Monday, August 25, 2014

Infinity times infinity: updates

Remember this post? The one about infinity times infinity? The kindergarten kids? The infinities of dots? The pickets? That one.

Well, I have updates on that.


Just as I published the post, I received a notification from Google+. Weird, Google+ is a wasteland (sorry Google, I know you're doing your best). What is this scream from the void?

Google+ automatically uploaded this gif for me:

Very cool gif, Google+! Thank you! This explains my thoughts much, much better. I don't know how you came up with this.


How did you come up with this? Not only this animation makes much sense, but it is also named infinity-MOTION. Perfect name. Is there a brilliant man/woman at Google+'s that follows all my posts, looks up at the pictures and creates better ones? To make me feel inferior? Does he/she follows me around, stalking every keystroke?? You're not better than me, person!

Or maybe I am overreacting. It is just an automatic software. That crawls through all my stuff. And makes creepily appropriate gifs. Is it... is this how Skynet started? With gifs? What are you doing to my life, Google+?!


It turns out that infinity times infinity is a thing, and there is a symbol for it.

It seems a must for romantic things (as in "I love you infinity times infinity") and it appears in jewellery, tattoos, cards, etc... First, it seems like this blog will never be free from romantic stuff. This is great, as a good part of my intended audience is clearly made of teenage girls and pickup artists. Second, I am not really sure about this symbol. I mean, it is clearly one infinity (lemniscate) on top of the other, so it is more like infinity plus infinity. I imagine infinity times infinity as something more complex, a bit more mindblowing, something like this?
Well, this seems too much a pair of angry eyes. Maybe something more fractal? I am not a graphic designer (as you can see very well from my pictures), so please, somebody invent a better symbol! Infinity times infinity deserves it!



I have nothing to say about this. Everything he said it's right.

Did anybody said tattoo? Also, if you spend time on Google+, probably you have a lot of time for this.

Monday, August 18, 2014

These are infinities that count!

It's summer! Time to relax, chill out, go to the beach (at least this is what I heard normal people do, while I am in my without-air-conditioning city office explaining to various research centers the impact of my research on European society), listen to summer hits:

This is "Countably Infinite" by the A.G.Trio, an electrohouse band from Austria. In summer 2012 (I said summer hits, I didn't specified the year) it peaked the Austrian chart, and it also entered the German Club Chart. But I am not interested in the music now, I am interested in the lyrics. As far as I have understood, they're about a couple breaking up (of course, like every song ever written, so nothing to see here), the key sentence that gives the title of the song is this:

There are countably infinite things I'd like to say to you

In this cases I think: what does a person that doesn't know what "countably infinte" means feels hearing this? Which images come to the mind? An infinite... you can count? Because in fact "countably infinite" is a pretty specific term in set theory.

Something is countably infinite if it is infinite and as big as the natural numbers, or aleph 0. That is, if you can connect every object to one and only one number, the objects are countably infinite. In this way, you can count the objects: object 1, object 2, object 3, ...  and this is way is called "countably". One way to do this is a list: if you can write the objects, one every line, without ever stopping, than you have countably infinite.

So, is it possible for the singer to have countably infinite things to say? Let's try to imagine the possible things he has to say as a list:

1. I love you
3. Don't leave
2. I'll miss you
3. My, that is a BIG pimple
4. It's like... purple
5. Hey, did you know that there is no word that rhymes with purple?
6. The sky is blue
7. I kissed a girl, and I liked iiit (of course)
8. Some infinities are bigger than other infinities...
9. BAM! Romantic
10. There are more stars in the universe than grains of sand on all the beaches of Earth
11. I love dinosaur erotica
12. No, no, no this is not related to the first sentence, don't leave!
13. Where am I?
14. I am sitting in a room, different from the one you are in now
15. But seriously, how would you say your skin is scaly from 0 to 10?
16. ...

