Monday, July 28, 2014

A Pint of Aleph

Last week I showed you a frame of Futurama with a peculiar symbol. So, where did I already see this symbol?

That letter is the aleph, the first letter of the Hebrew alphabet, that seems to come from a hieroglyph of an ox. In the Kabbalah it refers to the origin of the universe, the "primordial one that contains all numbers". Such fascinating object of course caught the attention of the Argentine writer Jorge Luis Borges, an artist that wrote a lot on the concept of infinity. I can make many posts on his work, but for now let's consider his short story "The Aleph".

It tells the story of a man who has in his house the Aleph, a point in space that contains all other points. Borges himself sees it and comments it in this way:

All language is a set of symbols whose use among its speakers assumes a shared past. How, then, can I translate into words the limitless Aleph, which my floundering mind can scarcely encompass?
(Translation by Norman Thomas Di Giovanni)

This is all very nice, but Futurama's cinema had a 0 under the aleph, right? What does it mean?

Let's go back to Cantor, our hero. John Green He proved that there are infinities bigger than other infinities (in fact, he proved that for any infinity, one can find an infinity that is bigger), but are these infinities ordered? Let me explain:

suppose that I go to Cantor and I show him Infinity Leopold, then he will provide me Infinity Felix, an infinity bigger than Infinity Leopold. But of course, we can go on, and introduce Infinity Stanislaw, an infinity even bigger, and so on.

But now, Infinity Chad comes to disrupt the party with a "Hey, bros". Typical Chad. Where does Chad go? Is it bigger /smaller than the other infinities? Is it something completely different and not comparable (like apples and oranges)?

Under normal circumstances (*) it turns out that yes, the infinities are linearly ordered, i.e., any infinity is either bigger or smaller than all the others, and Chad (alas) has its place. As there is a limit of the names I can come up with, and nobody wants an Infinity Iqalussuarniartuqqasaagaluakagunnuuq, we can just call them First Infinity, Second Infinity, and so on. Better! thinks Cantor, let's use a symbol just for that. But which symbol can express the limitlessness of infinity, something that "contains all numbers", something that can translate into words things that a mind can scarcely encompass?

Good catch.

And so the infinities were called aleph 0, aleph 1, and so on. The smallest one is aleph 0, and it is the infinity of the natural number, the smallest infinity ever. Therefore the aleph 0-plex in Futurama is telling us that in that cinema, for any number, there is a screen. This opens up lots of questions on the physics of that universe, but let's not go there.

I am cheating a bit, here. Of course, the short story of Borges was written much later than Cantor, so they probably gained both inspiration from the Kabbalah. But we know that Borges was also thinking of Cantor's notation. He writes in a postscriptum:

For the Kabbala, the letter stands for the En Soph, the pure and boundless godhead; it is also said that it takes the shape of a man pointing to both heaven and earth, in order to show that the lower world is the map and mirror of the higher; for Cantor's Mengenlehre, it is the symbol of transfinite numbers, of which any part is as great as the whole.

Ehm. I don't know how to say this. Who am I to correct the great Borges? Well, but this is the aim of the blog, right? I... I will write it in a smaller text.

No! It's not! It is not true that in the transfinite numbers (aka what I called infinities, and now it's called infinite cardinals) ANY part is as great as the whole. Pick a transfinite number, and pick a finite part of it. See? You have a part that is not "as great as the whole". It is true that the infinite is characterized from having parts of it that are as great as the whole (remember 0 to 1 and 0 to 2?), but not ANY of them. You screwed up BIG TIME, Borges.

Phew, I am sorry Borges. But the first quote is so true. Just thinking about infinity, one can feel a sensation that it's not possible to explain in words, something so dizzying that the mind just boggles. I will try next time to show you some games that can give you that sensation, with the help of an advertisement and... a kindergarten.

Stay tuned!

