The question that drives a significant part of our research at Dvole is: what does it actually mean for us to understand something? The question is certainly not new, and perhaps not particularly surprising either. Descartes, in his Meditations, attempted to approach it by doubting almost everything he thought he knew, and then asking what could still be established from what remained. Some of the conclusions he arrived at have been questioned extensively, but what we find particularly interesting is the approach itself. Boyle, working in a much more experimental setting, confronted related questions about what it would actually take to explain the physical world. Newton, with his Philosophiæ Naturalis Principia Mathematica, later demonstrated how mathematical principles could account for phenomena that had previously appeared quite different from one another. These were not necessarily attempts to answer precisely the same question, but each, in its own way, changed how we could approach the problem of understanding the world.
Over the centuries, we have certainly become much better at explaining and predicting phenomena. But our descriptions of the world have also become increasingly involved. We have mathematical theories, physical models, and experimental results that allow us to reason about things that would have been almost impossible to study a few centuries ago. Yet there is still a distinction between having a theory that describes a phenomenon and understanding what that theory actually tells us about it.
Consider, for instance, two mathematical theories that describe related physical phenomena. They might begin with entirely different assumptions, use different mathematical structures, and still arrive at similar conclusions. What exactly allows us to relate them? Are we looking at different descriptions of the same underlying structure, or does their agreement tell us something more limited? And how much of what we conclude depends on the particular representation we have chosen?
This is where epistemology and mathematical abstraction become particularly useful. Epistemology gives us a way of questioning what we know, how we arrived at that knowledge, and which conclusions our observations actually justify. Mathematics allows us to take some of these questions further. We can identify structures, study equivalences, construct abstractions, and investigate which properties survive when we change our assumptions or representations. Sometimes the problem is not even finding an abstraction, but understanding which distinctions we should preserve in the first place.
And the same reasoning extends well beyond physical theories. Consider computational systems that communicate and make decisions. Understanding what information is available to a system does not immediately tell us what it can infer from that information, or what decisions it will make. Once several systems begin interacting, their behaviour may depend on their individual knowledge, their beliefs about one another, and the incentives under which they operate. Questions that begin with knowledge and information can therefore lead quite naturally to formal logic, computation, decision theory, and game theory.
There is also another level at which we can approach these questions: the development of knowledge itself. Scientific knowledge does not develop independently of the ways we preserve, communicate, and organize it. As our collective understanding grows, so does the amount of existing work that an individual researcher may have to learn before contributing something new. We continually develop new tools and institutions to make that knowledge more accessible, and those developments can change how scientific research itself is conducted. Understanding this process may also help us reason about how research collaborations, scientific infrastructure, and broader institutions should develop.
A lot of what we want to pursue at Dvole comes from moving between these different perspectives. Some questions may lead to purely mathematical investigations, others to theoretical physics, and others to formal models of computation, knowledge, or collective behaviour. Sometimes the result may be a theorem or a new mathematical connection. Other times, it may suggest an experiment, a protocol, or a different way of designing a system.
The important part is that we are not restricted to approaching a problem at only one level. An observation may suggest a mathematical abstraction, an abstract theory may reveal something about a physical phenomenon, and a practical problem may force us to reconsider the assumptions with which we began.