Elegance vs. Existence

Quality's not always more important than quantity.

I know that I've been growing learning slowly because I'm too meticulous. Some things you really only learn by doing. And if you work your whole life away perfecting one painting or writing the perfect program, you'll probably never get it just right. With some things you must go start to finish over and over and over and over.

And by doing more, you understand more, which leads to more (and better) applications of your particular art.

Maybe the quickest way to elegance is not by trying to have the most elegant solutions. And the quickest or best, or maybe only way towards that elegance is through relatively crude iterations.

The Quest for Random

Think of a sequence of five random numbers.
Got them?

Good.

Actually, hate to break it to you, but the numbers you just picked aren't truly random. In fact, I can probably tell you a good bit about all of the numbers you just thought of.. They're all most likely:

* Between 0 and 1000. And thus, positive.
* Whole integers, that is, none of them are fractional or have trailing decimals.
* Non-repeating.

To say all of those are probably true about your "random" numbers is pretty bold. But think about the numbers you generated. Are all three assumptions true?

In some respects, humans are poor random number generators. But don't feel bad because I have another surprise; sometimes computers are much worse!

Computers follow precise instructions. It's the nature of a computer. In fact, if it doesn't follow instructions perfectly, it's considered broken. There are formulas that a computer can execute in order to get a sequence of numbers that seem random. And for most purposes these pseudo-random numbers are close enough to random to use in gambling and simulations and such. But true random is hard and maybe even impossible to come by.

It really boils down to how we believe the universe behaves. Some people believe the universe, ever since its inception, is set upon a predetermined course. Others believe that the universe, at some level, perhaps at the quantum/sub-atomic level, is the way it is without cause.

If the universe is deterministic, and we are able to collect information about everything, then randomness becomes impossible to achieve. Even the things "True Random Number Generators" are based on, such as quantum physics, weather patterns, and atmospheric noise, become calculable, predictable, and thus non-random.

What's perhaps even more interesting, even if we were to have a perfectly random number generator, we could not formally prove its randomness! We would need a list of infinitely long numbers from it as well as an infinite amount of time to conduct statistical analysis on it.

Having said all this, I'd like to claim back the word 'random'. To be able to use it colloquially, we distinguish it from 'true randomness' by defining regular old randomness as simply being "unpredictable".

Reflections on the English language from the mirror of C++

The most common cause of security problems and run-time errors are "buffer overflow" errors (C++ Primer 4th ed).

Basically, this means that the programmer has written a program incorrectly. He/she has either tried to access an element of a container that doesn't exist, or tried to assign a value to a class that doesn't accept that type of value. Or something similar.

This sounds like an easy problem to fix--only assign numbers to number-type variables. Assign text to text-type variables. The real problem is that there are a lot of different "number-type" and "text-type" classes. And every different kind of class you can think of and a lot more that you haven't thought about. Even within the type "number" we can talk about whole integer type numbers, or floating point type numbers, imaginary numbers, Marsenne primes, etc.

Another problem; a lot of the types built into the C++ language are non-intuitive. Or rather, the vital differences are difficult to remember and cumbersome. Example, there are three different types of floating point classes, each with different storage capacities. And even the capacities are not set in stone, they depend on the machine that runs the source code a programmer writes.

So why don't we just change the language?

Because it's precise. Everything that C++ is, is that way for a good reason. It is a very powerful, efficient, and precise language. The fact that C++ seems "non-intuitive" really highlights how imprecise we are.

And I'm not saying it's a bad thing. Languages are developed for purposes. Programming languages are developed to solve problems and create data structures. English is a very general purpose language that I think succeeds in its goal. To communicate.

Sometimes its ambiguity is even a good thing. It helps succintly create multiple meanings in literature. We enjoy discovering and creating our own meanings from our experiences. Being able to communicate those meanings gives English--and other spoken languages--a power of their own.

Good Tidings

There are a surprisingly many kinds of loneliness, and they seem to creep up at very unexpected times. You can be lonely sitting next to good friends. Or lonely in a classroom full of people. It's even possible to be lonely and crowded at the same time. It can be trasient, or chronic. Excruciatingly painful, or just slightly annoying.