(*) People think that Mathematics is made of statements that have an universal value, and therefore extremely objective. It is true that mathematical statements have an universal value, but sometimes people don't agree on what the Universe is. There are thus many groups of people inside mathematics, with opinions that differ. For example, in this post I postulate the existence of infinite sets, therefore going against finitists (like Kronecker) and ultrafinitists. In the next post I define a way to compare infinities that not all people agree with. Well, just one or two don't agree with it, but their results are solid, so it is right to acknowledge them. Finally, for the result above I use the Axiom of Choice, that basically says that if you have infinite pairs of socks, you can pick one for any pair (Russel quote). Here I am going against intuitionists and constructivists. Of course the big majority of mathematicians agrees with me, but at each his own.

Every problem solved brings with it dozens of new problems to solve. Linearly ordered infinities bring the Continuum Problem, for example. Also, I would say, the aleph would make a cool tattoo.

Monday, July 21, 2014

Rendezvous with Futurama

Disclaimer: Very short post, today, since I am recovering from the Vienna Summer of Logic and I feel a bit dizzy from all that logic.

Enough teenager love stories! I really overindulged on that. It's a bit creepy. Let's see something more classical:

The opening title card for Futurama

Futurama is an animated TV show that was advertised at the beginning as "The Simpsons in the future", but was really a brilliant, humorus, post-modern cartoon. It was cancelled, revived again and put on hold another time. The authors were never afraid of going too deep into complex/nerdy matters, like physics or computer science, thus including mathematical subjects (most famously, a writer invented and published a new math theorem just for an episode), so it is quite expected to find a Set Theory joke. Where? Here it is, from the 21st episode (it appeared also on the 47th episode):

File:Number 9 Raging Bender.jpg

This is in fact pretty popular, and not obscure at all, but if you don't know what I am talking about, you would ask: "What is this? Where is the infinite here? There is a strange letter there, but what has this to do with Set Theory? Are you kidding me? You're wasting my time!"

To you, I'll tell you two things

1. I... I don't think this is how the Internet works. I am sorry. Please stop shouting at your screen, and in case write some comment below?
2. I am not going to answer this now, wait until next week, and I will ask help to a key figure in Spanish language literature...

So, see you next Monday!

Update: the answer is here, as well as Infinity Chad. And if you like Sci-Fi, maybe you'll want to know about this.

Monday, July 14, 2014

The Fault in Our Infinites (Part 2)

Previously on Cantor on the Shore

It turns out that in the book “The Fault in Our Stars”, by John Green, there is a reference on Cantor’s most famous result: there are infinities bigger than other infinities.  The reference was slightly wrong, but nonetheless alluring, and the message left a seed in the mind of one of the protagonists (no spoilers!) How did the seed developed? Let’s hear:

I am not a mathematician, but I know this: There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities. A writer we used to like taught us that. There are days, many of them, when I resent the size of my unbounded set.

NOOOOOOO! This is so wrong! Completely wrong! Abhorrently wrong. Wrong like socks and sandals. Like ketchup on pizza. Like “Smells Like Teen Spirit” on the ukulele. Like snow in August (in the northern emisphere). Like when two people say goodbye and then they walk in the same direction. When I heard this at the cinema I unleashed all my rightful fury (with the dismay of my fellow moviegoers. Sorry).

To be honest, John Green claims that he knew it was wrong (attention, spoilers in the link), but he wrote it wrong on purpose, to show that teenagers can and do reach incorrect conclusions, but find comfort in them anyway. Yet, there is no indication in the book (or in the movie) that it is wrong, so it’s easy to imagine a girl reading the sentence, thinking “There are more numbers between 0 and 2 than between 0 and 1? Duh, of course. And this is what Cantor said? That Cantor guy is useless”, then the girl becomes a researcher in the University, they ask her to evaluate a proposal for a grant, she reads “Cantor” and think “this research is nothing worthy”, and then I become unemployed.  And nobody wants that.

So, why there is the same amount of numbers between 0 and 1 and between 0 and 2?
First, let’s agree on the meaning of “same amount”, with an example. Let’s go against the timeless suggestions of my mother, and compare apples and oranges.

