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.

Contradictory.

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)."

[Infinite series: from history to mathematics education (PDF) by Giorgio T. Bagni. —Ed.]

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

anything

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"

Right?

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.

OK?

Plato: grr...

yes

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"

liar

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.