Saturday, May 1, 2010

"Seeing is not always believing" -MLK

My first instinct upon waking to find my eyelid sealed shut is to leap out of bed in panic, which leads to my second instinct which is to vomit all over the floor. The condition of my eyes has always been a sensitive topic to me. As a child, the routine trips to the eye doctor which did not result in me vomiting all over the floor required frequent intervals of floor-rest. This is a technique where I fall to the ground and lay very still until the cold sweats pass. Fortunately, on this morning, I manage to pull off this maneuver before I lose last night’s squid-stuffed squid across my carpet.

When my vision finally clears in my good eye, I blindly attempt to reach my arm over the edge of my desk to push off my cell phone so that I may seek help. I succeed only in knocking my computer speaker onto my face and pulling a muscle in my right shoulder. Fortunately, the combined pain from these two events actually distracts me from my nausea long enough to sit up and grab my phone with my functioning arm before returning to the safety of my carpet. I congratulate myself on my accomplishment, and then begin thumbing through my contacts list so that I can receive medical aid.

Here’s the thing about my contacts list: I hate all my friends, and they aren’t super keen on me either. I figure there are at least three names on the list however who are at least in some way responsible for my well being: my mother, my girlfriend Autumn, and Matt. Only two of these people are in the same state, so I start at the top of the list.

*ring, ring*

A groggy voice answers the phone. “…hello?”

“Autumn, I’m dying, I need you to drive me to the hospital.”

Autumn, unconcerned: “…what?”

Me: “I need you to drive me to the hospital.”

I imagine she rubs her eye or something at this point. “What time is it?”

I cannot see my alarm clock from my strategic position on the floor. I take a wild stab at it. “7:30?”

Several seconds of silence pass as she weighs the pros and cons of doing what I say. Finally: “Can’t you do it yourself?”

I actually consider this for a few moments before remembering why I’m still on the floor. “No, I really don’t think that’s a good idea.”

There are several more seconds of silence. She really, really doesn’t want to do this so I help her out. “I guess I might be able to take the bus…”

She is relieved. “Are you sure?” (this question is probably rhetorical)

No. “Yeah, I’m tough like that, you know?”

She knows the opposite, but she sort of owes it to me not to laugh so she mumbles something about hoping I feel better and then goes back to sleep. Next contact:

*ring, ring*

A groggy voice answers the phone. “…hello?”

“Matt, I’m dying I need you to take me to the hospital”

“…”

“…”

“…really?”

“Well…” It seems like too much work to explain. “Yes.”

“Shit…” A few moments. “Do I have time to shower?”

I tell him he probably has time to shower, and while he does I try to drag myself to the stairs but don’t quite make it so I wait until Matt arrives and helps me out to his Rav 4. “Beutel?”

“Beutel.”

Beutel is the name of our on-campus health clinic. Driving to the clinic I actually begin to feel a little bit better about my eye. It’s not that I know I’m going to be healed soon, it’s because I begin to feel afraid I’m going to catch something much worse: like the Ebola virus or mononucleosis (this actually happens later). The unofficial student motto of the place is “If you aren’t sick coming here, you sure as hell will be when you leave.” As long as it’s not my eye that is sick when I leave however I’ll be ecstatic.

Eventually we arrive, and I make it up the steps to the clinic without too much assistance. The smell of Lysol and rubbing alcohol hits me like a dirty mop upon entering the complex, but I shrug it off. Matt grabs a handful of free condoms from the Planned Parenthood bowl and sticks them in his jean pockets while I approach the student worker at the reception desk. “Hi, I’d like to get my eye fixed, please.” The girl looks up from her organic chemistry homework like I had been specifically employed to ruin her day.

“Fill out this form and have a seat over there.”

I feel like my case warrants immediate emergency care, but I am in a forgiving mood so I fill out her stupid forms anyway.

The waiting area is fairly typical, row of chairs against either wall separated by surfaces on which to stack magazines. Matt and I choose chairs on the far end of the room and try not to make eye contact with the other patients in order to avoid infection. One has the audacity to cough as we pass him, so we make some offhand remarks to each other graphically depicting his murder.