Loneliness can come even during the warmest time of the year: Christmas. To everyone experiencing the holiday blues; I send my heart and all my warmth.

n-dimensional Learning.

A lot of the technical learning I do seems to come from books. Textbooks. Reference books. Manuals. Etc.

But I'm tired of learning in one dimension.

The author(s) organize the text to be read, generally, from front cover to back. Reference manuals are a little more generous. The authors of such books usually assume the reader will jump from topic to topic and there may be more of a guide than a simple table of contents and index. But no matter how the information in a book is organized, I feel like the job of connecting all the topics is up to me.The problem, is that I get it wrong initially and it takes me a while to get it right.

So I want to introduce a new way of learning. One that's more conducive to spatial understanding. A way of organizing information so that the connections between topics is obvious.

This won't be the first time graphical, spatial interfaces are used to organize information. Visuwords is a free online graphical dictionary. You can type in a word and see a word's hyper/hyponym, mero/holonym associations, related verbs, nouns, adjectives, adverbs, topic/region/usage domains, etc.

Essentially, my system would consist of a number of orthogonal axes 'n' and an origin. For instance, in learning C++, the origin might be the following: "C++ is a programming language. The language is made up of: Classes, Functions, Variables, (etc.)." Then to the right would be a column-like space. The top sub-space of the column might be Classes, below that, Functions, below that, Variables, then Enumerators, etc. So we might have a column of 7 'sub-spaces'. In one of the seven sub-spaces we might read "Enumerators group sets of integral constants." To the right of that, a more detailed explanation of enumerators, specific syntax, limits to how we can use enumerators. To the right of that would be a space for concrete examples of enumerated lists in C++, more detailed explanation, problems and solutions. To the right of that might be more challenging exercises, abstract relations problems, etc.

In the example we deal with two axes. The horizontal axis would be depth of one topic, and the vertical axis would be breadth of all topics given a certain depth. The interface for a book-space of C++ might look like:

The grand idea might be to have a single, universal Information-Space (IS). Something like Wikipedia but with more detailed information as well: textbook-depth information, academic journal level information. You could navigate the IS through a customizable digital interface. An IS browser. Your browser could track where you spend your time learning/studying. The breadth you cover as well as the depth of individual topics. You might chart it out and see that "Oh, while I was in school I spent a lot of time in the Math sector of the IS, now I'm spiralling outwards, hitting computer sciences, chemistry, and physics. But compared to last year, my average depth in topics of the IS has increased."

The goal of this all is just to help people learn. And maybe to help us see something about how we learn.

Punctuation.

I have a real problem with quotations in written English. I'm not sure this problem persists when writing most other languages, but I believe it to be quite a nuisance in mine:

QUOTATIONS

The period ( . ) indicates the end of a complete thought. Quotation marks ( " " ) delimit text you've taken from somewhere and inserted somewhere else. But if I'm making a list of topics and one of the the elements is "bracket," then according to proper style I have to put whatever (if any) punctuation inside the quotation marks.
If my list is:

List:
full stop
bracket
quotation marks

There might be an important reason I didn't capitalize or add punctuation to the elements in my list. However, when I mention this element, "bracket--" I have to include punctuation inside the quotation as if it were a part of where it came from. Even when it looks ridiculous, as in the previous example! I propose that punctuation outside of the quotation marks becomes standard.

Time again;

#include <iostream>
#include <string>

int main( ) {
std::cout <<  "Hello World." << std::endl;
return 0;
}

Lip balm.

You never run out of chapstick. It's one of the few things in the real world that is infinite. No matter how long you have a stick or how often you use it or how heavily you apply it, there's always more.

The only reason chapstick manufacturers make money is because I keep losing sticks and have to buy more.

-

Winning

I think about the future a lot.
I think about all the material things I want. The way I want my relationships with people to be, what I want to know and be able to do.

