Summer in San Francisco has been creating some (quintessentially American) questions. What are my big goals? What are the important problems worth solving? How does what I am doing feed into what I will work on eventually?
Most of these rabbit holes reduce to two questions:
- What 'things' are worth my time?
- How do I do these 'things'?
The first is a question of value, the second of strategy (both short-term and long-term).
The danger of the value question is that it invites a very artificial answer. Artificially, it would be supremely valuable in my life to build a theory of how concepts and language emerge in humans and machines, collect scientific data while making first ascents, and become a great writer. These answers seem big, important, and familiar because they are related to what I already do. The only problem is: knowing the answer does not entail knowing the solution. And before I know the solution, I really do not have any idea how meaningful each goal is to me or other people. There are certain goals that are obviously valuable (e.g. establishing P vs NP, curing cancer, unifying quantum gravity), but their intense value means that many people have either tried already and failed, or are still making steady but incremental progress. Attempting these mega-goals therefore decomposes to pursuing individual angles of attack, each of which suffers from the original problem of unknowability. Moreover, even if I do encounter one solution which is not valuable, it is quite possible that the value of a goal is only revealed through attempting another one of its many solutions. In other words, the value problem is like Turing's halting problem. It is undecidable in general. You will never find out the value of a goal until you attempt it (whether the Turing machine will halt), but there is no guarantee if you do (perhaps the Turing machine runs forever).
The halting problem may not seem heart-wrenching, but the value problem is real. Part of the problem is the subjectivity of value. Every person has a different value system: some combination of societal contribution, personal joy, external feedback etc that I do not claim to understand. But subjectivity and undecidability are separate issues. Even if value were subjective, the value of a goal is still undecidable: because value is coupled with the solution, not the goal. Take curing cancer, for example. It is valuable insofar as the solution enables a person to live longer. Living longer is valuable insofar as it makes a person happier. Happiness is valuable insofar as it is, well, valuable. Philosophers have tried forever to decompose or quantify the value of happiness (especially the utilitarians) or construct other proxies for value (fulfilment, contentment, etc), and I would argue not to much avail. Like halting on a Turing machine, value is an irreducible concept that must be experienced to be known. The solution to a goal is what provides that experience. Since a goal carries no inherent solution or experience, its value is undecidable.
Issues with the value question extend further: it is uninformative even when solved. Not only is it undecidable in general, even if it could be decided for a particular problem, I believe it is rarely informative towards the second problem of strategy. Assume for now that we know in advance which goals are valuable. I have often imagined achieving value as the process of first picking a valuable goal, then applying a top-down approach to achieve it. However, most valuable goals are fairly complicated and require significant ingenuity; otherwise they would be solved goals. This is true for goals in academia (unknown discoveries), but also for seemingly standard goals. In the latter case, consider a sub-18 5k. There are standard regiments I can follow to achieve a sub-18 minute 5k, but the goal is far from solved: how I fit it into my daily schedule, how much my body can handle, and how motivated I am are all variables I have to consider, because I am breathing in the real world, not running on a machine. More generally, anything that requires exploration and fitfoing on the fly cannot, by definition, be planned, and surmounting that uncertainty is arguably where most of the value lies in a solution. Because the true value of a goal is coupled with its particular solution, it is dangerous to choose goals based on their perceived value.
Consider a more concrete analogy. When I plan a mountaineering objective, the value problem is like deciding which peak to summit. No amount of theorising about the objective facts of a peak (its name, location, height, etc) allows us to establish its value: its name is a proxy for history and reputation; in turn, location is a proxy for remoteness and climate; height is a proxy for altitude and difficulty. Each of these (history, remoteness, difficulty, etc) are really factors of the solution: the particular route you take up. Indeed, in mountaineering, the value of an expedition really lies in the choice of route. This is why helicoptering onto the summit of Aconcagua is not very valuable, while summiting the Eiger through the North Face is an especially sought-after feat. Often as well, an expedition is made all the more valuable because of the mishaps that occur and your solutions to them, such as Tommy Caldwell inventing the downclimb to sidestep the dyno problem on pitch 16 of the Dawn Wall. In other words, the value of an expedition goal is unknowable before you attempt it (the value problem is undecidable). And quite clearly in mountaineering, knowing where you want to go does not really help you get there at all (the value problem is also uninformative). On El Capitan, firm faith does not entail firm footing.
At the end of the day, all goals, in and of themselves, are equally meaningless. Maybe this is why people who think 'if I do X, I will be happy' never become happy. Conversely, a great goal does not make someone great; a great solution does. But even though the value problem is undecidable except in hindsight, that does not mean the second problem of strategy is also undecidable. Perhaps there is a way of doing valuable things without first knowing what those are. Good mountaineers do it, so do good scientists, and people who find a lot of fulfilment in their lives.
I think the strategy problem admits a natural solution. The best mountaineers confront the difficulties of one face, then push forward just past the brink of the possible. Good scientists are limited by the contemporary techniques and information available to them, but combine them to create magic. The trick is not to artificially search for the most valuable thing I could possibly work on, but to think about the most valuable thing I could currently work on. The former is undecidable even approximately; the latter is doable within a finite time. Of course, the most valuable thing I can currently work on is bottlenecked by both my ability (known unknowns) and knowledge (unknown unknowns), but the only way to reliably make progress on either bottleneck is to work on the most valuable thing I can currently work on. What I said was obvious, but it creates concrete consequences: it means that the best way to 'search' for myself is simply to 'be' the best idea I currently have of myself. Similarly, the former is generally an undecidable problem, whereas the latter is doable within a finite time.
There is a benefit, however, of constantly asking the value question. In the process of deliberation, I tend to reduce it to debating the value of my current work, which helps me prune options that are not valuable to me and develop faith in the ones that are. In turn, data from working on valuable things informs how I assign value to my goals. Footing and faith are symbiotic.