Reducing Problems to Impossibilities
02-19-2022 • Ryan Prendergast
Originally written February 19, 2022. Edited May 15, 2025.
Certain society-wide problems are impossible. "People are stupid", "People are selfish", "People are not good people". However, other society-wide problems are solvable. "Minimize snow removal response time given this much money". The difference is how well specced they are. Sometimes, specifications are based on sentiments and not metrics. "Make the city clean, make it safe, make it functional". These can be somewhat, but not fully, rationalized. Some problems like "human selfishness" can never be summarized by a metric, and are an impossible problem.
In algorithms class in school, I was taught the concept of a Hard problem for computers.
Certain problems are really hard for a computer to solve. They take exponentially more time as you increase the size of the problem. There's no easier way to solve them than "try every single possibility". You need to brute force a solution. Hence, we call this problem Hard. At large scales, hard problems become impossible. Like, "all the computers in the world would take a million years to crack this" hard.
Hard Math Problems are like Hard Society Problems in that they are basically impossible to solve. "Make people less selfish" is about as difficult to solve as "find the unique prime factorization of this extremely large number". Theoretically possible, but not practically.
What's interesting about Hard Society Problems is that since they are succinct, they sound easy. They sound so easy that people prepend a "just" to it. "We just need people to be less selfish!" "We just need people to be a little bit more empathetic!". As if solving it is as simple as writing the sentence!
People seem drawn to Hard Society Problems. So much so, that when presented with a non-impossible problem, they propose solving it by ditching the problem entirely and instead solving impossible, Hard Society Problem! Say your city has potholes. I can't tell you how many people will say "well maybe if these politicians just weren't so corrupt, we'd have smooth roads!"
What? You've reduced a problem into an impossibility!
Road pavement is a relatively straightforward problem with engineering and budget constraints that can be optimized. You know what can't be optimized? The depth and breadth of human corruption! So why would you propose solving a (relatively speaking) easy problem, by throwing your hands in the air and saying "Well, this easy problem wouldn't exist if we solved this impossible problem, so I guess there's nothing we can do". Seems idiotic, yet this kind of commentary is by far the most common attitude of citizens towards government online.
We see this with regards to Covid, with something along the lines of "if people weren't so selfish then we wouldn't still be in this mess." What does that mean? How do we measure selfishness, what is the acceptable level? Of course asking such questions would be foolish, because it's not meant to be precise. It's a platitude. Similar common platitudes abound. "The administration doesn't care about our mental health". "The admin only cares about money." "Greedy corps are ruining society for selfish gains." These are statements that don't really mean anything precise.
Does that mean solving Covid is impossible? No! It's a dumb reductionist way to look at things. What is the use of reducing a problem to an impossibility? You prove something is Math-Hard by reducing an impossibility to a problem, NOT by reducing a problem to an impossibility!
Why do so many people do this?