But now I'm asking myself why I want that future. I guess I have this notion these things and relationships will bring me peace of mind.

I see that a portion of this 'notion' is illusion. Maybe peace is found. Maybe you have to just get lucky.

Siddhartha's Struggle

I never feel like I'm telling the truth.

Life can be complex. Feelings and human things especially. By the time I describe precisely how I feel, my emotions will have changed.

My pains lie in discrepancies. The world I see and the world I want are different, and the discrepancy gives me a driving pain. The difference I see in the way I want to be and the way I really am induces a sinking pain.

I thought being more honest and self-aware would make the discrepancies clearer and easier to deal with, but that's not necessarily the case. I just see more and more differences between ideal and actual.

Why is it that discrepancy causes me such disturbance? Why can't I just accept discrepancy?

Paths

I've recently had the opportunity to work with power tools.

A lot of my studies seem abstract. Beatiful, but I work a lot with my head. To work with my hands trying to construct something almost as beautiful out of wood or metal is refreshing.

It's making the division between engineer and mathematician more evident I think. Part of me is interested in application. I find real-world problems fascinating. But part of me is curious. Mathematics does not limit itself to what is real, or even what's possible. Math seems limitless. And the more I see of it the more curious I become.

Unfortunately, it may be a dangerous road. You know what happened to the cat who got too curious.

Complexity.

Sunset, Sunrise.

The atmosphere is constantly being barraged by meteoric material.

During the day we only see blue skies and white clouds.
It's only after dark that we can see what sunshine suppresses.
We can see meteorites-- "shooting stars." They trace blazing holes in the deep-sea-blue-blackness of night.

In this hour, even my stay-up-late friends are tucked in.
I'm tired, but sleep does not come.
There is so much unvisited material. Unnamed systems. Secret satellites orbiting hidden bodies. Even their trajectories tease of mysteries.

The thoughts I stifle during the day have little competition now.
I feel their constant, burning volley and I trace the holes they leave.
I'll wake up without memory or evidence of punctured consciousness. The balm of morning and noise of day will avert my attention.

They slow the bombardment like a lake slows speeding bullets.

The Calculus of Identity

I've been thinking of boats and rivers lately.

Particularly of Heraclitus' river and Theseus' boat.

They both deal with identity. Heraclitus claims that you can not step into the same river twice because the waters are constantly flowing. The river holds different water, and is thus, a different river.

On Theseus voyage, the young Athenians he carried with him replaced every decaying board with new and strong timber. By the time they returned to home port from their long journey, they had replaced every board leaving philosophers to ponder--was it the same ship that had sailed out, or was it an entirely different one?

To answer whether they're the same ship or river, let's find out what identity is. From a philosophical perspective, identity, is whatever makes something definable or recognizable. Heraclitus argued that change of components made it impossible for us to step into the same river. Several centuries later, Newton and Leibnitz Calculus. I feel that Calculus is an apt perspective to take on the problem since it is "the mathematics of change" and I think change is the core variable to study when looking at identity.

I agree that, for the most part, we identify entities based on the sum of their components. A forest is its trees, a river is its waters. Or precisely:

x = all parts of x

Heraclitus argues that after time, the identity is broken:

x =/= x'

(Where x' is x after some time has passed.)

I argue however, that the river remains the same if you observe it as it goes through sufficiently small, continuous intervals. Precisely:

1 ~ 1.000000001 ~ 1.000000002

(Where ~ means approximately equal to.)

The closer the change in x gets to 0, the closer we get to being able to say x and x' are the same. Our question really becomes, "When is the change in x insignificant?" 1 meter of water flow? 1 millimeter? 300 kilometers? So the ability to step in the same river twice depends on your perspective of the changing water in the river.

Applied to Theseus' ship: If we stand on the dock as the Athenians leave in the first ship (ship y) and the next time we see the ship sail in, it's made entirely of new timber (ship y'), then it would seem to us that the ship is not the same (that is, y =/= y'). However, if we embark along with Theseus on ship y, we see the ship change very slowly and continuously. It would seem that we've been riding the same ship the whole time.

