Problem sets. We'll have a few problem sets -- they're useful for you, and also they give me feedback about how much you're understanding. But here's the rule: you won't have to solve all the problems. You're allowed to write down, "yeah, Problem 4's a real stumper. I thought about it, I don't know the answer. Here are some easier problems that I can solve." I'll evaluate that the same way I'd evaluate a research paper that said a similar thing. On the other hand, if you have no idea how to solve the problem but you pretend to know -- for example, if you write gibberish that goes on and on and on, hoping that something vaguely resembling an answer will be buried in there just by chance -- that counts negatively. You'd do much better by leaving the question blank.
Grades. I should tell you that I hate grades, I hate GPA's. I'll tell you why. Some people think grades are a bad idea because they reduce a complicated human being to a number. That's not what I think. I think, if you're going to reduce a human being to a number, at least do it in a statistically sensible way! "This person scores one standard deviation above the mean on this task within this population." An A-: what does that mean? It means the teacher decided to you give you an A-. Because maybe he said, well, this student got a B+ according to my arbitrary made-up algorithm that's different from everyone else's, but B+ is really on the borderline of A-, so I'll make it an A-. And these are math PhD's. And they don't notice the problem of infinite regress. And of course, for a less likable student, the B+ gets just as arbitrarily downgraded to a B.
All the same, I suppose I have to give you grades. So what can I say? If you show up, participate, do the scribe notes, do the problem sets, you'll do fine. If you want an A or A+, you might have to argue and disagree with me. (I mean, intelligently disagree with me.)
Alright. So why Democritus? First of all, who was Democritus? He was this ancient Greek dude. He was born around 450BC in Abdera, which was sort of this podunk town. where people from Athens said that even the air causes stupidity. He was a disciple of Leucippus, according to my source, which is Wikipedia. He's called a "pre-Socratic," even though actually he was a contemporary of Socrates. That gives you a sense of how important he's considered: "Yeah, the pre-Socratics -- maybe stick 'em in somewhere in the first week of class." (Incidentally, there's a story that Democritus journeyed to Athens to meet Socrates, but then was too shy to introduce himself.)
Almost none of Democritus's writings survive. (Some of them apparently survived into the Middle Ages, but they're lost now.) What we know about him is mostly due to the fact that other philosophers, like Aristotle, brought him up in order to criticize him.
So, what were the ideas they criticized? Democritus thought the whole universe is composed of atoms in a void, constantly moving around according to determinate, understandable laws. These atoms can hit each other and bounce off, and they can stick together to make bigger things. They can have different sizes, weights, and shapes -- maybe some are spheres, some are cylinders, whatever. On the other hand, Democritus says that properties like color and taste are not intrinsic to atoms, but instead emerge out of the interactions of many atoms. For if the atoms that made up the ocean were "intrinsically blue," then how could they form the white froth on waves?
Remember, this is 400BC. So far we're batting pretty well.
Why does Democritus think there are these atoms surrounded by void? He gives a few arguments, one of which can be paraphrased as follows (following Carl Sagan). Suppose we have an apple, and suppose the apple's not made of atoms but is instead this continuous, hard stuff. And suppose we take a knife and cut the apple into two pieces. It's clear that the points on one side go into the first piece and the points on the other side go into the second piece, but what about the points exactly on the boundary? Do they "disappear"? Do they get duplicated? Does the symmetry get broken? None of these possibilities seem particularly elegant.
Incidentally, some of you might know that there's a debate raging even today between atomists and anti-atomists. This time, the headquarters of the atomist side aren't in Abdera; they're a mile down the train tracks from here, in a certain sleek black building. At issue in this debate is whether space and time themselves are made up of indivisible atoms, at the Planck scale of 10-33 centimeters or 10-43 seconds. Ironically, the physicists have almost no experimental evidence to go on, and are basically in the same situation that Democritus was in 2400 years ago. If you want an ignorant, uninformed layperson's opinion, my money is on the atomist side. And the arguments I'd use are not entirely different from the ones Democritus used: mostly they hinge on inherent mathematical difficulties with the continuum.
One passage of Democritus that does survive is a dialogue between the intellect and the senses. The intellect starts out, saying: "By convention there is sweetness, by convention bitterness, by convention color, in reality only atoms and the void." In my book, this one line already puts Democritus shoulder-to-shoulder with Plato, Aristotle, or any other ancient philosopher you care to name. But the dialogue doesn't stop there. The senses respond, saying: "Foolish intellect! Do you seek to overthrow us, while it is from us that you take your evidence?"
I first came across this dialogue in a book by Schrödinger. Ah, Schrödinger! -- you see we're inching toward the "quantum computing" in the course title. We're gonna get there, don't worry about that.
But why would Schrödinger be interested in this dialogue? Well, Schrödinger was interested in a lot of things. He was not an intellectual monogamist (or really any kind of monogamist). But one reason he might've been interested is a certain equation he was involved with, which you've probably heard about:
i dψ/dt = Hψ
(Did I get it right, Ray?)
