This is a strictly nonreligious piece [1]. Instead, it’s about how, to do things, you have to have faith in their feasibility. This is also a gripe about olympiads and how I never pursue the ideas.

Pushing hard on an idea, and actually going deep into the “idea tree,” an abstraction I will use to refer to the tree of possible arguments, is valuable. This is because even unpromising ideas can be valuable — most ideas look unpromising, but there are enough of them that you shouldn’t rule them out.

My canonical example for this is this time I was sitting in the front passenger seat of a car and dropped my phone. I was pretty sure it fell between the center console and my seat. I stuck my hand down and fished for the phone, forwards and backward, passing over the length of the console a few times. And in all this time, I didn’t feel my phone. Having lost faith that my phone was actually there — and not under my chair, I pinged it with my Apple Watch [2]. This told me that it was in fact in the thin ravine, so I stuck my hand in again and this time, with genuine full faith that it was there, I immediately felt the distinctive edge of the phone and quickly fished it out.

And this matters to me because the idea of faith has applications in mathematics. I have plenty of examples of problems where I swore I had the idea but just didn’t push hard enough on it. The biggest (aka most painful) example, which I’m still mad about, is USAMO 2022/2.

I had the idea of using vectors/complex numbers to represent everything and then using lin-alg stuff to get area (I don’t want to spoil). But writing this all out seemed ugly and alg-manip heavy, so I never actually wrote stuff out [3]. This is a common occurrence, so much so that I repeatedly tell myself to write down every technique that I considered.

Now, back to the tree metaphor where the nodes are ideas/claims/patterns, and the edges are techniques (imagine the root at the top). This gets to my mental model of how olympiad skill works: it’s a mix of how quickly you can move around [4], faith a.k.a whether you prematurely cut off branches, and then how well you combine ideas from different branches. When I got stuck on IMO 3/6s, it felt like I had all of the necessary low-level ideas and observations, but just wasn’t able to put it all together. I started saying, as a joke, that I was getting “IQ locked.” Eventually, it stopped being a bit, but there was definitely some kind of phase transition between doing “easier” problems (I got a 14/15 on AIME and 5/6 solves on JMO), and doing hard, late-USAMO problems. I have a theory that the feeling I had toward these IMO problems is the same feeling that most people have towards calculus.

But, faith might just be a derivative of the “how fast you can move along the tree” factor. Speed can help the assessment and synthesis of branches since you can rediscover things quickly. Additionally, it is not always true that more faith is better. On the Wikipedia page “List of disproved mathematical ideas,” there is an entry saying “Erik Christopher Zeeman tried for 7 years to prove that one cannot untie a knot on a 4-sphere. Then one day he decided to try to prove the opposite, and he succeeded in a few hours.” 

I really enjoyed writing down every single one of my ideas, but I no longer do this and instead jot down promising ideas, especially those that reframe the problem in a different way with different degrees of freedom. Math-wise, I think this idea will one day help me decide whether olympiads are really as orthogonal to research mathematics as adults claim. And non-math-wise, I think lowering the bar on what’s a reasonable idea is generally a good thing [5]. Maybe in conspiracy theories, but definitely in choosing your model of the world. [6]

[1] I’m not religious (although I wish I was). It would’ve been nice to have started out with a pre-made moral framework instead of building it out on my own, synthesizing ideas from a multitude of sources. 

[2] An amazing feature which lets you basically do the find my iPhone alert, but it only plays the sound once.

[3] aka I’m bad at formulas (https://web.evanchen.cc/static/otis-samples/synopsis.html) problems when I don’t know they’re formulas

[4] https://terrytao.wordpress.com/career-advice/advice-on-mathematics-competitions/

[5] I’m thinking of Simulated Annealing here, and this is kind of arguing for exploration over exploitation.

[6] Which is a funny thing to say because I’m philosophically conservative on institutions, I give them (perhaps inaccurately) the benefit of the doubt, assuming they were designed like that for a reason.