On Wandering

Cambridge’s A.L. Dictionary defines the word ‘wander’ as:

  1. to walk around slowly in a relaxed way or without any clear purpose or direction;

If you stop to think about, much of our lives is spent on wandering around. We are born into this world without a clear purpose, with no guidelines, and without truly knowing ourselves. That can only lead to much time spent trying to find our path, and hence, wander.

And I don’t just mean to wander physically, or from job to job, but also wander in our way of thinking. How many times we change our minds about something, only to change it back again some time later.

Have you ever felt that strange desire to just leave your ordinary life, your steady-pace and run into the wilderness or live from place to place?

I believe that, to some people more than others, there is still a residual urge from our nomadic past that runs very deep. I’d go as far as saying that it’s the reason why they never seem to find a place, and prevents them from bringing their lives into a focus.

Drifting from one job to the next, focusing on today’s assignment, dispersing one’s energy rather than having a single long term goal are all traits of people who are bound to wander the rest of their lives. Somewhere along the line society stepped on the right of these people to exist as they were.

And there is the danger of not realizing in time that one is drifting too much. This leads to poorly conducted lifes, often with suffering as an end result.

But that doess’t necessarily mean a bad thing. It’s more often a compromise between how one’s dream about his life and how it actually flows. You can’t judge a person’s life based on how aimless it was, as some remarkable life experiences come exactly from wandering around.

As J.R.R Tolkien beautifully wrote in this much tatooed sentence:

Not all those who wander are lost;

The apparent contradiction of these words in fact reveal this truth. To not seek a particular goal can be a goal on itself. There is a fine line between wandering to explore the world and it’s things, and wondering without any purpose – to be simply lost.

[IMAGE: Mt. Cook in New Zealand, image source]


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 abtraction 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 a Sunset

Few things can rival the beauty of a sunset, particularly after a storm when the sky is filled with clouds. Too many clouds and it spoils the view because the sunlight can’t get through.

Ordinary beauty, yes. But still beauty.

I could probably go on discussing how the red-orange color of a sunset is due to Rayleigh scattering, but as a scholar, it’s important to know when to just sit back and admire the view.

One phenomena involving sunsets is the so called ‘green flash’, a flash of green light moments before the sun disappear in the horizon. Living so far from the ocean I must admit I’ve never seen it with my own eyes.

Big_green_flashA green flash in Santa Cruz, California image source

This was depicted in the Pirates of the Caribbean: At World’s End. There are many other atmospheric phenomena, some of them quite rare to be seen and registered, this source provides a list of them with photographic records.

But one of the most intringuing aspects of a sunset is how it’s beauty is fleeting. The sun goes down and the red light touches the clouds making them orange and then pink, and finaly blue. The sky in general, is always in constant change. You will never get to see the same sky again.

It is a privilege to be alive to witness a sunset.

[IMAGE: Sunset near Swifts Creek, Australia. image source]

On Ethics

We are often plagued by that “Is it wrong?” question. It’s the human nature to be inquisitive about the ethics of things. It’s uncertain if that is something that we were born with or if it was acquired along the way, enforced, for example, by religion.

Rarely we find a definite answer to a moral dilemmas, however. We tend to think that there is a ‘yes or no’ solution to everything, when problems can be so complex that can only be tackled in very specific conditions.

Just as an example: should you save person A or B from drowning? What if A was you son, and B a complete stranger? See how the problem changes shape, just by adding more information. For some people, it falls under a completely different algorithm while for others, nothing changes.

This is applied ethics, asking specific questions to the purpose of determining a policy or a decision in real life. If this was an easy task the world would not be so troubled. In fact, in the heart of every major conflict lies an ethical ambiguity, a problem that at least on the surface seems to have more than one acceptable solution.

But don’t be tempted to think that the answer to all ethical dillemmas is to ensure conformity. Many practical disasters have come from attempting this, including the Second Great War. There are two separate movements in society, one leading to conformity and other to dissidence of opinions. Somehow, the balance of these opposing forces reaches an end result that is society itself.

This affects ethical decisions because our own decisions are based by the mainstream views of the society, its values and beliefs. If my entire family bans abortion, for example, it is lot more unlikely for me to support it.

It’s important to know that although there are little definite answers, much work has been done in thinking about the most common ethical dilemmas. This material should be used to guide the most difficult decisions.