Physics

I ran across an interesting problem in my Physics homework today.

Find the torque caused by forces F1 and F2 about point A given distances between all points and the angles between F1 and the y-axis and F2 and the x-axis.

The actual calculations are rather boring, so I'll spare you. But it's neat to think that, given the mass of my legs and hips, angle between the ground and my feet, my weight, rotational ability of my hips and legs, I can calculate the force of a round kick. And if you can write an equation and calculate it--with some help from calculus--you can maximize it.

Also, I finally understand what a flywheel is for. It stores mechanical energy in rotating mass. That's so brilliant and yet so simple.

Control

I've been thinking about reality vs. beliefs and illusions. If you'd asked me a few months ago what I thought about the whole situation, I'd probably have given you "Reality is that nobody can control anything. Nobody can know anything and we're only flailing in vain at coincidences and things beyond our control."

But as I sat last night, getting my character Belethor to level 16 in Puzzle Quest, I realize we do have control. Similar to the moves I make in a chess game or the jewels I switch in a video game puzzle, there's a degree of control I have over my immediate environment. More interestingly, I can create vast change in my global environment depending on how I act. Like a pawn move that opens lines of attack and protects vital squares, like a combo in my puzzle game, like a spider in a web, like a butterfly who flaps his tiny wings that pushes some wind which pushes other wind and 100 years later, there's a hurricane on the other side of the planet.

I can't calculate that far with each of my actions, but just like in Chess, there's an intuition, a strategy, and a limited amount of calculation that makes my control real.

Big Crazy System

Every time your heart beats, it's because your Sinoatrial Node is pumping electricity to other nodes and muscles within your heart. Every square inch of your skin is covered with nerves that send electrical impulse to your spine and up to your brain. Inside your brain is a storm of electrical activity that we have only recently begun to measure and understand.

Did you know that every electrical current induces a magnetic field? Furthermore, the hemoglobin in our blood is a metalloprotein and has a lot of iron it. So every pulse of electricity up our spine, every firing neuron, and every activated nerve applies a (very slight, perhaps insignificant) force on even the tiny blood cells in your body.

I'm not sure what to think or say. The System is unpredictable. How can modern man be surprised by anything? Life is crazy. How can anyone believe anything?

In order to even communicate meaning, you have to isolate. Choose a specific word, give me a time, supply context. Eventually, with enough chosen restrictions, you can start to create meaning with words. Maybe we can do the same thing with the Universe. In fact maybe we have been on our way ever since the days of Pythagoras. Maybe now we're just working our way up--or around--and as we understand more and more, one day we'll approach that inifinity of Universe understanding.

For now we can only choose to survive by feeling things out the best we can.

Methodology

I want to share what I believe is possibly the most important thing I have learned in college--or elsewhere.

It's the learning/doing methodology.

By "learning/doing methodology" I mean the systematic breaking up of bigger objectives/problems into more immediately possible bits. The most convincing example for me is probably that of computer programming.

We can program computers and other devices--using a large amount of simple instructions--to perform a very complex task. Like graphically display our geographic location as we're driving, or simulating molecular reactions or complex electrical circuits. Thanks methodology.

Road map to doing something:
1.) Identify your objective. - What's great is that your objective doesn't have to seem possible! The tricky part is defining parameters of success. Keeping with the computer motif, let's say we want to create a system that tracks our location via satellite. Great. But how accurate do we want to be? Down to the mile, to the millimeter? When have we achieved what we set out to do?

2.) Break up the problem. - Learn ballet. That's incredibly difficult. At least for me, I've been struggling to understand for the past few weeks. So how do I learn?

Systematic deconstruction of objectives. Primary goal is to learn ballet, this includes learning terminology, being aware of my body, understanding what good ballet looks like when others do it.

Being aware of my body? What does that even mean? Well the cool part I think, is that there are different roads to the same place. I can concentrate and get to know my stomach/leg muscles. Case two, I could read an anatomy book. Case three I could have a discussion with my ballet instructor. If I ever get stuck, I just break the problem into a smaller piece.

