Collatzian issues in politics today

Photo by Casey Gomez

In society, there are several problems that are easy to state in ideological terms, but nearly intractable to solve policy-wise. These difficult problems mostly relate to structural inequalities that exist in our society — housing, taxes, infrastructure, foreign policy — where corrective actions must be balanced against one another.

In these balancing games, there are only compromises. Whenever someone waltzes in and says  they can make everything great (again), suspicion should be the logical first response.

This reminds of me of an interesting problem in mathematics. The Collatz conjecture, proposed in 1937 by the German mathematician Lothar Collatz, is an infamous example of a deceptively difficult problem. It sounds so simple, like a problem that a 6th grader would solve in a county math competition. “Take a positive integer n. If n is even, divide by two. If n is odd, multiply by three and add one. If you keep on doing this, will you always eventually reach 1?”

A proof or disproof of the Collatz conjecture eludes even the most talented mathematicians. Because the problem definition can be understood by anyone with basic knowledge of algebra, the Collatz conjecture has become the mathematical equivalent of the Sirens, drawing in naive graduate students eager to find a proof only to find their long hours wasted once they are slain by its difficulty. Established mathematicians warn people to avoid it. In a quote that’s become dogma, Paul Erdős, one of the greatest (and most eccentric) mathematicians of the 20th century, said “Mathematics is not yet ready for such problems.”

Several social issues are Collatzian in nature:  easy to state, hard to solve. These problems seem like they might have easy solutions, and if you are not an expert on the matter yourself, you can be easily tricked into thinking a charlatan has solved the puzzle.

Imagine that we plan to elect a person to run an Institute of Mathematics whose mission is to solve the hardest mathematical problems of our day.

Supreme of all these problems investigated by Institute of Math is the Collatz conjecture. Two candidates rise to the top of the popular ranks. One is well-published in academic journals and has been solving similar mathematical problems for decades (Alice); the other learned algebra in high school and stopped afterward (Bob).

Alice can appreciate the problem’s difficulty. When she talks in a public forum, she aptly notes all of the inherent challenges. Having spent years working on proofs and generating new mathematics, Alice knows that the Collatz conjecture is a doozer, and she wants funding to thrust more resources behind the issue. Bob, who is good enough at solving simple algebraic equations, runs through a few cycles of the Collatz conjecture — 5 goes to 16 to 8 to 4, 2, 1, check. Tries another obe, check. Bob figures there’s not much more to the problem; it seems to hold true for as many numbers he can be bothered to try. And when he gets up at the forum to talk, he is quite confident that he’s proved the issue, despite the fact that  his analysis is incomplete.

It is clear that many people can be persuaded by Bob’s misplaced confidence.

Ultimately, the media has a tremendous responsibility for communicating the complexity of Collatzian problems properly. The key is to hold discussions at an intermediate level in order to give a sense of depth: we might not be able to go down the rabbit hole with Alice, but we should at least get a sense of how deep it is.

Attempts to do this sort of bridging communication generally come up short. Newspapers and channels often choose to only flesh out the part of an issue that their writers and editors feel most passionately about — a natural tendency given space and time constraints — but this selective focus means that even through digesting a never-streaming arc of articles, it is almost impossible to get a broad sense of any matter. The analyses made by reputable news sources, albeit generally accurate, only illuminate bits and pieces.

Such bits and pieces will fail you when attempting to assess who is doing the best job on an issue. Instead, it is better to try to relate a topic at the level of intuition: to give people a broad sense of an issue, always keeping the analysis holistic, while providing ample resources for the curious reader to delve further.