\section{Introduction}
Mathematicians usually despise theoretical physics for lack of rigour (which does of course not, at least not logically, exclude admiration for theoretical physics on other grounds). Mathematical physicists are partly construed as conceding to this contempt by seeking to mathematically cleaning up the mess created by the theoretical physicists. However, in various areas of physics such as GR, mathematical physics does not only equip theoretical physicis with decisive tools: certain physical questions get recognized eventually as only studiable in a rigorous framework — of course, not because of the a priori conception of the mathematician on the importance of rigour but due to contextual opportunistic admissions that rigouros processing serves one’s purpose (and more so than non-rigorous one).\footnote{Sharma provides an instructive list of cases where more rigour would have saved one from epistemic and non-epistemic (say, time-consuming) fallacies.}
In the unchartered territories of quantum gravity, the few theorems that exist — whether at a theory-overarching level, or in an approach under construction — impress one as decisive and robust signposts\footnote{This language is inspired by Schneider’s `signpost’ talk.}. Sure: a theorem is only as generic as its conditions get — but the theorems one has in mind here take in rather generic conditions (it is therefore that they can run under the names of no-go theorems, no-loss theorems, etc.).
The relevance of theorems as signposts is found across all sorts of physics. In a way this is analytic in `theorem’ generally introduced to students of maths as high-key mathematical claims. We get a concrete idea of this signpost role when we look at the study of black holes: all kind of concrete modeling (analytic or numerical), with the specific idealisations (such as symmetries at play), seems suspect unless some general proven claims (such as the singularity theorems) back ups that there is indeed something to explore in the chosen terrority of exploration.
At a general level, one can of course discuss whether rigour and informativity are not in some sort of trade-off relation; and at some very general level it is surely true that a claim will somehow range between super accurate but limited in scope, and less likely to be true but general (think of the theory of guessing here which formalizes guesses as a trade-off between accuracy, and precision). But this explains why general theorems are so important: they are informative and rigorous (a strong, strong proxy for accuracy\footnote{Rigour does not guarantee full accuracy, but it promotes it.}) at once — and so of immense epistemic value.
To understand the character of `signposting via theorems’, we will consider various theorems in quantum gravity. We start with a survey of important theorems in quantum gravity in section 2. Then, we consider important theorems that are outside specific approaches, and thus count as `theory-overarching’ (section 3); in particular, we will ask how It comes that most of them seem to be of no-go theorem character. Next we consider important theorems specific to specific approaches to quantum gravity (section 4). A decisive distinction here seems to be that between theorems outside of the approach that however motivates and directs one to consider that approach in question (we will talk of motivational theorems), and theorems within that approach which seem important in grounding that approach. We will focus here on loop quantum gravity.
Within the philosophy of physics literature, one has often discussed the role of principles in theory construction, and their various roles for motivating (guiding principles), weakling confirming (confirmatory principles), etc. Some of this what is said about theorems echoes this about principles. However, theorems come with a certain epistemic force (they have been proven) that principles usually do not have; so I would rather see it the other way around: what I say about theorems could be attributed as roles to principles — but under much more qualification.
Mathematicians usually despise theoretical physics for lack of rigour (which does of course not, at least not logically, exclude admiration for theoretical physics on other grounds). Mathematical physicists are partly construed as conceding to this contempt by seeking to mathematically cleaning up the mess created by the theoretical physicists. However, in various areas of physics such as GR, mathematical physics does not only equip theoretical physicis with decisive tools: certain physical questions get recognized eventually as only studiable in a rigorous framework — of course, not because of the a priori conception of the mathematician on the importance of rigour but due to contextual opportunistic admissions that rigouros processing serves one’s purpose (and more so than non-rigorous one).\footnote{Sharma provides an instructive list of cases where more rigour would have saved one from epistemic and non-epistemic (say, time-consuming) fallacies.}
In the unchartered territories of quantum gravity, the few theorems that exist — whether at a theory-overarching level, or in an approach under construction — impress one as decisive and robust signposts\footnote{This language is inspired by Schneider’s `signpost’ talk.}. Sure: a theorem is only as generic as its conditions get — but the theorems one has in mind here take in rather generic conditions (it is therefore that they can run under the names of no-go theorems, no-loss theorems, etc.).
The relevance of theorems as signposts is found across all sorts of physics. In a way this is analytic in `theorem’ generally introduced to students of maths as high-key mathematical claims. We get a concrete idea of this signpost role when we look at the study of black holes: all kind of concrete modeling (analytic or numerical), with the specific idealisations (such as symmetries at play), seems suspect unless some general proven claims (such as the singularity theorems) back ups that there is indeed something to explore in the chosen terrority of exploration.
At a general level, one can of course discuss whether rigour and informativity are not in some sort of trade-off relation; and at some very general level it is surely true that a claim will somehow range between super accurate but limited in scope, and less likely to be true but general (think of the theory of guessing here which formalizes guesses as a trade-off between accuracy, and precision). But this explains why general theorems are so important: they are informative and rigorous (a strong, strong proxy for accuracy\footnote{Rigour does not guarantee full accuracy, but it promotes it.}) at once — and so of immense epistemic value.
To understand the character of `signposting via theorems’, we will consider various theorems in quantum gravity. We start with a survey of important theorems in quantum gravity in section 2. Then, we consider important theorems that are outside specific approaches, and thus count as `theory-overarching’ (section 3); in particular, we will ask how It comes that most of them seem to be of no-go theorem character. Next we consider important theorems specific to specific approaches to quantum gravity (section 4). A decisive distinction here seems to be that between theorems outside of the approach that however motivates and directs one to consider that approach in question (we will talk of motivational theorems), and theorems within that approach which seem important in grounding that approach. We will focus here on loop quantum gravity.
Within the philosophy of physics literature, one has often discussed the role of principles in theory construction, and their various roles for motivating (guiding principles), weakling confirming (confirmatory principles), etc. Some of this what is said about theorems echoes this about principles. However, theorems come with a certain epistemic force (they have been proven) that principles usually do not have; so I would rather see it the other way around: what I say about theorems could be attributed as roles to principles — but under much more qualification.