And those two steps are really all you need to do the impossible. When you get stuck apply step 1 if you can, if not apply step 2. Repeat ad nauseum.

Evaluation vs. Approximation

The difference between the two, for the purpose of this post, is that "evaluation" means perfect appraisal while "approximation" is just close to perfect.

It's natural to ask which is better, and I believe most people would say "perfect" looks better on paper. So why approximate anything? Don't we have calculators and computers that can crunch the numbers for us and give us perfect answers for everything? The answer, actually, is no.

Probably the most famous example is of our friend pi. Nobody has ever perfectly evaluated pi in decimal form and no one ever will. My TI-83 can evaluate pi to about the ninth digit place. Better equipped computers and computational engines can provide even more. As many as you want so long as you a.) have the time to wait for the computer to calculate pi that far and b.) don't want all of the digits of pi. It would take an infinite amount of time to calculate the infinitely many unique decimal place values.

In a way, this is as close as we'll ever get to our friend:

And as handsome and perfect as the Greek vision of our friend is, it's rather useless; we measure everything in decimal. We compute (with computers) in binary. We encode in hexadecimal. Pi, in it's perfectly evaluated state, is worthless for purposes other than observation and pontification.

My first analogy relates evaluation to moral perfection and approximation as effort to be morally good. Specifically within certain divisions of Christianity, there sometimes seems to be a belief that even if you don't try to be the best you can, if you just reach the point of "good/holy enough" then that's okay. It's okay specifically because you can never be perfect anyways. So really evaluation becomes worthless yet again. A perfect person is not something we can all objectively agree upon. Approximation then becomes of great importance.

My second analogy relates evaluation to Truth and approximation to mere observation. I've talked to friends and peers and there seems to be a common discomfort with the possibility that we may never reach an absolute Truth no matter how far Science or Philosophy advances. "How can we even define progress? How can anything mean anything?" And on a larger level I'm, indeed, not sure if anything has meaning. But if we choose to assume that anything has any significance, then we can start making meaningful observations. Just because we can't reach infinity, it doesn't make the whole number line insignificant.

Finally I present a more concrete example of the significance of approximation. As we all may have noticed, the lifespan of a storage medium is finite. As a culture, we have moved quite rapidly from 8-tracks to cassettes to CDs, DVDs, Blu-Ray, etc. If we assume the lifespan of the CD as a popular medium is 20 years, then how long should we spend on making CD's better? Should we perfect the CD as a medium? Of course not, 20 years would pass before we make the CD able to fit an infinite amount of data (and that would be a perfect CD!). So let's just spend 3 months designing our medium, the CD. Well then we can use this medium for 19 years and 9 months, but it may only be possible to store 100 MB on this poorly designed medium. Very unfortunate given how expensive plastic is these days. Perhaps you can see that there is a point somewhere between 0 and 20 years that we should spend designing the CD for maximum efficiency. In order to achieve the maximum--and therefore perfect amount of efficiency--we must design our medium to be less than perfect.

The State of Mathematics

The primary objective of this blog is to record thoughts/ideas I have.

I'd also like to inform and entertain readers. It'll probably center around math but there's no reason the laypeople shouldn't be able to read and enjoy.

I guess a good first post would be to address the state of Mathematics as it is today. I asked my professor today after class if Mathematics is dying--as one of my friends majoring in Math would have me believe. The answer I got surprised and relieved me.

Yes, in a sense Mathematics is dying. "All the big questions left will take a lot of time." My professor went on to tell me though, that the area covered under the umbrella of 'Mathematics' is expanding rapidly. No one person knows everything about Geometry, or even a division of Geometry. There are plenty of dark corners left in the study of math. I feel and hope that there always will be.

Just as poets relate our world to feelings, words, etc. through peoms, mathematicians relate the world we experience to precise models through equations, variables, etc. We will as soon run out of math to study and make as we will songs to sing and literature to study.

The same professor is writing a play about Calculus now, actually.