Actually, let me write it in a more correct form:
|ψt+1〉 = U |ψt〉
What is this equation? Well, maybe you have to add a few details to it -- like the physics -- but once you do, it describes the evolution of a quantum pure state. For any isolated region of the universe that you want to consider, this equation describes the evolution in time of the state of that region, which we represent as a normalized linear combination -- a superposition -- of all the possible configurations of elementary particles in that region. So you can think of this equation as the sophisticated, modern version of Democritus's "atoms and the void." And as we all know, it does pretty well at the atoms and the void part.
The part where it maybe doesn't do so well is the "from us you take your evidence" part. Where's the "us"? Remember, the equation describes a superposition over all possible configurations of particles. So, I don't know -- are you in superposition? I don't feel like I am!
Incidentally, one thing I'm not going to do in this class is try to sell you on some favorite interpretation of quantum mechanics. You're free to believe any interpretation your conscience dictates. (What's my own view? Well, I agree with every interpretation to the extent it says there's a problem, and disagree with every interpretation to the extent it claims to have solved the problem!)
Anyway, just like we can classify religions as monotheistic and polytheistic, we can classify interpretations of quantum mechanics by where they come down on the "putting-yourself-in-coherent-superposition" issue. On the one side, we've got the interpretations that enthusiastically sweep the issue under the rug: Copenhagen and its Bayesian and epistemic grandchildren. In these interpretations, you've got your quantum system, you've got your measuring device, and there's a line between them. Sure, the line can shift from one experiment to the next, but for any given experiment, it's gotta be somewhere. In principle, you can even imagine putting other people on the quantum side, but you yourself are always on the classical side. Why? Because a quantum state is just a representation of your knowledge -- and you, by definition, are a classical being.
But what if you want to apply quantum mechanics to the whole universe, including yourself? The answer, in the epistemic-type interpretations, is simply that you don't ask that sort of question! Incidentally, that was Bohr's all-time favorite philosophical move, his WWF piledrive: "You're not allowed to ask such a question!"
On the other side we've got the interpretations that do try in different ways to make sense of putting yourself in superposition: many-worlds, Bohmian mechanics, etc.
Now, to hardheaded problem-solvers like ourselves, this might seem like a big dispute over words -- why bother? I actually agree with that: if it were just a dispute over words, then we shouldn't bother! But as David Deutsch pointed out in the late 1970's, we can conceive of experiments that would differentiate the first type of interpretation from the second type. The simplest experiment would just be to put yourself in coherent superposition and see what happens! Or if that's too dangerous, put someone else in coherent superposition. The point being that, if human beings were regularly being put into superposition, then the whole business of drawing a line between "classical observers" and the rest of the universe would become untenable.
But alright -- human brains are wet, goopy, sloppy things, and maybe we won't be able to maintain them in coherent superposition for 500 million years. So what's the next best thing? Well, we could try to put a computer in superposition. The more sophisticated the computer was -- the more it resembled something like a brain, like ourselves -- the further up we would have pushed the 'line' between quantum and classical. You can see how it's only a miniscule step from here to quantum computing.
I'd like to draw a more general lesson here. What's the point of talking about philosophical questions? Because we're going to be doing a fair bit of it in this class -- I mean, of philosophical bullshitting. Well, there's a standard answer, and that's that philosophy is an intellectual clean-up job -- the janitors who come in after the scientists have made a mess, to try and pick up the pieces. So on this view, philosophers should sit in their armchairs waiting for something surprising to happen in science -- like quantum mechanics, like the Bell inequality, like Gödel's Theorem -- and then swoop in like vultures and say, ah, this is what it really meant.
Well, on its face, that seems sort of boring. But as you get more accustomed to this sort of work, I think what you'll find is ... it's still boring!
Like most of you, I'm interested in results -- in finding solutions to nontrivial, well-defined open problems. So, what's the role of philosophy in that? I want to suggest a more exalted role than intellectual janitor: philosophy can be a scout. It can be an explorer -- mapping out intellectual terrain for science to later move in on, and build condominiums on or whatever. Not every branch of science was "scouted out ahead of time" by philosophy, but some of them were. And in recent history, I think quantum computing is really the poster child here. It's fine to tell people to "Shut up and calculate," but the question is, what should they calculate? At least here at IQC, the sorts of things we like to calculate -- capacities of quantum channels, error probabilities of quantum algorithms, etc. -- are things people would never have thought to calculate if not for philosophy.
Alright, I promised you a puzzle. Earlier I mentioned inherent mathematical difficulties with the continuum, so I've got a puzzle somewhat related to that. If it's too easy, let me know and I'll give you a harder one.
You know the real line, right? Suppose we want a union of open intervals that covers every rational point. Question: does the sum of the lengths of the intervals have to be infinite? One would certainly think so! After all, there are rational numbers pretty much everywhere!
[Richard Cleve immediately solves the puzzle.]
Alright, I guess that was too easy.
[Solution: Not only can the sum of the lengths of the intervals be finite, it can be arbitrarily close to zero! Simply enumerate the rational numbers, r0, r1, etc. Then put an interval of size ε/2i around ri for every i.]
Here's a harder one: we have the unit square, [0,1]2. Consider a function S, which maps every real number x∈[0,1] to a countable subset S(x) of [0,1]. Can we choose S so that, for every (x,y)∈[0,1]2, either y∈S(x) or x∈S(y)?
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