I’ve created more pictures based on the previous post. I’ve represented the different approximations as surfaces, and put all of them in motion so that you can better see how they work. The page is at http://blog.dutempstrouve.com/math-art/contour-approximations/.
You might be thinking, what is that? And what does it have to do with me? Or maybe you’re just thinking: Gosh, that looks beautiful – I wish I knew how to make such pretty pictures.
For the latter sort, you could get a math degree, or you could take a shortcut and read my tutoring blog, where I explain other marvels and will soon get into some more technical detail about today’s work of art. But for the former, read on.
We start off with some really weird curve (read: I mashed a few keys in a graphing program and worked with what came up). Now, if you were an ant on there looking close up, how would you visualize that curve?
Maybe you’d think that it was just a line, a line heading off at that same angle that you currently find yourself. Maybe, as you walk along, you’d constantly revise your view of the world:
Such an ant could manage themselves just fine. As long as they keep looking at where they are, they can make it around the curve. But their conception of the world at large is completely off. Almost every single judgement they would make about other points on the curve would be wrong. The ant can live in the “now” as long as they keep it to themselves.
But our ant could be a little more sophisticated:
This second ant is still using data merely from their local environment; they just estimate a curve of best fit, instead of a line (in mathese: they find a tangent conic, instead of a tangent line. In worse mathese: they use a truncated Taylor series as calculated at that point, the general technique for all estimations in this post). This curve still doesn’t match the overall pattern very well, but it does a darn sight better at making decisions about the neighbourhood.
A third ant might be a little more resourceful, and come up with this:
Now, this ant can start making broader claims about the world. The ant will still be off, but significantly less so than the previous two. And finally, we have genius ant here:
Again, this ant’s knowledge is not perfect – but she absolutely nails entire portions of the world, using just the information that is at her immediate fingertips. She doesn’t have a larger view of the world, she has a deeper view. Maybe she still lives in the “here and now”, but she doesn’t rest content with mere appearances. She doesn’t just calculate how things are, or how they are changing, but how change changes, and so on (technically, up to 8th derivatives). By assuming that everything is changing, including change itself, she can understand.
And as a bonus (which I’m not sure has any analogical value, but which looks cool), here’s all of the approximations together in one animation:
Why drive on the right side of the road? The left works for the Brits. Why start question-words with wh- instead of k-, as does Sanskrit, Farsi, and certain dialects of Ancient Greek? Why read left to right? We could go right-to-left, like Hebrew and Arabic, or even top-to-bottom as is sometimes done with Chinese. Why is i the square root of -1, when (-i)² also equals -1? Why does a clock go clockwise, when making it go anti-clockwise would work just as well?
All of these choices are purely arbitrary. The alternatives would have worked equally well. But refusing to choose one between equivalent choices would have left confusion; it would have erased the possibility of meaning and cohesion.
Meaning starts from the meaningless; from de-cision, that is, cutting something away.
The paradox of “Buridan’s ass” is about a donkey that is faced with two equal succulent bales of hay, both the same distance away. Does the donkey starve since there is no reason to choose one bale over the other? Al-Ghazali similarly writes,
Suppose two similar dates in front of a man, who has a strong desire for them but who is unable to take them both. Surely he will take one of them, through a quality in him, the nature of which is to differentiate between two similar things.
So the rational rests on the irrational; because without this symmetry-breaking, nothing happens.
Picture Data from Star Trek. The quintessential robot: all reason and logic, no emotion. Not that he’s heartless, but rather, he just doesn’t understand this side of human behaviour, no matter how hard he tries. (Except when it makes for a better plot point otherwise.)
But what are emotions, other than our own preprogramming? Those who are at the mercy of the passions are the ones blindly following their hardwiring. Those who can’t take a step back and look at why they really are doing what they do, those who are wounded by the suggestion that we are organic machines with modules shaped by our biological and sociological histories, are those who end up being the most robotic. Machines running on chemicals masked as “spirituality” and “humanity.” It is at our most irrational that we are most programmed.
Even the cold logic of economists and engineers follows the same route: a search for logical efficiency and consistency, but stuck in the rut of an initial question that is not itself questioned. The precision of gears, working toward an initial impulse to categorize, simplify, and control. The dominance of their evolutionary firmware is less apparent, but ultimately the same.
Being human is neither revelling in feeling for its own sake, nor in mere logical analysis (which is much the same); rather, it consists in the ability to look behind the curtain at who we are and where we come from, to understand, to make decisions based on this; it is to be able to make sense of our own owner’s manuals and fiddle, reprogram, rewire, and, in the end, simply accept what sort of beings we are.
