Sunday, February 7, 2016

Who Cares What Old, Dead White Guys Thought?

The title of this post is inaccurate if you don't consider the ancient Greeks to have been white. But that's probably not a discussion I want to get into right now. Anyway, today we're discussing my ancient philosophy course from last semester, or more precisely, my Socrates, Plato, and Aristotle course.

There are two main points I'd like to articulate: (1) if philosophy has made objective advancements in the last 2,400 years, why should we care what philosophers thought 2,400 years ago, and (b) man, I had a really annoying classmate in my ancient philosophy class. In essence, I'm wondering whether it was worth it to take this class, just as I had similar concerns about the value of paper writing in my philosophy in literature class from last spring.

To think about the first point, there are two paths you can go down. First, you can go the "philosophy is the mother of science" route and wonder where that leaves philosophy nowadays. That is, there used to be essentially no distinction between being a philosopher and a scientist. Science is a relatively new word, and people like Newton were referred to as "natural philosophers." Science was just doing philosophy about nature rather than philosophy about justice or god or what have you.

The usual argument you see here is that philosophy birthed the sciences we're familiar with today, and where it's done so, philosophy is obsolete and the science is all that's left. There are still philosophers of physics today (after all, I took a class on that, too), but they're not doing physics. Philosophers of physics no longer ask whether the world is made of four fundamental elements, or if all matter is composed of atoms, or if the planets travel in perfect circles, because physicists have definitively answered those questions (no, depends, no).

So the domain of philosophy has shrunk. Where philosophy about the natural world is still relevant, it's in asking questions about physical models, rather than coming up with the models themselves. (Metaphysicists might disagree, but a lot of modern philosophers don't hold metaphysics in particularly high regard, as I understand it.) Similar shrinkage has occurred in the other sciences, with psychology being one of the latest disciplines to squeeze philosophy further.

Here I want to look at a particularly egregious example from my ancient philosophy course, Plato's tripartite soul. Plato reasoned that a statement and its contradiction cannot both be true at the same time. This is reasonable and one of the foundations of classical logic. Take a statement like, "The sun is yellow." Either that statement is true, or the statement, "The sun is not yellow" is true. They can't both be true, because one implies a contradiction of the other.

So then let's look to the soul. We've all had the experience of simultaneously wanting and not wanting the same thing. "I want to eat that chocolate cake" and "I don't want to eat that chocolate cake" are thoughts we can have at the same time. In the first instance, it's our carnal desire for the cake, but in the second instance, it's our willpower that's talking. But if the law of non-contradiction holds, it can't possibly be true that we can both want and not want a piece of chocolate cake simultaneously.

...unless we have a divided soul, as alluded to above. Plato identifies three different competing interests in the human psyche that can produce contradictory desires. Roughly, these are the appetitive, passionate, and rational parts of the soul. They are distinct and incompatible, Plato argues, otherwise the law of non-contradiction is contradicted.

And that's all well and good, and proceeds from some reasonable assumptions, but it's baloney as far as modern neuroscience and psychology are concerned. What a hundred years of research into the brain have taught is that the brain is really complicated, possibly the most complicated three pounds in the universe, and it's decidedly not true that you can chop it up into distinct, one-pound chunks.

(I've cleverly switched from talking about the soul to talking about the brain, but a distinction between the two was not necessarily important to Plato, and science says that "the mind is what the brain does.")

There are two main ways in which Plato's tripartite soul fails as a theory. The first is that there are probably many components to the human psyche, far more than three. The second is a subtle problem that has plagued philosophers for thousands of years, which is that it's possible for words and concepts such as "want" to have different meanings depending on the context. So you can want something, and you can want* something. The former may mean "desire enough to actively pursue," whereas the latter might be "like thinking about but have no inclination to pursue." In that case, you can not want something, and also want* it, and there is no contradiction.

This is a tricky problem that crops up all over the place, which is why analytic philosophers spend large chunks of their time trying to tease apart just what we mean when we talk about seemingly plain concepts such as free will or beauty or truth.

But if all we have to go on is what remains of a large, sometimes disjointed collection of Plato's writings, it's easy to find flaws in his logic. His work cannot defend itself. It's also possible that those old, dead white guys were just wrong about stuff. They had a limited amount of data and lacked the thousands of years of philosophical tradition (that they began) to draw upon.

Which brings me to my annoying classmate. During lecture, he frequently raised his hand and asked the instructor questions such as, "But doesn't that produce a contradiction?" and "But wouldn't that mean nothing is beautiful?" and "But didn't Plato condone slavery?" And every single time, the instructor would engage with him and answer his questions in a thoughtful manner.

Terrible, right? Provoking the instructor into discussing philosophy with us. Well, yes. We had two lecture periods and one discussion period per week, and he brought up his objections during the lecture period. His interruptions were so frequent that there was material we were never able to cover in class. And all of this was possible because, yes, duh, Socrates and Plato and Aristotle were wrong about stuff. It was very frustrating, but I suspect I'm coming off as kind of petulant here, so let's go back to Plato for a moment.

While Plato did divide the mind into three different parts, he had particular affection for one of those parts: the rational mind. It was through employing the rational mind in dialectic that truth could be revealed. This is where Plato's allegory of the cave comes in. Plato conceived of a metaphor where the reality we perceive is just shadow puppets lit by torchlight that we are forced to watch in some kinky Clockwork Orange setup.

Philosophers, however, have broken out of the cave and can see real objects illuminated by the pervasive, powerful sun. So there's a distinction between the ever-changing, distorted, and 2-dimensional shadows we think of as reality and the constant, colorful, 3-dimensional objects that actually compose reality. When we see a chair, we are only seeing an indistinct, imperfect shadow of a chair that does not fully encompass the essence of true chairness.

At first blush, this whole idea seems patently ridiculous. We all accept that our eyes can deceive us and that reality is maybe actually electrons and protons, but it seems laughable to suggest that in some eternal, unchanging realm there exists the true forms of the objects we behold here. Where is this realm? Is there a form of the electric fan there, the cell phone, the credit card offer?

Well, it's unclear how diverse Plato intended his realm of forms to be, but he almost certainly thought it was populated by mathematical objects. Many ancient Greeks (including Plato) took math and geometry as the model of a priori knowledge, knowledge we could come to know just by thinking logically and without relying on evidence from our senses. To Plato, this meant accessing Platonic forms.

So there's some ideal triangle out there, as well as a perfectly straight, infinitesimally thin line, and also the true form of the number 5. Again, this sounds plainly absurd. But let's look at a particular number, such as the ratio between the circumference and diameter of a circle: π.

In a little over a month, it will be Pi Day, which means the internet will be stuffed with memes about π pies and whether ϕ is the true constant and how π is a magical number that contains everything in the universe.

That last one relies on a conjectured property of π, that it is a normal number. A normal number is one that has an endless sequence of digits in a non-repeating pattern that are distributed perfectly randomly, with no particular numeral being more likely than any other. Assuming that’s true, then if you peer deep enough into the digits of π, you will eventually find your telephone number, or a bitmap of your face, or your life story written out in ASCII code.

But you'll also find a lot of nonsense, and there's no way to tell the true from the false, so this is more like Borges' Library of Babel than, say, the Encyclopedia Galactica. It’s true that highly random data has a lot of information in it, but there’s nothing profound about that; that’s numerology, not number theory.

Additionally, it turns out that almost all (real term) the real numbers are normal, but it's not easy to pick out any particular number and say that it's normal. Currently, there is no proof that π is normal, although the evidence suggests that it is.

But what if there is no proof? What if it turns out to be impossible to demonstrate rigorously that π is a normal number? (You can often prove that it's impossible to prove something in math, but maybe a proof is just never found.) In math, a statement is only taken to be true if can be proven via deductive logic. So if there is no proof that π is normal, is it normal?

Well you're probably thinking, it's either normal or not, duh. Its being normal doesn't depend on whether or not we're smart enough to prove it. The Earth was four and a half billion years old long before we were able to show, scientifically, that it was. But look what's happened here. We've asserted that π has definite properties independent of our conception of it. That is, we're saying π is real, as real as the Earth, and that it has a form beyond our crude and incomplete perceptions.