If Cantor has a bowl of apples and Kronecker, his eternal nemesis, has a bowl of oranges, how can they know who has more, since they don’t trust the counting capabilities of each other?
They pick some string, and start connecting every apple to one orange, one by one. If in the end the bowls are empty, there was the “same amount” of oranges, of course. (*)

And this is what we are going to do with the infinities.  If I can connect any number between 0 and 1 to a number between 0 and 2, finally without leaving out numbers, it is clear that there is the same amount of numbers. Here we go: connect every number between 0 and 1 with its double, like this

And so on. Did we leave out some number? Nope. Every number between 0 and 2 is connected with its half, so 0.3 is connected with 0.15, 1.33 is connected with 0.665 and so on.
Once you see it, it is pretty clear (even for a teenager? Yes? Please tell me so, I have such a good faith in teenagers).  So that’s it: I righted the first error.

Yes, because there is another mistake! The parade of the errors seems to have no end. Maybe. I don’t know, the problem is that I simply don’t understand the last sentence. It’s like many pretty words with no real meaning. So, since probably John Green won’t come here to explain this, I have to guess, and it seems to me that the protagonist is lamenting that there are too many numbers between 0 and whatever. But then the word “unbounded” is wrong! That set is not unbounded, it is perfectly bounded, by whatever. But it’s infinite. Yep, unbounded and infinite are two different things.

Another thing! Like Columbo, I want to add just a small thing before leaving: even if the premises are wrong, in the end the protagonist is right in finding comfort in the infinite. There is an infinity of moments even in a short period of days, so you can project an eternity in there. So romantic (and, most importantly, correct)!

(*) One can, in fact, refuse to use this as a method of measuring the "same amount". But then, what can we use? Some mathematicians came out with a definition such that, yes, there are more numbers between 0 and 2, but the definition is incredibly complex and unintuitive. I am open to the possibility that in the future there will be an intuitive definition that satisfy our intuition, but until then, we do with what we have. If you have better ideas, please write them in the comments!

Literature is in fact a great resource for infinity. You enjoyed apple and oranges? Here is something even more challenging, courtesy of David Foster Wallace. And if all this infinities counfuse you, Borges has a nice way to put some order.

Monday, July 7, 2014

The Fault in the Fault in Our Stars (Part 1)

Science fiction and ”serious and intellectual” novels sometimes contain references to Set Theory, but starting the blog with them would be cheap. Of course there are references, these writers are geeks, what do you expect? I want to begin with a blast! Somewhere unexpected! So, what about… a young adult romance novel?

John Green is a popular writer of young adults novels, mostly heartbreaking teen romances, and he’s pretty good at it. If you like the genre, it doesn’t get any better than that. Despite the constraints of the genre, he always tries to put something new in its stories.  My attention goes to “The Fault in Our Stars”, a novel published in 2012, and then adapted in film in 2014. And yes, he went there. This is an excerpt:

 “So Zeno is most famous for his tortoise paradox. Let us imagine that you are in a race with a tortoise. The tortoise has a ten-yard head start. In the time it takes you to run that ten yards, the tortoise has maybe moved one yard. And then in the time it takes you to make up that distance, the tortoise goes a bit farther, and so on forever. You are faster than the tortoise but you can never catch him; you can only decrease his lead.
Of course, you just run past the tortoise without contemplating the mechanics involved, but the question of how you are able to do this turns out to be incredibly complicated, and no one really solved it until Cantor showed us that some infinities are bigger than other infinities.”

So, is this correct or not?

It goes very, very close, I am tempted to say that is correct, but no, it isn’t.

Let’s start from the beginning: Zeno’s paradox of Achilles and the Tortoise. The quote does a good job to illustrate it, down to the big problem. Why “you can never catch him”? Because it would take an infinite amount of intervals of time? Or of space? Because the sequence never ends? 

Anyway, all of this was solved in the 19th century, during the complete overhaul of Analysis that has been made by, for example, Weierstrass and Cauchy: it simply is possible to have a sum of infinite terms that gives a finite result. So Achilles will catch the elusive tortoise in finite time (or space).

