An Answer to the Ultrafinitist Below

This post is an answer to this one (yes, I am answering myself, please wait before calling the mental house). If you read that one, you probably noticed that the style was, um, different. That is because that should have been an April fools' joke (you could guess it because the words fools and joke were bold, and because it was the 1st of April). It was a collection of arguments that people against infinity use. I wanted to answer immediately, but life finds a way... to kill your plans. So let's answer them now.

Who is right, then? I wrote 20 posts about infinity, but the arguments against it are pretty convincing, aren't they? Well, in fact thery are not really wrong, they are just too partial. Let's see them.

Q. Is it true that the universe is not infinite?
A. I am not a physicist, so I cannot answer this question in all the details (that's a bad start). In this interview Prof. Joseph Silk says that simply we don't know. It can be either way, with our current knowledge it's not possible to know. According to Wikipedia, one of the models of the Universe with most consensus is the Friedmann–Lemaître–Robertson–Walker metric, that appears to be infinite, but I have no idea. Anyway, it seems like it doesn't matter, The time we have here is finite, therefore even if the Universe is infinite, we will be able to observe just a finite part of it.

Q. But is it true that between two points there is just a finite amount of space?
A. Again, we don't know. I found this nice discussion on StackExchange, and the models we are using now to calculate stuff do have infinite points between two points (are continuous). The consensus seems to be that this is the case, but we cannot know.

Q. Ok, so we don't know if the infinite exists in this Universe, and maybe we will never know. But is it used in physics practice?
A. Constantly. The language of physics is mathematics, and there infinity is a necessary tool. If you think about it, nothing in mathematics really exists. There is no real number, there are no points, no lines. Triangles do not exist.

And yet we use them every day, because they work.

Q. Therefore infinity is used in maths.
A. That is not a question. Anyway, yes. math is practically founded on the infinite. Leibniz, Newton & co., while founding calculus, were using it just like any other quantity. Now we know better, and we actually can get rid of it, decide a maximum number and stay there. But the problem is: which one?

As an example, let's take computer science. In 1936 Turing invented the Turing machine, that is not a real machine, but a hypothetical one that is infinite and that can mimic the behaviour of any computer. It has been a fundamental tool for the understanding of computer science (well, it was the invention of the idea of computer). Now suppose that Turing, instead of conceiving an infinite machine, invented a finite one. How big do you think it would have been? 2000 possible numbers? 20000?

"Z3 Deutsches Museum" by Venusianer at the German language Wikipedia. Licensed under CC BY-SA 3.0 via Wikimedia Commons.

The Z3, built in 1941, could memorize only 64 words of 22 bit

It was surely unconceivable at that time more space, but now we can easily and cheapily have 137438953472 numbers hidden under our nails. If the Turing machine was finite, every year a bunch of people would have to meet to raise its size. Instead, we have a reliable and universal way to represent all computers, past and future, thanks of infinity. For example, now we now if something is just not possible to calculate with a computer. If we just deal with finite stuff, one can say "Heh, maybe if our computers are bigger/faster, we can calculate that", and spend millions of real world money for an impossible task.

The same is true for mathematics. Deciding a limit number is ridiculous. Also many concepts first start with infinity, and then become finite. Without the infinity part, we probably just wouldn't have such concepts, as "calculus" or "computer".

Q. Can we imagine the infinite?
A. That's an interesting question!
Q. Thank you.
A. I mean, even kids know perfectly well what infinity is, right? Yet, we cannot possibly think all the numbers. Our brain is finite. But we can think of the totality of the numbers. For example, think of a glass of water. In that glass, say, there are 8 x 10^24 molecules of water (that is more than 8 septillions for an American). Our mind cannot possibly think of each molecule, there are too many. Yet, you had no problem in thinking of a glass of water, am I correct?

So, that's it! Infinity is safe and alive! You can continue to infinity-and-beyond your stuff, people! Oh, one more thing...