So perhaps Plato's forms are not as crazy as they sound. Now, I'm not arguing that Plato is correct and that numbers are "real." This is a lively debate in the philosophy of mathematics (a subject I'll have more to say about at the end of this semester), with the other positions being "idealist" and "anti-realist." But Plato originated (or was the best, earliest articulator of) one tradition in this philosophical debate.

Which brings me back to my annoying classmate. If instead of a philosophy course, this had been a course on the history of Ancient Greece, at no point during the lecture would a classmate have interrupted the instructor with, "But teacher, weren't the Athenians wrong to butcher and enslave whole cities?" Of course they were wrong! That is clearly not up for debate. What's interesting, however, is why the Greeks did what they did, and how their actions propagated through history. That is, I want to understand the legacy they left behind, the traditions they began.

And that's how I look at an ancient philosophy course. To me, it's not primarily about finding all the myriad logical inconsistencies in the thoughts of some old, dead white guys, but in understanding how their thinking shaped humanity for millennia to come. In some cases, their ideas are obsolete and need to be discarded, while in others they represent the seeds of debates still flourishing in philosophy now. The greatest difference I see is that philosophers today strive for precision and nuance so as to avoid falling into the same old traps. But we couldn't have gotten here, couldn't have learned that lesson, without first falling in.

Thursday, January 21, 2016

Passwordocalypse

This is the second year in a row that I've seen an article decrying our collective cyber stupidity because of the awful passwords we use to protect ourselves. And this is the second year in a row that I've rolled my eyes very hard at the article because of its mathematical ignorance. This is the first year I've decided to blog about it, though.

The article linked to above lists the most popular passwords found in databases of stolen passwords. At the top of the list are groaners such as "password", "123456", and “qwerty”. How could those be the most popular, when everyone knows China is hacking its way into our country and we're using passwords to protect our finances, identities, and porn habits? How could everyone be so stupid?

Well, the truth is, very few people have to be stupid for those passwords to be the most popular. In fact, there's an easy to imagine scenario in which no one is so cyber-challenged. Let's see how.

The most popular password is the one that gets used more than any other individual password. This doesn't mean it's used by a majority of people, obviously, just as Donald Trump isn't supported by a majority of Republicans. Additionally, when we're ranking password popularity, we're doing so by login rather than by person, because that's how the data comes to us. So password popularity is measured as logins/password.

And what's being railed against in the above article is that the passwords with the highest login/password are bad ones. But what makes a password bad? Ease of guessing--those that take the least time to crack are the least secure.

This quality is quantified in a password's information entropy, which is a measure of the number of bits needed to specify the password. In other contexts, a piece of data's information entropy tells you how much that data can be compressed. The higher the entropy, the more bits needed to specify the data, the fewer bits you can get rid of and still preserve it.

When I think entropy, I think physics. Most people probably do, too, knowing it has something to do with thermodynamics and disorder. You probably know the second law of thermodynamics, which is usually stated as something like, "The entropy (disorder) of a system tends to increase."

The "tends to" there indicates that this is a probabilistic law. That is, if you have a box with octillions of gas molecules all bouncing around at different speeds and directions, it's hard to say exactly what they're going to do, but you can say what they're likely to do. And it turns out that a box of gas is more likely to go to a high entropy state than a low one. The reason is that there are many more high entropy states than low ones available.

This is where the connection to disorder comes in. The canonical example is probably an egg. An intact egg is a very ordered thing. It has a specific shape, and you can't change the shape of the egg without changing the fact that it's an intact egg. Thus order means low entropy, because there are only a small number of ways for an egg to be an egg.

Scrambled eggs, on the other hand, are disordered and high entropy. The high entropy results from the fact that you can rearrange your egg particles (eggs are made of egg particles, right?) in many, many different ways but still end up with the same basic breakfast: scrambled eggs.

How does this connect back to information and passwords? Because as the entropy of a system increases, it takes longer and longer to accurately describe the system in detail. With low entropy, high order systems, there might be one law of nature telling you why the system is shaped the way it is, which means it's easy to specify it in detail. But with a high entropy system, there are many microstates that are approximately the same, so you need to be more and more detailed if you want to specify a particular one. "No, the one with particle 1,036,782,561 going this way, not that way."

So high entropy data doesn't compress as easily because there are many high entropy systems, which means it takes a lot of bits to differentiate between two chunks of data. And this is also why high entropy passwords are more secure: because if you're randomly guessing a password, it takes you much, much longer to get through all the available high entropy passwords than it does the low entropy passwords.

But that's also why the least secure passwords will always be the most popular ones. Compared to the secure passwords, there just aren't that many bad passwords out there, because bad passwords are low entropy. The login/password for bad passwords is going to be high essentially by definition. Here's a toy model to demonstrate.

Mathematically, the entropy of a system (s) is proportional to the log of the number of microstates (n) that correspond to a single macrostate. Computer people like to do things in binary, so they use a log base of two: S = log2(n). Now let’s take some real data and see what we find. Using this website, I have found the entropy of each of the 25 most popular passwords. Their average entropy is 20.12. Using my password manager, I've found the average entropy of 10 randomly generated strong passwords (I got lazy, but the variation in entropy was low): 80.84.

So the average good password is ~4 times as strong as the average bad password. If we assume there are only 25 bad passwords (there are many more, but more makes the point even stronger), and that the population of logins (p) uses either good passwords or bad passwords, we can write an expression comparing password popularity (logins/password). For our model, let’s see what it would take for good passwords to be just as popular as bad passwords:

pbad/nbad = pgood/ngood

How do the number of good passwords compare to the number of bad ones? Well, from the log formula up there, if we multiply the strength of a bad password by 4, we get 4S = 4log2(n). From the rules of logs, we can take that 4 on the outside of the log and bring it in: 4S = log2(n4). So if you have n bad passwords, then you have n4 good passwords.

pbad/nbad = pgood/nbad4

Solving for the ratio of logins using bad passwords to good, we get:

pbad/pgood = 1/nbad3

Now let’s plug in nbad = 25.

pbad/pgood = 1/15625 = 0.000064

This means that as long as more than 0.0064% of all logins use bad passwords, they will be the most popular. Stating the converse, 99.9935% of all logins can use strong passwords, and the bad ones will still be more common.

Of course, in the real world, there are more than 25 bad passwords (and waaaay more than 254 good passwords), and people aren't divided up into binary good and bad password users. But I think this demonstrates that very few people need actually be stupid for the above article to be true.

And as I said, it's possible that no one is stupid because this is based on logins rather than users. All it takes is that more than 0.0064% of the time you need to pick a username and password for a site, it's a site for rating cat videos and you rightly don't care about security.

Tuesday, January 19, 2016

Quantifying Weirdness

Quantum mechanics is weird; there's no doubt about that. It’s got wave-particle duality, the uncertainty principle, and spooky action at a distance. Other fields have weird results, too, but although we might comment on the peculiarity of a particular finding, we do not indict other fields as a whole. With quantum mechanics in particular, though, it seems like its idiosyncrasies leave people with the feeling that it is either too weird to be right or too weird to be understood.

Well, today I'd like to help dispel those attitudes, particularly the first one—or at the very least put a number on just how weird quantum mechanics is. To do so, I'm going to be regurgitating material I learned in my philosophy of physics course.

In order to quantify the weirdness of quantum mechanics, we'll be exploring the phenomenon of quantum entanglement. Hopefully, we'll be able to unravel some of its mysteries and not get caught in a web of confusion.

I'm sorry, I promise there will be no more entanglement puns.

Entanglement first gained widespread awareness in physics after a 1935 paper by Einstein, Podolsky, and Rosen, henceforth known as the EPR paper. Einstein was unhappy with how that paper turned out, but he articulated his thoughts more clearly to his colleagues (especially Schrodinger) in private. Additionally, the thought experiment proposed then was more complicated than it had to be. The upshot is I'll be talking about this from a slightly more modern perspective; but historically, the EPR paper is one of the jumping off points for discussing quantum funny business.

So here's entanglement. In quantum mechanics, particles like electrons are described by a wave function which tells you the probability of finding the electron in a particular state. One such state is spin which, because of weird quantum mechanical reasons, can be either up or down. So the wave function could say there's a 50% chance the spin is up and a 50% chance it's down, for example.

You won't know what the spin is until you measure it. When you do so, the language is that the wave function “collapses,” so now it's just in one state, either up or down, instead of a superposition of both.