Matt: “Matt draws his katana and makes a single clean slice diagonally through the cretin’s mid-section.”

Me: “The victim pauses for a moment in shock, and tries to assess the damage as the top half of his body slowly slides off of the bottom half. His body erupts in a fiery explosion.”

Matt: “Matt coolly flicks the blood off his katana and re-sheaths it. The room applauds.”

This goes on for about 20 minutes until my name is finally called by an obese geriatric lady wearing a hideously noisy blouse, presumably so that she can be recognized as a legitimate healthcare professional. I tell Matt if I’m not back in 15 minutes to call the police and follow her to a small uncomfortable room where she begins taking my blood pressure and interrogating me overly loudly.

“WHICH EYE IS BOTHERING YOU?”

There is now a thermometer in my mouth, so I just point to the eye that is still crusted shut. She does not understand, although she is looking directly at me. So I mumble, “muh uft”. This works somehow, as she nods and scribbles something down and removes my thermometer.

“ALLERGIC TO ANY MEDICATION?”

“No.” There is something off about her.

“WHEN DID YOU FIRST NOTICE A PROBLEM WITH YOUR EYE?”

“About 40 minutes ag…”

“WHICH EYE IS BOTHERING YOU?”

I begin to have nightmarish fantasies about accidently getting a spleenectomy or something. “My left…”

“OH MY GOD…YOUR EYE LOOKS AWFUL!”

I think I notice her clipboard is upside down.

“OK, YOU’RE DONE”

There’s no way this was my doctor. “Wait…what’s wrong with me?”

“I THINK THERE IS SOMETHING WRONG WITH YOUR EYE. YOU’LL HAVE TO WAIT AND SEE A DOCTOR. YOU CAN GO TO THE WAITING ROOM NOW.”

“But I just came from the waiting room.”

“NO, THAT’S JUST THE PRE-PROCESSING HALL. THE WAITING ROOM IS OUT HERE.”

Sure enough there is another goddamn waiting room. This one is even more populated then the pre-processing hall, which I return to briefly to fetch Matt and inform him of the situation. He sees me coming, “Victory?”

I shake my head. “No, I just beat level one. It turns out the princess is in another castle.”

He nods, this requires no explanation. We enter the level two waiting room together and find some new isolated chairs. The time on my phone says it’s after nine o’clock now, and then my phone begins to vibrate. I have an incoming call from Autumn. I sort of remember that I’m supposed to be mad at her, but I don’t really have the conviction required to hold things against people.

“Hellooooo?”

“Hey cutie. Did you make it to the doctor?”

“Yeah. Matt drove me.”

“Good, I wanted to check and make sure you were ok. I’m driving to school now. I have an exam today. Am I going to see you later?”

“Yeah, they’ll probably call you to identify my body. I hope that’s not during your exam.”

“Haha, that’s not funny.”

“You should take me out for ice creams when I get out of here. This whole ordeal is very traumatic to me.”

“Aww…I would but I wanna look over my flash cards again a few more times before the test. How about tonight?”

“Yeah ok…”

“Bye lover!”

“Bye.”

After hanging up I remember that my eye is damaged and feel sick again, so I lie down in my chair. I want my mom.

Matt, who had been playing flash games on his iPhone while I was talking to Autumn, leans over to show me what level 67 of his tower defense game looks like, and isn’t his helicopter unit that transforms into a giant robot the coolest thing ever? I agree that it is, and then my name is called again by the loud obese woman. Matt wishes me luck and I am lead to a slightly larger but even more uncomfortable examination room to wait for the doctor. The wall is saturated with posters detailing genital diseases, and I have to lie down on the table to stop from vomiting.

Finally, the door opens and my doctor enters the room. I hate her immediately.

“So,” She looks down at my seven pages of information on her clipboard but then decides it’s too much to read. “What seems to be the problem, Michael?”

I point to my eye (which is still crusted shut because I haven’t been brave enough to touch it). “My eye is stuck.”

There must not still be twelve other students in the waiting room, because she begins wasting time having a conversation with me. “You are from the northeast, aren’t you?”

