The Lighthouse Problem

This is a piece of advice that I found myself giving to many early career researchers, students, and colleagues supervising and advising those as well. For years, actually.

A mathematician, the joke says, is a blind man, in a dark room, searching for a black cat that isn't there. I don't know about that, but I think we can still agree that a researcher, in most fields, is sailing a boat, at night, in the fog, searching for the harbor.

What I mean by this is that in research (at least in my experience and observation) we often don't know what we don't know, we aren't sure about what we do know, and we're just doing our best to get further. With time and experience we develop intuition to guide us, but even then, it's not always easy to get there. This is where a lighthouse can help, it will, if nothing else, provide some points of reference and perhaps a sliver of light.

So, what is a lighthouse in this analogy of mine? Well... it is a problem. An open problem.

A lighthouse is an open problem, so difficult for you (right now) that it might be another 20, 30, 150 years before it would be solved. Its goal is not to be solved. The goal is to think about it in terms of "Why can't I solve it? What techniques are missing? What knowledge is missing from the path from here to a solution?". The goal, is to provide food for thought from whence ideas grow more rapidly.

When I was working on my M.Sc., I remember thinking about the Partition Principle, and I remember thinking that if it had been open for so long, its very likely that it does not imply the Axiom of Choice. But how would I go about proving this? Well, one way would be to look at a pair \(A\) and \(B\) which are a counterexample (i.e., \(A\) maps onto \(B\) but there is no injection back), and then... fix it, and then do it again and again and again.

So, what does fixing a pair mean? It means adding an injection, but we want to be careful enough to ensure that we did not split at least one surjection. Otherwise, if we fix things by adding inverses to surjections we'll just end up with the Axiom of Choice again. I don't know how to be careful in forcing, generic objects code all kind of crazy sets, but I know how to get rid of it by taking a symmetric extension. But in 2011, there was no method to iterate symmetric extensions. There was no framework for that.

That became my Ph.D., that led to the development of the iteration framework, which I could then apply to the Bristol model. It led me to the point where questions about critical cardinals without choice became natural, it led me to the point where having a big research project on connections between the Axiom of Choice and large cardinals was possible. With smaller lighthouse problems coming naturally through that.

Of course, there's still a significant amount of work left in order to finish that original layout for solving the Partition Principle's problem. But that's the thing, the iteration framework was just the first step, and there are others.

The benefits of having a big lighthouse are many. Especially for early career researchers. It shows tenacity, it shows visions, it gives you research ideas, which in turn give you other research ideas, and that is important for [academically] young people whose time is generally much more devoted to research and learning than to teaching and admin. Of course, this is also good for faculty, it is always easier to come up with research grant ideas if you have a source from where to generate new ideas. And the more you do, the easier it gets to generate, your web of lighthouses gets bigger, denser, brighter. Project for 3 years? Got it. What about an 8 year fellowship? No problem. It's all good, the lighthouses are plenty and strong.

Young set theorists, young mathematicians, young researchers, researchers. Look for your lighthouse. Look for that problem that can inform your research ideas for the next 10, 20, 50 years. And if that one gets solved quickly, somehow, that's fantastic! Go get another!