If two electrons are hanging out, normally you have two wave functions to keep track of. But if two electrons get created together in a particular process, then they will be described by a single wave function. Once that happens, barring interference from the outside world, it is not possible to decompose that wave function into two separate ones.

Where before your wave function for a single electron said there was a 50/50 chance of spin-up or spin-down, now it might say something like there is a 50% chance that electron A is spin-up and electron B is spin-down, and a 50% chance that electron A is spin-down and electron B is spin-up. So if electron A is in your lab, and electron B is down the road at the chemist, and you measure electron A to be spin-up, then you know the wave function has collapsed to "A up, B down." This means you also know, without having measured it, that electron B is now spin-down. If you do later measure it, you will always find it to be spin-down if A was up.

Here's where things get weird. Again, as long as you prevent your electrons from being interfered with, they remain entangled until you measure the spin of one of them, no matter how far apart the electrons get. So if electron A is in your lab, and you send electron B to Alpha Centauri, when you measure the spin of electron A, you instantly know, across a distance that would take light 4 years to travel, what the spin of electron B is.

This is weird.

Here's another scenario. This one is totally going to blow your mind. Imagine you are playing a game with a street magician. He's got two hands and one coin. While your back is turned, he puts the coin in one of his hands and then asks you to guess where the coin is. There's a 50/50 chance for either hand. You say left hand. He opens, and reveals that there is no coin there.

Now here's the wacky part. Assuming the magician exhibits no trickery and that the coin is in one of his hands, you now know, as if by magic, that the coin is in his right hand. Even if the magician performs some real magic and sends his right hand to Alpha Centauri after hiding the coin, you know instantly, across a distance that would take light 4 years to travel, that the coin is in his right hand. Information has traveled faster than light—a clear violation of Einstein's special relativity!

Okay, no matter how hard I try, I can't make that second scenario sound as weird as the first one. But why not? Because you're saying, “Silly Ori Vandewalle (if that even is your real name), nothing spooky is going on here. The coin's location is a result of the magician's actions before the hands are separated. Revealing the hand doesn't decide the fate of the coin. Duh.”

This is essentially the argument that Einstien made in the EPR paper. If two electrons are entangled, and one of them is sent to Alpha Centauri, and measuring the spin of one tells you the spin of the other, then the only reasonable conclusion you can draw is that the spins were determined beforehand.

The name of the EPR paper is, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Following Betteridge's law, Einstein posited the answer was no. That's because quantum mechanics can only tell you the probability of the electron's spin being up. But just as with the magician's coin, Einstein argued, this probability represents nothing more than our ignorance, not any actual indeterminacy on the part of the coin or the electron.

So is the weirdness gone?

Well, let's see if we can't make this spooky action even more mundane. Another way to think of this result is that the two electrons are correlated. If two objects are correlated, they have a common cause. A caused B, or B caused A, or C caused both A and B. So we are suggesting that some common cause configured both spins beforehand but didn't bother to tell the wave function this.

In the 60s, physicist John Stewart Bell developed a theorem that must be true about any three binary properties of a single system. This theorem tells us something important about common causes. There are a few assumptions that go into the theorem, the most relevant of which is that, once you measure property A, that measurement can't affect properties B and C before you measure them.

Let's go through Bell's theorem with cookies so that I can distract you from the fact that we're doing math.

By Kimberly Vardeman from Lubbock, TX, USA (Perfect Chocolate Chip Cookies) [CC BY 2.0], via Wikimedia Commons
Say you've baked a batch of cookies, and the cookies can be large or not large (L, ~L), have walnuts or no walnuts (W, ~W), and have chocolate chips or no chocolate chips (C, ~C). Now say you want to know how many large, non-walnut cookies you have. We'll call that N(L, ~W). This number is the sum of all large, non-walnut, chocolate chip cookies N(L, ~W, C) and all large, non-walnut, non-chocolate chip cookies N(L, ~W, ~C). This must be true, because whether or not a cookie has chocolate chips does not affect its size or walnut content.

Similarly, the number of cookies with walnuts but no chocolate chips is N(L, W, ~C) + N(~L, W, ~C) because size doesn't matter. And finally, the number of large, non-chocolate chip cookies is N(L, W, ~C) + N(L, ~W, ~C) because walnuts don't matter.

Now let's add together the number of large, non-walnut cookies and the number of walnut cookies with no chocolate chips. That quantity is:

N(L, ~W, C) + N(L, ~W, ~C) + N(L, W, ~C) + N(~L, W, ~C)

If you notice, the second and third terms are also the terms for the number of large, non-chocolate chip cookies. That means our sum is always at least as great as the number of large, non-chocolate chip cookies.

Now let's make a slight shift and talk instead about probabilities. If you randomly reach out for a cookie, the probability that you get a particular one is directly proportional to the number of that cookie there is to take. This means we can reword Bell's cookie theorem thusly:

The probability of choosing a large, non-walnut cookie or a walnut, non-chocolate chip cookie is always greater than or equal to the probability of choosing a large, non-chocolate chip cookie.

This theorem is true regardless of how many of each cookie there actually is, because at no point in demonstrating this did we use numbers. It's also true no matter what kinds of properties we're talking about, so long as they are binary properties, because we could just as easily say L stands for lemon cookies or even something non-cookie-related.

But what's more, this theorem tells us about correlations. You see, if I give instructions to a thousand people to bake exactly the number of cookies I say and have each person randomly select and eat one cookie, we'll find that Bell's cookie theorem holds true. The probabilities will be maintained across all kitchens, because the cookie batches are correlated--spooky baking at a distance. The correlation is a result of the common cause known as me giving out instructions.

Now let's switch gears and talk about sunglasses—or as I prefer to call them, quantum shields. Polarized sunglasses only admit light that oscillates in a particular direction (up and down or left and right, for example). If you have horizontally polarized sunglasses, then only light waving from left to right (from the frame of the frames) will get through. But light coming from the sun is equally likely to be waving in any direction, so if you think about it, polarized sunglasses should only let a tiny, infinitesimal amount of light through—only light that is exactly horizontal and nothing at any other angle. Yet this isn't what happens. Polarized sunglasses will absorb roughly half the incident light and let the rest pass. Why is that?

Well, let's talk about the quanta of light, photons. A single photon doesn't have a direction it's waving, but it does have a polarization that is based on its spin. When a photon passes through sunglasses, the photon's spin is measured by the polarizing filter. Before the measurement, it's in a superposition of horizontal and vertical spin based on the angle of its spin (the direction it's waving).

When it's measured, that superposition collapses so that its spin is either horizontal or vertical. If it ends up being horizontal, it passes through. Otherwise, it's absorbed. The closer the angle of its spin is to horizontal, the higher the probability that it collapses to a horizontal spin. In this way, light from any polarization (except exactly vertical) can pass through, but the odds of it doing so go down the further away from horizontal you get, and anything that does pass through will subsequently be measured as horizontal. So sunglasses are quantum shields.

"Oakley half wire" by Jpogi at en.wikipedia.com. Licensed under Public Domain via Commons
This probability of getting a particular spin works for electrons, too, such as the two entangled ones in our EPR thought experiment. Instead of a polarizing filter, we use magnets to measure an electron’s spin. Before we talked about a 50/50 chance of an electron being up or down, but these odds can be adjusted by rotating our magnets in exactly the same way that light waves rotated away from horizontal have different odds of passing through sunglasses.

But this adds a new wrinkle to our thought experiment. Before, getting a spin-up on Earth meant the Alpha Centauri electron would be spin-down 100% of the time. If we rotate the Earth magnet by some angle θ, then that perfect correlation stops being 100%. It turns out that the odds of one being spin-up and the other spin-down are equal to cos2(θ/2), where θ is the angle between the two magnets.

We can carry out this experiment many times, creating entangled electrons and sending them to Alpha Centauri. A third of the time, we can measure with one magnet oriented at 0 degrees and the other at θ degrees clockwise, a third with one θ degrees and the other φ degrees, and a third with one 0 degrees and the other φ degrees. In this way, we are measuring three different binary properties of the system. Bell's theorem applies.

An entangled pair can be spin-up at 0 degrees and spin-down at θ degrees, spin-up at θ degrees and spin-down at φ degrees, or spin-up at 0 degrees and spin-down at φ degrees.

