Sunday, January 25, 2015

Infinite Mess (Part Two)

This is the second part of a post that started here. Go read it, if you still haven't done so. Or don't, and try to guess what is going on. It'll be enlightening anyway.

So, what did DFW wrote(*)? Here is the source, verbatim:

The Continuum Hypothesis gets characterized in all kind of different ways: [...](**) Is the same as 

Let me first try to explain to you what is the Continuum Hypothesis, let's see if I am better in this than Foster Wallace. First, let's get the objects straight:

Natural numbers are the numbers like 0, 1, 2 and so on. Basically, if you hear a number and you can imagine the same amount of zebras (or any other object, but I prefer zebras), it's a natural number. You can't imagine pi zebras, or 1.25 zebras. This are the numbers that we all learned in school, and I really shouldn't have spent 67 words on this.

And one picture

Real numbers are all the numbers. Period. Even with infinite digits after the digital point. 2, 3.45, square root of 2, pi are all real numbers.

Now, Cantor proved that some infinities are bigger than other infinities, right? Well, he was more specific: he proved that the real numbers are more than the natural numbers. Pretty cool. Then he asked, in his Ein Beitrag zur Mannigfaltigkeitslehere, whether the real numbers are immediately larger than the natural numbers, or if there is something in between, larger than the natural number but smaller than the real numbers. This is called Continuum Problem, because continuum was fancy -talk for real numbers. Then Cantor said: you know what? I think that there is nothing in between. This is called Continuum Hypothesis, and as it is formulated, it's just an opinion, a hypothesis (hence the name): we don't know the answer to the continuum problem, so let's suppose it's this.

So? Understood? If you read DFW's book I'd really like to know if it was easier to understand than this.

You can already see the first problem: the Continuum Hypothesis asserts something, so it cannot be a question! DFW is confusing it for the Continuum Problem. Already annoying. But let's go on: what are those strange symbols in DFW's quote?
 is the size of the set of the real numbers, i.e., how many real numbers there are.

is the size of the set of the sets of natural number (I'll stop you before you start Xzibit memes), i.e., how many sets of natural numbers there are.

DFW is showing something completely different than the Continuum Problem, then, he's asking if there is the same quantity of real numbers and sets of integers, like it is a great mistery.

It is not! It's Set Theory 101: they are the same! It's not a mistery, it's almost trivial! What were you thinking, DFW? He got everything completely confused, he wanted to show the Continuum Hypothesis and he showed an exercise for students that involves objects that have nothing to do with the Continuum Hypothesis.

You want to know why they are the same? Mmm... this is not immediate, unfortunately one has really to write down a mathematical proof. So put on your favorite thinking hat...

This is mine. Don't judge me, the situation at the office is awkward enough.

and concentrate on the following (***).

To prove that two sets have the same size, one should be able to connect every object of one set to only one other object on the other set,

Try as you might, this doesn't work in this case. What to do then? Mathematicians know a weird simple trick (doctors HATE it!!): what if we connect all the apples to the oranges and there are oranges left? This means that the oranges cannot be less than the apples, right? (****)  And if we connect, in another way, all the oranges to the apples and there are apples left, this means that the apples cannot be less than the oranges! So they are the same quantity!

Let's see first that the real numbers cannot be less than the sets of natural numbers. Pick a set of natural numbers

then draw it in the number line

write a 0 when it's empty, 1 when it's full

and finally add 0. at the front.

There you are, for each set of natural number, you can write a different real number. So real numbers cannot be less.

Now to see that the set of natural numbers cannot be less than the real numbers, I won't describe it, I'll just show it:

Therefore the two things do have the same size, and the climax of DFW's booklet is an epic fail.

Really, it's disappointing. It started so well, with Zeno's paradox of the turtle and everything...

Wait a minute, where did I see the turtle paradox and Cantor work together in the wrong way? John Green is a big fan of David Foster Wallace, right? Maybe...

Now that I think about it, the book-in-the-book An Imperial Affliction has many things in common with Infinite Jest, like the non-ending. But oh! Of course! The writer of AIA, Van Houten, is so similar to the prose of DFW! His obscurity, his way of talking encyclopedic but hard to understand... and it is Van Houten that connects (wrongly) the turtle paradox and Cantor's Theorem! Also, John Green has surely read Everything and More, he even reviewed it for Booklist Magazine!

