this is the prime numbers – is there a formula that can give any prime, given it’s position? This has haunted me for a very long time, it looks at first so simple. Solving it could render our cryptography useless, for those who want to see the world burn.


The mysterious pattern in prime numbers, known as Ulam’s Spiral. credits

There is much more that we can’t do than that we can. As an example, only a handful of integrals have a primitive in terms of defined functions. And the hardship of computing an integral increases dramatically if you start picking elaborate functions.

One might view this tendency as an evidence that the Math we developed is unsuitable to handle our intricate universe. We could be speaking Russian instead of plain English. Perhaps there is a different way of looking to things that is simpler and more effective. It’s hidden, waiting to be found.

From this freedom of creation comes the artistic aspect of Mathematics. In a carefully refined theory, one could leave a lasting work just as a painter or a composer does. I hope this has insipired some more courage when opening your next Calculus book.

[IMAGE: A computer generated fractal, Sierpinski. image source]


On Music

One of the oldest forms of art, music is natural to man. After all we are born with the ability to produce with our vocal chords a wide range of sounds. It’s like being born with an attached musical instrument. Being able to communicate with sound is obvioulsy an evolutionary advantage.

Music can be defined as the structured production of sound, with the purpose of transmitting though and emotion. In a non-trivial process, sound is received by our ears and decoded, stimulating certain regions in the human brain. It involves both acoustics and psychology.

Music has four major components: Melody, the set of sounds executed in succession; Harmony, the set of sounds executed simultaneously; Counterpoint is a subtle concept, perhaps best experienced; And there is Rhythm, the proportion of the sounds in melody and harmony.

Surprisingly enough, all the myriad sounds of music can be represented by only seven musical notes (and accidents). This has more to do with the fact that music as a subset of sound that is pleasant to hear, so it ends up having this pattern. Physically, every note has a specific frequence, and from one A to the next, the frequency is doubled.

Music is produced today in an industrial scale (it became mass art), with musical producers working side by side with performers and lyric writers. As Adorno and Horkheimer so brilliantly stated, we are tricked into believing we have a choice for music.

Formally, in Music Theory, a composition is written in modern musical notation. This is a rich creation, with somewhat simple rules, in which music is organized into a sheet composed of staffs.

Music Theory has many points of connection with Mathematics.

Since the music of an instrument is a sucession of frequencies, one can assign numbers to each note and obtain integer sequences. For example, by applying the number 1 to the the lowest pitch in the following sheet an integer sequence is obtained:


This entire song can be heard here.

Then one can, for example study the frequence of notes, or try to find a recurrency relation for the music. I’m sure that from all the possible sequences in the universe, only a handful are considered beautiful songs to us. Is there a pattern to those sequences?

If we understood better how some sequences are more pleasing than others we could improve the quality of our music.

The process described above doesn’t work if more than one note is played at the same time unless you separate it into different sequences. This is unfortunately quite common in music.

The other way around, taking a mathematical table or relation and turning it into music is also possible (although it’s kind of useless).

For example, the astronomer Johannes Kepler, studying the movement of the planets considered each planet as emmiting a  note proportional to the orbital velocity, that changes during a revolution around the Sun. There would be a celestial music to each planet. Something similar is done beatifully in an app here – complements to the designer.

A personal selection of songs, divided into pop and classical can be found in this page.


[IMAGE: Vinyl store. image source]