Bell's theorem tells us, then, that P(θ) + P(φ-θ) >= P(φ). Using the cosine formula up there, this comes out to cos2(θ/2) + cos2([φ- θ]/2) >= cos2(φ). Okay. Looks fine.

Except this isn't always true, depending on the angles you pick. Sometimes, the left-hand side will be less than the right-hand side. If you subtract the right from the left, then whenever Bell’s inequality is violated, the expression will be negative. You can see when that happens in this graph.

I am a Matlab Master.
So what does it mean for Bell’s inequality to be violated? Well, in the case of our cookies, the correlation was upheld because I sent out a common set of instructions to all the bakers. This is the common cause of the correlation. We saw that this common cause would lead to adherence to Bell's inequality for any set of three, binary properties of a system. This means that a common cause cannot be the origin of the correlation between entangled electrons. They aren’t deciding their configuration beforehand.

What Bell's theorem does permit is a non-local connection—the electrons instantly updating each other on their spin, or electrons that are governed by interactions across all of space. The other usual possible explanation for EPR and Bell is that electrons don't have any intrinsic reality, that realism itself is a foolish idea. No one likes either of these possibilities.

There are alternative ways of deriving, formulating, and generalizing Bell's theorem. When you do so via the CHSH inequality, you find that classical correlations can be no higher than 2. But quantum correlations violate this limit and can be as high as 2√2. And yet we can imagine other correlations, such as the Popescu-Rohrlich box, that are even higher than 2√2—correlations that you cannot reach even with entangled, non-local/non-real electrons.

So quantum mechanics is weird. But it's only weirder than regular spooky action at a distance by a factor of √2, or ~41%. Although √2 is irrational, so maybe quantum mechanics is unreasonably weird.

Thursday, January 7, 2016

Red vs. Blue

Sorry, but this post is not about the Halo-based web series. It's also not about quantum physics, like I suggested last time, except insofar as everything in the physical world is about quantum physics. Instead, this post is about my Hanukkah gift this year, a page-a-day calendar based on the show Are You Smarter Than a 5th Grader? Apparently my parents weren't sure I'd been learning anything this past semester and wanted a way to test me. Well, let's take a look at the Jan 4 entry.

Don't sue me, Fox, I guess?
First, let me be an astronomical pedant. Except for weird objects, stars are classified as either dwarfs or giants. Our sun is a yellow dwarf, and it's not clear that we should classify it as regular. There are many more small stars than big stars in the universe, with the consequence being that most stars are red dwarfs. That makes the sun more massive than most stars. On the other hand, stars can get a lot bigger than ours, both in terms of mass and size. So from that perspective, our sun is very small compared to what can exist. Does that make it a regular star? That question gets a shrug from me. But it's certainly not true that the sun is representative of stars in general.

Now, on to the question itself. The answer is that blue giants are the hottest. I suspect this is supposed to be something of a trick question. In everyday life we associate blue with cold and red with hot, but the exact opposite relation is true for hot, dense objects; red is relatively cool, blue hot. Why this discrepancy exists has to do with what color is really all about.

In general, there are three sources for the colors of objects: thermal radiation, reflection/absorption, and atomic spectra. That first one is the reason why blue stars are the hottest. Or rather, it's why the hottest stars are blue. Anything with a temperature emits a spectrum of radiation based on that temperature.

By Darth Kule (Own work) [Public domain], via Wikimedia Commons
This Planck spectrum has a peak wavelength inversely proportional to temperature, so the hotter an object is, the shorter its peak wavelength. Blue is a shorter (more energetic) wavelength of light than red is, so hot objects emit more blue light than red light.

For humans and most other room-temperature objects, the peak wavelength is in the infrared, which our eyes are not sensitive to. Everyday objects do emit some visible light, but as you can see from the Planck spectrum, the intensity drops off very quickly to the left of the peak, so our thermal emissions are essentially invisible to us.

Where we are most likely to encounter visible thermal radiation, not counting the sun, is the stovetop. Heat a piece of metal up to a few hundred degrees and it will start to glow red. Since it is difficult for us to achieve higher temperatures in everyday situations, this is probably where our sense of what hot looks like comes from. Compared to the most massive stars, a heating element is downright chilly, but it's much hotter than we are, so red = hot. Other Earthly examples include hot coals and lava (and some of the color of fire, with the rest coming from emission lines).

As far as why we associate blue with cold, the most likely explanation is that water and ice are blue-tinted. Another possibility is that blue is simply the opposite of red in our brain, but for the purpose of making this blog post longer, let's go with the first explanation.

The ocean is not blue because the sky is (nor is the sky blue because the ocean is). The ocean is blue because water preferentially absorbs red light and reflects blue light. The reason water in a cup is clear is because water transmits almost all light that touches it, but the light it does not transmit is either absorbed or reflected. So with small quantities of water, there is not enough reflection to notice. For an ocean, it's unavoidable.

Reflection and absorption account for nearly all the color we ordinarily see. The details of why objects reflect or absorb particular wavelengths turn out to be pretty complicated and not reducible to a clever function or graph. However, there are some relatively simple examples that demonstrate the importance of wavelength when it comes to the behavior of light.

The most obvious example is the blue sky. The sky is blue due to a process known as Rayleigh scattering. Rayleigh scattering occurs when the wavelength of light is significantly bigger than the particles that light is striking. In that case, the light is either transmitted or scattered, and the probability of scattering is inversely proportional to the 4th power of the wavelength. This means light at the blue end of the spectrum can be scattered up to 9 times as much as light at the red end (700 nm/400 nm)4.

When light is scattered, it bounces off the particle it strikes in a random direction. Eventually, this light will scatter such that it gets to your eye, but by then it's not likely to look as if it was coming from the source. So when we look at the sky, we see the sun no matter what direction we look. The difference is that the red and yellow light of the sun comes directly to us while the blue light bounces around a bit first.

When particle size gets much bigger, as happens for the complex molecules that make up people, shirts, and paint, the size and shape of the molecule plays a much more important and complicated role in which wavelengths get absorbed, transmitted, or scattered.

The final source of color is atomic spectra, which we observe as either emission or absorption lines. Each element on the periodic table is composed of electrons in orbit of a nucleus. The orbits an electron is allowed to have are prescribed by the number of protons, neutrons, and electrons present and the rules of quantum mechanics.

To occupy a particular orbit, an electron must possess a particular energy. If that electron moves from a high energy orbit to a low energy orbit, conservation of energy says it must release energy equal to the difference in energy levels between the two orbits to account for the transition. This energy is released in the form of a photon--light. The wavelength of that photon, and consequently the color, is inversely proportional to its energy. So big jumps produce energetic, blue photons, whereas small jumps produce red photons. (Gamma ray photons and radio wave photons and any other kind of photon are also possible depending on the energy levels involved.)

This process works in reverse, too. If light with enough energy to effect a jump hits an electron, then the electron absorbs the light and goes from a low energy to a high energy orbit.

Because these transitions only occur at specific wavelengths, we see these as emission and absorption lines rather than the spread out thermal spectra that hot objects produce. On Earth, the most common example of an emission line is a neon sign. An electric current passes through a gas, such as neon, exciting the electrons in the gas. When the electrons come down from their excited energy levels, they emit photons of a particular wavelength, giving off their characteristic orange-red glow.

There's not a great example of absorption lines on Earth that I'm aware of, but a particularly stunning example is the sun. While the sun has a nearly perfect blackbody spectrum, if you spread out its light with a spectrograph, you will notice gaps of color. These gaps are absorption lines and represent all the elements in the sun's outer photosphere (as well as some in our atmosphere, depending on where you take the light from), which is colder than the rest of the sun and absorbs light that passes through on its way to us.

Source: Nigel Sharp, National Optical Astronomical Observatories/National Solar Observatory at Kitt Peak/Association of Universities for Research in Astronomy, and the National Science Foundation. Copyright Association of Universities for Research in Astronomy Inc. (AURA), all rights reserved.
I feel it would be disingenuous of me to finish up here without noting that all of these sources of color are more connected than my discrete categorization would lead you to believe. Ultimately, light is emitted whenever an electric charge is shaken up. This is happening with thermal radiation in a messy, smeared out way and atomic spectra in a precise, limited way. And when light is absorbed or reflected by a surface, the ultimate reason is that quantum mechanical electronic energy levels are being messed around with, just like in atomic spectra. The difference is the former is much harder to calculate via quantum mechanics, so instead we label it with a simple refractive index that varies based on wavelength and is derived from observation.