John Green has put Cantor's Theorem in The Fault in Our Stars because he read it on Everything and More, and he did it wrongly because it was confusing already in the original book! That's where everything starts! That's a scoop!

What? What are you saying? You mean... he already admitted that in the FAQ page I have already linked once? 


I'll see myself out.

(*) Since I am taking for granted that you, reader, are a DFW fan, I am adding lots of footnotes. Have fun!
(**) I skipped the other three characterizations of Foster Wallace, without context it's pretty useless. For the curious: one is wrong and the other two are characterizations of the Continuum Problem, not the Continuum Hypothesis.
(***) Or don't. Really, you can just skip the whole paragraph, you, reader, are the king, because I am so post-modern.
(****) In the finite case, it means that the apples are less than the oranges, but infinity is weird.

Thanks to Gabriel for pointing this out to me.

Monday, January 12, 2015

Infinite Mess (Part One)

Uh-oh. That's it. I'm going to do it. I am going to criticize a very well-beloved author, an author that touched the hearts of millions of people, one of the most influential and innovative writers of the last 20 years, according to the Los Angeles Times, also one that met a tragic and untimely end, and therefore untouchable.

The reactions of literary fans are known to be sober

I'm not going to be only critical, I am going to destroy one of his works (well, at least some lines). I am talking about David Foster Wallace, the brilliant mind behind Infinite Jest, an encyclopedic, metamodernist, hysterical realist novel that almost single-handedly put him in the curriculum of English literature courses. He's edgy, irreverent, inventive and also sweet, how can I possibly go against him (especially since he cannot defend himself)? Not only that, but I'll even claim that I am better than him in explaining some stuff! Oh my, some little blogger really went over his head, now.

I will use his words to defend myself:

The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we can all use but he alone pays for. 

Well, here I am; as a Prometheus (and maybe Mentally Ill Mathematician, as this blog seems to attest) I can be forgiven if I bash a literary genius, as I am also bringing gifts for everybody, guys! They come from forbidden places!

Unfortunately, it seems that Foster Wallace didn't go where I've been. In 2003 he wrote a booklet, Everything and More, about the history of infinity, and especially the work of our good old Cantor. Great, right? Finally some popular recognition to our hero! So, is it any good?

Disclaimer: I haven't read it. What I read are the critical reviews of Rudy Rucker and Michael Harris, the second one being really interesting as it is more forgiving to the author, and some other snippet caught here and there in some preview, It was enough to bum me. There is always a misunderstanding when writers try to explain mathematical concepts: the literary way of dealing with concepts is through vagueness.  The beauty of a poem is that the words carry with them many, many meanings, and elicit in our mind different responses, therefore being able with this overlap to create sensations that would be impossible to explain in plain words. Mathematics is the opposite: its beauty is in the perfection of its concepts. It's a huge mechanism full of cogs and wheels, and yet everything works perfectly, every minimal part is on time. It's like juggling, or rock balancing: the slightest error can ruin everything.

Literature vs Mathematics
(Left: (c) Frank Grisdale)

You can see, then, how treating mathematical concepts with literary tools just doesn't work. It's even worse in this case: DFW is aggressively post-modern, and one of the staples of aggressive post-modernism is the unreliability of the narrator. And he admits that in this book! So, what's the point of reading a mathematical book where the writer always tries to trick you? Not only that, but he's trying to invent a new style, called "pop technical writing", that uses tons of footnotes (of course) and abbreviations that no one ever used. A complete mess.

Well, then, that is how it is written. But what about the content? Surely there is a lot to learn reading this book, right?


Unfortunately, there are many, many errors. There are websites that list all of them. There is enough material to publish one post per day, for a year. Now I cannot do that, I have already very few readers without actively alienate them with DFW marathons, so I'll choose just one. But it's a big one. Imagine: you spent hours and hours trudging through this book, through all the ridiculous fake notations, through hundreds of pages on convergent series, when finally you are realizing that the book had a direction, that all of this was to reach a particular point, You turn the page ready to read it, no, to experience it...

and it is wrong.

The musical equivalent

What? You don't expect me to explain the climax now, do you? Just go here to read about this crime against infinity.