Q. Are all finitists so obnoxious, like in the previous post?
A. Yes. Well, that's unfair, just the more vocal ones. I don't know what's wrong with them. At the mere hint of infinity, they unleash an anger and spite usually reserved to breast-feeding forums and Ben Affleck's Batman. They call people who use infinity "mystics", "religious" or even "fetishists". They stalk all famous mathematicians, ready to attack (I am not famous, but I do have a finitist that comes into my office). They feel like the whole world is wrong, and they're frustrated that no one listens to them, before it's too late. They are not stupid or ignorant, mind you, sometimes they are brilliant thinkers. It's just that we work on a different fundamental assumption, and there is no way to please both. But can we nonetheless aceept each other and be friends? Please?

Some infinities don't exist more than other infinities (an ultrafinitist point of view)

Attention! This was an April Fools' post, so it says the exact opposit of what I think. Still, it's food for thought, and I answer to this in this post.

Hahaha. Infinity.

What a joke.

Only fools would really believe that such a thing exists. Don't you see it? It's just a ruse to make you question the reality you are seeing with your own eyes! Wake up sheeple!

Infinity is just an illusion. God is dead, so why we don't do the same to infinity, huh? Time to get ride of it. Well, we are already doing it. We thought the universe was infinite, well, not so much. We thought the speed of light was infinite, not even close. We thought that between two points there were infinite points, or that between two instants there were infinite moments of time, wrong again and again. Everything around us is finite. You don't go to the pet shop and buy infinite kittens. That would be too cuteness for anybody.

Spoiler alert: this is not possible.

Then love is infinite? Sorry lovey-dovey couples, at the very least one of you will die, so you can just start to delete that tattoo right away. But then death is infinite? No, it's not: everybody is always dead for a finite amount of time. Julius Caesar has been dead for 2059 years and 17 days, Marvin Gaye for exactly 31 years. Give it up, infinitists!

So what about the numbers? Are they infinite? Of course not. Just pick the number of particles in the universe, its time and calculate all the possible combinations of them. It will be a finite number, the biggest on. What do you say? What about that number +1? Pfft, that does not exist. Just because you can imagine a number, it doesn't mean it exists. I can imagine a unicorn, but still not receiving it for Christmas. Because it does not exist.

Ok, so you're saying that one can imagine infinite numbers. Come on, try it. Start: 10, 100, 1000... At a certain point you cannot go on anymore. Some numbers are so huge, that we cannot conceive them, But then, where do they exist, if not in nature or in anybody's mind? Answer: they don't.

So you like tortoises and Achilleses, right? If you think about it, the solution is easy: the world is like a Monopoly table. There will be a moment when Achille and the Tortoise will be one step close. Then the tortoise should do half a step, but it cannot, so it stays, and Achille does one step and reaches the tortoise. Easy peasy!

Enough with this infinite foolishness, then! Eat your vegetables, and embrace the world as it is! (Vegetables are delicious, too! Especially the Brussels Sprouts.)

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,

In this way

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? 

Oh.

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.

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?

No.

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.

And the best tattoo of the year 2014 is...

Ok, I am a standard mathematician. I don't want to reinforce the trite stereotype of the nerd math-lover, but I have thick glasses, often the colours of my clothes don't match (and my clothes are too big/too small/perfect just for old people), I learned how to bike waaay after learning the difference between mass and weight, and of course I am terrified, terrified by needlespleasepleasepleasetakeoutfrommyviewthatinstrumentofterror. I'm hopeless, you really wouldn't expect me to dissert about the current fashion in tattoos, would you?

Yet, Buzzfeed made a list of the most trendy tattoos of the year 2014. And what do we have here?

Photo from here

The lemniscate really exploded this year. Even someone like me had to notice this trend in the skin of the people around. The most popular is probably the one with "love" embedded in it, or other writings (note the Buzz Lightyear reminder!)

Photo from here

and the combos "feather+infinity" and "finger tattoo+infinity" are also ubiquitous.