Anywho, that's all for now. And I didn't even touch on how the physics of color interacts with the biology of sight, which is also a fascinating subject. Next time, quantum physics. Unless I detour into my calendar once again.

Wednesday, December 30, 2015

The War on Stars

This post contains spoilers for both Star Wars: Episode VII The Force Awakens and my academic semester. Read on at your own peril.

As always, I must begin by apologizing for not having posted in months. My academic load this semester, combined with my work schedule, was probably about the limit of what I could handle and didn't leave me with a lot of time left over for blogging (or sleeping, for that matter). To remedy that, during winter break I'm going to try to find time to write about the classes I took, maybe posting every week or so. We're starting off today with my observational astronomy course.

But we're getting there through Star Wars. To begin, I enjoyed the movie a great deal (all three times). I also go into a Star Wars movie turning off the part of my brain that cares about scientific plausibility or consistency. In fact, I'm partial to the idea that Star Wars is science fantasy rather than science fiction, whatever that distinction may signify. Yet looking at media through a scientific lens is a fun way for me to analyze it, and it might even be educational. We'll see.

So, of course, TFA has a galaxy's worth of scientific errors, but there's one visual in particular I'd like to take a look at, because I think it gets at something important in astronomy. When the First Order fires the weapon from Starkiller Base at the New Republic, Finn on Takodana (Maz Kanata's planet) sees the beam split up and strike different planets in the Hosnian system. This is an impossible image, given the assumption that Starkiller Base, Takodana, and the Hosnian system all orbit different stars. The reason this image is so impossible is because, as the great Douglas Adams informed us, space is big, really big.

Now, I'm not thinking about the fact that light travels at a finite speed and there wouldn't have been time for the image to show up in Takodana's atmosphere. This is a universe with faster than light travel, so let's just mumble something about hyperspace and ignore that. Imagine that it did take years for the light of the beam to stretch across the lightyears; it still wouldn't look like it does.

The problem is that you can see multiple beams at all, that they can be resolved as striking different places. In astronomical terms, the angular separation between the beams is absurdly large. This point can be made with a simple trigonometric argument. If we imagine two lines connecting Finn's eyes and the planets struck by the beams, and another line connecting those two planets, we can make a little triangle.


What we're looking for is the angle between lines C and A. For our purposes, the relative lengths of A and C don't matter and we can just call one of those lines the distance between Takodana and the Hosnian system. Trig gives us the formula sin θ = B/C. But in astronomy we make use of the small-angle approximation a lot, which says that for very small θ, the sine of θ is approximately θ. So then we have θ = B/C.

The significant part of this formula is that, for astronomical purposes, staring up at the sky only gives us θ, not B (the size of the thing we’re looking at) or C (the distance to the thing we’re looking at). This means, without other factors, we can’t tell if we’re looking at a big object far away or a small object nearby.

Digging around Wookieepedia and starwars.com, it seems that Takodana is supposed to be in the Mid Rim of the galaxy and Hosnian Prime in the Core. If we assume that this galaxy is about the same size as ours (not necessarily a great assumption, but published maps show something like a spiral galaxy), then halfway out of the Core gets us a distance of 25,000 lightyears. We don't know the distances between the planets in the system, but if we make the very generous assumption that they are as far apart as Earth and Neptune, we get a distance of 4 lighthours. Plugging those numbers into the above formula (B=4 lighthours, C=25,000 lightyears), our angular separation is 2x10-8 radians, which converts to 4 milliarcseconds (mas). 1 mas is 1/1000 of an arcsecond, which is 1/60 of an arcminute, which is 1/60 of a degree. By comparison, the moon has an angular size of 31 arcminutes, over 400,000 times bigger.

So the beams wouldn't appear that far apart. In fact, you wouldn't be able to tell them apart at all. Okay, but why am I fussing about this? Because it gets into some interesting aspects of observational astronomy having to do with the wave nature of light. Specifically, when light waves enter an aperture, they diffract around the edges and form interference patterns. It's inevitable and must be taken into account no matter what type of observation you're doing.

When light diffracts through a perfectly circular aperture, it forms the following interference pattern, called an Airy disk.
"Airy-pattern" by Sakurambo at English Wikipedia 桜ん坊
That is, if you were to shine a laser pointer through a circular hole, instead of a dot on the other side, you would get the above pattern. However, trying this with a store-bought laser pointer and a hole punch won’t get you much, because the pattern is very sensitive to the wavelength of light used and the size of the hole.

In the case of the Starkiller beam, the aperture we're talking about is your pupil. The human pupil can change in size based on lighting conditions, but a good average diameter is 5 mm. The wavelength of the beam's light is based on its color. The red light of the Starkiller beam is at the long end of the visible spectrum, so let's call it 700 nm. These two variables play into the size and spread of the interference fringes.

In the 19th century, Lord Rayleigh proposed a criterion for determining the limits of image resolution. He said that if two images are closer together than the first minimum of the interference pattern, then you can't resolve them as two objects. This is arbitrary, but not entirely made up. If you add together the intensities of two interference patterns separated by less than that minimum, this is the difficult to interpret graph you get. Are you looking at one object or two?


The pattern of the Airy disc is described by a Bessel function, which is a special function invented to be the solution to some common differential equations. The first minimum of the Airy disc is the point where the function goes to 0 for the first time and happens at an angular distance of θ = 1.22λ/D, where λ is the wavelength of light, D is the diameter of the aperture, and 1.22 is a rounded-off figure for a number that goes on forever, because Bessel functions aren't very nice functions.

In fact, my observational astronomy professor explained that if we're going to use 1.22, we might as well memorize a few more digits because that number only comes up with perfectly circular apertures anyway, and 1.22 is not much greater than 1, so you're not gaining much precision as it is. In most cases, making the approximation that θ = λ/D works well enough. The interesting thing to note about this criterion is that fine angular resolution results from small wavelength or large aperture. This is why radio telescopes are much bigger than optical telescopes. Radio telescopes are looking at very large wavelengths (centimeters to meters compared to hundreds of nanometers), so to be able to resolve images, they need much larger apertures.

Since I just made up the wavelength of our beam and I'm assuming the pupil is exactly 5 mm, let's leave off the .22. In that case, our minimum angular resolution is 700 nm/5 mm = 1.40x10-4 radians, which comes out to 29 arcseconds. This limit is ~7000 times higher than our estimated angular separation of 4 mas for the Starkiller beams. To our eyes, the split beams would look like one beam.

...if they looked like anything at all. If you remember, Finn also saw the beams during the daytime. And as you may also remember, the only celestial object we tend to see during the day is the Sun (and the moon depending on its phase, and occasionally some planets and stars near sunrise and sunset). We intuitively know why this is: the Sun washes out dimmer objects. Even the reflected light of the Sun in the atmosphere is bright enough to wash out dim objects.

But why should that be? If the point where a star is has star and atmosphere, shouldn't it be a smidgen brighter than atmosphere alone? And shouldn't we be able to tell the difference? It turns out we can't, and the reason why is preserved in an ancient system for judging the brightness of stars that has persisted to this day with a few modifications.

The Greek astronomer Hipparchus set about cataloging the fixed stars a little more than two thousand years ago, managing to compile the position and brightness of several hundred of them. He called the brightest ones “stars of the first magnitude,” the second brightest “stars of the second magnitude,” and so on down to the dimmest stars visible to his naked eye, which he placed at magnitude six. Many an astronomy student today curses Hipparchus for giving lower numbers to brighter stars, but the system has stuck nonetheless.

In the 19th century, the English astronomer Norman Pogson realized that with a little fudging, it looked like 1st magnitude stars were 100 times brighter than 6th magnitude stars. You can divide this up a little further and discover that a magnitude jump of 1 represents a change in brightness of about 2.5 (2.55 ~ 100). But to our eyes, 1st magnitude stars don't seem to be 100 times brighter than 6th magnitude stars. They're not necessarily 6 times brighter either, but that's much closer to what we perceive than the physical reality. That's because human eyes don't respond to light in a linear fashion, but on a logarithmic or power scale instead (the details are messy and beyond my understanding).

