Infinite Amateurism on Maths
I stumbled upon Thomas Thurman's post on his blog (here) where he comments about a discussion on The Guardian about how .99(recurring) is or is not the same as 1.
Of course to anyone who knows what a rational number is, .99(recurring) is simply a very long way to write 1. Hell, to anyone who bothered learning his fractions, that should be obvious!
But anyway, one of the comments mentions Hilbert's Hotel, which is a pet toy of mine.
If you are uncomfortable or annoyed by the concept of infinity, you may want to avoid the rest of this post.
Hilbert's Hotel is this paradox (Thanks Wikipedia!):
A hotel with an infinite number of rooms (1, 2, 3 and so on, so it's a numerable infinity) is full. A guest arrives. Yet he still gets a room. How?
The answer is that you ask the guest in room 1 to move to room 2, from 2 to 3 and so on. Then the new guest goes to room 1, which is now free.
Because of the nature of infinity, this works, while on a finite hotel it wouldn't.
Unintuitive things about infinity: If you add any number to it, the result is the same infinity.
Now assume an infinite (numerable) number of guests arrives. You have to ask the guest in room 1 to go to room 2, the guest in room 2 to go to room 4, and guest n to go to n*2.
Now you have an infinite (numerable) number of free rooms: all the odd rooms.
Unintuitive things about infinity: If you multiply it by any number, the result is the same infinity.
Unintuitive things about infinity: If something is infinite, it contains a part the same size as the whole (follows from the previous two things, and is, in fact an "if and only if").
However, here's where it gets tricky. You could get a certain number of new guests and there could be no way to fit them in the rooms even if the hotel was empty.
Because there's infinity, and then there is infinity. You saw that whenever I mentioned the number of rooms I mentioned they were infinite (numerable)? That's because you can put an integer number to each, and number them all.
There are infinite sets of things that are bigger, they are literally uncountable. You can't put a number to each, even with an infinite amount of time (and yes, I know that infinite amount of time there is a big problem).
The simplest set imaginable that large is that of the real numbers. The real numbers are all the numbers you can imagine, allowing for infinite decimals, and allowing that those decimals may not ever be recurring (so you have things like 2, 1/3, and pi).
Showing there are more of those that there are integer numbers is not simple enough for this but go along with me for a while.
Unintuitive things about infinity: There are different sizes of infinite. Go blame Georg Cantor.
Now it gets really weird. Suppose we call the size of the infinite in Hilbert's Hotel A0 (I have no idea how to do an Aleph, sorry), and the size of the real numbers C. Cantor showed how to build, once you have an infinite set, a larger infinite set called his power set.
That means we now have a whole infinite (numerable) "sizes" of inifinite things. Those are called the transfinite numbers.
Unintuitive things about infinity: There are infinite different sizes of infinite. Go blame Georg Cantor some more.
Which brings a lot of questions:
Are there only those? Isn't there something between A0 and C which is some in-between size?
Is there an infinite set that's smaller than the integers?
Ok, infinite infinites... infinite (numerable) infinites, or infinite (something else) infinites?
Well... I have no idea. And last I checked, which was long ago, and my memory is no good, noone else knew.
This is the kind of things that will tell you whether you could be a mathematician. Do you find all this talk about transfinite numbers intriguing and mysterious, or just dull and boring and impractical?
If you find it dull and boring, it may be my writing, or you may be unsuited for maths.
If you find it intriguing and/or mysterious, it certainly is not my writing, and you would probably enjoy maths in your life. Where else are you going to run into "noone knows that" this quickly?
The problem is, of course, that most of the fun math has already been done at least a century ago, but there is always a chance of something fun and intriguing and new coming along.
The last I know of was Gödel's theorem, which is really simple enough for anyone with knowledge of arithmetic to follow, but weird enough for 99.99% of the people to go crazy about (and for those who don't really understand it to write whole books about it applying it to totally improper subjects).
But you know, noone really had thought of such thing as "larger than infinity" quite as Cantor did, and noone thought about Gödel's subject quite as he did before him.