 Photo from here

Photo from here

There are tumblrs devoted to it and all sorts of websites. But I am a set theorists, not a fashion designer, so what can I say to you about it?

Well, now I am about to dump to you a nugget of information that will surely make you popular with your friends. You can ask: why there is one symbol of infinity, when there are many infinities? The reason is that the lemniscate, in mathematics, always refers to the potential infinity, i.e., the infinity you cannot reach, (I explain this here) while each different actual infinity, i.e., an infinity considered as size of an infinite set, has a different symbol. Cool, huh?

Pick up artist suggestion
So, you are at a New Year party and you see a cute girl with an infinity tattoo. What do you do? You go close to her, you say "Hey, did you know that this symbol only means potential infinity? But the actual infinity..." Pause. "... is in my pants".

And then you pick from your pants' pocket a piece of paper with an aleph on it. Mark my words, she'll throw herself to your arms. You're welcome.

You want a symbol that it's even more than infinity? Here you are. 

Teoría de conjuntos (a poem for Christmas)

Christmas is back! The most... depressing period of the year. Really, while the peak of suicides over Christmas is a myth, it surely can bring the worst in our souls: extremely overcrowded shopping streets (with the included unnecessary touching, keep your heels far from my feet, woman, thanks), creepy Santa Claus(es?) grinning at every corner, sad, pale light decorations that always miss a row of burned bulbs and that in the most flamboyant cases will surely induce a seizure. And what about the perfect present you have to choose for your Nazi sympathizer cousin that you managed to avoid for one year?

For this reason, this time I'll try to avoid the hard mathematics and I'll post something... warmer. A poem, whose title is "Set Theory"

Teoría de conjuntos

Cada cuerpo tiene

su armonía y

su desarmonía.

En algunos casos

la suma de armonías

puede ser casi

empalagosa.

En otros

el conjunto

de desarmonías

produce algo mejor

que la belleza.

 (taken from here)

This is a poem by Mario Benedetti, a Uruguyan poet, probably one of the most important writers in Latin America in the last century, even if he is not really known in the English-speaking world. 

Unfortunately, my knowledge of Spanish and English is not enough for a good translation, so if you don't know Spanish (hello, 93% of the world population!), I can just give you the meaning. In the end, it has nothing to do with maths: "conjunto" in English is "set", but also "collection", "whole". Benedetti says that every body has a harmony and a disharmony. somethimes the sum of harmonies can be almost loathsome, while in other cases the collection of disharmonies can produce something even better than beauty.

So, nothing to say about math here, the "set theory" of the title was just the inspiration for Benedetti to write a small and wise poem about the beauty of the imperfection of human nature. Keep it with you this days: what if the collection of all the horrors we will have to endure for the holidays, in the end really forms something beautiful?

I'll take Benedetti's word here.

Thanks to Luz for having pointed me out this poem! See? The suggestions box works!

Looking for something light? Then naive Set Theory is for you. 

Intermission

It is time for this blog to take a little pause. I mean, you wouldn't think that finding set theory instances in pop culture would grant the same rapid fire coverage of, say, Hello Kitty Hell, would you?

So, today no pop culture. Instead here is a video:

This is the Full Program: Infinity part of the World Science Festival. A Philosopher/Theologician, a Mathematician, a Set Theorists and a Physicist meet together, and instead of starting a joke they discuss for more than one hour about infinity. If you are seriously interested in infinity, this is a nice video to be introduced to, with different perspectives. In parts it is a bit technical, so take your time, don't sweat it.

The introduction is fantastic, with a parade of people talking about infinity in absurdely vague and personal terms, all thinking that infinity is central in their work, like a yoga man, a librarian (!) and of course, an ultrafinitist (that, frankly, doesn't really come out well from the comparison).

Sit back and enjoy the ride!

Here is some more connection between Set Theory and Theology. Oh, and the above is divulgation done right. Wanna see divulgation done wrong?