Wait, this is taking too long. Let's do this more systematically. How many thoughts there are, at all? Better: let's count how many sentences one can write. For example: how many with one character? we have the letters (26) and the space (1). Who cares about punctuation. So 27. How many with two characters? 27 times 27, that is 729. And so on. So let's code any sentence with a number: a=1, b=2, z=26, _=27, aa=28, ab=29, a_=54, ba=55 and so on. Think about it: all the words you are seeing are saved in the computer as binary numbers, so of course to every sentence is associated one and one only number, and to every number one and one only sentence. For example: the number 1234567890 is the sentence "ceaantr" (almost Cantor!), while "I_love_you" is the number 76387629278892. So there are countably infinite possible things to say, one for each number!

But does the singer have really infinite things to say? Our brain is pretty big, but not infinite, it has somewhat around 2.5 petabytes (or a million gigabytes). So the thoughts he has in mind have to be finite! Come on, A.G. Trio! How could you even think that one guy had infinite thoughts! It's like you are purposedly exaggerating for...

Wait a minute.

That's it, isn't it? They didn't really mean infinite things. It was just a way to say "more things that I can say", wasn't it? And I am making a fool of myself meticulously analyzing a simple hyperbole?


Blimey! I always fall for this! I... I need time to think.

(Anyway, Cantor is 43868727)

Countable infinity is probaby much, much bigger than what you have in mind. Still, other things are even bigger.

Monday, August 11, 2014

Advanced Thinking & Thinking: Infinity times Infinity

Warning: the procedures in this post, if followed correctly, can lead to vertigo, psychological intoxication, and air-headedness. In other words: awesomeness.

I wanted to give you a tiny, little glimpse of the sensations that I experience daily on working in Set Theory. I want to take you on a stroll in the paradise that Cantor provided us. I am going to describe some really simple infinite set, and the only thing you have to do is to try to imagine it. Sometimes the mental image just don't come: don't worry, just keep thinking about it, and return back to it in another moment. Also, take your time. Don't just say "Yes, I understood" and go on, try to explore with your mind every nook and cranny of the infinite set.

And have the right attitude. Which attitude, you say? This one:

This advertisement is incredible. It manages to introduce effortlessy a nice amount of infinite sets. But can you really image infinity times infinity? Try it now. I'll wait.


What took you so much? So let's see if my description is the same as yours.

This is one.

This is two (yeah, I know, stay with me).

And so on. Just adding one on the right, you are making +1.

Once you finished the numbers, you have the first infinite set! Congratulations! We call it ω. (Infinity! like the girl said).

Now, you know how to do ω+1: after ω, add one point.

Well, there is not much space there, let's go to a new line.

And now add another one (ω+2).

And so on, until you finished the numbers again. We have ω+ω! (Infinity plus infinity, like the guy said).

Now add another point, and continue. We have ω+ω+ω, i.e. three times infinity.

Go on, with four times infinity, five times infinity, and so on. At the end we have ωxω (Infinity times infinity! like the other girl said).

Great, this was not that difficult. So let's go deeper. I want you to imagine ωω, i.e. Infinity to the power of infinity! Let's do it visually.

Visualize again ω.

Now, between each two points, add ω. We have ωxω, i.e. ω2.

Do it again: between any two points in ω2, add ω. We have ω3.

Of course I am not going to make a drawing for that, don't be silly. This is the moment when we have to use just our minds, the screen does not have enough resolution for that. Now do it again (add one ω for each point in ω3), and again (ω4), and again! In the end you have in front of you ωω!

Let's try another way. Imagine you are on a road, with pickets on the side of the road. The first picket you see is one meter high. You start counting them: one, two, three...

When you counted them all, one picket two meters high appears. Then again the short ones, start counting again: one, two, three...

counted them all the second picket two meters high appear. Continue like this, and a third, a fourth will appear, and so on. When you counted all the two meters high pickets, one three meters high appears!

And then the small ones again, etc. After the second batch of two meters high there is a second three meters picket, after the third batch a third, and so on. Once all the three meters high are finished, here it is one of four meters. Continue like this, and you will have passed ωω pickets.