Maybe we are missing something absolutely simple, incredibly elegant, awesomely shocking somewhere in basic maths. Not likely. But possible. Wouldn't it be fun to find it?
BTW: Gödel starved himself to death and Cantor "suffered poverty, hunger and died in a sanatorium".
Hi, I really couldn't get pass the hotel thing.
Obviously I´m not suited for maths, which can be a explanation why I became a lawyer. Or maybe it´s just because I'm a woman. I read once that women are less suited for maths than men, because they tend to be more practical, while men are better with abstract ideas. That sounds quite sexist, but It proves right on me, at least.
But what I can surely say is that I wouldn´t spend two minutes in a hotel that pretended to be changing my room indefinitely. So the whole idea is pointless or it´s just a bad example. Where is the person who thought about it pretending to get an infinite bunch of nothing-to-do-with-their-time people to try such an experiment?
I know that this is just theoretical, nothing pretended to be done really, and that´s is mostly what Í don't understand about maths. What's the use of thinking about things that have no practical use???
No wonder those mathematicians starved to death, no matter how simple, elegant or shocking their findings where...
Anyhow I guess maths must be important in some kind of way. I'm just glad to leave the task to others.
"Isn't there something between A0 and C which is some in-between size?"
well, the reason nobody knows is because it is proven that it can't be proved. In the usual mathematical axiom system (ZFC - zermelo-fraenkel and axiom of choice) the question
2^(aleph0) = aleph1 ?
(with aleph1 the first set bigger than aleph0 in ZF, and 2^(aleph0) the number of subsets in aleph0, which is the number of real numbers i.e. your C)
is equivalent to the continuum hypothesis, meaning that it cannot be proven in ZFC. Either you accept it, or you don't.
It's like the axiom of choice: you can't prove nor disprove it with ZF (only zermelo-fraenkel), and some mathematicians don't accept the axiom of choice and develop their own math system, the same with the continuum hypothesis.
"Is there an infinite set that's smaller than the integers?"
assuming you work in ZFC, no there isn't.
@Roflech - You said "What's the use of thinking about things that have no practical use?"
If you think that way, all art is useless. Whats the practical use of having Monalisa?
Feynman once said about Physics (same applies to Math): Physics is like sex. Both may produce practical results, but that's not why we do it.
@Roflech: I could tell you why maths are important, but I have already tried. It works about as well as you explaining me politics ;-)
@Eimai: Nice to see someone who actually knows the subject around ;-)
So, within ZFC, we know some things, and not everyone accepts ZFC. Sounds a lot like "we have no idea" to me, for some value of "we" (and some value of idea ;-).
Anyway: I am still waiting, maybe someone can figure out an axiomatic system as simple as ZFC where all this makes more sense.
And anyway "the continuum hypothesis" is one of my top 5 math thing names. It's so startrekky.
There are other set theories besides ZFC. If you want one where the continuum hypothesis is true, you can just add it on as an axiom, and then ask deeper questions -- does there exist yet another infinty between C and the newly added one? (Yes, and all the infinities you could ever want.)
You can also work in another mathematical universe -- a different topos -- where things can be wildly different.
@Alsina: It's not so much that some mathematician's don't accept ZFC, it's just that sometimes it makes sense to work with or without some axiom, such as the axiom of choice. In this case, since choice is independent of ZF, we are free to accept or reject it whenever necessary. ZFC is logically consistent, so it is not rejected on a basis of being incorrect.
@Roflech: As far as why these things are important -- you need uncountable infinities to make calculus work, and last time I checked calculus was being used to make just about everything.
And for infinities of infinities of infinities... 2^infinity is always a bigger infinity, and you can take infinities^infinities, and infinite number of times if you want, and it turns out that there is no largest infinity. See large cardinals on Wikipedia.
Well, another reason some mathematicians do not include the axiom of choice in their reasoning is that where it is not necessary, a proof is strengthened by its absence - that is, it requires a smaller set of axioms (ZF) to prove whatever is at hand.