Problem solved? Nuh-uh. You wish. Yes, the solution made perfect sense and the calculations worked, but it was not sound, because everybody was scared by infinite sums. “Infinity is just for God! How can we puny humans deal with this!” 

(Imagine Kronecker with a shrill voice saying this)

It was Aristotles fault, of course, like it always is: he practically said that the only way to think about infinity is to think of finite sets larger and larger, but we cannot deal with an infinite set.

And here comes our hero Cantor to save us, daiquiri in hand! With his study, he proved that it was mathematically sound to deal with infinite sets, therefore giving the theoretical support necessary to Analysis to solve Zeno’s paradox. I am not saying this, Russell said it. Also, at the same time with his brilliant work he canceled centuries of superstition, advanced the knowledge of the entire humanity and validated all the smart kids that to win at the game of the biggest number shouted "Infinity plus one!"

So, Zeno -> Cantor, right? Where is the error? Well, John Green quotes Cantor’s Theorem:  yes, even if it goes against our intuition, there are infinities bigger than other infinities. This is the starting point of modern Set Theory, the pedestal upon which the magnificent crystal castle is built. The proof of this is practically everywhere, it is called Cantor’s Diagonalization. But! It is not the solution to Zeno’s paradox. His treatment of infinity (that lead him to his Theorem) is.

Ok, I admit it, that was very nitpicky. I forgive John Green, because he couldn’t write the history of 19th century mathematics in one line, right? But I won’t forgive what he did next, because, believe it or not, that was not the only occurrence of infinity in the book. Something much worse is coming…

Stay tuned for the next episode!

Update: in another post I explain in more details why Cantor's approach solves the tortoise paradox. If you are unsatisfied by the explanation above, go here. And there is also Buzz Lightyear! But I stress: Cantor's Theorem has nothing to do with it.
You're wondering why John Green made this error? The behind-the-scenes answer is here!

Wednesday, July 2, 2014

This Blog Will Blow Your Mind! (a.k.a. Mission Statement)

Hello everyone, my name is Vincenzo Dimonte and I am a researcher in Set Theory (as long as the contract holds, so not for very long). In other words, I spend my time writing applications for grants, project proposals, CV and so on, and in the spare time I do some research in Set Theory. “But come on!” I hear you blurt. “What there is to research in sets?  Aren’t they just colored potatoes?”

Above: Set Theory?

Well, that is the starting point, just like learning to write the letter A is the starting point for literature.

A a B b C c D d E e

Above: Literature?

Set Theory grew very much from its roots, and now it is one of the most complex and difficult fields of Mathematics, capable of giving headaches even to the most skilled mathematician. And I work on that! Amazing.
Where does the difficulty come out? Infinity. As long as the sets are finite, things are pretty much what we expect (not trivial, though, it is incredible the amount of work we can do with finite sets), but when infinities come into play, then all the world as we know goes upside down.
Anyway, as humanity has been fascinated by infinity since the dawn of time, it happens that Set Theory creeps in popular medias: books, movies, etc. But is it done correctly? How are writers dealing with such a difficult topic? This is why this blog exists: it is a collection of instances of Set Theory in pop culture, with comments on whether the quotation has been done surprisingly well, or disastrously wrong.
I suspect I will update sparsely: I will just post anytime I find something. So, if you are reading/watching/hearing something about infinity and you wonder if it is correct, comment here, or contact me! I will solve your doubt.

One last thing: the title of the blog. The impulse for its creation comes from the belief that there is no contest between science and humanities, both concur in the progress of mankind, and yet both have a lot to learn from each other. This is a very metamodernist concept, and for that reason the title of the blog is a variation of “Kafka on the Shore”,  the book from the most metamodernist author I know, Haruki Murakami. Also, I enjoyed the view of Cantor, the founder of Set Theory, chilling on the shore with a daiquiri in his hand, and solving doubts in his spare time.