Because we are, after all, organic machines. Our emotional operating systems are part of us, and must be taken as such; just as a computer cannot run without some sort of system to boot it up and manage resources, so too must we rely on our own frameworks, imperfect as they may be.
But we don’t worship Windows because we have to use it. Valorization of the emotions as distinctly “human” or as some royal road into reality is similarly misplaced.
Currently in the middle of watching a Korean martial arts flick (Legend of the Shadowless Sword), and enjoying it. But while watching people flying and fighting off entire armies, I started thinking about what sort of truth fantasy might be telling. (Yep, welcome to my brain. But honestly, do you expect to believe that talking about football and the weather is more interesting than epistemology?)
The genre is purposefully fantastic, not even trying to make the feats look realistic. And yet, there is something lacking in a purely realistic film. I’ve punched someone in martial arts, only to have them suddenly not be there. It’s not that I registered them moving, so much as they were no longer in the way of my fist. And suddenly, there was something striking my solar plexus, or hitting a pressure point and making my arm go numb. I’ve watched someone dodge a sword strike – the assailant was attacking from behind, with no warning, and without letting the person know the specific strike they were going to use. I’ve felt the force of a true martial artist merely looking at me with serious intent.
If I were to depict all of this in purely physical terms, to make it “realistic,” it would fall short of the actual experience. By exaggerations, I can accurately portray what “really” happened.
Of course, at the same time, this only works because it’s on the level of human emotion. It’s not reality by itself; it’s how human beings reconstruct events. So maybe one sort of distortion is necessary to transcribe another sort of distortion.
Just some random thoughts. Now, back to watching this confident, poised warrior chick kiss some a$$.
Der König Ödipus hat ein Auge zuviel vielleicht.
King Oedipus has perhaps one eye too many.
- Hölderlin, In lieblicher Bläue blühet
When do we know too much? Oedipus was a figure who was insistent on finding out the truth, despite being warned repeatedly by the seer Tiresias that it would bring misery. He had to know what was causing the plague on his kingdom; and upon finding out that it was himself, who had killed his father and married his mother, Oedipus blinded himself and ran off. He went from having two eyes which searched too much, to having none.
And seeking understanding can bring suffering. There is a wisdom that is woe. But, at the same time, there was already a plague on Oedipus’ kingdom. The objective consequences of his actions were already there; by refusing to confront the matter, he could have put off his own suffering at the expense of the world around him. And in the play Oedipus at Colonus, Sophocles tells of how Oedipus dies a blessed death after all because of this, despite his being the lightning rod of the gods’ wrath earlier.
Seeking the truth is not always rewarding. Dispelling delusions can rob us of comfort and alienate us from others. But if those really are delusions, then they are already at work twisting us, warping us. There is something natural about keeping that one eye closed; something tragic and heroic about the excess of light followed by darkness. And who can say whether the natural or the heroic is the better path?
I’m going to split time between this blog, and the (rather similarly named) http://blog.dutempstrouve.com. I might get rid of this one entirely, or merge the two. I might put more personal and controversial stuff here, and stuff that I’m more willing to give to a broader audience there. But anyhow, the other blog is up now, and I have some entries already planned out for next week. (So the 30 Day blog challenge is going to transmute somewhat into a 5-day a week regular blog on different topics.)
But today, Hölderlin (full text here: http://300daysinberlin.wordpress.com/2013/05/26/lebenslaufthe-course-of-life-holderlin/; see Lebenslauf II toward the bottom of the page):
More you also desired, but every one of us
Love draws earthward, and grief bends with still greater power;
Yet our arc not for nothing
Brings us back to our starting place.
Whether upward or down – does not in holy night
Where mute Nature thinks out days that are still to come,
Though in crookedest Orcus,
Yet a straightness, a law prevail?
This I learned. For not once, as mortal masters do,
Did you heavenly ones, wise preservers of all,
To my knowledge, with foresight
Lead me on by a level path.
All a man shall try out, thus say the heavenly,
So that strongly sustained he shall give thanks for all,
Learn to grasp his own freedom
To be gone where he’s moved to go
I’m thinking about getting a line from this poem as my next tattoo, curving both randomly yet gracefully up my upper right arm. It describes my life pretty well – I keep having plans, things I intend to do, and they almost never seem to work out. Things change, in one way or another, sometimes in ways I like, sometimes in ways I dislike. But somehow, through the twists and turns, something seems to come out of it all, year after year. Beyond that, I’ll just let the poem speak for itself.