I nod.

She waits for me to ask her how she knew that. I don’t, so she asks for me. “Do you know how I knew that?”

“Umm…it has my place of birth on that information sheet?”

Disappointment shows on her face as she glances back down at the clipboard. “But I could tell before I saw that.”

“Oh. Ok, how?”

She beams. “From your social security number! The first 3 numbers are a regional code. Most people don’t know that.”

I start to wonder if she is really my doctor or just an autistic woman who memorizes social security numbers playing dress-up. “That’s pretty cool, I guess.”

Satisfied, she gets down to business by putting on some spectacles. “So what seems to be the problem with this eye of yours?”

This information must not be encoded in my social security number, or in the fluid visibly dripping from my eye socket. “I think it’s infected.”

The part where she examines my eye goes by too quickly for me to notice. The waiting room must be full again. To save time, she simultaneously begins to perform some medical procedure on my eye while explaining to me what the problem may be. A vial of terrifying red liquid appears in her hand and I find myself pinned to the examination table with the other. I am afraid. “There may be something stuck in your eye.” The cap on the vial has been removed, and I instinctively recoil. “I am going to drop this dye into your eye, it will let us see if there is anything stuck inside.” The vial is directly above my eye, I try to edge away but she pins me tighter. “Open your eye.” Although I try my very best to do this, I am psychologically completely unable so she takes the liberty of prying it open herself. I stifle a scream, and tears begin to flow as I feel the dye enter my eye and she releases her hold on me. It’s over. She shakes her head and writes me a prescription for antibiotic eye drops. “I don’t see anything in your eye, just put these drops in your eye every day and come and see me in a week if it’s not better.” Before she exits, she turns around and says to me: “Jesus…I’ve never seen anyone who is so squeamish about their eyes before.” I am left on the table, broken and crying.

I recover myself and stagger, in a stupor, back to the waiting room. Matt gets up and follows me to his car. “Ice creams?” He says.

I smile. “Ice creams.”

Saturday, April 17, 2010

The End is Near

On the flight back from Monaco I read Bozo Sapiens between glasses of neat Scotch. I would read some, then I would stare out the window of the plane at an altitude of several miles and wonder how we (Humans) have managed to achieve anything. Then I would remember that we haven't.

The book points out carefully how nearly everything you ever do, or see anyone else doing, is no different from anything you might catch a monkey doing if you watched it long enough. Reading Bozo Sapiens puts you in pretty much the same mood as that Monkeysphere article that came out a long time ago - before Sarah and the contest and New York and the Project made it so my leisure time was rearranged and timeshared between a handful of personalities. The gist of the book is: you're forgiven - you're only human, and it's inevitable that you will exhibit these destructive tendencies. Don't worry about it. And: you're damned - you're only human, so ... well, you're only human.

I don't buy it.

It's really easy to be a cynic. Trust me. It's easy to look at civilization and think to yourself, "I didn't know they stacked garbage that high." I'd venture that if you haven't ever thought that, then you probably haven't seen enough of the world. But if that's where your thought process stops, well, you still haven't seen enough of the world.

It's not like there aren't people working toward the goal. And I mean this beyond Michio Kaku-esque platitudes , feel-good but totally innocent of content. There are people with ideas and direction.

Ben Goertzel may embody the best mixture of pragmatism and ambition. The project he manages, OpenCog, is an Open Source artificial intelligence project. The A.I. field has long been plagued by researchers looking for the Holy Grail, the single concept that is the Secret of intelligence. Goertzel seems to know that the secret to building a mind is already known by any infant human being - long, patient days or work and attention, structuring the thing layer by layer and training it painstakingly by experience. His weighted labeled hypergraph structure doesn't rely on any Deus Ex Machina trickery, it is merely a semantic web where the nodes and links contain semantic information. But if you look at the architecture of OpenCog you quickly see that a lot of what it is designed to do (and a lot of what humans do) is a lot more basic than concept formation and manipulation, and just as important to "intelligent" behavior. Dogs and chimps are fairly rotten at the algebra of "meaning" but still pretty good at solving basic practical problems and keeping themselves alive. This outline contains quite a bit of viewpoint-altering information about what our sensory modalities actually are and what they are good for in an A.I. context. My favorite example is the triangular lightbulb. You just imagined a triangular lightbulb when you read that sentence, even though you've probably never seen one before. You did it automatically, and I didn't even give you any instructions on what I meant. The layers between our "mind" and our senses do a lot of work for us of which we're not even aware.