If one star is twice as bright as another star, the above relation tells us that the magnitude difference is less than 1. In other words, Hipparchus might not even have noticed. The gist is that very small changes in brightness don't register to us if they are below a threshold called the just-noticeable difference. So while star+atmosphere is slightly brighter than atmosphere alone, it's not enough of a difference for our eyes to notice. And if the Starkiller beams shine with the brightness of a star (which seems about right given that Starkiller Base seems to explode into a star), then we wouldn't be able to see the beams at all during the day, let alone tell them apart.

But this isn't a problem just for human eyes. We don't point our telescopes at the sky during the day for the same reason. Modern telescopes pipe their images down to CCDs, digital devices that convert photons into electrons and count them up at each pixel. We can tell we've found something in a CCD if there's a signal that is significantly more intense than the background. But the background is noisy, and if the fluctuations from noise are greater than the difference between the background and the signal, then we can't tell if we've actually found anything at all.

Returning to Hipparchus for a moment, early astronomers noticed that brighter, lower magnitude stars appeared bigger than dimmer stars. We now know that the biggest stars are about a thousand times wider than our Sun. Yet we don’t see any stars in the sky that are a thousand times bigger than any other stars. In fact, it turns out the star with the largest angular size is R Doradus at 0.057 arcseconds. This is still tiny, with the moon about 30,000 times wider. But it doesn’t seem plausible that we could line up 30,000 stars as we see them in the night sky across the face of the moon.

The answer goes back to diffraction. To the naked eye, all stars are too small to have a resolvable disk. Instead, while the width of the central peak of a diffraction pattern is a function only of the aperture size and wavelength, the intensity across that width depends on the overall brightness of the object. As such, brighter stars appear bigger to our eyes, because diffraction means the whole Airy pattern is brighter, and that pattern is not point-like. Thus the size of a star in the night sky is not directly related to its physical size except insofar as bigger usually means brighter. We are not seeing the physical disk of the star itself, only the illusory Airy disk that results from diffraction.

Anyway, I think that’s enough nerding out over Star Wars and astronomy for now, what with me passing the 2000-word mark. I'll have more to say about observational astronomy later because I want to touch on image processing, which was a big chunk of the class. Next up will probably be quantum physics, though, because that’s a demon I’ve yet to exorcise.

Friday, September 4, 2015

Here's Where the Fun Begins

Hey guys. Remember me? Yeah, I haven't done any writing (fiction, blogging, or otherwise) in quite a while due to life being somewhat chaotic of late. I'd like that to change, so here's a quick blog post just to make sure I haven't forgotten how to type.

So I'm almost done with my first week of class, and I have now been to (or watched) at least one lecture for all of my classes. In the order in which I did so, here's a brief summary of said lectures followed by some general commentary. Man, this sounds exciting. I wish it were possible for Statcounter to track the exact paragraph in which my readers decide to leave the page.

Monday morning I had Quantum Physics I, which is an introductory course in quantum mechanics. Intro QM courses often seek to get students to develop some intuition for the quantum realm, which is quite counter-intuitive compared to the well known land of blocks sliding across incline planes. To develop this intuition, professors have students solve the Schrodinger equation again and again and again until their dreams are nothing but operators and wavefunctions.

To that end, the textbook we're using is Griffiths, which is apparently the text almost all intro QM classes use. Page one of that book writes down the Schrodinger equation and simply plows ahead from there. My professor thinks this is actually kind of a dumb way to go about things, so we're beginning the semester with the story of how quantum mechanics came to be.

Now, having been a science dude for quite some time, this is a story I've heard a lot. I'm getting some math to go along with it this time, but in general the story of quantum mechanics goes something like this:

Near the end of the 19th century, the kingdom of physics was at peace. Two centuries earlier, the father of physics, the great Sir Isaac Newton, had discovered the Stone of Counting, Calculus (this is a really funny joke), and used it to tame the very moon itself. Later, Maxwell forged together electricity and magnetism to bring light to the world. And Boltzmann conquered heat with entropy. Plus maybe some other things happened in the intervening two centuries.

But then evil blackbody radiation from the quantum realm brought about the ultraviolet catastrophe. Using classical thermodynamics, physicists predicted that hot objects would emit an infinite amount of energy at low wavelengths. Oh no! But then Planck saved the day by creating oscillators that only emitted and absorbed radiation in discrete chunks. Forced to obey a Boltzmann distribution, these oscillators were too few in number at low wavelengths to bring about divergent infinities.

Yet this was a false peace. Where did these quantized oscillators come from, and why did they only act in multiples of Planck's constant? Tune in next time to find out. (That's as far as we got in lecture. The rest of the story involves the photoelectric effect, emission lines, and some other stuff, but this post is already 8 paragraphs long and I'm only on my first class. Maybe I'll write a children's book about quantum mechanics.)

Tuesday morning (I have another class on Monday, but it's a discussion section and didn't meet the first week) I had Philosophy of Physics. This class is taught by a Distinguished University Professor who got a PhD in Mathematical Physics several centuries ago but then decided to go into philosophy for some reason. It turns out this class is mostly going to be talking about the "weirdness of quantum mechanics," which should make it a nice complement to that other class where I'm just going to "shut up and calculate."

Weirdness, though, is not about how maybe we're all really connected and you can change the world just by looking at it and other quantum woo like that. To this professor, the weirdness of quantum mechanics arises from an SAT-like analogy. Relativity is to space-time as quantum mechanics is to information. That is, Einstein taught us that space and time aren't what our intuition leads us to think they are, and QM does the same for information. Information, which has roots in probability theory, works differently than we think it does and the consequence is that quantum stuff can be correlated in ways that classical stuff can't. I think this is going to be pretty interesting.

Both my quantum classes were prefaced with a quote from Feynman about how nobody understands quantum mechanics. My QM professor thinks this isn't really true anymore and that the results of QM speak for themselves, whereas my philosophy professor thinks we might be getting close to an understanding via thinking about information theory.

Right after that I had Ancient Philosophy. I'm taking this class mostly because I need a history of philosophy credit for my philosophy minor, but also because I want to learn about some of the lesser known ancient Greek philosophers (Pre-Socratics, Stoics, Epicureans, etc.). And the text is chock full of readings from/about those philosophers. It was a shame, then, to learn that the instructor will mostly be teaching us about the moral philosophies of Plato and Aristotle. Yeah, that's good stuff. But doesn't everyone know that Plato's utopia is a dictatorial city-state run by wise philosopher kings? Sigh.

After a morning of philosophy came Observational Astronomy, which is the next required course in the astro sequence. This course is less about what's out there in the universe and more about how we come to learn about what's out there. We'll be studying optics, image processing, celestial coordinates, statistics of signal and noise, and how CCDs work. The biggest chunk of this class grade-wise is some observational projects where we have to take data from the observatory and process it into something useful and meaningful. That's pretty awesome.

Wednesday morning was another lecture of QM. Wednesday afternoon I had Solar System Astronomy. Like ancient philosophy, I'm taking this course mainly because I need a number of upper level astronomy courses to fulfill my major. I'm not super-interested in solar system stuff, but for some reason I'm trying to graduate next spring (it might have something to do with me turning 30 in a couple months...), which means I kind of have to take what's available. Also like Ancient Philosophy, I learned during the first class that this won't be a wide-ranging course about all aspects of the solar system, but will focus mostly on planetary geology, delving into the planets and other rocky bodies that inhabit our sun's domain.

After some thought, I realized I'm actually pretty okay with this. The one novel for which I have something approaching a rough draft spends a lot of time on Ceres and Europa, two big spheres about which I am not all that qualified to say much, despite the number of Wikipedia articles I've read. So, you know, getting a grounding in how these kinds of worlds really work might improve my ability to write about the things I'm already writing about. Or it just might make my infodumps that much more painful. We'll see. Either way, this course involves a term paper about some topic in solar system astronomy, so I'll definitely be writing.

Thursday was identical to Tuesday, and I'm writing this Friday morning, but Friday is essentially identical to Wednesday. The only class I haven't talked about is an online one, Theory of Knowledge. This is an intro philosophy course in epistemology. The course probably technically started Monday, but due some technical glitches (the course is being hosted on the professor's personal website, which he coded himself), I wasn't able to watch the first video lecture until Thursday evening.

