21 January 2010

Infinity, part 2: Zeno's paradox

(An ongoing series. See part 1.)

In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...

—Bertrand Russell, The Principles of Mathematics (1903).

The previous post involved two different kinds of infinity. There's the infinite on, off, on, off... of Thomson's lamp. And there's the infinite division of time: one minute, then half a minute, then a quarter of a minute, etc.

Maybe your reaction to the paradox was, “Oh, that's impossible, there's no such thing as infinity in the real world.” Perhaps not. Perhaps once you get down to tiny enough fractions of a second, you see that light is emitted in tiny, discrete quanta, and to emit even one quantum of light each time the lamp turned on would require more than enough energy to burn it out.

But math refuses to make that excuse. Math deals with abstracts, ideals. Math must deal with infinity somehow. From the moment you start counting, it is always staring you in the face.

Shortly after I posted part 1, I ran across this old chat log. Oddly enough, it discusses the same two infinities, but in a different guise.

Eudoxus: Hey, are you there?
Plato: yeah
is this about that 0.999... = 1 thing, because I'm kind of busy
Eudoxus: no no
check this out, I ran across this paper about teaching math to children, and the example they were using was infinite series.
The question was, 1 - 1 + 1 - 1 + 1 - 1 + ... = ?
Care to guess?
Plato: 0
Eudoxus: Right, obviously
because all the terms cancel out
only there's another way to see it...
you start with 1, and then all the terms *after* that cancel out with each other. see?
So the answer is 1.
Plato: hmm...
Eudoxus: You with me so far?
Plato: yup
Eudoxus: So here's the snippet from this paper that jumped out at me.
"It is important to point out that it is not enough to consider at the same time two conflicting statements in order to develop in pupils' minds the awareness of an inconsistency and the necessity of second thoughts (Schoenfeld, 1985): the perception of some mutually conflicting elements does not always imply the perception of the situation as a problematic one (Tirosh, 1990)."
This surprised me because it seems so obvious that inconsistency is a sign something is wrong.
Plato: yes
kids are dumb
from a contradiction, anything follows. Everyone knows that
Can I say something?
Eudoxus: shoot
Plato: this thing about the infinite series
Maybe this is dumb
but I don't see how you can get an answer
i mean it keeps going on and on
where would you put the equals sign?
Eudoxus: :)
Plato: which is a joke
Eudoxus: sure
Plato: but I think that, obviously, you are going to come up with weird answers when you start assuming that if an infinite series "stopped" and you could make an equation out of it, etc.
Eudoxus: But the answers work out just fine, and it's hard to avoid. I don't think you can do calculus, for example, without infinite sums.
Plato: that's why calculus is dumb
Eudoxus: Pfft. Calculus is going to be huge. I'm going to write a book about it, as soon as I get some technical issues resolved. I just hope no one else gets to print first.
Plato: you're all talk
Eudoxus: I am going to try and convince you that infinite sums can work, because if I'm wrong then all my work is contradictory and useless.
Have you heard of Zeno's paradox? The one with Achilles and the hare?
I mean, tortoise
Plato: Is that the half of half of half one?
Eudoxus: Yes.
Zeno says that for Achilles to catch the tortoise would require an infinite number of moves. Clearly impossible. Therefore motion must be an illusion--because there's no way to make sense of it.
Plato: Right. Well, it makes sense...if you assume that the dude is infinitely small...
I mean, in real life you wouldn't be able to do it
because at some point you would just be too big
to go that small a distance and have it mean anythign
and zeno is such a jerk anyway
Eudoxus: Well, hang on.
Would you accept, arguendo, that very small distances exist, even if they are way too small for Achilles to see?
Plato: sigh, yes, Socrates
Eudoxus: :) Then suppose I have two marks on the ground, exactly 1 stadion apart.
Euclid could construct the line between them. And the midpoint. OK?
Plato: yes
Eudoxus: And as many points as you like, successively closer to point B. Right?
Plato: of course
Eudoxus: So you accept that there are (at least in principle) infinitely many points there, getting closer and closer to B?