There are many schools of ethics, some that favor the balance such as the Stoicism, other more pragmatic as the Utilitarism. Simply knowing what they are about is a enough to allow you to use them in a rudimentary way, and leads to a better justified position. One needs to draw the line about using them and embracing them as life styles, though.

In very simple terms: Stoicism is inside the moral absolutism (ie murder is always wrong) and anything outside can’t be judged; In Hedonism, right is what gives pleasure (bound to have a fan or two); To Epicureans, enduring pain makes it right; Consequentialist states that one must compute the pros/cons of a moral action; Utilitarism chooses something to maximize, say welfare, and defines right accordingly; Deontological ethics look beyond consequences, and find right in the actions taken.

I’ve found with years that one should navigate between these views, none of them alone is enough to face every decision. One needs to see how arbitrary a solution to a problem is, if only to be more willing to listen to another opinion. If the purpose is not debating, but to reach a consensus, one will find Consequentialism an usefull tool.

On top of that, one has to process these ethical considerations with his religious belief system. For some people religion takes precedence, and for other is the logical approach. It’s a wonder we can reach any decision at all.

[IMAGE: Plato and Aristotle in ‘The School of Athens’, by Raphael Sanzio. image source]

On Gods

Talking about God is hard because you are always stepping on eggs, is something too close to personal core beliefs. You could easily offend someone without intent simply by phrasing something in a bad way. With that in mind, let’s make sure to note that these are my thoughs on God. It goes without saying that you need not to agree with me.

By God I mean the entity that created, somehow, this reality that we experience.

The way I see it, there is nothing wrong with believing in one God, or two, or many. Usually what’s wrong, in my opinion, is what comes next. I’ll get back to this later.

New Hubble image of NGC 2174


For some, the natural beauty of the cosmos is seen as evidence of God; NGC 2174, ESA/Hubble. image source

There are many ways of believing in a God. It can be supreme (omnipotent and omniscient) or not; It can be benign or evenly capable of kind and cruel acts; It can have a special connection with each living thing or be completely unattached.

It can be a force of nature, an animal, personify a human or be completely alien. It could have only a divine nature or also have a human nature, including having lived among us at some point.

Or, to some people who don’t need that hypothesis, it might even not exist at all. Believing in a God requires a leap of faith that not everyone is wanting or willing to do.

Although each religion ends up enforcing a particular kind of God, it’s important to realize that you are free to choose your own conception of the creator. In my opinion you should go beyond taking for granted a belief set and instead, build your God upon questioning what makes more sense and what your instincs tell.

I understand that God in some religions is tied to eternal salvation, or more accurately to not being damned. But you shoud not let fear of punishment be the only basis for choosing a belief. Personally, I refuse to believe in a God that gives us the reasoning skills but doesn’t allow us to use them.

Logic will only take you so far though. Believe me, I’ve tried. For example, many philosophers have either proven or disproven God using reasoning.

Personal life experience can contribute either way to having a Deity. To some people, the sucession of failures leads to dismissing the idea of a compassionate being. To others, it the overcoming of challenges that leads to a stronger faith. It depends on the individual.

My concept of God is not carved to stone. It could change tomorrow, but for some time I’ve been stuck with a notion.

To me, when God created this universe, he became the universe, with all it’s clockwork precision. So we are stepping on Him right now. He was not omnipotent though, so, for example, in order for life to exist, he would compromise with all the bad things we usually see in this world (that are not caused by us, of course).

In that view, God is only the spark of creation. The details of its evolution followed from the set rules that he laid out in the beginning. Thus, for me, God did not create the world but rather the mechanism of its creation. I’ve found this notion to reasonably end the conflict between science and religion.

My personal experiences led me to believe that he has a keen eye for beauty, but that he doesn’t value life the way we do – which is not as to say he doesn’t care. For me, there is no afterlife tied to this existence (such as you paying for past life transgressions), but there might be another after this one.

On some days, I like to believe in the ‘Game Theory’, that reality is a simulation done to gain something or an experiment. But it’s not every day that I think that.

But what can go wrong with believing in God? As long as you keep it to yourself, nothing, but should you not…


A painting portraying an episode of the Spanish Inquisition image source

The problem usually begins when one tries to interpret what are God’s best intentions. Soon after, that person is led to believe that this interpretation is correct while all others are false. Doens’t take much for persecution to start.