This segues nicely into Geoffrey Hinton, whose wonderful Google talk showcases the closest thing I've witnessed to an example of a machine convincingly thinking. You can even duplicate this and play with it if you have Matlab.

I would be remiss to neglect mentioning Ray Kurzweil who has been popularizing the Singularity since way before it was cool. I haven't read so much Kurzweil since I stopped caring if anybody else was convinced about this stuff. I met a woman in a hotel bar in Santa Fe who argued with my for forty minutes that the Singularity didn't make any sense, no matter what logical approach vector I used to describe it. People have difficulty conceptualizing exponentials - grains of rice on chessboards and such. Didn't change things one way or the other that we made a "Singularity" in my hotel room later.

There are literally way too many authors cashing in on the same narrow band of popular futurist ideas (strong A.I., radical genetic alteration, advanced nanotechnology, make it "weird" and force it all through the die of an uncreative and half-baked plot) but I will unhesitatingly recommend Blindsight by Peter Watts. I feel like Blindsight is going to be the Snow Crash of the next thirty years. Or maybe just the next five years, what with exponential growth.

People have concrete plans to achieve our apotheosis. We are working toward it. Watch this space.

Lesson 1: On Rigor

It was a very long time before I saw my first "rigorous proof". Geometry in 8th grade purported to be a proof-based class, what with lists of theorems and lemmas, but it felt completely inorganic. There was no deep result made more meaningful by beautiful argument, no cleverness.

We literally listed theorems and lemmas on one side of the page, with definitions on the other.

In college, however, you very quickly run in to the type of people who swing entirely the other direction. They desire perfect rigor--infuriating pedantry that similarly detracts from what, in my opinion, makes mathematics so interesting. Does it matter if every line of a proof is written in flawless logical notation, upside-down As and backwards capital Es abound? Or is it more powerful to understand the proof itself, and to eschew formality in favor of clarity?

I would say that whatever path leads you to understand why a given statement is true, or why a proof actually is a proof is more useful--more powerful. I find that pedantry in notation obfuscates the truth of things, but I've known people that wouldn't have it any other way--they would literally fill chalkboards with excruciatingly detailed proofs of honestly very simple concepts.

Much of this is besides the point, though, because you probably don't even have a good idea of what perfect mathematical rigor even is. But don't worry, mathematicians didn't for a couple thousand years, either.


Lesson One: A Formula You Probably Memorized
A Preface to Infinite Series

Throwing aside rigor for the moment, it shouldn't be difficult to convince yourself there's an infinite number of colored triangles along the band in the image to the left.

There's an infinite number of purple-shaded triangles, and an infinite number of yellowish-orange-dirt colored triangles. I don't really know what to call either of those colors.

Regardless, the square's area is clearly a2. It follows from the formula for an area of a triangle (1/2 * base * height) that the bigger, white triangle has an area of (1/2) a2, and the smaller "big" white triangle has an area of (1/4) a2.

This leaves a total area of (1/4) a2 for the entire strip of infinitely many colored triangles.

Now consider the largest pair of colored triangles. One has an area of (1/2)*(1/2)*a*(1/2)*a, or (1/8)a2, and the other has an area of (1/2)*(1/2)*a*(1/4)*a, or (1/16)a2. The goofy looking quadrilateral formed by the union of two largest colored triangles then has an area of (3/16)a2.

Notice now that each triangle of the same color has roughly the same shape--their dimensions are just scaled down. Again, throwing aside rigor, it should not be difficult to convince yourself that successively smaller triangles differ in area by a factor of 1/4. If you're having difficulty seeing this, write out the areas for the first handful of triangles of either type (which isn't terribly rigorous), or simply notice that a triangle of a given size can be constructed by 4 triangles of the next smallest size, like so:

We can now construct an expression for the total area of any given pair of colored triangles.

