Do Infinite Pencils Exist??
Does infinity exist? This comment from MathOverflow riled my feathers a bit:
I’ve heard a worse story. A college instructor claimed in Number Theory class that there are only finitely many primes. When confronted by a student, her reply was: “If you think there are infinitely many, write them all down.” She was on tenure track, but need I add, didn’t get tenure.
What a dumb teacher, right? Everyone knows there are an infinite number of primes! Haven’t you heard of Euclid?
Except I think she’s right. (Depending on what she meant, exactly.)
I believe this professor was attempting to make a subtle point that many students of mathematics tend to miss. That is, they mistakenly believe that infinity actually exists. Like, in the real world.
If I tell someone to imagine an “infinite number of pencils,” usually they picture a bunch of pencils. Like, more than they could ever count. An empire state building made out of pencils. An ocean full of pencils. Mars. But pencils.
That’s not really what infinty means, mathematically. When a mathematician says infinity they mean a repeatable process that we can keep doing forever.
For example, let’s imagine I’m standing at the blackboard and I ask the class to give me a pencil. Now I have 1 pencil. That’s our repeatable process. I do it a second time, and I have 2 pencils. A third time, and three. And so on and so forth.
Imagine I never stop asking for pencils. Ever. How many pencils do I have, at that indeterminate point in the future? A mathematician would say an infinite number of pencils.
But you see, I’d never really get to an “infinite” quantity of pencils. Infinity is purely a work of imagination. Eventually I’ll have to stop asking for pencils. I’ll get hungry, or get old or die. Or we’ll run out of wood or something. Or we’ll exhaust all the matter in the universe and there will be nothing but pencils floating around in the vastness of space. Then who would ask for the pencil, and who would give it?1
Anyway, back to the primes. Yes, the student is correct in that there is an infinite number of primes. (And by that we mean there is a repeatable process2 for generating primes which we can repeat until the heat death of the universe.)
But the teacher is also right, in her way. If we built a computer to count all the primes, it would eventually run out of memory. Even if we turned the whole universe into a computer, eventually we’d run out of stars and junk to fuel our prime-counting computer. Thus the number of primes we’ll ever be able to count is finitely bounded by the size of our universe.3
That’s… funny? Sad? I don’t know. Why did I even write this?
If infinity doesn’t really exist, then why do we talk about it so much? Because it’s a useful approximation of reality. Of course, it doesn’t make sense to ask, “if I keep asking for pencils how many pencils will I have?” But suppose instead of asking for 1 pencil each time, I asked for 1/4th a pencil, and then 1/9th, and then \(1/n^2\) of a pencil for ever and ever. How many pencils would I have? The answer is that I will get real close to 1.645 pencils, but never more than that. Why’s that useful? Well, ever try to build a rocket? No? Me neither. ↩
Euclid’s proof is famously not constructive, so it doesn’t directly give us a method for constructing a new prime. I prefer the proof by Filip Saidak that does. ↩
I wonder how many digits it has? ↩