If one doesn’t see God as the spark of creation, but rather as the creator of Genesis, believing in a literal interpretation, one must be ready to admit that he will never be able to put science and faith in peace in his mind. It’s a matter of choice.

[IMAGE: The iconic Pillars of Creation, as seen by the Hubble Telescope. NASA/ESA image source]

*Note: The content of these notes is not endorsed or affiliated with NASA/ESA, and express solely the author’s view.

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, estimulating 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]





On Star Trek

For those of you that are trekkies, this post is to remember the good old TV show Star Trek. For those who are born in this century or lived in a nuclear bunker, this will be a short introduction into the Star Trek universe, and to my favorites:

Note that I don’t care for TOS or the new movies. Just a matter of personal taste, nothing to it.

Star Trek creation is usually credited to Gene Roddenberry, although that seldom does justice to the many talented people who worked in all phases of Trek.

It’s a sci-fi series set on the future, where mankind is part of a Federation of Planets, along with many other alien species. The Federation is a highly techological, utopian society. The humans explore the unknown galaxy, engage in stellar warfare and face dilemmas ranging from ethical to technological problems.

There is an interpersonal aspect in Star Trek, in the characters interplay, but that is often in a second plane. What really gets you to sit and watch it are the open questions that are brought with every episode.

You often find yourself wandering how a warp drive could be build, or if Janeway should murder to save another life, and so on. It’s pure fuel for though, carefully inspired in many actual theories of science and philosophical debates.

Note that you don’t need to know how the stardate system works, or what is exactly warp 6 to follow the series. In a short span you get so used to it it doesn’t bother you anymore. Nothing wrong with being a hardcore trekkie though.

Star Trek can provide you with humanistic core values and a general regard for life, as a result of watching.

But Star Trek is also a bit farfetch. It often streches the limits of possible science (aka ‘Heisenberg Compensator‘)  and solutions are often over simplifications of reality. Ending is 99.9% of time merry. Everyone is a multi pro. It had to adapt to the format of a TV series, in the end.


The USS Enterprise on orbit of the planet Haven. credits

ST: TNG is remarkable series depicting the adventures of the starship USS Enterprise, to coin a phrase, to boldy go where no man has gone before.

The ship runs in a tight military hierarchy, under the command of captain Jean-Luc Picard (Patrick Stewart). Picard and his crew travel among the stars inside the Federation running missions, or in the outskirts exploring unfamiliar species and phenomena.

In no other series the humane side of ST is better depicted. Picard is a moral example to such an extent that I often wonder is tight situations ‘what would Picard do’. Committed to peace, cunning and forceful in battle, is the perfect military strategist.

In January, ST:TNG completed 30 years.  And it is still timeless.


Picard is the ‘living’ example of the power of self-education. image source

ST:VOY is a more recent series (better FX).

It has the most perfect plot motto of all times (spoiler alert): a starship that got stranded in a a remote part of the galaxy and needs to make its way back into the Federation. It was completely new ground, new species (some old acquaintances), as a result a very fresh series.

The USS Voyager is commanded by Kathryn Janeway (Kate Mulgrew). Janeway is resorcefull, driven and workaholic. It got me to become addicted to caffeine too.

The ship encounters many perils on the journey home, many times comes close to destruction. But the determination of the crew to continue (or will of the writers) always seem to keep the ship together.

A memorable character developed  late in the series is Seven of Nine (Jeri Ryan).


Captain Janeway (left) and Seven of Nine, probably arguing. image source

The last series we will discuss is ST:ENT.

A short-lived project. Enterprise depicts the adventures of the first starship capable of travelling in warp 5. This is the beginning of the Federation, a time where technology was a lot less developed than in the former series.

There is nothing much I can say without spoiling the series, but it has a different flare from the others: it’s more crude, more close to our reality. The challenges reflect this, they are more of a basic nature.

The series didn’t last long. In my opinion, this was due to the long story arcs (covering half a season in one case).  Still, there are memorable episodes.

Considering the number of episodes in all ST, it’s bound to have plot repetitions. There are at least 7454457 hostage situations alone.  One can’t help to wonder if is there anything new to be imagined in the universe. I certainly hope so.

[IMAGE: USS Voyager, from the opening scene. credits]