Let's try even another way. Imagine a tally counter. But instead of having just numbers between 0 and 9, every disk has ALL the numbers. And instead of having finite disks, it has infinite disks. There: the tally counter can count exactly ωω numbers.

Homework! How to visualize Infinity to the power of infinity to the power of infinity? And Infinity to the power of infinity to the power of infinity to the power of infinity? And Infinity made to the power of infinity infinite times? Have fun!

One personal note: it is disarming how practically all the children manage to think about infinity plus one, like above pretty much. And yet many times they are stopped by adults, that tell them that you cannot do infinity plus one, because infinity is infinity. Of course you can! This reminds me of a schoolmate at kindergarten that once told me that 100 is the biggest number. I asked her "What about 103?" and she answered "It does not exist". In the same vein, my teacher at primary school told me that you cannot do 4-6. "Isn't it -2?" I asked. She told me "That does not exist, and that operation is FORBIDDEN". I am still recovering from that. So please, if you are reading this, think of the children! Don't limit their fantasy, don't close them the gate to Cantor's paradise!

I think that the people that make the most fascinating jobs have still inside them the kid that they were. Every astronaut is driven by the "childish" fascination of the stars, every zoologist by the passion kids have for animals, and what about paleontologists and dinosaurs! Well, si parva licet, I am still playing the game of "who says the biggest number".

And I am winning.

The story does not end here. You can find an update at this page. Also, did you know that there is a dance song about all of that?

Monday, August 4, 2014

Don't be naive, Google!

And now this blog is as contemporary as ever!

Google's doodle today is about the 180th birthday of John Venn, the "inventor" of Venn diagrams (their history, in fact, is much more complicated).

Now, I know I have been a bit harsh in my manifesto about them. I should apologize: they really are important.

Just think of the concept of set. It's something very extravagant and abstract: it splits the world exactly in two, the things that are in the set, and the things that are not. How could we arrive at that? The real world is much more vague: think for example of the set of chairs. Can you really say which objects are in it and which are not? Is a stone in it? You can sit on it! And a doll chair? You cannot sit on it, why do you call it a chair? Bah!

Is this a chair? You tell me.

I mean, Plato would say that there is an idea of the chair, and then the physical objects adhere more or less to that idea. There is no sharp line between "chairs" and "not chairs".

Yet a set-theoretic approach is exactly what is needed for the abstract thinking. Every concept entails the existence of the set of objects that apply to that concept, and (here is the master stroke) you can use that set as an object, manipulating it in different ways, creating new concepts even in transcendental ways. One can make intersections, unions, complements, talk about sets of sets or sets of sets of sets. The possibilities are endless. This is what is called Naive set theory: not formal, but effective in practice. And the Venn diagrams give an immediate way to imagine it.

You can find naive set theory everywhere. Think about linguistics: when you say that somebody is beautiful AND smart, you are saying that he is a member of the intersection of the set of beautiful people and of the set of smart people. If you say that tonight you are eating Chinese OR Indian food, you are sayin that what you are going to eat is in the union of the set of Chinese food and the set of Indian food. So AND is in fact an intersection, and OR is a union.

But then what about Google search. If you search "Marilyn Manson" "cute bunnies" (the AND is implicit), it (should) give you the intersection of the results for Marilyn Manson and the results for cute bunnies, while if you search for "apocalypse" OR "Facebook down", it will give you the union of the two results. This is clearly just an example of the action of filtering a database, and without it the world as we know it would just collapse.

Things can get weirder: there is something called musical set theory. The earliest example I know (but I am not a musicologist, so take it with a grain of salt) is Herma by Iannis Xenakis, where he established a set of notes and then used set-theoretic operations to change it in different ways.

In short: yes, Venn diagrams are great. But when you have to do things that are more complicated, they become useless and more formality is needed.

So the next person that tells me "so you work with Venn diagrams", he's going to feel the intersection of my fist with his mouth.

A naive approach to sets can lead you to beautiful poems. But not to iconic phrases.