It is simply (3/16)a2 * (1/4)n, where n=0 would correspond to the first pair of triangles, and so on.

If we wish to represent the area of that entire strip, it follows that we simply sum every single pair of colored triangles.

But we already know what this area is! We've constructed our strip such that it only takes of (1/4)a2 of area of our square.

This must mean that the sum

(3/16)a2 * (1/4)0 + (3/16)a2 * (1/4)1 + (3/16)a2 * (1/4)2 + (3/16)a2 * (1/4)3 + ... must equal (1/4)a2.

If you're clever, you'll notice the quantity (3/16)a2 is present in every term, so it can be factored out, leaving us with:

(3/16)a2 *[1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ] = (1/4)a2

Which, by a little rearrangement, implies that:

[ 1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ] = (1/4)a2 * (16/3) * (1/a2) = (4/3)

Now, the mathematician must ask, what have we actually accomplished? All we did was cut a square into a bunch of pieces, and subsequently show that the sum of these pieces was in fact the total area of the square, and then rearranged everything a little bit. Is this really surprising at all?

How about we take it a little bit further.

Consider, again, the infinite series:


[ 1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ]

We've already demonstrated what it equals, but let's play with it a little bit. We know each successive term differs only by a factor of (1/4), just by inspection. Then, it follows that:

[ 1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ] - (1/4) * [ 1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ] =

[ 1 + (1/4)1 + (1/4)2 + (1/4)3 + ... ] - [ (1/4)1 + (1/4)2 + (1/4)3 + (1/4)4 + ... ] = 1

Well that's all well and good. But isn't this little relation true if we replace 4 with any natural number?

Let's see:

[ 1 + (1/m)1 + (1/m)2 + (1/m)3 + ... ] - (1/m) * [ 1 + (1/m)1 + (1/m)2 + (1/m)3 + ... ] = 1

This appears to be alright, as long as m behaves properly, but well get to that in a second. Our sum appears twice on the left hand side, so let's factor it:

[ 1 + (1/m)
1 + (1/m)2 + (1/m)3 + ... ] * (1 - (1/m)) = 1

And, by division:

[ 1 + (1/m)
1 + (1/m)2 + (1/m)3 + ... ] = 1 / (1 - (1/m))

Which might be more familiar as

[ 1 + (1/m)1 + (1/m)2 + (1/m)3 + ... ] = 1 / (1 - r)

Where r is just that common factor by which successive terms are related. This is just the formula for a geometric series, which we appear to have derived from an intuitive geometric argument, along with some algebraic hand-waving. What isn't clear from our argument is under what conditions this formula fails.

For instance, it doesn't really make sense that this formula would work if (1/m) where greater than or equal to one, because our infinite series would grow infinitely large, whereas our formula predicts it would it would either be a particularly infuriating indeterminate, 1/0, or it would be negative.

It's also unclear if this formula would still hold for negative numbers greater than -1 (it does), and if it would fail for negative numbers less than -1 (again, it does). Does it work for complex numbers as well? Interestingly enough, you might have heard the phrase "Radius of Convergence" with regards to when infinite series like our example above are convergent. This phrase doesn't make much sense when you're talking about Real numbers, because you can only really go two directions, more positive, or more negative. On the complex plane, however, the Radius of Convergence is actually a radius, centered around 0, and in this case, means that all complex numbers of modulus or "length" less than 1 satisfy our nifty little formula.

But, again, we haven't really proved why our formula works in the region that it does. It's just intuitively clear that it would work in that region, and not outside of it. Unfortunately, the machinery required to rigorously prove all of this would require several more pages of theorem proving, so we'll just put that off...for a while.

More interesting, to me at least, is that such a profound result can be constructed so simply. You don't need a command of any math higher than basic algebra. And if I have any single objective in writing about mathematics, it's to prove (there's that word again) that there is no topic in mathematics, physics, or anything really, that can't be explained in an intuitive and simple way.

Sunday, April 11, 2010

First

The conceit of the blogger is that there exist people on the internet that care about blogs.