Monday, June 22, 2015

The Truth about The Zero Theorem

The other day I have seen The Zero Theorem, a recent movie by Terry Gilliam. The movie was not really appreciated by the critics, but if you are reading here, probably you have already seen it, so it doesn't really matter. Also, it was greatly imaginative, and Gilliam's fans will appreciate it. But if you've seen the movie, then you could ask: how much of it is mathematically true? Of course, it's all pretty surreal and it must not be taken at face value, but is there a kernel of truth? Maybe. Let's see.

Q. What is the Zero Theorem? Is it something mathematicians are really trying to prove?
A. The movie is rather vague, so I cannot know for certain. I have three hypotheses:

The rational zero theorem, or rational root theorem, is a classical algebra theorem that indicates all the possible rational solutions to a polynomial equation. Now there is a rule: if there is a proof of it on Wikipedia, it means that in the spectrum of sureness, it is between death and the protests following a new Facebook layout:

so probably this is not what Gilliam had in mind.

In the movie, Bob tries to explain what is going on: proving the Zero Theorem would be proving the Big Crunch, that is the cosmological idea that the universe at a certain point will stop expanding and will start to collapse, ultimately disappearing. This is opposed to the Big Freeze, where the universe will continue to expand and therefore cool off, at a certain point too cold for life, and the Big Rip, where the expanding universe will rip, and two points will be at infinite distance between each other. Whatever notorious B.I.G. the scientists are trying to prove, probably it cannot be solved just with a theorem, it needs practical experiments. So, yeah, probably that is what Gilliam had in mind, even if it doesn't hold much water. Yet there is space for another hypothesis...

At a certain point we learn that Qohen managed to prove that 0 equals 93.78926%, and the objective is 100%.

The Mainframe! I must access the Mainframe!

100% of what? What about... everything? Then yes, that would be bad, it would destroy all mathematics. There is a catch, though: it would not be a slow climbing like in the movie, one percentage point at the time, but a sudden death! One just needs a very small percentage of things to be 0, and then everything would be 0. In fact, one needs just 1=0. Think about it: if 1=0, then 2=1+1=0, and 3=0, and so on, every number is 0. Also, if you have a triangle, what would be its area? 0, so it's a point, so any geometric shape would not exists. Using the principle of explosion (yes, I linked a xkcd comic and not the Wikipedia page, good enough) one could falsify any theorem. In fact, it's an inside joke between set theorists that 1=0 is the strongest axiom, because it can prove everything and its opposite (and this also answers the final question of one previous post of mine! this is an eight-months Chekov's gun!)

Q. Is it possible to prove that everything is 0? Can it really happen?
A. Yes, it can definitely happen. If you read this post, then you'd know that mathematics is a list of things we say are true (axioms) and everything that can be logically derived by them. Gödel proved with his Second Incompleteness Theorem that we cannot be sure that such a list doesn't prove a contradiction, ever. Tomorrow one smart guy can wake up, and prove that all mathematics is 0.

Q. What would happen if the Zero Theorem is true? Everything would be meaningless, like in the movie?

A. Well, my life surely. Your life will probably be fine. Sure, a lot of the things we do depend on mathematics, but there is maybe a way to save it, it depends how deep the contradiction is. What brings a contradiction? Is it very complex formulas? Then maybe limiting the complexity of formulas we are safe (again, we can never be sure). Too big numbers? Then we can limit the numbers (here you are again, ultrafinitist! You happy, now?). A contradiction is like gangrene (yikes, what a bad metaphor): sure, it can infect all the body, but you can cut just the part affected. If it is the little toe, then mathematics will be pretty much unscathed. Of course, it can happen that the problem is in the heart, for example in the basic arithmetic of numbers. That would mean that the basic mathematics does not work like we predict, for example computers could not work as expected and we would need a team of mathematicians that put order in the disorder (like in the movie) and rethink mathematics. But it is very implausible,

Q. Is the name of the protagonist relevant to mathematics?
A. There is a Cohen that is extremely important in mathematics, Paul Cohen, his work on the continuum problem (see this post) was unprecedented and changed mathematics completely, thanks to his technique, called forcing. But this has nothing to do with the Zero Theorem, and as the surname of Qohen is Leth, probably Gilliam was referring to Qoheleth, the Ecclesiastes.

Q. What is the Transfinite Paradox?
A. Yes, at the beginning Qohen solves the Transfinite Paradox. Transfinite is how mathematicians call an infinite that is not absolute, but all such paradoxes are already solved, so I have no idea what he meant.

Also, can we please, please, please avoid every time there is someone doing something mathematical avoid to picture him (because it's always a male) as a disturbed, asocial and in general a few cards short of a deck? Of course, mathematicians are a odd bunch, just like musicians, artists, writers and every job where you need a lot of passion to survive. But there are many ways to be odd, and not only this Asperger-like grumpiness: probably the worst case, bordering libel, was in the recent The Imitation Game, where they used the usual palette to paint Turing, that in reality was a fun, sociable and likeable person. Come on, moviemakers! De-Sheldonize mathematics!

Infinity is a staple of science fiction, like in Futurama or Artemis Fowl.  But if you want to know more about its limits, then go no further then here.