An Open Letter to Invisibilia
Hi Lulu, Alix, Hanna, and everyone else at Invisibilia,
I’m writing in regards to your recent episode, “The Problem with the Solution,” which I thought was fantastic.
However, the beginning part nagged at me a little bit. The part where you basically set the stage for the mental illness “non-solution solution” by having the hair-trap guy tell his story and say, “Every problem has a solution! Try to solve all the things!”
I know exactly why it bothered me, but before I jump into that I just want to reinforce how great I think Invisibilia is. I don’t want this to come across as a criticism, but rather as excitement to share cool ideas with you. I need this caveat because I’m a mathematician. And mathematicians tend to think and speak in a way that emphasizes consistency, precision, and nuance of language over things like, whether you come off as an arrogant prick or whether someone’s feelings are hurt. So again, I love the show, I respect you all for your fantastic work, and also here are some cool things I like to think about.
My thoughts about this episode culminated in a snarky tweet
It turns out that all mathematicians know that there are problems that cannot be solved. Period. I don’t just mean the sort of silly, “you can’t divide by zero,” rules that we’re made to memorize in school (though that is a very simple example). Rather, I can give you an example of many problems that everyone would agree would be a fantastic problem to solve, but there is a mathematical proof that it cant be solved definitively, period.
Here’s my favorite example, which comes from a field called “theoretical computer science.”
Computers are a big deal. They’ve revolutionized the world we live in, and the software engineering discipline accounts for trillions of dollars of value across all industries that use it worldwide.
Consequently, bugs in computer programs regularly result in millions of dollars of lost value. From that perspective, it would be extremely useful if we could design computer programs that can automatically tell whether the programs engineers write have bugs or not.
Turns out, this problem can’t be solved. And you can prove it mathematically. In fact, this problem was known to be unsolvable before computers were invented! It’s one of the crowning achievements of Alan Turing, along with cracking Nazi codes in WWII and recently being portrayed by Benedict Cumberbatch in a movie. People also tend to focus on his persecution as a gay man, which was bad, but that focus often comes at the expense of spreading his intellectual ideas, which I think is worse in some ways. Turing is the forefather and the epicenter of the intellectual revolution of our time, the transition to thinking about problems in terms of algorithms and computing.
I could go into a detailed exposition about why the computer-bug-finding problem is unsolvable, and I think that it’s a really interesting (albeit technical) story. But instead I’ll just give the sort of philosophical pseudoscientific reason, which boils down to the fact that the expressive power of computer programs as a whole is too vast for a single computer program to comprehend.
On top of that, Alan Turing is famous for this “thesis” that was named after him and another influential mathematician Alonzo Church. It’s called the “Church-Turing thesis.” What it says is that the powers and limitations of algorithms is a universal feature of computing.
That was unclear so let me try again: computations happen in lots of different ways. Computers crunch numbers, cats compute when exactly to pounce on an unsuspecting mouse, you compute whether you want a grande or a venti chai latte this morning. Scientists have even noticed that soap bubbles are effectively computing things as they float around. So when I say “computing” in this context, I really just mean decision making by any process that has some sort of principle (even random choices count).
The Church-Turing thesis says that anything which can be computed, can be described by an algorithm. That is, if it can be done, it can be done on a computer, so long as that computer is big enough and has enough time to run. And vice versa, any problem that cannot be solved by algorithms also cannot be solved in general.
And if you believe Turing’s thesis—at least, you can’t point to any evidence against it because nobody has found any—if you believe that the human mind has the same fundamental limitations as a computer program, you get some interesting consequences.
Take the justice system, for example, which essentially encodes a computation for deciding legality. As we’ve seen those processes can draw out for unbounded lengths of time, with complicated logic, appeals, and reversed decisions after many years. If the computations that can be expressed by the legal system are sufficiently expressive, then it follows that (1) no computer can definitively reason about the legality of an action, and (2) no human can definitively do so either!
Admittedly, it would take an awkward contortion of the legal system to, say, multiply two numbers. But the problems that result from over-expressiveness of laws recently bit a group of people in the butt, and collectively cost them over 50 million dollars. It was this company called the DAO, which wrote a program to define the rules by which the company operates. It wasn’t possible for the DAO founders to definitively say there were no bugs in the rules of their business, or whether some user’s bad behavior would result in a loss of money. And wouldn’t you know, their business rules were used against them and they lost millions. It was colloquially called a “hack,” but the “hacker” simply played by the rules.
Stepping back from all this, what’s most astounding is how invisible mathematics is to most of the world. And how a huge part of mathematics isn’t just solving equations, but rather taking some invisible pattern you discovered in your head, and getting that insight into another person’s head so they can understand it just as lucidly as you do. That’s how mathematics grows in practice. It’s this evolving network of conversations, drawings, and gesticulations. Mathematics is like this thick underbrush that takes a machete to get a peek inside. It grows and expands while some branches whither and die. But a select few offshoots slowly but surely pierce through the canopy into the sunlight, and their fruits change the world.