During that video, the professor talked about the benefits of online courses, such as the freedom to edit lectures into conveniently sized chunks by excising parts that aren't helpful. Also during that video, the professor gave instructions on how to access his site in a video that his students could only be watching if they had successfully accessed his site.

Anyway, I'm pretty excited about this course. Epistemology is a fascinating subject to me as it acts as a bridge between thinking about the world and knowing about it. The basic stance of modern epistemology is that knowledge is "justified true beliefs." But how do we know if a belief is true? And how can we justify our beliefs? And what does it actually mean to believe something? Epistemology asks and attempts to answer all these questions, and it does so in surprisingly technical ways, invoking psychology, neuroscience, Bayesian statistics, and other pretty modern tools.

Week 9 of the course examines the philosophy of psychedelic transformations. (But it's a 15 week course, so I'm okay with a brief excursion into eye-rolling territory.)

And that about does it. This is going to be my busiest, toughest semester since I returned to school for real in 2012. I've got 19 credits of 300 and 400 level classes. Plus I'm working.

As far as general commentary, I have two things to say. The first is a pattern that may be a coincidence or may be indicative of what happens at this level. My observational astronomy, quantum physics, and epistemology classes are all prereqs for more advanced topics. And all of those professors are covering a lot of ground that is necessarily going to be somewhat outside of their precise areas of expertise.

On the other hand, my ancient philosophy, philosophy of physics, and solar system astronomy courses mostly stand on their own and don't lead explicitly to anything else. And my instructors in those classes have chosen to focus on a particular branch of each field that happens to coincide with their research interests. Coincidence? Probably not. But it does mean I may want to pay more attention to which teachers are teaching which classes when I decide to take free-standing, upper level courses.

My other comment is that most of my instructors (this semester and previously) talk pretty openly about pedagogy, which I think is a good sign. One of the stereotypes of college is the ancient professor who stares at the blackboard with chalk in hand, talking nonstop for the duration of the lecture and paying little heed to any students who might also be occupying the classroom. My college career thus far has been largely absent that phenomenon, and I suspect the apparently institutionalized focus on pedagogy is partly responsible for that. So yay.

Thursday, May 14, 2015

Why Am I Writing This Paper?

I went another month without posting. Sorry about that. I have half a dozen things I'd like to write about, but instead I've been swamped with end of semester stuff--term papers, lab reports, studying, etc.

So instead, like I did before to keep your attention, here's one of my philosophy papers. I did not want to write this paper because it deals with a question that (a) I think the answer to is plainly true, (b) is depressing, and (c) brings to mind a lot of the terrible arguments I had with those close to me when I was super depressed.

Consequently, I procrastinated writing this paper and wasn't able to get started on it before I found a way to make it funny. But I did manage to conceive of a fairly novel (to me) argument while writing it, which is kind of the point, so that's good. Unfortunately, in the first draft (which I turned in), that novel argument was kind of muddled. I cleaned things up a bit for this post. So here's hoping my TA tries to find out whether or not I plagiarized anybody and ends up stumbling onto my second draft.