Plato: yes, of course
Eudoxus: (thinking)
And between every point and its successor, there's some finite distance. That is, between A and the midpoint is 1/2 a stadion; between that point and the next is 1/4 stadion, and so on?
Plato: yes, a measurable distance, agree
Eudoxus: Now, none of the distances overlap. And taken together, they cover the entire line segment AB. Right?
Plato: yes
Eudoxus: So doesn't it make sense to say the sum of all these lengths is 1 stadion?
Plato: you couldn't take a sum
I mean, yes, together they do equal one stadion
But, um, you couldn't really measure each part and add them all up
Eudoxus: I couldn't construct all the points, either, as a practical matter. But that doesn't stop them from existing.
Plato: okay, true
So, yes, all of them together equal one stadion; I guess that's a sum
Eudoxus: I say it's a sum by analogy to the finite case. Other than a fear of infinity, I see no reason not to call this a sum, and say that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1.
Plato: yes, yes, it's a sum
Eudoxus: Ah, then an infinite sum is possible.
Plato: See, here's the thing
You are being tricky
You are using infinity two different ways
On the one hand, infinity is just blah plus blah plus blah, etc.
On the other hand, you are saying blah/this finite thing + blah/this finite thing + blah/this finite thing...
Listen every finite thing in the world could be cut into infinite "parts"
Eudoxus: Sure.
Plato: Okay, but
that doesn't mean that you can just start with an infinite number of things and decide to multiply and add them and assume that in the end you'll get a finite thing
Eudoxus: Hmm.
I need to think about that.
Incidentally, what would you say about the sum 9/10 + 9/100 + 9/1000 + ...?
Plato: do you mean, an infinite series, the next being 9/10000?
Eudoxus: Yes
Plato: Well, that's the "other" sort of infinity
the type that is just an infinite series of numbers, added, that you are assuming will result in a finite sum
Eudoxus: I don't mean to "assume" that it will or won't. I want to find out if it will or not.
Plato: oh, okay
it won't
Eudoxus: No? But Euclid can also construct a point, call it C, that's exactly 90% of the way from A to B.
Plato: grr...
Eudoxus: By design, AC is 9/10 of the whole.
What's left, CB, is the other 1/10, right?
Plato: yes
Eudoxus: So repeat with CB. You'll draw a point D that is very close to B.
CD is 9/100 of the whole. What's left, DB, is the other 1/100.
Plato: okay
Argh, so then, they equal a finite thing?
Eudoxus: You see where I'm going? You're constructing a picture of the statement "9/10 + 9/100 + 9/1000 + 9/10000 + ... = 1"
Plato: yes
but, okay, yes, I see that
Eudoxus: And what's another way to write 9/10?
Plato: no, I won't do it; because it is dumb
Eudoxus: (laugh)
when did you see it coming?
Plato: when you said "no no"
Eudoxus: OK, the point of all that mess was of course that 0.999... = 1.
Plato: yes, I know
Eudoxus: I wanted to make an argument in a way that would appeal to a Platonist.
Plato: no name calling :)
Eudoxus: Hey, that's what you are. You should be proud. :)
Plato: I am, sort of. It's just that...okay, I still don't understand exactly. I mean, I get the proofs, but I still feel like we are adding apples and then adding oranges and then saying, voila, apples are oranges.
Eudoxus: Yes, yes, I know...
I'll think about it some more later...
You are right to say that not every infinite sum is as... easy to deal with as 0.999...
Plato: yes, exactly
But it's scary, isn't it
Eudoxus: ?
Plato: I mean, even just mathematically in general
Just the idea that you can get two contradictory answers to an equation
Eudoxus: Uh, yes, it's very troubling.
Plato: I mean, mathematics deals in absolutes
There's no room for contradictions, right?
From a contradiction, everything follows
Eudoxus: Yes.
Plato: Right, so if these infinity things are correct as is, what does that mean?
Eudoxus: Oh, you mean 1 - 1 + 1 - 1 + ...?
Plato: yes
it proves that 0=1, right?
Eudoxus: Well...
that sort of problem doesn't really arise in practice though.
Plato: o rly?
i can prove 0=1 and it doesn't affect your work? what kind of logic is this?
the consequences, man1`
Eudoxus: Sorry, I have to go. I think someone's calling me.
Plato: what???
come back here you coward
Eudoxus is offline.

You might be relieved to know that the paradox has been resolved.

The cure was transformative.