\section{Introduction}
Mathematicians often look down on theoretical physics for its lack of rigour (which, of course, does not—at least not logically—exclude admiring it on other grounds). Mathematical physicists are sometimes seen as conceding to this critique, stepping in to mathematically clean up the mess left by theoretical physicists. Yet in many areas of physics, such as general relativity, mathematical physics does more than just equip theoretical physicists with powerful tools: certain physical questions eventually come to be seen as only properly addressable within a rigorous framework. This shift isn’t usually driven by a mathematician’s a priori commitment to rigour, but rather by a more opportunistic realization that a rigorous approach better serves one’s purposes—often more so than a non-rigorous one.\footnote{Sharma provides an instructive list of cases where greater rigour could have prevented both epistemic and non-epistemic pitfalls (e.g., wasted time).}
In the uncharted terrain of quantum gravity, the few theorems that exist—whether overarching or internal to a developing approach—stand out as striking and robust signposts.\footnote{This language is inspired by Schneider’s “signpost” talk.} To be sure, a theorem is only as general as its assumptions allow—but the theorems we have in mind here tend to rest on relatively generic assumptions (hence they often earn labels like “no-go” theorems, “no-loss” theorems, and so on).
The significance of theorems as signposts extends across all domains of physics. In a sense, this is analytic in the notion of “theorem” itself, which students of mathematics encounter early as markers of high-level, formally secured claims. This guiding role becomes especially vivid in the study of black holes: all sorts of concrete modeling—whether analytic or numerical, with various idealisations such as symmetry assumptions—can seem suspect unless anchored by broad, proven results (like the singularity theorems) that assure us there is, in fact, something to be explored in the chosen theoretical terrain.
At a more general level, one might ask whether rigour and informativeness stand in some sort of trade-off relation. And at some highly abstract level, this is surely true: claims tend to range between being highly accurate but narrow in scope, or broad but less likely to be true. (Think here of the theory of guessing, which formalizes guesses as a trade-off between accuracy and precision.) This is precisely what makes general theorems so epistemically powerful: they are both rigorous and informative—rigour being a strong proxy (though not a guarantee\footnote{Rigour does not guarantee perfect accuracy, but it significantly promotes it.}) for accuracy.
To better understand this idea of “signposting via theorems,” we will explore several theorems in the context of quantum gravity. Section 2 offers a survey of key theorems in quantum gravity. In Section 3, we consider theorems that lie outside any particular approach—so-called “theory-overarching” results. A central question here is why so many of these seem to take the form of no-go theorems. In Section 4, we turn to theorems that are internal to specific quantum gravity approaches. A key distinction will be drawn between theorems external to an approach but which serve to motivate it (we will refer to these as motivational theorems), and theorems internal to the approach which serve to substantiate or anchor it. Our focus in this section will be on loop quantum gravity.
Within the philosophy of physics, the role of principles in theory construction has been widely discussed—principles that serve variously to guide, motivate, or weakly confirm a theory. Some of what can be said about theorems resonates with what has been said about principles. However, theorems carry a distinct epistemic weight: they have been proven, while principles typically have not. So, rather than seeing the roles of theorems as derivative of those of principles, I would suggest the reverse: the roles I ascribe to theorems might also apply to principles—but only under qualified circumstances.
Mathematicians often look down on theoretical physics for its lack of rigour (which, of course, does not—at least not logically—exclude admiring it on other grounds). Mathematical physicists are sometimes seen as conceding to this critique, stepping in to mathematically clean up the mess left by theoretical physicists. Yet in many areas of physics, such as general relativity, mathematical physics does more than just equip theoretical physicists with powerful tools: certain physical questions eventually come to be seen as only properly addressable within a rigorous framework. This shift isn’t usually driven by a mathematician’s a priori commitment to rigour, but rather by a more opportunistic realization that a rigorous approach better serves one’s purposes—often more so than a non-rigorous one.\footnote{Sharma provides an instructive list of cases where greater rigour could have prevented both epistemic and non-epistemic pitfalls (e.g., wasted time).}
In the uncharted terrain of quantum gravity, the few theorems that exist—whether overarching or internal to a developing approach—stand out as striking and robust signposts.\footnote{This language is inspired by Schneider’s “signpost” talk.} To be sure, a theorem is only as general as its assumptions allow—but the theorems we have in mind here tend to rest on relatively generic assumptions (hence they often earn labels like “no-go” theorems, “no-loss” theorems, and so on).
The significance of theorems as signposts extends across all domains of physics. In a sense, this is analytic in the notion of “theorem” itself, which students of mathematics encounter early as markers of high-level, formally secured claims. This guiding role becomes especially vivid in the study of black holes: all sorts of concrete modeling—whether analytic or numerical, with various idealisations such as symmetry assumptions—can seem suspect unless anchored by broad, proven results (like the singularity theorems) that assure us there is, in fact, something to be explored in the chosen theoretical terrain.
At a more general level, one might ask whether rigour and informativeness stand in some sort of trade-off relation. And at some highly abstract level, this is surely true: claims tend to range between being highly accurate but narrow in scope, or broad but less likely to be true. (Think here of the theory of guessing, which formalizes guesses as a trade-off between accuracy and precision.) This is precisely what makes general theorems so epistemically powerful: they are both rigorous and informative—rigour being a strong proxy (though not a guarantee\footnote{Rigour does not guarantee perfect accuracy, but it significantly promotes it.}) for accuracy.
To better understand this idea of “signposting via theorems,” we will explore several theorems in the context of quantum gravity. Section 2 offers a survey of key theorems in quantum gravity. In Section 3, we consider theorems that lie outside any particular approach—so-called “theory-overarching” results. A central question here is why so many of these seem to take the form of no-go theorems. In Section 4, we turn to theorems that are internal to specific quantum gravity approaches. A key distinction will be drawn between theorems external to an approach but which serve to motivate it (we will refer to these as motivational theorems), and theorems internal to the approach which serve to substantiate or anchor it. Our focus in this section will be on loop quantum gravity.
Within the philosophy of physics, the role of principles in theory construction has been widely discussed—principles that serve variously to guide, motivate, or weakly confirm a theory. Some of what can be said about theorems resonates with what has been said about principles. However, theorems carry a distinct epistemic weight: they have been proven, while principles typically have not. So, rather than seeing the roles of theorems as derivative of those of principles, I would suggest the reverse: the roles I ascribe to theorems might also apply to principles—but only under qualified circumstances.