Before writing a paper, one should always figure out why one is writing it. However, to save time, I have decided to answer this question while writing it. More broadly put, the question I’m considering here is whether writing this paper is in some sense a meaningful thing to do. In asking this question, I will also be forced to wonder whether anything at all—up to and including being alive—is meaningful. A cursory examination of my thoughts reveals three potential reasons why I might want to write this paper: to get a good grade, to have some sort of positive impact on the world, and to give my life value. A detailed exploration will reveal that none of these are sufficient reasons for paper-writing and that it’s overwhelmingly unlikely that completing this assignment could be considered at all meaningful. And yet there is no possible way for me to reach this conclusion without analytically contemplating the question itself—without writing the paper. I could have come to a different conclusion, so it would appear that any necessary first step in finding meaning in life is looking for it.
The most compelling reason for writing this paper is that I want to get a good grade on it. When we ask whether something is meaningful in this sense, we’re inquiring as to the point or purpose of doing it. Here I am asking to what ends paper-writing is a means. While it may not always be easy to elucidate the motivation for any particular action, it seems clear that anything we end up doing was motivated by something. Thus the motivation for writing this paper—the meaning in doing so—is that I wish to excel academically. Within the context of academic excellence, it is easy to find meaning in paper-writing.
Where trouble arises is that the goal of getting good grades is itself embedded in broader contexts. So we might be tempted to ask why doing well in school is a meaningful activity. After all, if maintaining my GPA is not meaningful, it’s hard to argue that any task geared toward GPA maintenance is also meaningful in a deep sense. So we can follow a causal chain up from paper-writing that goes something like this: I’m writing this paper to get a good grade; I want good grades so that I can get a degree; I want a degree so that I can find a satisfying, well-paying job; I want a satisfying, well-paying job so that I can live a happy, moral life; I want a happy, moral life so that… well… here’s where our chain runs into some problems.
Why do I want to live a happy, moral life? It might be so that I can raise happy, moral children who will raise happy, moral children, and so on. There’s no escape from the chain in that direction. I might want to live this kind of life because I am motivated to do so psychologically. If I am merely a machine in a clockwork universe, then my desire to live such a life can be understood as a tool of biological evolution for producing viable offspring, like the kind of animal life described by Taylor in “The Meaning of Human Existence.” Happiness is meaningful only insofar as I am a more efficient tool when happy; morality is meaningful because social cohesion provides a better environment for rearing children.
We might be tempted to stop here and find meaning in being happiness-generating biological machines, but doing so forces us to admit other features of the natural world we find less palatable. We are also motivated to kill competitors, to steal mates, and to enslave our inferiors. In fact, any action we take can be rationalized as psychologically-motivated and thus ultimately stemming from biological urges. Not only does this seem to grant legitimacy to terrible actions, but it also doesn’t leave room for degrees of meaningfulness. If writing this paper is just as meaningful as binge-watching House of Cards (consuming popular media signals to others that I am a member of the group, increasing my social status and apparent reproductive fitness, or something), then there’s no positive reason to perform any particular action at all.
If we continue on down the causal chain, we must engage in some reductionism. Biology is nothing more than the chemistry of self-replicating, homeostatic, organic molecules. Chemistry is nothing more than the physics of very large chunks of atoms. And physics is nothing more than a fundamental description of reality. From this vantage, why we engage in any particular action such as paper-writing can be summed up rather neatly: because thermodynamics, or because the fine-structure constant is 0.0072973525698.
While these might be accurate descriptions of why we do what we do, they are not altogether satisfying as explanations. The reason is that there doesn’t appear to be any deeper significance to the laws of physics. It’s difficult to say that the purpose of writing a paper is to conserve angular momentum. In fact, such a statement hardly even seems intelligible, which casts doubt on it being meaningful. At the end of this causal chain, we’re left not with motivations for actions but abstract descriptions of them.
The way out that many take here is to suppose that the underlying rules do exist for a reason, and that reason is God. If there is a transcendent entity who makes all the rules, including the rules that govern what is meaningful or moral, then acting in accordance with the purpose laid out by this being would be a meaningful way to spend one’s life, as Wolf alludes to in “The Meanings of Lives.” In that case, all I have to do is figure out whether or not me writing this paper is part of God’s plan.
Ah, but which God? Throughout the span of human history, we have described (either via revelation or invention) a great many possible gods. It’s unlikely that I’m going to be able to settle on the correct one before completing this paper. In fact, it’s not even clear how one might go about proving that a particular god is the correct one, because many who profess such knowledge claim that it is a subjective matter of faith. I might be tempted to find one specifically devoted to paper-writing, but that seems somewhat self-serving.
In the absence of any definitive proof about which gods are real, I am forced to abandon my search for meaning down the path of purposes and points. While there is certainly meaning within limited contexts, there is not a clear way toward objective meaning by focusing on the reasons for acting a particular way.
Perhaps the meaning of a thing is not found in the reason for it but in the significance of it. Perhaps me writing this paper will have an impact on the world or be felt in some way. This sense of meaningfulness is divorced from notions of what is good about paper-writing and instead focuses on the lasting effects of paper-writing. Something is meaningful if its creation adds to the world, changes the course of things, or leaves a mark. Here, meaning is found in the positive features of a thing—its extent and shape.
From this perspective, it’s easy to see how my paper will be meaningful. It will have a significant impact on the way its grader spends a half hour. Rather than binge-watching House of Cards, the person deciding my grade will read my paper, mark it up, complain about its inanity to sympathetic ears, and be forced to wrestle with ELMS in order to record my grade for all time. There are two possible objections one might make to this conception of meaning: it’s rather permissive, and our intuitive sense of meaning is of something grander.
Meaning as impact is permissive in that significance is lacking qualification. Everything I do has an impact on the world. Every breath I take rearranges the positions of billions and billions of air molecules. Given the sheer number of states that can be occupied by the atoms around me, everything I do ensures a permanent change. That is, after I act, nothing will ever be exactly the way it was before. Every tap of the keyboard makes microscopic changes in the structure of the keys themselves. These are all lasting changes to the world brought about by my direct intervention, but few would describe any of it as meaningful. Yes, from this perspective, writing papers is meaningful, but so is scratching my head or yawning.
So then we must be discerning about what qualifies as significant if we wish to exclude the trivial. One possible criterion is that actions must be noticed for them to be significant and meaningful. Because I have no direct awareness of how my actions change the molecules around me, my breathing is not noticeable and thus not significant. This qualification still permits my paper to be meaningful because someone else will be forced to read it, which might okay. We can say that my paper would be more meaningful if it were read by more people, if its brilliant philosophical insights changed the way millions thought, if it were referenced in Wikipedia articles, if undergraduate students taking introductory philosophy courses a thousand years from now were required to read it. This sense of meaning gets at the grandeur lacking from simply capturing the attention of a grader for a short while.
We can object to this notion of meaning in two ways. First, meaning as a noticeable impact on the world is grounded concretely in the limitations of human awareness. These limitations can be overcome by advances in observational tools. For example, we could imagine a world in which robots with exquisite sensors monitor the microstates of air molecules in my house and broadcast that information across the internet for all to consume. Under such a scenario, my breathing has once again become meaningful. But in the opposite direction, that which too few of us are aware of is not meaningful. We can imagine another world in which the prosperity of our civilization rests on slave labor that is hidden from us. We would all find it to be very significant indeed if the weight of our world were carried on the backs of the impoverished, and it seems incongruous to believe that the meaningfulness of this notion depends on our being aware of it. It’s also reasonable to believe a hidden slave population should be meaningful to more than just the slaves, especially because it is easy to conceive of a world in which they are unaware of why they labor.
The second objection picks away at the seeming grandness of what we are capable of doing. Having my paper appear on the reading list of future generations is about as significant as paper-writing can get. We can move up in scope and ask what possible significance my life in general could have. History is certainly peppered with great men and women who have done awesome and terrible things that echo in the present. Many historians might quibble with the idea that great people are ultimately responsible for the changes we see, but it’s probably possible to have a lasting impact on human civilization.
Yet here we are faced with the inevitable absurdity of human life. History is doubtless populated by countless significant figures we remain forever unaware of. But beyond that, the extent of our possible significance is quite literally infinitesimal. Virtually every human event in history has taken place inside a sphere with a radius under 6,400 km. The distance to the nearest star is 6 billion times that; the distance to the nearby Andromeda galaxy is half a million times that; the known size of the universe is a hundred thousand times as large as that; and the universe in all its unknown extent may be infinite. Geological records indicate that most species don’t persist longer than a few million years. Even if we beat the odds, in five billion years the Sun will swallow the Earth. If we somehow manage to escape that, the heat death of the universe will eventually erase any contribution we make. And long after we are gone, the universe will continue to exist for a span that is possibly trillions of times longer than its current age.
In “The Absurd,” Nagel objects to this notion of absurdity by pointing out that if nothing we do now will matter in a million years, then it doesn’t matter now that nothing we do will matter in a million years. But this misses the importance of meaning as significance. What’s important about this conception of meaning is persistence, whether through time or space. Binge-watching television isn’t meaningless because it happens not to be important years from now, but because its effects don’t persist through those years. It captures my attention while I am engaged in it but has no effect beyond its limited scope. So if the condition for meaningfulness is persistent significance on a large scale, then everything we could do ultimately fails.
Finally, we are left with a definition of meaning that most closely resembles more traditional meanings of the word meaning. It is possible that writing a philosophy paper could give my life value. That is, writing this paper may be an expression of who I am, a tool that others could use to gain knowledge about me. This is what it means for something to have meaning. The dictionary definition of a word tells you what a word is about; similarly, this paper may tell you what I am about and consequently be meaningful. In this sense, something is meaningful if it builds up some representation of an object that lets us understand something about that object.
From this notion alone, we can again naively conclude that paper-writing is clearly a meaningful activity. Anyone who reads this paper will gain some measure of insight into how my mind works. Similarly, anything I end up doing with my life can be meaningful if the events of my life create a narrative which tells you about me. Yet that presents us with a problem, because our intuition tells us that some lives might be more meaningful than others and that this should depend on what you end up doing with your life. It shouldn’t depend on the quality of the representation that can be built up based on your life.
As an example of why not all representations we can construct about a thing are meaningful, consider lightning. We could image a picture of lightning as being a manifestation of Zeus’ anger over the fact that we build skyscrapers. You could even argue that Zeus has reason to be mad at trees and sometimes even people. This is a description of lightning which may match what we observe, but we would not say that it is a meaningful description of lightning. It does not correspond to what we now know lightning to really be about—electricity, ions, and the like. So what we might say is that some things you do with your life—such as going to the bathroom or watching television—might not be meaningful because they don’t correspond to what we really know life is about.
Once again we are confronted with our sense of what is meaningful. That is, if we sense that some life activity is meaningful, our belief is that the activity accurately maps on to the person. If we follow the sense analogy, we can consider two ways in which we can sense what is out there in the world. On the one hand, our eyes see in color. It might seem obvious to believe that color inheres in objects, but the mechanism by which eyes work suggests something else. Rather, our eyes detect the intensity of light around three wavelength bands and then construct colors based on that information and a variety of other contextual clues. Color is not something that really exists but something our minds make a posteriori because it is useful for distinguishing between objects.
On the other hand, we also sometimes see objects that resemble triangles. Triangles, rather than being something we experience, are things we can construct a priori based on the formal rules of geometry. When we see a triangle in the world, we are comparing it to the Platonic triangle that is a product of our reason.
The parallel with our sense of meaning is this: is meaning a useful tool we build up from experiences, or is it an abstract entity that we see reflected in the world? If I write a philosophy paper and others see something meaningful in it, does that meaning arise from a psychologically-motivated heuristic about what’s important in life, or from a formal system that deductively defines human experience? If it is the former, then that meaning may not necessarily connect to what’s out there in the world—namely me. If the latter, then perhaps my paper is a true reflection of me and the sense of meaningfulness accurately signals this.
Unfortunately, there are no problem-free theories about what humans really are. Are we invested with souls? Are we rational agents or just animals possessing the illusion of control? What is the essence of being a human? What is consciousness? Does personal identity persist over time? Many of the questions regarding what it means to be human come down to what Nagel calls the subjective character of experience, a problem some consider unsolvable. We can never really know what is going on inside another person’s head because qualia are simply not objective. This leads us to the conclusion that it is very unlikely our haphazardly constructed brains have stumbled upon a sense of meaningfulness that is logically sound, so our sense is not a reliable indicator of whether what someone does with their life reflects who they really are. This does not rule out the possibility that people can do meaningful things, but it does rule out our knowing about it. And it might not make sense to say that something can be meaningful if no one gets the meaning, in which case nothing is meaningful.
From all this I can conclude that there is no point to writing this paper, that doing so will have no lasting impact on the world, and that it does not say anything meaningful about who I am. I clearly shouldn’t have wasted any time on it. However, it cannot be ignored that I could not have reached this conclusion without carefully considering what it means to be meaningful. While the arguments I present show that life does not appear to be meaningful, they do not prove that life could not be meaningful. This leaves open the possibility that we may discover some meaning in the future, and the only path toward that meaning is through thinking about it. So writing papers about meaning is not meaningful, but it might be a prerequisite for meaning.