As far as the attempt to describe mathematics research to others:
Doing mathematics is like helping to build a beautiful house. I am drawn to help because I have a love for what the finished house will be, and beyond this, because I enjoy the challenge of building. But my role is small. It may only consist of choosing the right location for the foundations, or determining how much insulation to put within the walls, or finding the right kind of wood for the exterior. If I speak with someone who admires architecture, but has never helped to build a house themselves, I worry that focusing on technical issues will blunt their interest in what I am doing, or rather, obscure the larger picture.
Yet the analogy is not absolute, because I find that the technical issues of mathematics are inevitable in a way that I do not believe specific choices about location, insulation, wood, etc. are always inevitable. Building a real-life house leaves a great amount of room for chance, arbitrariness, non-necessity. Maybe it is more revealing to say that the mathematical technicalities that have interested me have held a naturalness that inheres without my needing to keep in mind some vaster project. However, I cannot even describe them without first explaining a good deal of the rest of that vastness.
In the end, I usually resort to showing sketches of what the house will look like from afar, where the details become blurry. But these sketches convey nothing of how it feels when one is building — placing one’s fingers to the grain of the wood. I do not want people to believe that drawing a picture is the same as building, though the former may well be a step of the latter. The question is how to bring you a small piece, or perhaps a model, that you can touch, touch every way.
Tell me you trained at a top US program without saying you trained at a top US program. (J/K) I work in a "synthetic" subfield, too.
While I can relate, I have to confess that I've never been able to take the "house-building" or "cathedral-building" analogy to heart. I'm well aware that my own work is but a minuscule contribution in the grand scheme of things, but I've never found it inspiring to feel like I'm even working on a wing of the cathedral, much less choosing the right material for the facade. It's a personal hangup. I chose theory in youth because it rewarded independent thinking, and I wasn't happy to just build what people told me to build. Maybe I'm just too immature to be satisfied with the implication of authority and hierarchy.
I much prefer the "exploratory" approach, that you're going on a journey somewhere new, at least to you. Sometimes there's a train that can take you there, or at least a well-trodden path, but sometimes all you've got is a hastily scribbled map from some old guy at a pub (or more likely, tea room), what you've got in your pack, and your wits to guide you. You follow your nose, and while you usually just find the things found by those who there before you, every now and then you stumble on something new. Sharing your discovery with folks back home depends on the audience; some people are OK with a sketch or snapshot, others might want a full detailed map and itinerary, while most people just want to hear a good story.
Of course, this is all just a matter of perspective. No outside observer can tell the difference. But when I'm trying to prove some tricky lemma, I'm almost never thinking of the bridge itself, but of the possibilities that its existence opens up, even if it's just a matter of replacing the old hastily made version with modern materials, so that it's sturdy and safe and occasionally even beautiful.
Your points are well-received!
In my writings on this site, I’ve been recording the dates of first composition in the subtitle field. This piece was first written in 2017, in the middle of my grad-school years. It was easier back then to believe in noble causes, not only in mathematics: I did not perceive as much the apparatus of authority and hierarchy supporting them, which you identified. At the same time, I was more timid. I had not yet completed anything of length or sophistication. So, I could not imagine houses of my own, to take leadership of building or to hold in stewardship.
If I had to revise the piece beyond minor changes of wording, I’d replace the sentence: “But my role is small.” I now know that it is possible to take the lead role in building. It may even be important, for our sense of purpose, to strive for a research program that is ours from its foundations: complete in itself, however humble. I think that to work in a “synthetic” subfield, as you call it, is inherently to believe in the beauty of houses: at heart, of those thoughtfully furnished homes where conversation between friends is heightened by an awareness of a surrounding prevailing order.
I would now write instead: “But each step I take is small.” Because I still think that the smallness of the individual steps can distract from the larger picture: in your metaphor, the vastness opening on the path ahead.
But another thing I now appreciate better is the importance of surprise. To me, this is what you are really pointing out — that we don’t want our work to feel rote, to feel pre-ordained; that research should feel at least a little like an adventure, with no small element of danger along the way. Perhaps the image of the adventurer and the image of the architect can be reconciled. What is surprising in mathematics only gains in surprise when it also feels natural, as if the architecture had been lying in wait for us all along. (The child to the sculptor: “But how did you know that a statue was hidden in that block of marble?”) And what is natural, in the architecture of mathematics, only impresses us all the more in naturalness when the pathways and bridges are precisely those which we did not expect.
Ah, well, all but the most self-assured of us are hesitant in grad school. Everything makes a lot more sense with that context. I'm glad to hear of how your own research philosophy and perspective has evolved since then. I most certainly believe in the beauty of houses, though if we're comparing with grad school, I think I've come to enjoy the griminess of the construction process the more I've been able to forge my own path. At least for me, it's fun to garden, it's not as fun when your dad is telling you to mow the lawn before guests arrive.
You're bang on with that last paragraph as well. The combination of surprise with that sense of inevitably is perhaps the reason I find mathematics so fascinating, and what I assume is a big reason why an overwhelming majority seem to say that research tends to feel like "discovery" in contrast to "invention." I don't know if you've seen this already, but there's this short clip by Dick Gross (RIP) where he expresses this beautifully.