There is a beautiful description of the mathematical endeavour by Andrew Wiles that goes like this:
Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark.
I would describe my work in terms of a journey through the illuminated rooms of Wiles’ mansion in order to save the world of shadows evoked by Jun’ichirō Tanizaki in his essay In Praise of Shadows. I enter such rooms and I trace the steps and stumbles of the mathematicians who were there from their various testimonies, broken or displaced things, scratches and dents. What part of the mansion they came from when they entered a room, what tools they had with them, or what the weather was like outside while they were in the room, are some of the issues that are also relevant to my work.
Then, I gradually restore the shadows that preceded the lights on. I believe that it is perhaps only in this way that the true beauty of each of the steps taken by the mathematicians can be grasped, as Tanizaki said about the Japanese lacquerware and its gold and silver decorations, the sitting room’s walls “of clay textured with fine sand”, “the costumes of the Nō theatre”. From this standpoint, understanding the journey of the mathematicians does not consist in shedding light on it; instead, shadows must be cast back onto it.
And yet, the restoration of the shadows may in turn shed light on the rest of the mathematicians’ journey through the mansion called mathematics. As was the case when Wiles entered Fermat’s room, sometimes mathematicians not only carry with them tools that were not available to those who ventured into a room before, but they also carry with them a substancial understanding of previous unsuccessful explorations. Hopefully, my work can contribute to improving the study of their journey and our understanding of it, but also to the mathematical endeavour.